NMR petrophysical cross correlations for partially saturated reservoir ...

NMR petrophysical cross correlations for partially saturatedreservoir rocks

Author:Alghamdi, Tariq

Publication Date:2012

DOI:https://doi.org/10.26190/unsworks/15439

License:https://creativecommons.org/licenses/by-nc-nd/3.0/au/Link to license to see what you are allowed to do with this resource.

Downloaded from http://hdl.handle.net/1959.4/51883 in https://unsworks.unsw.edu.au on 2022-06-24

http://dx.doi.org/https://doi.org/10.26190/unsworks/15439

https://creativecommons.org/licenses/by-nc-nd/3.0/au/

http://hdl.handle.net/1959.4/51883

https://unsworks.unsw.edu.au

https://unsworks.unsw.edu.au

NMR PETROPHYSICAL CROSS CORRELATIONS FORPARTIALLY SATURATED RESERVOIR ROCKS

by

TARIQ ALGHAMDI

MSc Petroleum Engineering

A thesis submitted in fulfilment

of the requirements for the degree of

Doctor of Philosophy

in

Petroleum Engineering

School of Petroleum Engineering

The University of New South Wales

2012

ii

DECLARATION

I hereby declare that this submission is my own work and that, to the best of my

knowledge and belief, it contains no material previously published or written by

another person nor material which to a substantial extent has been accepted for

the award of any other degree or diploma of a university or other institute of higher

learning, except where due acknowledgement is made in the text.

Tariq AlGhamdi

iii

PUBLICATIONS AND CONFERENCE PROCEEDEINGS ARISING FROMTHIS THESIS

[1] Tariq M. AlGhamdi and Christoph H. Arns ;

”Two-phase relative permeability predictions from simulated NMR relaxation-

diffusion responses of partially saturated tomographic images”, Transport

in porous media, in-submission.

[2] Tariq M. AlGhamdi, C. H. Arns and R. Y. Eyvazzadeh;

”Correlations between NMR relaxation response and Relative permeabil-

ity from tomographic reservoir rock images”, proceeding at the Society of

Petroleum Engineers (SPE) Annual Technical symposium, AlKhobar, Saudi

Arabia, April 8-11, 2012.

[3] Tariq M. AlGhamdi, C. H. Arns and R. Y. Eyvazzadeh;

”Correlations between NMR relaxation response and Relative permeability

from tomographic reservoir rock images”, SPE Journal of reservoir evalua-

tion and engineering, submitted March, 2012.

[4] Tariq M. AlGhamdi, Jiyoun Arns and C. H. Arns;

”Relative permeability estimation from NMR responses: a numerical study

using X-ray micro-tomography”, The 11th International Conference on Mag-

netic Resonance Microscopy, Beijing , China August 14-18, 2011.

[5] Christoph H. Arns, Tariq AlGhamdi and Ji-Youn Arns; Numerical analysis

of NMR relaxation-diffusion responses of sedimentary rock, New Journal of

Physics, 13:015004, 2011.

[6] C. H. Arns, Tariq AlGhamdi, Ji-Youn Arns, Lauren Burcaw, and K. E.

Washburn. Analysis of T2-D relaxation-diffusion NMR measurements for

partially saturated media at different field strength. Number SCA2010-17

in The 24rd International Symposium of the Society of Core Analysts, pages

1-12, Halifax, October 4-7 2010. Society of Core Analysis.

iv

ABSTRACT

This thesis considers the relationship between NMR relaxation-diffusion re-

sponses and permeability for partially saturated rock samples on the basis of Xray-

CT images.

The NMR relaxation response is often used in petroleum engineering applica-

tions to estimate a pore size distribution, providing a length scale for the estimation

of permeability downhole, where direct measurements of permeability via well log-

ging are not available. This application generally relies on the introduction of a

constant surface relaxivity, and assumes that the dominant relaxation mechanism

is surface relaxation. Extending this concept to partial saturations, e.g. rocks

saturated with two immiscible fluids, this thesis analyses the relationship between

relative permeability and the NMR relaxation response of the fluids saturating the

pore space directly on tomographic images of the rock samples.

Segmented tomographic images of Berea and Bentheimer Sandstone and a

Ferroan-Dolomite are used to calculate the permeability and relative permeability

at various saturations. Here the fluid saturations are derived by simulating the

drainage of these samples assuming strongly water-wet conditions. The NMR re-

sponses are calculated using a random walk method which takes account of internal

gradient effects on the basis of mineralogy, which was derived by a combination of

Xray-CT imaging and XRD analysis. Literature values for the susceptibility of the

minerals present were used.

To test the correlations between NMR relaxation response and relative per-

meability for the individual fluids, we set the hydrogen index of one of the fluids to

zero, allowing a numerical partition of the two fluids, while calculating the correct

internal magnetic field distributions. There is excellent agreement between relative

permeability predictions from NMR and lattice Boltzmann calculations for all three

samples. Remarkably, this includes the relative permeability prediction for the oil

phase with a zero surface relaxivity between oil and all other phases, implying that

the correlation is based on internal gradient effects and saturation alone. We finally

extend the analysis to NMR relaxation-diffusion responses and show that a parti-

tion of the response into the different fluids like in conventional NMR fluid typing

allows the prediction of relative permeability.

ACKNOWLEDGEMENTS

First of all, I thank ”Allah” for giving me the strength and inspiration to

come here and work on my research that I hope to deliver the benefit from this

work to my country. I always thank my father who has been always pushing me to

get my PhD and has been always supporting me and my mother who always prays

for me to succeed and their continous love to me has been a great motivation for

my success. Without their support, I would not be here. I also thank my brothers,

sisters and uncles who always support me with their inspiring words and great

motivative gestures.

I would like to thank my wife who has accompanied me since the start of my

journey of my PhD. Her true love and endless support has reflected on my comfort

and I owe it to her being so patient for me not being there most of the times. Our

son Mohammed who brought fullness of Joy and love to our life since the beginning

of this year. My Parents and my family have so much expectations from me and I

devote this work and dedicate it to them.

I would like to thank all my friends at Saudi Aramco, who fully supported

me and always chasing my news and accomplishments. I thank Margerat Dyer who

came to me and pushed me to go for PhD, her inspiring words to me that day

are unforgotten. I thank Saudi Aramco management for giving me this chance to

come and enrich my knowledge. I am grateful for their financial and emotional

support during the whole period of my PhD. Special thanks to Saudi Aramco Hong

Kong (Yacoub, Mohammed and Antonio) for their amazing support and motivation

everytime they visit me at campus. They have facilated all the needs for my research

interms of resources and their follow up on my accomplishments has driven me to

exceed their expectations.

I would like to thank Dr. Christoph Arns who was been very patient with me

and has been excellent supervisor on my work and always has been working closely.

He has facilated softwares and providing me with access to numerical analysis code

which has been extremely essential to accomplish my results and reach a milestone

in my career. His endless support throughout this research is remarkable. I thank

Ji-Youn Arns for her extreme support for teaching me segmentation and providing

vi

me with support all the times. I thank Prof. Val, Prof. Sheikh Rahman, and Dr.

Yildiary Cinar for their support to this school and the program during my time.

I thank my friends (Mohammed AlHamdan, Sefer Yenici, Furqan Hussain,

Ahmed Khalili, Mostafa Feali, Jafar, Nural, Fahad, Altaf, Igor, Mehdi, Saeed) who

are studying PhD with me as we have been always supporting each other. I also

thank my friends in Sydney who have been very supportive since the beginning of

my arrival specially (Haitham Alawi).

...

DEDICATION

I wish to dedicate this work

to my parents and my wife;

for their guidance,

support,

and unconditional love.

CONTENTS

CHAPTER

1 INTRODUCTION 1

1.1 Nuclear Magnetic Resonance . . . . . . . . . . . . . . . . . . . . . . 2

1.2 NMR analysis in fully saturated porous media . . . . . . . . . . . . 3

1.3 Permeability correlations from NMR . . . . . . . . . . . . . . . . . 5

1.4 Analysis of NMR in partially saturated porous media . . . . . . . . 6

1.5 NMR analysis for fluid typing T2-D . . . . . . . . . . . . . . . . . . 8

1.6 Internal Gradients . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.7 Numerical NMR Analysis . . . . . . . . . . . . . . . . . . . . . . . 13

1.8 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 METHODS 17

2.1 High resolution micro-CT . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 CT scan tomographic image processing . . . . . . . . . . . . . . . . 17

2.3 Transport and petrophysical properties calculation . . . . . . . . . . 18

2.3.1 Porosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3.2 Mercury intrusion capillary pressure (MICP) . . . . . . . . . 24

2.3.3 Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3.3.1 Relative Permeability . . . . . . . . . . . . . . . . 25

2.3.4 Formation factor . . . . . . . . . . . . . . . . . . . . . . . . 26

2.4 Simulation of NMR response on high resolution 3D tomographic images 26

2.4.1 NMR Simulation . . . . . . . . . . . . . . . . . . . . . . . . 26

2.4.1.1 Surface relaxation . . . . . . . . . . . . . . . . . . 26

2.4.1.2 Dephasing due to internal gradients . . . . . . . . . 27

2.4.1.3 Modeling parameters . . . . . . . . . . . . . . . . . 28

3 RELATIVE PERMEABILITY CORRELATIONS FROM NMR T2 34

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2.1 Sample characterization . . . . . . . . . . . . . . . . . . . . 37

ix

3.2.2 Image processing and analysis . . . . . . . . . . . . . . . . . 37

3.3 Numerical simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3.1 Image based fluid saturations . . . . . . . . . . . . . . . . . 38

3.3.2 Permeability and relative permeability . . . . . . . . . . . . 38

3.4 NMR Permeability correlations . . . . . . . . . . . . . . . . . . . . 39

3.5 Results and analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.5.1 NMR response of samples saturated with single fluid . . . . 39

3.5.2 NMR simulation response of partial saturations . . . . . . . 40

3.5.3 Relative permeability from NMR relaxation measurements . 42

3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4 NUMERICAL ANALYSIS OF TWO DIMENSIONAL NMR T2-D 53

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.2.1 Image acquisition and processing . . . . . . . . . . . . . . . 56

4.2.2 Internal magnetic field calculation . . . . . . . . . . . . . . . 57

4.2.3 NMR response simulation . . . . . . . . . . . . . . . . . . . 60

4.2.4 Restricted diffusion and diffusion averaged internal gradients 61

4.3 Diffusion-relaxation analysis . . . . . . . . . . . . . . . . . . . . . . 62

4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5 RELATIVE PERMEABILITY CORRELATIONS FROM NMR T2-D 68

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.3 Results and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.3.1 T2-D NMR analysis for partially saturated samples . . . . . 71

5.3.2 T2-D NMR analysis of individual fluids . . . . . . . . . . . . 72

5.3.2.1 T2-D analysis of water phase . . . . . . . . . . . . 72

5.3.2.2 T2-D analysis of oil phase . . . . . . . . . . . . . . 76

5.4 Relative permeability from NMR T2-D analysis . . . . . . . . . . . 82

5.4.1 Relative permeability from NMR T2-D of wetting phase . . . 82

5.4.2 Relative permeability from NMR T2-D of non-wetting phase 82

5.4.2.1 Non-wetting phase correlations with known bulk re-

laxation . . . . . . . . . . . . . . . . . . . . . . . . 83

5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6 RECOMMENDATIONS AND FUTURE RESEARCH 97

x

BIBLIOGRAPHY 99

TABLES

Table

2.1 Summary of Petrophysical Properties of Samples . . . . . . . . . . . 26

2.2 Volume magnetic susceptibility (χvol) of minerals and fluids used to

calculate effective susceptibility. . . . . . . . . . . . . . . . . . . . . 29

2.3 NMR Simulation input for surface relaxivity values . . . . . . . . . 29

2.4 NMR Fluids Input parameters. . . . . . . . . . . . . . . . . . . . . 29

3.1 Summary of Petrophysical Properties of Samples . . . . . . . . . . . 38

4.1 Mineral and fluid susceptibilites used to calculate effective suscep-

tibilities [101]. The effective susceptibility of Bentheimer sandstone

is calculated assuming a mixture of Kaolinite and brine with equal

volume percentage. For Berea the XRD analysis gave 86.4% Quartz,

6.2% Kaolinite, 2.5% Ankerite, 1% Rutile, 2.9% Illite and/or Mica,

and 1% Feldspar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

FIGURES

Figure

1.1 Relaxation process of [a] longitudinal relaxation time (T1) and [b]

transverse relaxation time (T2). In [a], the net magnetization vector

is aligned in the z-direction followed by a 180 degrees pulse is applied

and the magnetization is tipped to the z-direction. Later, magneti-

zation is recovering to its initial z-direction. Finally, magnetization

reaches its initial equilibrium condition. In [b], the net magnetization

vector is initially aligned in the z-direction, the Mxy component is

zero, followed by a 90 degrees applied pulse and the magnetization

is tipped to the xy-plane. Next, the Mxy is recovering to its equilib-

rium state and then Mxy reaches its initial equilibrium condition at

zero. [59] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 NMR magnetization decay and pore size distribution (Westphal et

al. 2005). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 NMR Two-Dimensional Measurements during Drainage and Imbi-

bition [a] Bentheimer Sandstone, and [b] Berea Sandstone (Fully

Water on top, Fully Oil in middle and spontaneous-Imbibition in

bottom(Huerlimann et al. 2002) . . . . . . . . . . . . . . . . . . . . 10

1.4 Classification of wide range of susceptibility measurements on min-

erals and reservoir fluids(Ivakhnenko et al. 2004) . . . . . . . . . . 12

2.1 X-ray CT apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2 Intensity histogram of X-ray CT scanned image after applying anistropic

diffusion (AD) filter, which is the main filter to smooth uniform re-

gions. The intensity of solid and pore is presented by the high peak

representing solid and low peak for the pore phase. . . . . . . . . . 19

xiii

2.3 Processing of tomographic images. [a] Raw image captured on Savon-

niere limestone and [b] three-phase segmented image of Savonniere.

Here black is the pore space, green represents the intermediate phase

where solid and pore co-exist, and red presents the solid phase of the

rock sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4 Processing of tomographic images. [a] Raw image captured from

Bentheimer sandstone and [b] three phase segmented image of Ben-

theimer sandstone. Here black is the pore space, green represents the

intermediate phase and clay regions where solid and pore co-exit, and

red presents the solid phase of the rock sample. . . . . . . . . . . . 21


Berea sandstone and [b] three phase segmented image of Berea sand-

stone. Here black is the pore space, green represents the intermediate

phase and clay regions where solid and pore co-exit, and red presents

the solid phase of the rock sample. . . . . . . . . . . . . . . . . . . 22


Ferroan Dolomite and [b] three phase segmented image of Ferroan

Dolomite. Here black is the pore space, green represents the inter-

mediate phase and clay regions where solid and pore co-exit, and red

presents the solid phase of the rock sample. . . . . . . . . . . . . . . 23

2.7 Simulated Drainage of water by oil using capillary drainage mecha-

nism numerical technique, [a-c] Bentheimer sandstone ([a] Sw100%,

[b] Sw50%, and [c] Sw25%), [d-e] Berea sandstone ([d] Sw100%, [e]

Sw80%, and [f] Sw44%), [g-i] Ferroan-Dolomite Carbonate ([g] Sw100%,

[h] Sw75%, and [i] Sw25%); (White is the invading non-wetting phase,

Black is the defending wetting-phase, Gray is the solid-phase, red is

the clay region). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.8 Image based relative permeability from numerically simulated fluid

distribution validated with experimental results on Bentheimer sand-

stone samples (Hussain 2011). . . . . . . . . . . . . . . . . . . . . . 31

2.9 Slice through internal magnetic fields of 8003 simulation domain in

units of B0 for [a] Bentheimer sandstone, [b] Berea sandstone, and

[c] Ferroan Dolomite. . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.10 NMR simulation response of Bentheimer sandstone using random

walk method: [a] Magnetization decay response and [b] Inverted

transverse relaxation time distribution. . . . . . . . . . . . . . . . . 33

xiv

3.1 Segmented tomographic images of [a] Berea Sandstone, [b] Bentheimer,

and [c] Ferroan Dolomite Carbonate. . . . . . . . . . . . . . . . . . 38

3.2 NMR response of fully saturated samples of oil and water. [a] Ben-

theimer sandstone, [b] Berea sandstone, and [c] Ferroan-Dolomite. . 41

3.3 NMR simulation of water phase in Bentheimer sandstone sample at

different saturations. [a] Normalized magnetization decay and [b]

transverse relaxation time responses. . . . . . . . . . . . . . . . . . 43

3.4 NMR simulation of water phase in Berea sandstone sample at differ-

ent saturations. [a] Normalized magnetization decay and [b] trans-

verse relaxation time responses. . . . . . . . . . . . . . . . . . . . . 44

3.5 NMR simulation of water phase in Ferroan Dolomite sample at dif-

ferent saturations. [a] Normalized magnetization decay and [b] trans-

verse relaxation time responses. . . . . . . . . . . . . . . . . . . . . 45

3.6 NMR simulation responses of oil phase drainage into water at dif-

ferent saturations in Bentheimer sandstone sample. [a] Normalized

magnetization decay and [b] transverse relaxation time responses. . 46

3.7 NMR simulation responses of oil phase drainage into water at differ-

ent saturations in Berea sandstone sample. [a] Normalized magneti-

zation decay and [b] transverse relaxation time responses. . . . . . . 47

3.8 NMR simulation responses of oil phase drainage into water at differ-

ent saturations in Ferroan Dolomite sample. [a] Normalized magne-

tization decay and [b] transverse relaxation time responses. . . . . . 48

3.9 Logarithmic mean T2lm response as function of saturation. [a] Berea

sandstone, [b] Bentheimer sandstone, and [c] Ferroan Dolomite. . . 49

3.10 Relative permeability comparison for [a] Bentheimer, [b] Berea and

[c] Ferroan Dolomite samples. . . . . . . . . . . . . . . . . . . . . . 50

3.11 Experimental validation and match of relative permeability derived

from NMR drainage experiments with numerical NMR calculated

relative permeability from tomographic images on [a] Bentheimer and

[b] Berea sandstone samples. . . . . . . . . . . . . . . . . . . . . . . 51

4.1 Tanner NMR pulsed field gradeint stimulated spin-echo sequence for

an applied variable field gradient ga over a constant time interval δ

in the presence of constant background gradient(s) gb, see 4.1-4.4. . 54

xv

4.2 Slices through tomograms and derived phase distributions of the

sandstone samples used in this study. [a-c]: Bentheimer sandstone

(FOV: 9602 voxel, resolution: 2.89μm, total image porosity 0.239,

resolved image porosity 0.232). [d-f]: Berea sandstone (FOV: 9602

voxel, resolution: 2.84μm, total image porosity 0.18, resolved image

porosity 0.179). Left: grey-scale tomograms. [b,c,e,f]: tomograms

segmented into quartz (grey), clay region (dark grey), and pore space

(black and white). The pore space is partitioned into two fluids using

a morphological approximation to fluid distributions, white being the

non-wetting fluid. Wetting fluid saturations are ≈ 50% (middle) and

≈ 25% (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.3 [a] Dipole profile calculated numerically using 4.9 for a sphere dis-

cretised on a cartesian regular grid with spacing ε = 115

a for a given

susceptibility contrast and static magnetic field in excellent agree-

ment with theory 4.6. [b] Sketch of the material distributions and

material property assignments inside the porous partially saturated

rock. Here χi stands for the susceptibility of the ith mineral, which

in general can be a tensor. . . . . . . . . . . . . . . . . . . . . . . . 58

4.4 Internal gradient distributions for Bentheimer sandstone at image

resolution for a 2MHz system (B0 = 470 G). [a] brine saturated (Sw =

100%), [b] dodecane saturated, [c] Sw = 23%, [d] Sw = 50%. . . . . 59

4.5 Internal gradient distributions for Berea sandstone at image resolu-

tion for a 2MHz system (B0 = 470 G). [a] brine saturated (Sw =

100%), [b] dodecane saturated, [c] Sw = 26%, [d] Sw = 50%. . . . . 63

4.6 Effective internal gradient distributions for Bentheimer sandstone at

B0 = 470 G. [a] water saturated (Sw = 100%), [b] dodecane satu-

rated, [c] Sw = 23%, [d] Sw = 50%. . . . . . . . . . . . . . . . . . . 63

4.7 Effective internal gradient distributions for Berea sandstone at B0 =

470 G. [a] water saturated (Sw = 100%), [b] dodecane saturated, [c]

Sw = 26%, [d] Sw = 50%. . . . . . . . . . . . . . . . . . . . . . . . . 64

4.8 Diffusion coefficient distributions for Bentheimer sandstone over dif-

ferent diffusion times. [a] water saturated (Sw = 100%), [b] dodecane

saturated, [c] Sw = 23%, [d] Sw = 50%. The vertical lines indicate

the bulk diffusion coefficients of brine (black) and dodecane (red). . 64

4.9 Diffusion coefficient distributions for Berea sandstone over different

diffusion times. [a] water saturated (Sw = 100%), [b] dodecane satu-

rated, [c] Sw = 26%, [d] Sw = 50%. . . . . . . . . . . . . . . . . . . 65

xvi

4.10 Diffusion-relaxation correlation maps for Bentheimer sandstone for

a diffusion time Δ = 80 ms (top) and Δ = 320 ms (bottom). [a,d]

brine saturated (Sw = 100%), [b,e] Sw = 23%, [c,f] Sw = 12%.

The horizontal lines indicate the diffusion coefficients of water (solid

line) and dodecane (dashed line). The vertical line notes the bulk

relaxation time for both fluids used in the simulation. The sloped

line indicates diffusion-relaxation correlations for alkanes [91]. Note,

that for crude oils a distribution of diffusion coefficients and bulk

relaxation times present in the complex fluid would cause it to follow

the sloped line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.11 Diffusion-relaxation correlation maps for Berea sandstone for a dif-

fusion time Δ = 80 ms (top) and Δ = 320 ms (bottom). [a,d] brine

saturated (Sw = 100%), [b,e] Sw = 50%, [c,f] Sw = 26%. The addi-

tional lines are as in 4.10. . . . . . . . . . . . . . . . . . . . . . . . 66

5.1 Diffusion-relaxation correlation maps for Bentheimer sandstone for

diffusion time of Δ = 480 ms (left) and Δ = 80 ms (right). [a,b]

brine saturated (Sw = 100%), [c,d] Sw = 55%, and [e,f] oil saturated.

The horizontal lines indicate the diffusion coefficients of water (solid

line) and dodecane (dashed line). The vertical line notes the bulk

relaxation time for both fluids used in the simulation. The sloped

line indicates the relaxation-diffusion correlations for Alkanes. . . . 73

5.2 Diffusion-relaxation correlation maps for Berea sandstone for diffu-

sion times of Δ = 480 ms (left) and Δ = 80 ms (right). [a,b] brine

saturated (Sw = 100%), [c,d] Sw = 78%, [e,f] Sw = 40% and [g,h] oil

saturated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.3 Extracted profiles of logarithmic mean relaxation times as function

of water saturation for individual fluids and partial saturated fluids

from NMR T2 − D responses in [a] Bentheimer sandstone, and [b]

Berea sandstone at low field (2 MHz) and echo spacing (tE = 0.4msec). 75

5.4 T2-D maps for water phase drainage response in Bentheimer sand-

stone at low field (2 MHz) and echo-spacing time (tE = 0.4 msec) at a

diffusional time of 480 msec [a] Fully water saturated, [b] Sw = 55%,

[c]Sw = 50%, [d] Sw = 25%, and [e] Sw = 14%. . . . . . . . . . . . . 77

5.5 T2-D maps for water phase drainage response in Berea sandstone

at low field (2 MHz) and echo-spacing time (tE = 0.4 msec) at a

diffusional time of 480 msec [a] Fully water Saturated, [b] Sw = 78%,

[c] Sw = 50%, [d]Sw = 40%, [e]Sw = 34%, and [f] Sw = 17%. . . . . 78

xvii

5.6 Observed NMR relaxation-diffusion response of water phase at dif-

ferent echo-spacing times and low field (2 MHz) in [a] Bentheimer

sandstone, and [b] Berea sandstone. . . . . . . . . . . . . . . . . . . 79

5.7 T2-D maps for capillary drainage of oil phase into water at different

saturations for Bentheimer Sandstone at low field (2 MHz) and echo-

spacing time (tE = 6 msec) at a diffusional time of (tD = 480 msec).

[a] Fully oil saturated, [b] Sw = 55%, [c] Sw = 50%, [d] Sw = 33%,

[e] Sw = 25%, and [f] Sw = 14%. . . . . . . . . . . . . . . . . . . . . 80


saturations for Bentheimer Sandstone at medium field (12 MHz) and

echo-spacing time (tE = 3.5 msec) at a diffusional time of (tD = 480

msec). [a] Fully oil saturated, [b] Sw = 55%, [c] Sw = 33%, and [e]

Sw = 14%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81


saturations for Bentheimer sandstone at high field (400 MHz) and

echo-spacing time (tE = 3 msec) at a diffusional time of (tD = 480

msec). [a] Fully oil saturated, [b] Sw = 55%, [c] Sw = 33%, and [d]

Sw = 25%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.10 Sensitivity analysis of observed NMR relaxation-diffusion response

of oil phase in Bentheimer sandstone at different echo-spacings. [a]

Low filed (2 MHz), and [b] medium field (12 MHz). . . . . . . . . . 85


saturations for Berea sandstone at low field (2 MHz) and echo-spacing

time (tE = 6 msec) at a diffusional time of (tD = 480 msec). [a]

Sw = 17%, [b] Sw = 34%, [c] Sw = 40%, [d] Sw = 50%, and [e]

Sw = 78%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86


saturations for Berea sandstone at medium field (12 MHz) and echo-


[a] Fully oil saturated, [b] Sw = 17%, [c] Sw = 40%, [d] Sw = 50%,

and [e] Sw = 78%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87


saturations for Berea sandstone at high field (400 MHz) and echo-


[a] Sw = 17%, [b] Sw = 40%, [c] Sw = 50%, and [d] Sw = 78%. . . . 88

5.14 Sensitivity analysis of observed NMR relaxation-diffusion response

of oil phase in Berea sandstone at different echo-spacing times and

magnetic fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

xviii

5.15 Relative permeability predictions from NMR relaxation-diffusion mea-

surements at low field (2 MHz) and short echo spacing (tE = 0.4 msec)

for [a] Bentheimer sandstone, and [b] Berea sandstone. . . . . . . . 90

5.16 Relative permeability of the wetting phase from relaxation-diffusion

measurements at different echo-spacings and low magnetic field (2

MHz) for [a] Bentheimer sandstone, and [b] Berea sandstone. . . . . 91

5.17 Relative permeability of non-wetting phase from NMR relaxation-

diffusion measurements for Bentheimer sandstone at different echo-

spacings [a] low field (2 MHz), and [b] medium field (12 MHz). . . . 92

5.18 Relative permeability of non-wetting phase from relaxation-diffusion

measurements for Berea sandstone at different echo-spacings [a] low

field (2 MHz), and [b] medium field (12 MHz) . . . . . . . . . . . . 93

5.19 Relative permeability of non-wetting phase from NMR relaxation-

diffusion measurements for Bentheimer sandstone at longest echo-

spacing allowed and utilizing SDR model with power of 1 at [a] low

field (2 MHz), and [b] medium field (12 MHz). . . . . . . . . . . . . 94

5.20 Relative permeability of non-wetting phase from relaxation-diffusion

measurements for Berea sandstone at longest echo-spacing allowed

and utilizing SDR model with power 1 at [a] low field (2 MHz), and

[b] medium field (12 MHz) . . . . . . . . . . . . . . . . . . . . . . 95

CHAPTER 1

INTRODUCTION

Low magnetic field Nuclear Magnetic Resonance (NMR) measurements both

in wireline logging and conventional core analysis are an essential tool in reservoir

characterization, providing information about pore size distribution and fluid type.

The basic mode of investigation of NMR in porous reservoir rock is the use of

relaxation and diffusion measurements to learn about the structure of the reservoir

rock (”pore size”) and the type of fluids contained within. This eventually leads to

the estimation of the absolute permeability of hydrocarbon bearing reservoirs and

fluid typing. Logging tools developed continuously, with early tools using the earth

magnetic field. Current tools use permanent magnets to assess the NMR response

of hydrogen protons in the reservoir rock. The interpretation of NMR responses is

complicated by the fact that it involves the inversion of a Laplace transform, which

is an ill-conditioned problem. To make progress in the understanding of NMR

responses, random walk algorithms have been developed as early as 1954 [35] to

simulate the NMR response. Today, the availability of high-resolution tomographic

images allows the simulation of NMR responses of reservoir rock with increasing

accuracy.

In this work, we use this capability and numerically test correlations as func-

tion of saturation between NMR response and relative permeability from digital core

analysis. We test three reservoir samples, Bentheimer and Berea sandstones and

a Ferroan-Dolomite carbonate. All samples contain clay minerals and we perform

XRD analysis to identify mineralogy and consider mineralogy effects in the simula-

tion. The remainder of this chapter covers an introduction about NMR in porous

media and a literature survey about uses of NMR to characterize reservoir rock

saturated by multiple fluids, considering both experimental and numerical work,

followed by a detailed problem statement.

Literature survey 2

1.1 Nuclear Magnetic Resonance

Nuclear Magnetic resonance (NMR) is frequently used in the petroleum in-

dustry for reservoir description and characterization as a permeability and fluid

identification tool [26, 2, 97]. The NMR signal of reservoir rock originates from the

proton content of the reservoir fluids, e.g. hydrocarbons and water, and is weighted

by the hydrogen index (HI). When an external magnetic field is applied, the hydro-

gen nuclei preferentially align in the direction of the static magnetic field, resulting

in a net magnetization. When this balance is disturbed, e.g. with a radio frequency

magnetic field radiation applied perpendicular to the static magnetic field, the sys-

tem reacts with a range of typical relaxation times to regain equilibrium. The

frequency here is referred to as Larmor frequency [43], which depends mainly on

the gyromagnetic ratio that is a property of the nuclei; allowing one to differentiate

different nuclei types according to their Larmor frequencies for a given magnetic

field. Hydrogen for example has a fixed gyromagnetic ratio that is different that

other nuclear species. In the case of hydrogen, the larmor frequency will be function

of the strength of the static magnetic field. Following the applied radio frequency

pulse to the system, the nuclear spin system will eventually relax to equilibrium

satisfying the resonance condition [71]. There are typically two relaxation times

recorded from NMR logging tools see (Fig-1.1), namely the longitudinal-relaxation

time T1 and the transverse-relaxation time T2. Industry practice often acquires both

T1 and T2, but T2 measurements are much faster than T1 [80].

The T2 relaxation of a single pore in the weak coupling limit is described by:

1

T2

=1

T2b

+ ρS

V+

D

12(γGtE)2 . (1.1)

Here T2b is the bulk fluid relaxation time , ρ is the surface relaxivity , S/V

is the surface area to pore volume ratio, D is the diffusion coefficient, γ is the

gyromagnetic ratio of the proton, G the magnetic field and tE is the CPMG echo-

spacing [29, 75]. The magnetization decay M(t) of the initial magnetization M(0)

of a reservoir rock can then - in the weak coupling limit – be written as a sum of

exponentials as in:

M(t) =∑

i

aiM0 exp

(−

t

T2i

). (1.2)

Here, ai are the amplitudes of the corresponding relaxation times T2i.

The inversion of the magnetization decay, written as the sum of exponentials,

results in a relaxation time distribution. This distribution is frequently converted

into a pore size distribution using Eqn.1.1 while neglecting bulk relaxation and

Literature survey 3

[a]

[b]

Figure 1.1: Relaxation process of [a] longitudinal relaxation time (T1) and [b] trans-verse relaxation time (T2). In [a], the net magnetization vector is aligned in thez-direction followed by a 180 degrees pulse is applied and the magnetization is tippedto the z-direction. Later, magnetization is recovering to its initial z-direction. Fi-nally, magnetization reaches its initial equilibrium condition. In [b], the net mag-netization vector is initially aligned in the z-direction, the Mxy component is zero,followed by a 90 degrees applied pulse and the magnetization is tipped to the xy-plane. Next, the Mxy is recovering to its equilibrium state and then Mxy reachesits initial equilibrium condition at zero. [59]

dephasing by diffusion. The integral of the distribution gives porosity [43]. A

robust average of the relaxation time distribution Tlm is the logarithmic mean:

Tlm = exp

(∑i ai × log(Ti)∑

i ai

). (1.3)

Tlm of the relaxation time is frequently used in conjunction with porosity

to establish NMR permeability models from both T1 and T2 [112, 126, 76]. The

relaxation time distribution provides a range of short and long NMR T2 that can be

used to classify fluids, where shorter relaxation times allow one to predict regions of

bound fluids from free fluid(Fig-1.2). The mechanism of NMR relaxation responding

mainly to surface area to pore volume ratio implies a direct relationship between

faster relaxation in small pores and slower relaxation in larger pores, and therefore

the two fluid regions of bound and free fluids are clearly identified.

1.2 NMR analysis in fully saturated porous media

In well-logging and reservoir characterization, NMR is often the best logging

choice to predict the absolute permeability of the reservoir if a physical correlation

Literature survey 4

Figure 1.2: NMR magnetization decay and pore size distribution (Westphal et al.2005).

can be established [49, 59, 16]. Other information that can be derived from NMR

analysis includes porosity, irreducible water saturation, free fluid index (FFI), and

residual oil saturation [80, 1]. Furthermore, NMR measurements have been used as

wettability indicator [31, 30, 56, 52]. NMR relaxation measurements are a function

of three parallel processes as described in Eq.1.1. In fully saturated water-wet rock,

we normally neglect the diffusion term and the relaxation times become a function

of the bulk relaxation and the ratio of surface area to pore volume where surface

relaxivity dominates [43].

In fully water saturated reservoir rock, bulk relaxation from NMR core anal-

ysis where external gradients are applied is typically neglected due to low viscosity

of water and the response is highly controlled via surface relaxivity. However, in

fully oil saturated rocks, the relaxation is dominated by bulk relaxation [80]. NMR

measurements have been also adopted in core analysis to calibrate and measure

petrophysical and transport properties [127]. Comparison between porosity, irre-

ducible water saturation, free fluid index and permeability have been successful to

match results from conventional core analysis and centrifuge measurements.

Surface relaxivity is highly related to mineralogy as paramagnetic minerals

tend to have a higher surface relaxivity compared to diamagnetic minerals. In

Literature survey 5

particular, minerals such as iron and chromium have a huge impact on the surface

relaxivity due to their high susceptibility values. Generally, carbonates tend to

have lower surface relaxivity than sandstone reservoir samples due to a lack of

paramagnetic mineralogy. The values of surface relaxivity observed in sandstone

samples are typically larger by about a factor of 3 compared to those found in cabinet

samples [1]. The combination of surface relaxivity and transverse relaxation time

have been used to establish pore throat definition in porous media which relates to

permeability estimation [21].

1.3 Permeability correlations from NMR

NMR has a proved capability of predicting formation permeability indirectly

from well logging and core analysis. A wide range of correlations have been estab-

lished and tested to relate permeability to NMR relaxation responses. The first

attempt to establish a permeability correlation from NMR relaxation was made by

Seever in 1966 [112]. He used the combination of Kozeny-Carmen equation and

incorporated T1 relaxation time to measure permeability from NMR on samples

consisting of quartz powder and water. His correlation showed good agreement

between NMR derived permeability and measured core permeability.

In 1969, Timur [142] has proposed a new term called producible porosity

which according to his argument would improve NMR permeability correlations.

He used the spin-lattice relaxation time T1 used by Seevers and established perme-

ability correlations that employ the effective porosity which is producible based on

the accuracy of free fluid index calculations. Later, Sen et al. in 1990 [115] analyzed

and statistically established permeability correlations from 100 sandstone core sam-

ples. One of these correlations is a function of T1 relaxation in rocks saturated with

water. The correlation also gathers other rock information such as the conductivity

exponent (m), and tortuosity factor defined as φm. The NMR permeability correla-

tion they established has a high regression value (R =0.94) which justified linking

NMR T1 to absolute permeability.

In 1989, Thompson et al. [140] published their work on use of NMR for ab-

solute permeability correlations. Katz and Thompson previously had established

permeability correlations from morphological properties of reservoir rock samples

namely the critical diameter lc [72], arguing that the percolation threshold of injec-

tion from mercury intrusion experiments is directly linked to permeability. Since

the critical diameter presents a pore diameter characteristic, They established cross-

correlations between lc and the spin-lattice relaxation time T1 from NMR measure-

ments. From those correlations, they were able to correlate T1 to absolute perme-

Literature survey 6

ability by replacing the critical diameter with relaxation time measurements. In this

work, several experiments were conducted on cemented glass beads, a sandstone,

and a carbonate sample.

It is favorably now to measure T2 relaxation time since it is much faster and

gives better correlation to permeability for sandstone samples compared to early

seevers models using T1 [80, 59]. The transverse relaxation time T2 permeability

model from Schlumberger-Doll-Research (SDR), or Kenyon SDR model, has be-

come widely used for estimating permeability from NMR logging and core analysis

[75, 76]. The majority of recent published NMR permeability work involves use of

T2 relaxation time as basis of permeability calculations [26, 127, 21, 15]. Kenyon’s

model was further applied to carbonate samples [152] by modifying the exponents

and resulted in good agreement between NMR permeability with laboratory mea-

surements. Importantly, some carbonates are vuggy and a careful cut-off analysis

should be carried out to subsequently improve NMR permeability correlations.

kNMR = a × T 22lmφ4 . (1.4)

In general, cut-off analysis is applied on NMR relaxation measurements to

eliminate the bound fluid region from the T2lm calculated value for permeability

estimates, as it would tend to enhance the permeability correlation [107, 59]. This

cut-off analysis and appropriate values have been derived by laboratory core analysis

measurements based on capillary pressure curve analysis. The industry is using

currently around 33 ms as cut-off value for bound fluids mainly in clastics [127, 107],

other reported the value of cut-off between 10 ms to 30 ms as they identified other

influential factors like iron contents [122].

1.4 Analysis of NMR in partially saturated porous media

Considerable amount of published work focused mainly on NMR relaxation

responses and cut-off analysis for the sake of absolute permeability estimation.

In addition, there has been a reasonable amount of research published on partial

saturations. Banavar and Schwartz in 1987 [26] were the pioneers to perform NMR

measurements for partial saturations and emphasized that the oil industry should

analyze NMR on mixtures of multiple fluids as in oil and water due to the nature of

hydrocarbon reservoirs. Their work on partial saturations illustrated the behaviour

of ratios of longitudinal relaxation times at multiple saturations to the fully water

saturated NMR relaxation response, which they called relative lifetime. The sample

was drained by simply injecting air to displace water from a sandstone sample.

Straley et al. in 1991 [125], were the first to systematically measure NMR responses

Literature survey 7

at partial saturations. They used a 10 MHz NMR instrument on four clay rich low-

permeability sandstone samples and measured the longitudinal relaxation times at

multiple saturations by allowing drainage using non-hydrogen fluids such as air and

Kerosen. From this experiment, it was noted during injection of air to drain water

that the long peak of T1 disappears as saturation decreases and the short T1 remains

the same. At the end of the experiment, good agreement was reached between the

derivation of free fluid index by NMR against centrifuge experiments.

In 1993, Chen et al. [38] conducted a drainage experiment on a Bentheimer

sandstone sample and measured NMR spin-lattice relaxation measurements T1 as

function of saturation. They injected nitrogen at different pressures to drain water

and measure the relaxation time at each corresponding saturation. The experiment

focused mainly on the effect of surface relaxation while draining the sample to

obtain a relation between pore size distribution and T1 relaxation time. It was

observed that during drainage, the amplitude of peaks at long relaxation times

reduces corresponding to largest pores, while the amplitudes get smaller at smaller

relaxation times. This led to a power model describing the decrease of relaxation

times with decreasing saturation. No relative permeability was derived directly

from the wetting phase, but it was suggested that this is feasible.

In addition to conventional NMR experiments, NMR saturation imaging was

adopted into petrophysical applications and reservoir core fluid assessment [106,

25, 42, 83]. Kulkarni et al. [83] utilized NMR imaging apparatus to visualize the

saturation profiles of multi-phase displacements experimentally. They performed a

drainage experiment on a Texas Cream Limestone sample and used hexadecane as

oil phase and deuterium oxide (D2O) as the wetting phase. Since deuterium oxide

does not give an NMR signal, the experiments only observed the oil signal. The

results from their experiment led to rough estimation of oil-water flow functions

like relative permeability and capillary pressure. However, the experiment had

limitations to water saturations above 40 percent.

Relative permeability predictions from NMR was initially predicted experi-

mentally by Chen et al. in 1994 [41]. In their work, the wetting phase relative

permeability from three reservoir samples, Bentheimer and Berea sandstones in

addition to a limestone carbonate sample. Using a 2 Tesla NMR equipment, the

previous experiment [38] was repeated and the relationship by Katz and Thom-

spon [73] in combination with Archie’s law [8] was utilized as basis model for the

predictions. In Katz model, spin-lattice relaxation times was related to critical di-

ameter and this utilized instead in the correlation. This has successfully led to the

derivation of sensible wetting phase relative permeability from NMR measurements

in the Bentheimer sandstone sample used. Similarly, Xue [155] used a 50 MHz

Literature survey 8

NMR imaging experiment to determine fluid saturations from a sandstone sample.

Spatially resolved longitudinal relaxation times T1 for full and partial saturations

were measured by displacing water with nitrogen from the sample. Xue adopted

Chen’s model described earlier [41] to extract the relative permeability curve of

the wetting phase. Ioannidis et al. in 2006 [69] repeated Xue’s experiment using

transverse relaxation time T2. Using a 500 MHz NMR on a glass beads model filled

with degassed and deionized water, NMR transverse relaxation measurements were

collected as gravity drainage took place allowing water to drain from the system

and saturation was measured accordingly. They used the correlation found by Chen

et al. in 1994, but replaced spin-lattice relaxation times with T2 as a linking pa-

rameter. Their work remarkably provided evidence that NMR is highly capable

in predicting relative permeability of just the wetting phase in partial saturated

porous media samples.

1.5 NMR analysis for fluid typing T2-D

The use of NMR interpretations for reservoir description and fluid typing has

been of great interest due to its non-destructive and passive means of providing in-

formation about pore size distribution, permeability, bound and free fluid indices in

addition to differentiating water from hydrocarbon fluids. For multi-fluid systems

e.g. water, oil and gas, several techniques were used to optimize and distinguish

each fluid. One of the earliest works was by Akkurt et al. in 1995 [2, 4], who

highlighted the impact of NMR in gas reservoirs due to the fact that porosity mea-

surements in gas bearing reservoirs tend to read lower than neutron and density

logging tools. Using MRIL-C NMR logging tool wihch has an external fixed mag-

netic field gradient (around 17 Gauss/cm), they introduced approaches for fluid

typing mainly to distinguish gas or oil from water by differential spectrum method

(DSM) and shifted spectrum method (SSM). In the DSM method, fluids could have

overlapping signals from spin-lattice NMR measurements but different responses

from spin-spin relaxation measurements in the presence of magnetic field gradient

due to the distinctive properties of fluids from their diffusion properties. The SSM

method is more recognized which consists of having two passes of CMPG sequence

with fixed waiting time and different echo-spacing times. The results from SSM

showed that gas was easily separated from water signal and more accurately quan-

tified. This application is very important in the case of mud invasion that would

influence the NMR fluid typing in dual-fluid systems.

Later, a new methodology of NMR logging was proposed [3] to account for

longer echo-spacing and diffusion effects that would essentially assist in separa-

Literature survey 9

tion of fluids. Akkurt et al. utilized MRIL NMR logging tool which has a broad

choice of gradient strengths and variable echo-spacing time (1 ms and 4 ms). This

Enhanced-Diffusion-Method ”EDM” offers higher accuracy for fluid separation due

to investigation of diffusion effects by obtaining multiple long echo-spacing times to

visualize the effects of separating water and oil signals. The correlation between vis-

cosity and diffusion-coefficients for many reservoir fluids like hydrocarbons is inverse

[159]. Thus, one limitation of this technique might arise for hydrocarbon reservoirs

with viscosities exceeding the range of 1-50 centi-poise, the latter of which is typi-

cal range for intermediate to light oils. The phenomena of EDM is that oil is more

viscous and less diffusive than water and for longer echo-times; water will tend to

have shorter relaxation times making oil separate. similarly, Akkurt et al. analyzed

EDM in core laboratory with an external static field gradient (around 17 Gauss/cm)

on fully and partially saturated Berea sandstone at multiple echo-spacing times and

observed more distinguishable oil phase by increasing the echo-time. This method

is applied in modern logging tools to account for diffusion effects that are decisive

in quantifying and establishing a method for fluid determination [53].

Since last decade, NMR fluid typing techniques featured two-dimensional

maps, initially by plotting spin-lattice relaxation time T1 against spin-spin T2 re-

laxation time [18, 104]. This technique uses the contrast of spin-lattice relaxation

times between water and hydrocarbon fluids and allows better recognition in fluid

typing. More recently, fluid typing by NMR T2-D maps became popular for geosci-

entists and petrophysicists due to higher accuracy in distinguishing fluids via the

relaxation-diffusion NMR responses of fluids [130, 114, 134, 133, 146, 104]. In this

method, fluids often tend to have unique identities such as diffusion coefficient. The

principle lead in NMR fluid typing is to acquire multi-repeats of diffusional time

and echo-spacing time tE. Some limitations in NMR T2-D maps might occur in the

presence of heavy oil and irreducible water saturations as they would be challenging

to distinguish.

In flow measurements such as core drainage and imbibition, NMR T2-D mea-

surements have been utilized to characterize static fluid distribution, and thus to

provide insight of fluid behavior during mechanisms of fluid flow [63], with Toumelin

et al. [147] also examining wettability effects from the same measurements. Huer-

limann et al. have published a significant amount of work on NMR fluid typing

including associated effects like restricted diffusion [64, 66] and measurements in the

presence of inhomogeneous fields as in internal gradients which will be explained

[65]. Their work has been well-recognized in this field. Using CPMG sequence at

different echo-times on a variable choice of Berea and Bentheimer sandstone sam-

ples, they obtained NMR response during static flow measurements on core samples

Literature survey 10

while using low-field NMR measurements (2 MHz). They employed the same prin-

ciples of diffusion coefficients and relaxation time and constructed two-dimensional

NMR maps of relaxation-diffusion which enabled visual to phase behavior during

dynamic flow of oil and water (figure-1.3).

[a] [b]

Figure 1.3: NMR Two-Dimensional Measurements during Drainage and Imbibition[a] Bentheimer Sandstone, and [b] Berea Sandstone (Fully Water on top, Fully Oilin middle and spontaneous-Imbibition in bottom(Huerlimann et al. 2002)

An important aspect in NMR relaxation-diffusion measurements are internal

gradients and restricted diffusion. Both may cause an encoding time dependence in

the diffusion coefficients resulting in apparent diffusion coefficients. The latter may

be used to characterize pore geometry since the restriction of fluids to diffuse occurs

due to geometrical restrictions from the surrounding walls of pore space [64, 66].

1.6 Internal Gradients

In porous media including reservoir rock samples, the application of an exter-

nal magnetic field leads to internal gradients. These result from the differences of

magnetic susceptibility of minerals found on the surface or formed within the matrix


of the solid phase to the susceptibility of reservoir fluids. These internal gradients

are important in the interpretations of Nuclear Magnetic Resonance (NMR) mea-

surements of sandstone or carbonate samples. As stated by Hunt et al. [62] ”all ma-

terials have magnetic susceptibility”. Materials to be grouped into three categories

when exposed to a magnetic field: diamagnetic, paramagnetic and ferro-magnetic.

Diamagnetic materials show a weak response to the magnetic field and have neg-

ative susceptibility values. In contrary, paramagnetic material are those of strong

positive susceptibility response. Ferro-magnetic behavior which is strongly param-

agnetic when exposed to a magnetic field can be found in iron bearing minerals in

addition to Magnesium (Mn) and Copper (Co) [1].

For minerals as found in composition of reservoir rocks such as (quartz, calcite,

dolomite) and clay samples like (illite, siderite, chlorite, hemetite), a wide range of

susceptibility values can be observed. Fluids also exhibit magnetic susceptibility

and it was confirmed that all reservoir fluids and formation waters are diamagnetic

as reported in [70, 101] where a comprehensive survey covering most minerals and

fluids found world–wide in hydrocarbon reservoirs (figure-1.4)

Internal gradients have been investigated in NMR relaxation and diffusion

measurements. Published works [131, 65, 66, 154, 40] highlighted benefits of pres-

ence of this inhomogeneity to better characterize fluids in fluid typing applications

and to better overcome issues in NMR interpretations such as restricted diffusion.

The induced local magnetic field from susceptibility contrast will cause enhancement

of transverse relaxation time as David Bergman theory [86] is widely used to model

the relaxation rate change due to existence of both external and internal magnetic

field. It was noted from Bergman theory that small grain samples need only small

contrast in susceptibility for internal gradient effects to dominate the relaxation

rate. For larger grain size samples, small external gradient field are needed to affect

the relaxation time changes. This highlights that internal gradients are dominant

in small pores rather than larger ones. In the diffusion regime, it was observed

that fluids need short distance to travel in smaller pore space to be influenced by

magnetic field whereas in large pores external fields are dominant and diffusion of

fluids will take longer. In NMR measurements, the choice of avoiding effects from

internal gradients is controlled using a very short echo-spacing to minimize or elim-

inate that effect on relaxation time measurements [154]. Lauren Burcaw in 2010

[33] modeled and simulated the internal field gradients on pack of sphere of around

49 micro-meter radius. Her analysis and model incorporated the model by Q. Chen

in 2005 [36], who simulated the internal gradients on partially and fully saturated

Berea sandstone samples and observed that internal gradients are independent of

saturation. The effects of restricted diffusion and internal gradients were presented


Dolomite

Lepidocrocite

Calcite

Vermiculite

Chamosite

Nontronite

Kaolinite

Ilmenite

Magnesite

Crude oil

Formationwater

Halite

Quartz

Montmorillonite

Glauconite

Muscovite

Illite

Siderite

Gypsum

Feldspar

Anhydrite

Chlorite BVS

Chlorite CFS

-30

-20

-100102030405060708090100

110

120

130

Min

eral

s an

d F

luid

s

Mass Magnetic Susceptibility (10 -8 m3/kg)

Dia

mag

neti

cP

aram

agne

tic

Fig

ure

1.4:

Cla

ssifi

cati

onof

wid

era

nge

ofsu

scep

tibility

mea

sure

men

tson

min

eral

san

dre

serv

oir

fluid

s(Iv

akhnen

koet

al.

2004

)


on Bentheimer and Berea sandstone samples [20] where their magnitude for each

sample was calculated according to mineralogy and incorporated such effects in a

sophisticated NMR simulation software. An inversion model was constructed to

assess the effects of internal gradients in NMR T2-D fluid maps and resulted in

clearer separation of gas and water [89, 40].

1.7 Numerical NMR Analysis

Numerical NMR analysis has a huge potential in reservoir characterization to

model and analyze porous media samples to carry out a multi-sensitivity analysis

and to match experimental results performed from conventional methods. In the

latter case, the mechanisms generating a particular NMR response are directly

accessible. Numerical techniques to model NMR response include the random walk

method (RW) [95, 119, 45, 149, 11, 147], which is widely accepted and tested on

tomographic images and pore network representations of porous media. Random

walk method employs a physical principle of spreading random walkers in the pore

space with life time associated with relaxivity of that walker when it reaches such a

boundary as in pore space walls. Applications of numerical simulations of NMR have

exceeded expectations starting from permeability modeling from NMR response on

digital core analysis [59, 21, 15, 17] and fluid typing [66, 147, 17].

Due to an increase in interest to model NMR simulation on partial saturations

and multi-phase fluids, several numerical work have been published [138, 92, 144]

highlighting observations of the NMR relaxation behaviour in two-phase flow. Ta-

labi [99] analyzed NMR responses on pore network models of sandpacks, Foun-

tainebleau and Berea sandstone samples at partial saturations. His main focus was

to examine and study wettability during drainage. Nevertheless, his work provided

insight to results obtained from numerical approaches such as network modeling

and image based calculations on 3D representation of porous media.

NMR numerical simulations models should consider internal field gradients

effects which can be roughly estimated from the knowledge of mineralogy. Most

of NMR work discussed here inferred to internal gradients by either minimizing or

eliminating effects which were found problematic for the NMR experiments or sim-

ulations. It is crucial to incorporate susceptibility contrast effects in the simulation

as we are showing how significant the influence is to the NMR simulation response.

NMR simulation models for the relaxation responses in heterogeneous magnetic

fields using the CPMG sequence are described [156, 157]. For relaxation-diffusion re-

sponses, constructive NMR simulation models using PFG-CPMG sequence [20, 10]

provided enhanced results on fluid characterization via T2-D measurements due to


analysis of effects of restricted diffusion and internal gradients. In 2011, Oliver

Mohnke et al. [96] simulated numerically a combined Lattice-Boltzmann (LBM)

method and NMR in a dynamic modeling were utilized to assess multi-phase flow.

They analyzed a micro-CT image of a reservoir rock and obtained the NMR re-

sponses at multiple saturation stages following drainage experiment. In their work,

an uncalibrated NMR permeability model of Kenyon [76] was used to establish the

relative permeability curve of the wetting phase.


1.8 Problem statement

For NMR relaxation measurements, a literature review on partial saturations

has addressed some results on correlations between NMR response and relative

permeability of only the wetting phase. The correlations were dependent on the

observation of surface to volume ratios that usually dominate water response char-

acteristics. In addition, the results from these analysis did not refer to presence or

effects of internal gradients in the interpretations which are crucial. When internal

gradients in literature survey was incorporated, there was no interpretations of NMR

relaxation measurements with respect to establishing correlations such as to relative

permeability. In particular, we found no work on NMR responses at partial satura-

tions considering non-wetting phase relative permeability predictions. The effects

of susceptibility contrast emanating from minerals and fluids on relative permeabil-

ity are poorly understood. For the relaxation-diffusion measurements, extensive

literature outlined the internal gradients and restricted diffusion effects, but NMR

simulations were found limited in quantifying some input such as internal gradients

magnitude as part of simulation methodology. In NMR T2-D measurements, there

is an absence of work related to establishing correlations between NMR diffusion–

relaxation measurements and relative permeability for both water and oil phases.

Due to the high accuracy in fluid identification and separation offered by T2-D NMR

measurement, one can find an approach to relate fluid relaxation responses to rel-

ative permeabilities. The remainder of this thesis will progress as follow: Chapter

two introduces the numerical methods used in this thesis including the derivation of

morphological properties from image analysis in addition to transport properties of

the reservoir rock such as porosity, permeability, relative permeability and Nuclear

Magnetic Resonance (NMR) responses at low field (2 MHz).

Chapter three considers correlations between NMR response from spin-spin

relaxation (T2) simulations using CPMG sequence and relative permeability as func-

tion of saturation. We perform low field (2 MHz) NMR numerical analysis on

high resolution tomographic images from three reservoir samples, Bentheimer and

Berea sandstones and a Ferroan Dolomite carbonate. In addition to external ap-

plied magnetic field gradient, we also account for internal magnetic gradients from

susceptibility contrast between solid and fluid interfaces. The Schlumberger-Doll-

Research (SDR) NMR permeability correlation is modified to accommodate the

effective NMR responses of each phase. For absolute permeability, NMR response

is simulated for each phase separately at full saturations. For partial saturations,

we switch off the hydrogen index of one phase at a time and obtain a profile of each

phase at multiple saturations. The ratios from partial saturations (SDR) model


to the absolute ones shall provide the relative permeability predictions from NMR

response for all three samples.

In Chapter four, NMR relaxation-diffusion measurements numerically quan-

tify the effects of restricted diffusion and internal gradients in multi-fluid saturated

media. Similar NMR simulation code as to the previous chapter will facilitate

the measurements with an additional kernel for fluid-encoding. Pulse-field-gradient

(PFG) is added to CMPG sequence in succession and effects of internal gradients

and restricted diffusion discussed and presented.

In Chapter five, we test correlations between NMR response from relaxation-

diffusion (T2-D) measurements and relative permeability as a function of satura-

tion. Initially we simulate low field (2 MHz) NMR responses for T2-D analysis on

the samples at short echo-spacing time (tE = 0.4 msec). We simulate drainage for

two-phase flow to provide insight for the profile of relaxation time responses as func-

tion of saturation. Later, we simulate the NMR responses for each phase separately

and calculate the relative permeability using SDR model explained earlier. The

intention from using short echo-spacing time is to initially see the strength of cor-

relations between NMR relaxation time responses with relative permeability when

the effects of internal gradients are minimal. Following that, we start increasing

the echo-time spacing and observe the dependency of diffusion induced relaxation

from the presence of internal gradient fields for both phases. Finally, we conduct

the same numerical experiment using higher fields of (12 MHz) and (400 MHz) to

investigate the survival of the correlations for both phases. The findings from this

work provide a much better understanding of NMR relaxation responses in partial

saturated porous media when influenced by internal gradients and high magnetic

fields with the aim of accurate characterization of static and dynamic reservoir prop-

erties and potentially hydrocarbon recovery. Finally, Chapter six closes the thesis

with conclusions and recommendations for future work and study.

CHAPTER 2

METHODS

2.1 High resolution micro-CT

In reservoir characterization, conventional core analysis is routinely used to

calibrate wireline logs and also to measure important transport and petrophysical

properties such as permeability and relative permeability, formation factor, Archie’s

exponents of cementation and saturation. A fair amount of uncertainty arises from

conventional core analysis because measurements are global for the full core and it

is difficult to understand the mechanisms of correlations from single measurements.

Furthermore, for heterogeneous rocks it is often impossible to access whether the

core measurements are representative, since heterogeneity is not accessed. To do

so, medical scanners were introduced in the petroleum industry to visualize the

density of the core plugs by slicing core plugs and measuring porosity; however, the

resolution obtained by these medical scanners is in the range of a couple of millime-

ters. This has led to the introduction of purpose-build high resolution micro-CT

scanners into the reservoir characterization workflow, which can construct 3D im-

ages of reservoir rock samples at resolutions of up to about 2 microns [108, 81]. In

modern CT scanners (Fig-2.1), high resolution projections of the sample, which is

mounted on a rotation state, are captured at different angles via a CCD camera. A

reconstruction software is used to process these images to reconstruct the sample

resulting in a tomogram, a 3D volume of x-ray CT intensity [108]. Once the tomo-

gram is reconstructed, further steps of filtering and phase identification are carried

out with a specialized software.

2.2 CT scan tomographic image processing

In order to numerically calculate the transport and petrophysical properties

on the CT images, accurate image processing must be performed to eliminate arti-

facts and define the pore and solid phases. Starting with the raw image, filters are

applied on the image to remove noise and sharpen the image [118]. X-ray intensity

histograms are used to validate filtering and processing of raw images. Initially, an

Methods 18

Figure 2.1: X-ray CT apparatus

anisotropic diffusion (AD) filter is applied to reduce the noise level on the image

while preserving phase boundaries. After utilizing filters, the intensity histograms

of original and filtered images are analyzed to examine differences and to judge

whether the application of the chosen filters is likely to result in a better defini-

tion of pore space and solid phases (Fig-2.2). When the pore phase is accurately

estimated, a two-phase segmentation of the tomographic image is carried out. Fur-

thermore, a three-phase segmentation is required when more phases are present such

as clay, microporosity, and/or dolomitization. The solid phase resulting from the

two-phase segmentation will undergo further analysis and an intermediate density

phase is split off, representing e.g. clay regions, lower density minerals, or micro-

porous regions [118]. This division is reached by growing from known regions of low

or high density using a watershed algorithm with a speed function incorporating

X-ray density and gradient. This will ultimately result in the three-phase segmen-

tation of pore, solid and intermediate phase (Fig-2.3, 2.4, 2.5, and 2.6). Here we

consider 50% of the intermediate phase porous. Once the confidence level is high

that image processing is complete, the sample is ready for the further analysis of

morphological and physical properties. This stage is computationally intensive.

Two main approaches are used in the industry to calculate petrophysical and trans-

port properties from digital core images: one is the direct analysis on the images

[14, 54, 137, 22, 82] and the other is by extracting pore network [100, 24, 23]. Our

approach is to work directly on the analyzed segmented tomographic image.

2.3 Transport and petrophysical properties calculation

In our numerical analysis performed on high resolution tomographic images

of reservoir rock samples, the computations of physical properties on large tomo-

graphic images directly is a specialized one. The software was developed in over the

past 15 years, in a collaboration between the School of Petroleum Engineering of

the University of New South Wales (UNSW) and the Department of Applied Math-

ematics of the Australian National University (ANU). The majority of the methods

Methods 19

Fig

ure

2.2:

Inte

nsi

tyhis

togr

amof

X-r

ayC

Tsc

anned

imag

eaf

ter

apply

ing

anis

trop

icdiff

usi

on(A

D)

filt

er,w

hic

his

the

mai

nfilt

erto

smoot

hunifor

mre

gion

s.T

he

inte

nsi

tyof

solid

and

por

eis

pre

sente

dby

the

hig

hpea

kre

pre

senti

ng

solid

and

low

pea

kfo

rth

epor

ephas

e.

Methods 20

[a]

[b]

Figure 2.3: Processing of tomographic images. [a] Raw image captured on Savon-niere limestone and [b] three-phase segmented image of Savonniere. Here blackis the pore space, green represents the intermediate phase where solid and poreco-exist, and red presents the solid phase of the rock sample.

Methods 21

[a]

[b]

Figure 2.4: Processing of tomographic images. [a] Raw image captured from Ben-theimer sandstone and [b] three phase segmented image of Bentheimer sandstone.Here black is the pore space, green represents the intermediate phase and clay re-gions where solid and pore co-exit, and red presents the solid phase of the rocksample.

Methods 22

[a]

[b]

Figure 2.5: Processing of tomographic images. [a] Raw image captured from Bereasandstone and [b] three phase segmented image of Berea sandstone. Here black isthe pore space, green represents the intermediate phase and clay regions where solidand pore co-exit, and red presents the solid phase of the rock sample.

Methods 23

[a]

[b]

Figure 2.6: Processing of tomographic images. [a] Raw image captured from FerroanDolomite and [b] three phase segmented image of Ferroan Dolomite. Here black isthe pore space, green represents the intermediate phase and clay regions where solidand pore co-exit, and red presents the solid phase of the rock sample.

Methods 24

of analyzing and calculating petrophysical properties from 3D image analysis have

been published. We will introduce the methods relevant to this work to illustrate

the feasibility of numerical analysis as applied to petroleum research and industry

areas.

2.3.1 Porosity

When tomographic images are analyzed and segmented into two or three

phases, porosity can be directly defined. The porosity is easily determined and

calculated by counting the number of voxels directly on the image [11, 12]. For

voxels of intermediate phase, the individual voxels will count toward the porosity.

For voxels in the clay region, we use φclay = 0.5.

φt =1

N

N∑i=1

φi . (2.1)

2.3.2 Mercury intrusion capillary pressure (MICP)

Mercury intrusion experiments on cores result in the determination of cap-

illary pressure curves, which contain detail about the complex pore structure of

the reservoir samples. Digital core analysis via tomographic images makes the

microstructure directly available and a capillary drainage transform [60] can be

used to simulate partial saturations on dry images (Fig.2.7). Choosing a particular

entry pressure akin to a drainage radius allows one to calculate a target satura-

tion if utilizing the capillary pressure curve. The drainage simulation proceeds by

the non-wetting phase (NWP) invading increasingly smaller pores in a percolation

type approach. The porous medium system is initially fully saturated with wetting

phase (WP) and the capillary pressure at this condition is zero. Once the system is

drained (from WP), it is saturated completely with NWP. This approach employs

zero contact angle between the solid and the non-wetting phase, which makes the

system totally wetting. Also, the approach assumes no irreducible or trapped water

e.g. WP is connected through wetting films. The optimum use of image based

saturations is essential in the numerical calculations of dynamic transport proper-

ties such as relative permeability [22]. Mercury injection capillary pressure (MICP)

curve from digital core analysis is essential in validating segmented images of core

samples before conducting numerical analysis by achieving a match in the Pc curves

between experiment and simulation for the resolved pore space.

Methods 25

2.3.3 Permeability

The permeability of the rock to single phase flow of a Newtonian fluid is nu-

merically calculated on the high resolution 3D digital core images using the lattice

Boltzmann method (LBM) [57, 117, 37, 28]. While LBM methods are compu-

tationally expensive, they have the advantage that complex geometric boundary

conditions can be dealt with effectively. Furthermore, the computations can easily

be carried out in parallel if long-range interactions can be neglected. consequently,

LBM remains the method choice for flow and transport at the pore scale. The

LBM method here is used on D3Q19 (3-dimensional lattice with 19 possible mo-

menta components) [103]. The implementation of the algorithm is similar to that

detailed by Martys et. al. [93]. The physical boundary condition at the solid-fluid

interfaces with no-slip boundary conditions, which is LBM method is most simply

realized by the bounce-back rule [94]. The pressure gradient acting on the fluid is

simulated by a body force [51].

The Permeability and porosity for the samples used here are in (Table-2.1).

The values for Bentheimer and Berea sandstone samples agree well with experimen-

tal data available in the literature.

2.3.3.1 Relative Permeability

Relative permeability is an important property for the recovery of hydrocar-

bon fluids from producing oil and/or gas reservoirs. The relative permeability in

numerical analysis is calculated using the same approach of permeability explained

earlier. While two-phase lattice Boltzmann methods exist (including our own), the

calculation of permeability or relative permeability on large systems is very de-

manding. Here we use a single-phase approach by deriving the distributions of the

fluids using a morphological approximation using the capillary drainage transform

explained in the previous subsection. Given the distribution of the fluids, for each

fluid a separate simulation is carried out with a body force applied only to the

fluid under consideration, using no-slip boundary conditions for all internal sur-

faces. The direct image based calculations of relative permeability was validated

experimentally on Bentheimer sandstone, leading to a good match between steady

state experiment and numerical calculations (Fig-2.8) using the capillary drainage

transform (CDT) [68].

krw =kw,eff

kabs

(2.2)

kro =ko,eff

kabs

(2.3)

Methods 26

2.3.4 Formation factor

The conductivity calculation numerically is performed using a Laplace so-

lution on the segmented tomographic images with charge conservation boundary

conditions. A gradient is applied in all coordinate directions of the tomographic

image and the conductivity is evaluated using a conjugate gradient technique. The

solid phase is assigned a conductivity value of zero and the pore phase which is

fluid-filled would have a conductivity value of σ = 1. Formation factor is defined as

the ratio of conductivity of fluid filled media at given water saturation to that of

the fluid it self [19]. An agreement was observed between analysis of conductivity

from sandstone samples matching the numerical conductivity results from digital

core analysis [21].

Table 2.1: Summary of Petrophysical Properties of Samples

Sample Class Size (voxel) Resolution Porosity K [mD] FFerroan-D Carbonate 1020 2.19 0.19 642 14.20CSI Berea Sandstone 1080 2.83 0.18 523 19.46Bentheimer Sandstone 800 2.90 0.23 2777 13.23

2.4 Simulation of NMR response on high resolution 3D tomographic images

2.4.1 NMR Simulation

The spin relaxation of a saturated porous system is numerically calculated

using a lattice random walk method [95, 27]. Initially the walkers are placed ran-

domly in the 3D pore space. At each time step i the walkers are moved from

their initial position to a neighboring site and the clock of the walker advanced by

τi = ε2/(6D0), where D0 the bulk diffusion constant of the relevant fluid, reflecting

Brownian dynamics and ε is a small fraction of the voxel size. We treat each random

walk as the movement of a spin packet with initial strength Mw(t = 0).

2.4.1.1 Surface relaxation

At each time step i of length τi, the strength of the walker is reduced by the

survival probability Si, with the strength of the walker at time t =∑

i τi given by

Mw(t) = Mw(0)∏

i

Si . (2.4)

Here Si = SbSs, where Sb = exp (−τi/Tb) for bulk relaxation and Ss = 1−ν for

surface relaxation. For steps within the same fluid Ss = 1. The killing probability

ν is related to the surface relaxivity ρ via

Methods 27

Aν =ρε

D0

+ O

((ρε

D

)2)

, (2.5)

where A is a correction factor of order 1 (here, we take A = 3/2) accounting

for the details of the random walk implementation. This leads to

Ss = 1 −6ρ τi

εA. (2.6)

2.4.1.2 Dephasing due to internal gradients

To capture the dephasing of random walkers caused by internal fields, we

model the phase accumulation of the random walkers for the CPMG sequence ex-

plicitly. For this, one needs to derive the local magnetic field strength. For the

cases considered here the internal field, which results from susceptibility contrast

between minerals and fluids, is accurately described by a dipole approximation,

since the magnetic susceptibilities of all components are small compared to one.

The internal magnetic field is numerically calculated on the tomographic image

by assigning an effective magnetic susceptibility to each voxel and convoluting the

dipole field around the susceptibility field [10, 9]. We calculate the dipole field when

the distance from the dipole center r is larger than the radius of the dipole a is given

as [20]

Bdip =μ0

4π

[3(mr)r − mr2

r5

]. (2.7)

For the dipole field inside the sphere when r is less than a

Bdip =2

3μ0m . (2.8)

Here μ0 is the magnetic permeability of the vacuum and m is the dipole

magnetic field for a unit volume of the lattice spacing (resolution of tomographic

image). If a porous sample is inserted in a static applied field of strength H,

the susceptibility field χ(r) results in an induced internal magnetic field Bint (Fig-

2.9). For sufficiently small susceptibilities this field can be calculated using the

convolution operation

Bint(r) = (χν ∗ Bdip)(r) =

∫R3

χν(r)Bdip(r − r′)dr′ . (2.9)

We calculate the internal fields directly on the tomographic image using a

discrete Fourier transforms implementing the convolution operation according to

Eq.4.9. The phase evolution of a spin with a reference to the Larmor frequency at

the staring position ω0 = g(Bz)(0) is given by

Methods 28

φD = φ − φ0 =n∑

i=1

γτi[Bz(tj) − Bz(0)], tj =

j∑i=1

τi (2.10)

The total magnetization decay including dephasing for an individual spin is

then given by

Mwt(t) = Mw(t) × cos φd . (2.11)

Here, the magnetization is recorded for the echo positions (maximal coher-

ence) of the CPMG sequence. An element-wise sum over the magnetization decays

of the individual walkers finally results in the total magnetization decay.

2.4.1.3 Modeling parameters

The important solid and fluid material parameters for NMR modeling are

surface relaxativities, and fluid properties like hydrogen index (HI), diffusion coef-

ficients (D), and bulk fluid relaxation times. Also, magnetic susceptibilities of the

minerals and fluids are essential for estimating internal field gradients. For surface

relaxivities we use the values published by [135, 90]. We use as susceptibilities of

the minerals and fluids the results of a recent literature survey [101]. The values for

the different properties incorporated in the simulation are reported in (Tables 2.2,

2.3, and 2.4). To derive the volume susceptibility of the minerals and in particu-

lar the intermediate clay phase, we carried out XRD analysis for all three sample.

Since most reservoir fluids are diamagnetic [70], we used dodecane as analogue for

the simulation of fluids [10, 20]. We obtained fluid properties from service company

logging catalog [43]. In reality, fluid properties are temperature and pressure depen-

dent [61]. The relaxation time measurements for oil phase for instance is viscosity

dependent and thus highly influenced by temperature. For this work, we assume

ambient conditions and fluid properties are chosen accordingly. In our simulations,

we define the lattice spacing as being a fraction of the resolution of the sample. The

resolution is about 3 microns and we typically fine-grained the system by a factor

of 10 to achieve a good time resolution of the CPMG pulse sequence. We use be-

tween 60000-100000 walkers per simulation. An example of a magnetization decay

and an inverted T2 relaxation time distribution from our simulation is illustrated in

(Fig-2.10).

Table 2.2: Volume magnetic susceptibility (χvol) of minerals and fluids used tocalculate effective susceptibility.

Material χvol [SI, ×10−5]Quartz -1.641Calcite -1.311Dolomite -1.37Kaolinite -1.68Feldspar -0.1695Ankerite 36.1Illite 4.16Water, H2O -0.9035Dodecane, C12H26 -1.2650kppm NaCl brine -0.935Bentheimer Clay region, 50% brine -1.31Berea Clay Region, 50% brine 4.71Ferroan Dolomite Clay Region, 50% brine 2.38

Table 2.3: NMR Simulation input for surface relaxivity values

ρ [micron/sec] Water Clay Solid OilWater 0 10 3 0Clay 10 10 3 0Solid 3 3 0 0Oil 0 0 0 0

Table 2.4: NMR Fluids Input parameters.

Parameter Water Clay OilHI 0.94 0.47 1.04D0 [cm2/s] 2.30E-05 4.10E-06 8.00E-06T2b 1 0.1 1

[a] [b] [c]

[d] [e] [f]

[g] [h] [i]

Figure 2.7: Simulated Drainage of water by oil using capillary drainage mecha-nism numerical technique, [a-c] Bentheimer sandstone ([a] Sw100%, [b] Sw50%, and[c] Sw25%), [d-e] Berea sandstone ([d] Sw100%, [e] Sw80%, and [f] Sw44%), [g-i]Ferroan-Dolomite Carbonate ([g] Sw100%, [h] Sw75%, and [i] Sw25%); (White isthe invading non-wetting phase, Black is the defending wetting-phase, Gray is thesolid-phase, red is the clay region).

0 0.2 0.4 0.6 0.8 1Sw

0

0.2

0.4

0.6

0.8

1

Rel

ativ

e Pe

rmea

bilit

y - K

r

Krw_ExpKro_ExpKrw_imageKro_image

Figure 2.8: Image based relative permeability from numerically simulated fluiddistribution validated with experimental results on Bentheimer sandstone samples(Hussain 2011).

[a]

[b]

[c]

Figure 2.9: Slice through internal magnetic fields of 8003 simulation domain in unitsof B0 for [a] Bentheimer sandstone, [b] Berea sandstone, and [c] Ferroan Dolomite.

[a]0 1 2 3 4 5 6

Time (Sec)0

0.2

0.4

0.6

0.8

1

Mag

netiz

atio

n M

(t)

[b]0.1 1 10

T2 (Sec)0

0.05

0.1

0.15

0.2

Freq

uenc

y

Figure 2.10: NMR simulation response of Bentheimer sandstone using random walkmethod: [a] Magnetization decay response and [b] Inverted transverse relaxationtime distribution.

CHAPTER 3

RELATIVE PERMEABILITY CORRELATIONS FROM NMR T2

NMR is typically used in the petroleum industry to characterize pore size and

identify fluids in fully and partially saturated reservoir samples. While the NMR

relaxation response can be used to estimate the permeability of the rock, it may also

provide information about the fluid distribution for multi-phase systems. This may

lead to the estimation of effective permeability of fluids at partial saturations and

thus relative permeability to assess hydrocarbon recovery. We simulate the NMR

response as function of saturation on tomographic images of Bentheimer and Berea

sandstone as well as a Ferroan Dolomite utilizing a random walk method. Fluid

distributions are simulated using the capillary pressure curves of these samples via

capillary drainage transform (CDT) allowing the calculations of the saturations

directly on the images. The magnetic susceptibility of minerals and fluids is used to

calculate the internal magnetic fields from the material distributions of solids and

fluids. We show that the logarithmic mean of the NMR T2 distribution is a robust

measure of permeability and results in strong correlations between NMR response

and relative permeability of both fluids. The observed relative permeability from

NMR in our work is in excellent agreement with relative permeability calculations

on direct image based using the lattice Boltzmann method (LBM). We compare our

NMR results for the wetting phase to published experimental results on Bentheimer

and Berea sandstone samples and observe strong agreement. Using NMR numerical

calculations, we demonstrate that internal gradients aid the establishment of relative

permeability correlations for the non-wetting phase.

3.1 Introduction

Nuclear magnetic resonance (NMR) is increasingly used to estimate pore size

distribution and to identify fluids [79, 128, 3, 132]. One of the main advantages of

NMR is its ability to give an estimate of permeability [46, 49, 59]. An essential tool

in reservoir description is the estimation of permeability from NMR response by

empirical formula. NMR relaxation time measurements (T2) for CPMG sequence

Relative permeability correlations 35

are a function of parameters such as saturating fluid bulk relaxation time (T2b),

surface relaxivity (ρ) , surface area to pore volume (S/V ), diffusion coefficient (D),

the gyromagnetic ratio of the proton(γ) , the magnetic field (G) and the echo-

spacing (tE) which is the time spacing for the CPMG sequence [29]. For a single

pore in fast diffusion, the NMR T2 relaxation time is described [127, 2] as the

following:1

T2

=1

T2b

+ ρS

V+

D

12(γGtE)2 . (3.1)

In the weak coupling regime, the process of NMR relaxation will generate a

magnetization decay as a function of relaxation time of the individual pores. The

magnetization decay resulting from spin-spin relaxation then can be expressed as

M(t) =∑

(ai)M0 exp

(−

t

T2

). (3.2)

Correlations between NMR responses and permeability were first established

by Seevers [112] using spin-lattice T1 relaxation times, both for laboratory mea-

surements and in the borehole. NMR T2 correlations with permeability have been

introduced by Kenyon [78] and superseded spin-lattice relaxation measurements be-

cause transverse relaxation responses gave better results for sandstones and can be

measured much faster. A wide range of relationships between NMR spin-lattice

relaxation responses and physical properties including permeability are given by

[115, 26, 49, 14]. Kenyon NMR permeability model using transverse relaxation time

T2 became much recognized [127, 152] even for carbonates due to the advantage of

faster acquisition to obtain relaxation measurements.

Jayanth Banavar and Lawrence Schwartz [111] were the first to measure NMR

relaxation responses at partial saturations, but suggested that industry should per-

form NMR measurements for mixtures of oil and water to match the reality of

hydrocarbon reservoirs, which frequently are partially saturated. Their partial sat-

uration measurements depicted the behavior of ratios of spin-lattice relaxation times

at multiple saturations to the NMR response at 100% saturations. Straley et al.

[125] were among the first to address partial saturations directly. Using a 10 MHz

NMR instrument, they performed drainage centrifuge experiments on clay-rich low

permeability sandstone samples by air and kerosene so that only the water phase

contributes to NMR spin-lattice relaxation. He noted during water drainage that

the long peak of T1 disappears as saturation decreases and the short T1 relaxation

time remains the same. These observations led to the derivation of free fluid index

by NMR in agreement with centrifuge experiments.

More recently, NMR responses of partially saturated rocks have been used to

give information about fluid flow [38, 139, 144]. Chen et al. [38] conducted drainage


experiments on Bentheimer sandstone using nitrogen to displace water from the

sample and measure NMR spin-lattice relaxation T1 at different saturations. He

introduced a power law model to describe the decrease in T1 relaxation time with

decreasing water saturations. While not directly deriving a relative permeability

curve for the wetting phase, he suggested that this is feasible.

Further process in the assessment of dynamic flow through reservoir cores

was made with NMR saturation imaging [106, 39, 83]. Kulkarni et al. [83] used

NMR imaging to map saturation profiles of oil and water for a limestone carbonate

sample. This led to an estimate of two-phase flow functions like relative permeability

and capillary pressure. The experiment showed the possibility of deriving relative

permeability via NMR measurements, but the experiment was limited to water

saturations above 40 percent.

In 1994, Chen et al. [41] reported the wetting phase relative permeability

from water-nitrogen experiments at 2 Tesla magnetic fields and room temperature

conditions on Bentheimer and Berea sandstone samples in addition to a limestone

carbonate reservoir sample. From the experiments, profiles of spin-lattice relaxation

time at multiple saturations were generated. The relationship by Katz and Thomp-

son [73] relating T1 to morphological length scales was used in combination with

Archie’s law [8] to derive the relative permeability for the water phase from NMR

T1 measurements. Xue [155] used a 50 MHz NMR imaging setup to determine the

fluid saturations and used Chen’s model [41] to derive the wetting phase relative

permeability on a sandstone sample. Spin-lattice NMR T1 measurements in full and

partial saturations were gathered by displacing water by nitrogen from the sample

and resolving T1 spectra locally. She concluded that her analysis is not validated

and might be subject to flow stability conditions governing relative permeability.

Ionnidis [69] repeated the NMR experiment of Chen and Xue on glass beads for

transverse relaxation and predicted relative permeability successfully using high

field at 500 MHz frequency. He modified Chen model by replacing spin-lattice with

transverse relaxation time T2 in the permeability correlations.

The interpretations of NMR relaxation responses is complicated by the nature

of the Laplace inversion needed. Simulations of NMR responses assist these inter-

pretations and can provide additional insight. The most accurate NMR response

simulations are carried out on high resolution CT images, mainly utilizing random

walk techniques [45, 149, 20]. The work presented by [20, 135, 145] highlighted

advancement in NMR numerical modeling for porous media on high resolution to-

mographic images and 3D pore network models at multiple saturations. Earlier

simulation methods did not account for internal gradients, which would lead to

erroneous results in particular if iron minerals are present [86, 74]. Newer NMR


simulations include internal gradient effects by explicitly calculating the internal

field distributions, e.g. [149, 20, 40].

In this work, we use digital images of a set of samples (Bentheimer, Berea and

Ferroan Dolomite) and calculate the petrophysical properties porosity, permeabil-

ity, and relative permeability directly on the tomographic images. NMR relaxation

responses are simulated at low field (2MHz) as a function of saturation and internal

gradients accounted for. First, NMR relaxation responses are simulated for both

fluid phases at full saturation for both phases (100% water and 100% oil) to calcu-

late absolute permeability of both phases. Then, the NMR response of each phase

at different saturations is simulated. Following the extraction of spin-spin relax-

ation times behavior with saturation, we adopt and modify the Schlumberger-Doll

research (SDR) NMR permeability equation to accommodate the relation of NMR

T2 relaxation response as a function of saturation. Excellent results are observed

matching NMR relative permeability to image based (LBM) relative permeability

in particular the non-wetting phase. Finally, we compare our wetting phase relative

permeability results for Bentheimer and Berea sandstone to published results.

3.2 Methodology

3.2.1 Sample characterization

In this work, we consider three samples: Berea and Bentheimer sandstone

and a Ferroan Dolomite from Middle East. XRD results on these samples revealed

that Berea comprises of 86% quartz and a mixture of different clay minerals (6%

Kaolinite, 3.5% Ankerite, 2.9% Illite and 1% feldspar). The Bentheimer sandstone

has 2-3% of Kaolinite clay mineral. The Ferroan Dolomite contains 90% dolomite

and around 10% Ankerite; this type of dolomite is called Ferroan due to existence

of iron-bearing minerals. All the three samples display a relatively homogeneous

micro-structure with high porosity and permeability. The samples (Fig-3.1) were

imaged utilizing a high resolution X-ray computed tomography facility [109, 110]

at a field of view of 20483 voxel size with an image resolution of around 3 microns.

3.2.2 Image processing and analysis

The sample processing of high resolution tomographic images is a critical

step [118] for the accurate definition of pore and solid volumes respectively. Raw

tomograms exhibit a wide range of different noise types and image artifacts and

set of sophisticated filters are utilized to accurately quantify the solid, pore and

intermediate phases. The final process is the segmentation of the sample into two

or three phases by defining the volumes of pore, solid and clay or micro-porosity from


[a] [b] [c]

Figure 3.1: Segmented tomographic images of [a] Berea Sandstone, [b] Bentheimer,and [c] Ferroan Dolomite Carbonate.

intermediate phase. Sample dimensions and characteristics are given in (Table-3.1).

The reported porosity corresponds to the porosity derived from the tomographic

images after segmentation, assuming that the clay regions have a porosity of 50%.

Table 3.1: Summary of Petrophysical Properties of Samples

Sample Class Size (voxel) Resolution Porosity K [mD] FFerroan-D Carbonate 1020 2.19 0.19 642 14.20CSI Berea Sandstone 1080 2.83 0.18 523 19.46Bentheimer Sandstone 800 2.90 0.23 2777 13.23

3.3 Numerical simulation

3.3.1 Image based fluid saturations

The fluid distributions in the samples are numerically simulated directly on

the voxelated tomographic images via capillary drainage transform [60]. The simu-

lation performs a fixed capillary pressure corresponding to a pore entry, mimicking

the standard mercury intrusion experiment. Partitioning fluids utilizing numerical

analysis provides information that is not accessible by conventional core analysis.

(Fig-2.7) shows different saturations profiles simulated on the tomographic images

for the three reservoir samples used in this work.

3.3.2 Permeability and relative permeability

Absolute and relative permeability are calculated using he lattice Boltzmann

method [57, 117], which is well suited to deal with multiphase flow in complex

geometries. Using the static fluid distribution with no slip boundary conditions,

permeability for each phase is calculated at different saturations. This allows rela-

tive permeability for each phase to be calculated according to


krw =kw,eff

kabs

(3.3)

kro =ko,eff

kabs

(3.4)

The direct image based calculations of relative permeability using simulated

fluid distributions was validated experimentally on a Bentheimer sandstone [68].

The absolute permeability values for the samples (Table-3.1) agree well with the

experimental data for Bentheimer and Berea available in the literature.

3.4 NMR Permeability correlations

NMR is a practical petrophysical tool especially in determining permeability

[142, 115, 102]. Since NMR measurements has been widely used to calculate the

absolute permeability of reservoir rocks; the information that we can get from NMR

response as a function of saturation would enable the calculation of effective perme-

ability of multiphase in partial saturated media. The Schlumberger-Doll-Research

(SDR) NMR permeability correlation found by Kenyon [78] is described as

kNMR = a ∗ T 22lm × φ4

1 . (3.5)

where φ1 is the total porosity of the sample

We modify the NMR permeability equation of Schlumberger-Doll-Research

(SDR) to account for partial saturated samples by calculating the logarithmic mean

T2lm from NMR T2 responses at different saturations. The modified version of SDR

equation becomes

kNMR(Sw) = a ∗ T2lm(Sw)2 × φ42 . (3.6)

Here φ2 denotes the fractional porosity of the effective phases (water/oil) and

T2lm is calculated as function of saturation.

3.5 Results and analysis

3.5.1 NMR response of samples saturated with single fluid

We first simulate the response on Bentheimer, Berea and Ferroan-Dolomite

samples fully saturated with water or oil respectively. The response of the water

phase includes the NMR response of the clay regions, which are considered 50%

water saturated. Consequently, the NMR response of water includes a fast relaxing

component in the clay region and the region around it, which can exchange. The


NMR water response is dominated by surface relaxation, while the oil phase re-

sponse is mainly due to bulk relaxation [1]. (Fig-3.2) illustrates the NMR responses

of the saturated rocks. Bentheimer sandstone exhibits only a small amount of clay

and at the same time shows a weaker susceptibility contrast. Accordingly, the short

relaxation time peak attributed to water saturated clays is weak. For Berea sand-

stone both susceptibility contrast and clay fraction are higher, leading to a much

more noticeable clay peak. For Ferroan dolomite the short relaxation time peak

is caused by a combination of internal gradients and increased surface relaxivity,

making the peak position dependent on echo spacing.

3.5.2 NMR simulation response of partial saturations

NMR responses for partially saturated samples are derived using the satu-

ration methods (capillary drainage) presented before. For water and oil phases,

the simulation parameters are presented in Tables 2.2, 2.3, and 2.4. The wetting

phase (water) exhibits a non-zero surface relaxivity with the solid. However, for

the non-wetting fluid there are neither interactions with the solid surface nor the

wetting fluid. Thus, the non-wetting fluid can only relax by internal gradient effects

or bulk relaxation. We report the NMR response of the water phase at different

saturations in Fig-3.3, 3.4, and 3.5 and the oil phase (Fig-3.6, 3.7, and 3.8 ). For the

water phase starting at fully water saturated state, the relaxation time decreases as

drainage takes place until the samples are fully drained. This is due to large pores

being drained first then followed by the next larger. The decrease in relaxation

time is governed by the surface area to pore volume ratio that water phase signal

is dependent on.

Diffusional coupling effects between clay regions and resolved pore space were

examined in all three samples. As samples desaturate, the clay regions stay coupled

with the free water, since diffusion coupling effects increase. The shift to smaller

relaxation times is explained by the increase surface to volume ratio. As a result

of this diffusional coupling, larger amplitudes of short transverse relaxation times

were visible at lower water saturations. The diffusional coupling effects varied here,

as both Berea and Ferroan Dolomite exhibited larger peaks than Bentheimer and

this could be due to more paramagnetic clays being present compared to Ben-

theimer which contains diamagnetic Kaolinite. These remarks are very crucial in

the consideration of cut-off analysis defining the bound fluid index (BVI) and thus

the accuracy of calculating the irreducible water saturation. Considering a cut-off

value leads to more accurate permeability estimation in the free fluid region.

Consider now the NMR relaxation response of the oil phase. The Bentheimer

sandstone illustrated marginal increase in relaxation time as oil saturation decreases.


[a]0.001 0.01 0.1 1 10

T2 (Sec)0

0.05

0.1

0.15

0.2

Freq

uenc

y

Fully OilFully Water

[b]0.001 0.01 0.1 1 10

T2 (Sec)0

0.03

0.06

0.09

0.12

Freq

uenc

y


[c]0.001 0.01 0.1 1 10

T2 (Sec)0

0.05

0.1

Freq

uenc

y


Figure 3.2: NMR response of fully saturated samples of oil and water. [a] Ben-theimer sandstone, [b] Berea sandstone, and [c] Ferroan-Dolomite.


In the Berea and Ferroan Dolomite samples, incremental increase and wider sepa-

ration in relaxation time of oil phase is seen as oil saturation decreases. In addition

to bulk fluid relaxation in the oil phase, internal field gradients play a significant

role in the relaxation of the oil signal. The magnitude of the internal fields based

relaxation was observed to be weaker in Bentheimer and stronger in Berea and Fer-

roan Dolomite. The increase in relaxation time for Berea and the Ferroan dolomite

with decreasing saturation can be explained by loosing access to smaller pores and

crevices, where internal gradients are stronger. For both phases, we calculated

the logarithmic mean of relaxation time values as function of saturation in all the

samples and generated the profiles for each phase (Fig-3.9).

3.5.3 Relative permeability from NMR relaxation measurements

Both phases absolute permeabilities are calculated using (Eqn.5.2). To ac-

count for the influence of clay on the absolute permeability of the water phase, we

apply cut-off analysis on the logarithmic average mean of relaxation time T2lm the

same way as one would typically for experimental data. The cut-off value used

in Bentheimer and Berea sandstone samples is 33 msec, which is shorter than the

cut off value used for the Ferroan Dolomite sample 67 msec. Relative permeability

correlations from numerical NMR relaxation responses are calculated from the ef-

fective NMR permeability of each fluid at the different saturations from the NMR

response at partial saturations from the profiles of logarithmic mean of relaxation

times of each phase (Fig 5) using (Eqn.5.3). Relative permeability is calculated via

ratios of Eqn.5.3 to Eqn.5.2 for both water and oil phases. Strong correlations are

observed in all three samples (Fig-3.10) between the relaxation responses of both

phases against the relative permeability calculated directly on the tomographic im-

ages of samples by lattice-Boltzmann. This suggests that internal gradients played

a major role in establishing the correlations for both phases but powerfully the

non-wetting phase that was modeled with zero surface relaxation. We validate the

simulation results on Berea and Bentheimer sandstone samples with those measured

experimentally by Songhua Chen [41]. Our results are in excellent agreement to his

results and observed relative permeability of the wetting phase (Fig-3.11). This

shows that numerical simulations can be accurate in modeling fluid flow dynamics

by incorporating all important elements including the presence of internal magnetic

fields.


[a]0 1 2 3 4 5

Time (sec)0

0.2

0.4

0.6

0.8

1N

orm

aliz

ed M

agne

tizat

ion

M(t)

M(t) "Fully Saturated" M(t) 50% SwM(t) 25% Sw

[b]0.001 0.01 0.1 1 10

T2 (sec)0

0.02

0.04

0.06

0.08

Freq

uenc

y

100% Sw25% Sw50% Sw

Figure 3.3: NMR simulation of water phase in Bentheimer sandstone sample at dif-ferent saturations. [a] Normalized magnetization decay and [b] transverse relaxationtime responses.


[a]0 1 2 3 4

T2 (sec)0

0.2

0.4

0.6

0.8

1N

orm

aliz

ed M

agne

tizat

ion

M(t)

M(t) "Fully Saturated"M(t) 80% SwM(t) 44% SwM(t) 20% Sw

[b]0.001 0.01 0.1 1 10

T2 (sec)0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Freq

uenc

y

100% Sw80% Sw44% Sw20% Sw

Figure 3.4: NMR simulation of water phase in Berea sandstone sample at differentsaturations. [a] Normalized magnetization decay and [b] transverse relaxation timeresponses.


[a]0 1 2 3

T2 (Sec)

0

0.2

0.4

0.6

0.8

1N

orm

aliz

ed M

agne

tizat

ion

M(t)

M(t) "Fully Saturated"M(t) 44% SwM(t) 30% SwM(t) 10% Sw

[b]0.001 0.01 0.1 1 10

T2 (sec)

0

0.01

0.02

0.03

0.04

Freq

uenc

y

Sw = 100%Sw = 44%Sw = 30%Sw = 10%

Figure 3.5: NMR simulation of water phase in Ferroan Dolomite sample at differentsaturations. [a] Normalized magnetization decay and [b] transverse relaxation timeresponses.


[a]0 1 2 3 4 5 6

Time (Sec)0

0.2

0.4

0.6

0.8

1N

orm

aliz

ed M

agne

tizat

ion

M(t)

M(t) Fully Oil SaturatedM(t) 25% Oil

[b]0.1 1 10

T2 (Sec)0

0.05

0.1

0.15

0.2

Freq

uenc

y

Fully Oil SaturatedOil Response - Sw 25%

Figure 3.6: NMR simulation responses of oil phase drainage into water at differentsaturations in Bentheimer sandstone sample. [a] Normalized magnetization decayand [b] transverse relaxation time responses.


[a]0 1 2 3 4 5 6

T2 (Sec)0

0.2

0.4

0.6

0.8

1N

orm

aliz

ed M

agne

tizat

ion

M(t)

M(t) 50% SwM(t) Fully Oil Saturated

[b]0.1 1 10

T2 (Sec)0

0.05

0.1

0.15

0.2

Freq

uenc

y

Oil Response 50% SwFully Oil

Figure 3.7: NMR simulation responses of oil phase drainage into water at differentsaturations in Berea sandstone sample. [a] Normalized magnetization decay and [b]transverse relaxation time responses.


[a]0 1 2 3 4 5 6

T2 (sec)0

0.2

0.4

0.6

0.8

1

Nor

mal

ized

Mag

netiz

atio

n M

(t)M(t) Fully OilM(t) 75% Sw

[b]0.1 1 10

T2 (Sec)0

0.05

0.1

0.15

0.2

Freq

uenc

y

Fully OilOil Response 75% Sw

Figure 3.8: NMR simulation responses of oil phase drainage into water at differentsaturations in Ferroan Dolomite sample. [a] Normalized magnetization decay and[b] transverse relaxation time responses.


[a]0 0.2 0.4 0.6 0.8 1

Sw

0

0.5

1

1.5

2

T 2lm

(Sec

)

T2lm "Water"T2lm "Oil"

[b]0 0.2 0.4 0.6 0.8 1

Sw

0

0.5

1

1.5

2

2.5

T 2lm

(sec

)

T2lm "Oil"T2lm "Water"

[c]0 0.2 0.4 0.6 0.8 1

Sw

0

0.5

1

1.5

2

T 2lm

(Sec

)

T2lm "Oil"T2lm "Water"

Figure 3.9: Logarithmic mean T2lm response as function of saturation. [a] Bereasandstone, [b] Bentheimer sandstone, and [c] Ferroan Dolomite.


[a]0 0.2 0.4 0.6 0.8 1

Sw

0

0.2

0.4

0.6

0.8

1

Rel

ativ

e Pe

rmea

bilit

y - K

r

Krw_LBMKro_LBMKrw_NMRKro_NMR

[b]0 0.2 0.4 0.6 0.8 1

Sw

0

0.2

0.4

0.6

0.8

1

Rel

ativ

e Pe

rmea

bilit

y - K

r


[c]0 0.2 0.4 0.6 0.8 1

Sw

0

0.2

0.4

0.6

0.8

1

Rel

ativ

e Pe

rmea

bilit

y - K

r


Figure 3.10: Relative permeability comparison for [a] Bentheimer, [b] Berea and [c]Ferroan Dolomite samples.


[a]0 0.2 0.4 0.6 0.8 1

Sw

0

0.2

0.4

0.6

0.8

1R

elat

ive

Perm

eabi

lity

- Kr

Krw_NMRKrw_NMR_Exp (Chen 1994)

[b]0 0.2 0.4 0.6 0.8 1

Sw

0

0.2

0.4

0.6

0.8

1

Rel

ativ

e Pe

rmea

bilit

y - K

r

Krw_NMRKrw_NMR_EXP (Chen 1994)

Figure 3.11: Experimental validation and match of relative permeability derivedfrom NMR drainage experiments with numerical NMR calculated relative perme-ability from tomographic images on [a] Bentheimer and [b] Berea sandstone samples.


3.6 Conclusions

A random walk method is used to simulate NMR response as function of

saturation on Berea, Bentheimer and Ferroan Dolomite reservoir samples. We per-

form XRD analysis on all the samples to calculate magnetic susceptibility using

the reported literature values of magnetic response to minerals and fluids. Careful

modeling of susceptibility of minerals forming the sample including the clay is essen-

tial as internal gradients from susceptibility contrast between fluid and surrounding

solid interfaces enhances the NMR spin-spin relaxation measurements for both wet-

ting and non-wetting phases. Strong correlations between relative permeability and

NMR spin-spin relaxation response are observed on all three samples. The analysis

of diffusional coupling effects of clay micro-pores should be considered as it would

distinguish bound fluid from free fluid regions. As a result, use of cut-off analysis to

calibrate the logarithmic mean value leads to more accurate permeability estimates

which results in enhancements in relative permeability correlations for the wetting

phase. As oil phase does not relax via surface relaxation, internal gradients from

susceptibility contrast assisted in establishing a surface related/weighted relaxation

mechanism. Our results for Bentheimer and Berea sandstone are in excellent agree-

ment also to previously published work by Songhua Chen from the relation of NMR

spin-lattice and saturation for the water phase.

The work presented here comprise of a systematic approach that shall adopt

NMR experimental procedure. Mineralogy here plays a major role in the simulation

of surface relaxation and mainly the volume susceptibility of the samples at which

the magnitude of internal gradients shall be assessed. The result from this work

might provide hints for estimating dynamic flow relative permeability much faster

and with non-invasive methods like nuclear magnetic resonance.

This work demonstrates that NMR simulation is capable of estimating ac-

curately relative permeability in partial saturated samples. Relative permeability

experiments are very expensive and time consuming. It would be highly desirable

to extend this study to more complex rock, accompanied by experimental measure-

ments.

CHAPTER 4

NUMERICAL ANALYSIS OF TWO DIMENSIONAL NMR T2-D

The NMR relaxation-diffusion response of porous reservoir rock is frequently

used e.g. in oil field applications to extract characteristic length scales of the pore

space or information about the saturating fluids. External gradients are typically

applied to encode for diffusion. In rocks field inhomogeneities due to internal gradi-

ents can even at low field be strong enough to interfere with this encoding. Further-

more, the encoding for diffusion coefficients of the fluids takes a finite amount of

time, during which the diffusing fluid molecules can experience restricted diffusion.

Both effects can combine to make the interpretation of the diffusion dimension of a

relaxation-diffusion measurement difficult. We use Xray-CT images of porous rock

samples to define the solid and fluid phases of the reservoir rock and simulate the full

experimental pulse sequence, taking into account the static applied field, external

gradients, internal gradients as function of susceptibility of each component, and

surface and bulk relaxation properties of fluids and fluid-fluid and fluid-solid inter-

faces. We carry out simulations of NMR relaxation-diffusion measurements, while

explicitely tracking the time dependent diffusion coefficient in each fluid as well as

associated local gradients. This allows us to quantify the influence of restricted

diffusion and internal gradients for common choices of experimental parameters.

4.1 Introduction

NMR methods offer a non-destructive way to infer geometric characteris-

tics and fluid properties of the spatial distributions of proton containing fluids in

the pore space of porous materials. A particular experiment, the measurement

of diffusion-relaxation (T2-D) correlations, has been introduced relatively recently

to the petroleum industry [65, 66] and uses the fact that the diffusion coefficient

of fluids like oil and water are typically quite different, to separate the relaxation

response of the different fluids using the additional diffusion dimension. This ap-

proach combines the application of T2 relaxation analysis [151, 32, 77, 78, 75, 121]

with an encoding for the time-dependent diffusion coefficient of the saturating fluids

T2-D 54

[123, 44, 34, 113, 66], thus combining the advantages of T2 relaxation measurements

with diffusion measurements [52, 63, 154].

Figure 4.1: Tanner NMR pulsed field gradeint stimulated spin-echo sequence for anapplied variable field gradient ga over a constant time interval δ in the presence ofconstant background gradient(s) gb, see 4.1-4.4.

The T2-D experiment consists of two distinct pulse sequences in succession,

a diffusion encoding sequence, followed by a Carr-Purcell-Meiboom-Gill (CPMG)

sequence used for detection of the signal while also recording the T2 relaxation de-

cay. A short time spacing in the CPMG train recording the T2 relaxation is chosen

to suppress diffusion effects by frequent refocussing of the transverse relaxation. In

addition, the diffusion encoding sequence only includes short time intervals dur-

ing which T2-relaxation is present, resulting in a separable kernel describing the

physics of the problem. We consider in particular the pulsed field gradient stim-

ulated echo (PFGSTE) encoding sequence [136], which has advantages in keeping

the kernel separable even for longer diffusion intervals by lessening the effect of

relaxation during the diffusion encoding interval. It is useful for systems for which

the longitudinal relaxation rate T1 is larger than the transversal relaxation rate T2.

The diffusion kernel k1 of the experiment for a homogeneous background gradient

gb and a homogeneous applied gradient ga, assuming free diffusion, is given by

[136, 44, 154]

k1 =M(ga, td)

M(0, td)= exp

[−γ2D

(Cag

2a + Cabga · gb + Cbg

2b

)], (4.1)

Ca = δ2

(Δ + τ −

δ

3

), (4.2)

Cab = δ

[2τΔ + 2τ 2 −

2δ2

3− δ(δ1 + δ2) − (δ2

1 + δ22)

], (4.3)

Cb = τ 2

(Δ +

2τ

3

), (4.4)

where |ga| = ga, |gb| = gb, td = Δ + 2τ is the total diffusion encoding interval time

from the first π2

pulse to the first echo, γ is the magnetogyric ratio, τ is the spacing

between the first two π2

pulses, δ is the length of the gradient pulse, and δ1 and

δ2 are short time preceeding and following the gradient pulse with τ = δ1 + δ + δ2

[44]. Δ is the storage time between the second and third π2

pulse where only T1

T2-D 55

relaxation is present. The detection kernel k2 of the sequence is a standard CPMG

train described by

k2 = exp

[−

ktET2

], (4.5)

with tE being the CPMG echo spacing, and k noting the kth echo, resulting in the

combined kernel K = k1k2. This formulation allows to invert for diffusion coefficient

and relaxation times using a 2D inverse Laplace transform [50, 150]. Here we use

the L-curve method combined with a non-negative least square (NNLS) algorithm

solving an amplitude regularised inversion problem [87, 18].

It is well known that in porous rocks both restricted diffusion and internal

gradients play an important role. Restricted diffusion leads to a reduction of at-

tenuation in an applied gradient field by preventing the diffusing spins to test the

full variability of the inhomogeneous magnetic field [105, 98, 129, 58, 6]. Suscep-

tibility differences between rock and fluid components generate internal magnetic

fields increasing field inhomogeneity, [67, 116, 121, 48, 47, 154] contributing to the

loss of signal attributed to diffusion. Those effects lead to the measurement of

an apparent rather than absolute diffusion coefficient spectrum [154, 55]. The ef-

fects of internal gradients might be accounted for by explicitly encoding for them

[114, 18] or by experimentally suppressing the internal gradients (e.g. [44]), which

is not always possible or practical. We see from 4.1 that the balance between

internal and external gradients can be changed by either adjusting the timing of

the individual terms by influencing the coefficients Ca, Cb, or Cab, or by varying

the strength of the applied gradients or background gradients, the latter e.g. by

varying the static field strength. It should be noted that for internal gradients gb

cannot be assumed to be homogeneous and is a local quantity. This, in combination

with restricted diffusion, results in various subtleties in the interpreation of T2-D

relaxation-diffusion measurements. Significant ones include that the diffusion in

the internal field might concern a shorter time interval than the diffusion encoding

dimension, which includes a T1 relaxation interval and thus acts in general on a dif-

ferent scale if working with a device providing a permanent gradient. Likewise, the

presence of restricted diffusion can easily invalidate the assumption of the Gaussian

phase approximation often used in the interpretation of T2-D maps.

Very high internal gradients with distribution tails exceeding 1000 G/cm have

been reported for reservoir rocks at low fields [67]. It is clear that an accurate ana-

lytic treatment of the T2-D NMR response in porous rocks is not easy [88, 55]. To

aid quantitative treatment it is advantageous to model the NMR response numer-

ically. This allows for controlling physical and experimental parameters explicitly,

while attempting to match experiment as closely as possible. NMR random walk

T2-D 56

simulations go as far back as Carr and Purcell in 1954 [35]. Newer treatments

include effects of relaxation [95, 153, 144, 17], diffusion [156], and internal gradi-

ents [158, 13] in reservoir rocks. In earlier work, the surface relaxivity typically

was treated as a free parameter to match experiments and numerical simulation,

leading to much larger surface relaxivity in numerical simulations compared to ex-

periments. Recently, it has been shown that a consistent set of parameters with a

constant surface relaxivity can be used to match experiments and numerical simu-

lations at different field strengths by explicitly modelling the internal magnetic field

effect in complex geometries [13].

In this work we extend the approach of [13] for the T2 relaxation response

to T2-D responses. We use high-resolution Xray-CT images [109, 17] to define the

distribution of the solid and fluid phases of the reservoir rock, and simulate the full

experimental pulse sequence, taking into account the static applied field, external

gradients, internal gradients as function of the susceptibility of each component,

and surface and bulk relaxation properties of fluids and fluid-fluid and fluid-solid

interfaces. We directly compute the restricted diffusion coefficient spectrum and

apparent diffusion coefficient spectrum with and quantify the change to the diffu-

sion spectrum under the influence of internal gradients for Berea and Bentheimer

sandstone.

The chapter is organised as follows: in the next section we introduce the

samples used in this study, followed by a description of modelling of the sub-micron

structure. We then summarize the calculation of physical properties. Finally, we

present results followed by a discussion and conclusion section.

4.2 Methodology

4.2.1 Image acquisition and processing

In this study we consider two sandstone samples, Bentheimer sandstone and

Berea sandstone, which are often used as benchmark rocks in the petroleum indus-

try. Both samples were 5mm cylindrical plugs and were imaged on the ANU Xray-

CT facility at about 3μm resolution. In addition, XRD analysis was performed

to determine the mineral content of the samples. Slices through the Xray-density

fields of the samples are depicted in the left column of the 4.2. To simulate different

fluid saturations inside the pore space, we use the covering radius map [141] as

approximation for a long-time equilibrium distribution, which might be experimen-

tally achieved by a slow imbibition process and/or fluid redistribution by diffusion

(middle and left columns of 4.2). Alternatively, direct imaging of fluid distributions

might be used [148, 85, 84].

T2-D 57

[a] [b] [c]

[d] [e] [f]

Figure 4.2: Slices through tomograms and derived phase distributions of the sand-stone samples used in this study. [a-c]: Bentheimer sandstone (FOV: 9602 voxel,resolution: 2.89μm, total image porosity 0.239, resolved image porosity 0.232). [d-f]: Berea sandstone (FOV: 9602 voxel, resolution: 2.84μm, total image porosity0.18, resolved image porosity 0.179). Left: grey-scale tomograms. [b,c,e,f]: tomo-grams segmented into quartz (grey), clay region (dark grey), and pore space (blackand white). The pore space is partitioned into two fluids using a morphologicalapproximation to fluid distributions, white being the non-wetting fluid. Wettingfluid saturations are ≈ 50% (middle) and ≈ 25% (right).

4.2.2 Internal magnetic field calculation

The internal magnetic field is calculated in a dipole approximation [120] by

assigning each voxel in the image an effective isotropic magnetic susceptibility and

convoluting the dipole field with the susceptibility field. The dipole field Bdip for

r ≥ a, a radius of the dipole, r distance from the dipole center, is given as

Bdip =μ0

4π

[3(mr)r − mr2

r5

](4.6)

for a unit volume. Here μ0 = 4π×10−7 N · A−2 is the magnetic permeability of the

vacuum and m is the magnetic dipole moment for a unit volume V = ε3, with ε

being the lattice spacing

m = μ0

χν

1 + χν

B . (4.7)

Inside the sphere (r ≤ a) we have

Bdip =2

3μ0m . (4.8)

T2-D 58

If a porous sample is inserted in a static applied field of strength H, the susceptibility

field χ(r) results in an induced internal magnetic field Bint. For sufficiently small

susceptibilities this field can be calculated using the convolution operation

Bint(r) = (χν ∗ Bdip)(r) =

∫R3

χν(r)Bdip(r − r′)dr′ . (4.9)

[a] [b]

Figure 4.3: [a] Dipole profile calculated numerically using 4.9 for a sphere discretisedon a cartesian regular grid with spacing ε = 1

15a for a given susceptibility contrast

and static magnetic field in excellent agreement with theory 4.6. [b] Sketch of thematerial distributions and material property assignments inside the porous partiallysaturated rock. Here χi stands for the susceptibility of the ith mineral, which ingeneral can be a tensor.

Given the material distributions from Xray-CT analysis and fluid saturation

simulations (see 4.3.b), and the constituents of the samples from XRD analysis, we

can calculate the internal fields at a desired field strength 4.9. The relevant magnetic

susceptibilities are summarized in 4.1. We calculate the effective susceptibilities of

the clay regions assuming a brine fraction of 50% and using a volume weighted

average for χclay:

χclay =

∑Viχi∑Vi

. (4.10)

The resulting intrinsic internal gradient distributions at image resolution can be

calculated using

|gb| =

√(∂Bint,z

∂x

)2

+

(∂Bint,z

∂y

)2

+

(∂Bint,z

∂z

)2

, (4.11)

assuming that the applied static magnetic field B0 is oriented in z-direction and

|B0| � |Bint|. We show the internal gradient distributions for a field strength of

B0 = 470 G (2MHz) in 4.4 (Bentheimer sandstone) and 4.5 (Berea sandstone) for

different fluid saturations. The maximal internal gradients calculated for Berea

T2-D 59

Table 4.1: Mineral and fluid susceptibilites used to calculate effective susceptibilities[101]. The effective susceptibility of Bentheimer sandstone is calculated assuming amixture of Kaolinite and brine with equal volume percentage. For Berea the XRDanalysis gave 86.4% Quartz, 6.2% Kaolinite, 2.5% Ankerite, 1% Rutile, 2.9% Illiteand/or Mica, and 1% Feldspar.

Material χm(10−8m3kg−1) χν

Quartz, SiO2 -0.6191 -1.641Kaolinite, Al2[Si2O5](OH)4 -0.6474 -1.68Ankerite, Ca(Fe++,Mg,Mn)(CO3)2 120 36.1Rutile, TiO2 5.9 2.384Illite and/or Mica, Iron ¡ 2% 15 4.16Feldspar (albite) or orthoclase, KAlSi3O8 -0.67 -0.1695Water, H2O -0.9051 -0.9035Dodecane, C12H26 -1.68 -1.2650kppm NaCl brine -0.935Bentheimer clay region, 50% brine -1.31Berea clay region, 50% brine 4.71

[a]10-3 10-2 10-1 100 101

gb [G/cm]

0

0.01

0.02

0.03

0.04

0.05

freq

uenc

y

brine

[b]10-3 10-2 10-1 100 101

gb [G/cm]

0

0.01

0.02

0.03

0.04

0.05

freq

uenc

y

dodecane

[c]10-3 10-2 10-1 100 101

gb [G/cm]

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

freq

uenc

y

brinedodecane

[d]10-3 10-2 10-1 100 101

gb [G/cm]

0

0.01

0.02

0.03

0.04

0.05

0.06

freq

uenc

y

brinedodecane

Figure 4.4: Internal gradient distributions for Bentheimer sandstone at image res-olution for a 2MHz system (B0 = 470 G). [a] brine saturated (Sw = 100%), [b]dodecane saturated, [c] Sw = 23%, [d] Sw = 50%.

T2-D 60

sandstone of about 90 G/cm are consistent with experimentally derived maximum

gradients for Berea 400/500 of about 75 G/cm [63]. We emphasise that we used

an average susceptibility for the clay regions. Larger scale susceptibility contrast

and heterogeneity would result if the clay minerals are distributed differently with

potentially higher maximum gradients. No diffusion weighted average is taken here.

Both Bentheimer and Berea sandstone have some clay fraction, which is modelled

as a brine saturated effective medium with 50% porosity. For Berea this effective

medium is paramagnetic, while for Bentheimer the clay regions are diamagnetic,

leading to much stronger susceptibility contrast in the Berea sandstone. In addition

we note that the gradient distributions for pure fluids are quite similar in shape due

to the similar susceptibilities. Distinct partitions into low and high gradient regions

result from the partitioning of the pore space into wetting (brine) and non-wetting

(dodecane) fluid. The higher range of the internal gradient spectrum is of the same

order as tool gradients.

4.2.3 NMR response simulation

The spin relaxation of a saturated porous system is simulated by using a lattice

random walk method [95, 27]. Initially the walkers are placed randomly in the 3D

pore space. At each time step i the walkers are moved from their initial position

to a neighboring site and the clock of the walker advanced by τi = ε2/(6D0), where

D0 the bulk diffusion constant of the relevant fluid, reflecting Brownian dynamics.

We treat each random walk as the movement of a spin packet with initial strength

Mw(t = 0). At each time step i of length τi, the strength of the walker is reduced

by the survival probability Si, with the strength of the walker at time t =∑

i τi

given by

Mw(t) = Mw(0)∏

i

Si . (4.12)

Here Si = SbSs, where Sb = exp (−τi/Tb) for bulk relaxation and Ss = 1 − ν for

surface relaxation. For steps within the same fluid Ss = 1. The killing probability

ν is related to the surface relaxivity ρ via

Aν =ρε

D0

+ O

((ρε

D

)2)

, (4.13)

where A is a correction factor of order 1 (here, we take A = 3/2) accounting for the

details of the random walk implementation [95, 27]. This leads to

Ss = 1 −6ρ τi

εA. (4.14)

So far we discussed the classical approach of using bulk relaxation and surface

relaxation processes. In addition, we model in detail the phase accumulation of

T2-D 61

the random walker in the internal magnetic field for the CPMG sequence. The

evolution of the phase for a particular spin in relation to the Larmor frequency at

the starting position ω0 = γBz(0) is given by [35]

φD = φ − φ0=N∑

j=1

γτj [Bz(tj) − Bz(0)] , tj =

j∑i=1

τi, (4.15)

where τj is a variable time step of the random walk controlled by the local diffu-

sion coefficient and lattice spacing. This problem can still be conveniently solved

with the same random walk techniques. Note, that the Gaussian phase approxima-

tion is not invoked. The π and π2

pulses of the acquisition sequence are considered

instantaneous and either flip the sign of the accumulated phase or stop/start the ac-

cumulation of φD. All random walk calculation presented in this paper were carried

out using ≥ 106 random walks per parameter set on a 10003 voxel central domain

of the Xray-CT images with a grid resolution of ε = 110

a ≈ 290 nm (Bentheimer)

and ε = 120

a ≈ 140 nm (Berea), where a is the voxel size. A ≥ 4 s CPMG echo train

was acquired for each setting.

4.2.4 Restricted diffusion and diffusion averaged internal gradients

In porous media the diffusing spins encounter fluid-solid or fluid-fluid bound-

aries which in our simulation are considered impenetrable—we assume that no mag-

netisation transfer across fluid-fluid interfaces takes place. Such phase boundaries

lead to a time dependent diffusion coefficient while at the same time leading to

a better averaging of local fields. To reach an analytical expression for the mag-

netisation decay typically a homogeneous linear background gradient 4.1 and the

Gaussian phase approximation and/or the concept of effective gradients are used

([44]). In [88] a parabolic gradient is considered. Here we directly use 4.15 and

calculate an effective gradient of a particular random path in the following way

geff =1

τ

N∑j=1

τj

Bz(tj) − Bz(0)

|rj − r0|(4.16)

with τ = tN and where the sum extends over all terms for which rj �= r0. In the

limit of no diffusion or severely restricted diffusion this effective gradient approaches

zero. For partially saturated rock we expect stronger restricted diffusion effects and

better averaging of internal gradients for the fluid with the higher diffusion coeffi-

cient. The relevant time intervals where the magnetisation vector precesses in the

transversal plane are the encoding time δ for diffusion in the external gradient and

the “encoding time” τ for internal gradients. While in experiments eddy currents

and other artefacts need to be considered leading to finite but small values of δ1 and

T2-D 62

δ2 with δ1,2 � δ, numerically one can use δ1 = δ2 = 0, and therefore τ = δ. We show

in 4.6 for Bentheimer sandstone and in 4.7 for Berea sandstone the distributions

of 〈g2eff〉 averaged over the time intervals τ = 5 ms, Δ = 80 ms and Δ = 320 ms at

different saturations for the case of vanishing external gradient(s). We see that for

the fully brine saturated case the diffusional averaging of the internal gradients is

more effective than for the dodecane saturated samples. Furthermore, as seen for

distributions of the local internal fields the internal fields in the non-wetting dode-

cane phase are significantly lower. The shape of the distribution functions of |geff |

is markedly smoother than for |gb| distributions. Diffusional averaging is evident

both in the fully saturated and partially saturated samples. The effective gradient

distributions of Berea sandstone are in good agreement with measured values for a

Berea sandstone with higher susceptibility contrast (Berea 100, figure 4 in [67]).

To illustrate further our point about diffusional averaging, we calculate di-

rectly from the image the diffusion coefficient distributions

D(t) =||rt − r0||

2

6t, (4.17)

where r0 and rt denote the start and end positions of a random walk and t is time.

We consider the same time intervals as above (τ = 5 ms, Δ = 80 ms and Δ =

320 ms), depicted in 4.8 for Bentheimer sandstone and in 4.9 for Berea sandstone.

For the fully saturated samples the diffusion coefficient distributions are centered

around the bulk diffusion coefficient of brine D0b = 2.3 × 10−5 cm2/s and dodecane

D0d = 8 × 10−6 cm2/s. There is evidence for restricted diffusion particularly for

longer diffusion times. For the partially saturated samples we see a significant

shift of the diffusion coefficient distribution of the wetting brine phase, while the

dodecane diffusion coefficient distribution essentially stays unchanged for the partial

saturation range considered. At these partial saturations there is little contrast

between brine and dodecane in terms of diffusion coefficients. However, the product

of Dg2b will be different because of the higher internal gradients in the water phase

at partial saturations. For a 1D NMR derived diffusion spectrum, the diffusion

spectrum would be weighted by T2 relaxation. We now consider the full T2-D NMR

simulation in the presence of restricted diffusion and internal gradients.

4.3 Diffusion-relaxation analysis

We show in 4.10 and 4.11 the NMR T2-D relaxation diffusion response for the

classical PGSE sequence using τ = 5 ms, Δ ∈ {80, 320}ms, and a CPMG spacing

of tE = 0.4 ms. Using 4.2-4.4 we can calculate the values of Ca, Cb, and Ca,b.

For our particular choice we have Ca = Cb and Ca,b = 2Ca, e.g. the prefactors in

T2-D 63

[a]10-2 10-1 100 101 102

gb [G/cm]

0

0.01

0.02

0.03

0.04

0.05

freq

uenc

y

brine

[b]10-2 10-1 100 101 102

gb [G/cm]

0

0.01

0.02

0.03

0.04

0.05

freq

uenc

y

dodecane

[c]10-2 10-1 100 101 102

gb [G/cm]

0

0.02

0.04

0.06

0.08

0.1

freq

uenc

y

brinedodecane

[d]10-2 10-1 100 101 102

gb [G/cm]

0

0.02

0.04

0.06

0.08

freq

uenc

y

brinedodecane

Figure 4.5: Internal gradient distributions for Berea sandstone at image resolutionfor a 2MHz system (B0 = 470 G). [a] brine saturated (Sw = 100%), [b] dodecanesaturated, [c] Sw = 26%, [d] Sw = 50%.

[a]10-4 10-3 10-2 10-1 100 101 102

geff [G/cm]

0

0.01

0.02

0.03

0.04

0.05

0.06

freq

uenc

y

brine, δ=5msbrine, Δ=80msbrine, Δ=320ms

[b]10-4 10-3 10-2 10-1 100 101 102

geff [G/cm]

0

0.01

0.02

0.03

0.04

0.05

0.06

freq

uenc

y

dodecane, δ=5msdodecane, Δ=80msdodecane, Δ=320ms

[c]10-4 10-3 10-2 10-1 100 101 102

geff [G/cm]

0

0.01

0.02

0.03

0.04

0.05

0.06

freq

uenc

y

brine, δ=5msbrine, Δ=80msbrine, Δ=320msdodecane, δ=5msdodecane, Δ=80msdodecane, Δ=320ms

[d]10-4 10-3 10-2 10-1 100 101 102

geff [G/cm]

0

0.01

0.02

0.03

0.04

0.05

0.06

freq

uenc

y


Figure 4.6: Effective internal gradient distributions for Bentheimer sandstone atB0 = 470 G. [a] water saturated (Sw = 100%), [b] dodecane saturated, [c] Sw = 23%,[d] Sw = 50%.

T2-D 64

[a]10-4 10-3 10-2 10-1 100 101 102 103

geff [G/cm]

0

0.01

0.02

0.03

0.04

0.05

0.06

freq

uenc

y


[b]10-4 10-3 10-2 10-1 100 101 102 103

geff [G/cm]

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

freq

uenc

y


[c]10-4 10-3 10-2 10-1 100 101 102 103

geff [G/cm]

0

0.01

0.02

0.03

0.04

0.05

0.06

freq

uenc

y


[d]10-4 10-3 10-2 10-1 100 101 102 103

geff [G/cm]

0

0.01

0.02

0.03

0.04

0.05

0.06

freq

uenc

y


Figure 4.7: Effective internal gradient distributions for Berea sandstone at B0 =470 G. [a] water saturated (Sw = 100%), [b] dodecane saturated, [c] Sw = 26%, [d]Sw = 50%.

[a]10-9 10-8 10-7 10-6 10-5 10-4 10-3

D(t) [cm2/s]

0

0.01

0.02

0.03

0.04

0.05

0.06

freq

uenc

y


[b]10-9 10-8 10-7 10-6 10-5 10-4 10-3

D(t) [cm2/s]

0

0.01

0.02

0.03

0.04

0.05

0.06

freq

uenc

y


[c]10-9 10-8 10-7 10-6 10-5 10-4 10-3

D(t) [cm2/s]

0

0.01

0.02

0.03

0.04

0.05

freq

uenc

y


[d]10-9 10-8 10-7 10-6 10-5 10-4 10-3

D(t) [cm2/s]

0

0.01

0.02

0.03

0.04

0.05

0.06

freq

uenc

y


Figure 4.8: Diffusion coefficient distributions for Bentheimer sandstone over differ-ent diffusion times. [a] water saturated (Sw = 100%), [b] dodecane saturated, [c]Sw = 23%, [d] Sw = 50%. The vertical lines indicate the bulk diffusion coefficientsof brine (black) and dodecane (red).

T2-D 65

[a]10-9 10-8 10-7 10-6 10-5 10-4 10-3

D(t) [cm2/s]

0

0.01

0.02

0.03

0.04

0.05

0.06

freq

uenc

y


[b]10-9 10-8 10-7 10-6 10-5 10-4 10-3

D(t) [cm2/s]

0

0.01

0.02

0.03

0.04

0.05

0.06

freq

uenc

y


[c]10-9 10-8 10-7 10-6 10-5 10-4 10-3

D(t) [cm2/s]

0

0.01

0.02

0.03

0.04

0.05

0.06

freq

uenc

y


[d]10-9 10-8 10-7 10-6 10-5 10-4 10-3

D(t) [cm2/s]

0

0.01

0.02

0.03

0.04

0.05

0.06

freq

uenc

y


Figure 4.9: Diffusion coefficient distributions for Berea sandstone over differentdiffusion times. [a] water saturated (Sw = 100%), [b] dodecane saturated, [c] Sw =26%, [d] Sw = 50%.

4.1 are of similar order. We see that for both sandstones the peaks in the 100%

brine saturated samples shift to smaller diffusion coefficients for the longer diffusion

intervals Δ, implying that restricted diffusion has a much stronger influence on the

signal than internal gradients for the chosen Δ at 2 MHz static field. We remark

that we considered a sequence with constant δ, varying the strength of the applied

gradient ga. A sequence with varying δ, e.g. using a permanent gradient, would

give rise to stronger internal gradient effects.

T2-D 66

[a] T2 [s]

D [m

2 /s]

10−3 10−2 10−1 100 10110−11

10−10

10−9

10−8

[b] T2 [s]

D [m

2 /s]

10−3 10−2 10−1 100 10110−11

10−10

10−9

10−8

[c] T2 [s]

D [m

2 /s]

10−3 10−2 10−1 100 10110−11

10−10

10−9

10−8

[d] T2 [s]

D [m

2 /s]

10−3 10−2 10−1 100 10110−11

10−10

10−9

10−8

[e] T2 [s]

D [m

2 /s]

10−3 10−2 10−1 100 10110−11

10−10

10−9

10−8

[f] T2 [s]

D [m

2 /s]

10−3 10−2 10−1 100 10110−11

10−10

10−9

10−8

Figure 4.10: Diffusion-relaxation correlation maps for Bentheimer sandstone for adiffusion time Δ = 80 ms (top) and Δ = 320 ms (bottom). [a,d] brine saturated(Sw = 100%), [b,e] Sw = 23%, [c,f] Sw = 12%. The horizontal lines indicate thediffusion coefficients of water (solid line) and dodecane (dashed line). The verticalline notes the bulk relaxation time for both fluids used in the simulation. The slopedline indicates diffusion-relaxation correlations for alkanes [91]. Note, that for crudeoils a distribution of diffusion coefficients and bulk relaxation times present in thecomplex fluid would cause it to follow the sloped line.

[a] T2 [s]

D [m

2 /s]

10−3 10−2 10−1 100 10110−11

10−10

10−9

10−8

[b] T2 [s]

D [m

2 /s]

10−3 10−2 10−1 100 10110−11

10−10

10−9

10−8

[c] T2 [s]

D [m

2 /s]

10−3 10−2 10−1 100 10110−11

10−10

10−9

10−8

[d] T2 [s]

D [m

2 /s]

10−3 10−2 10−1 100 10110−11

10−10

10−9

10−8

[e] T2 [s]

D [m

2 /s]

10−3 10−2 10−1 100 10110−11

10−10

10−9

10−8

[f] T2 [s]

D [m

2 /s]

10−3 10−2 10−1 100 10110−11

10−10

10−9

10−8

Figure 4.11: Diffusion-relaxation correlation maps for Berea sandstone for a dif-fusion time Δ = 80 ms (top) and Δ = 320 ms (bottom). [a,d] brine saturated(Sw = 100%), [b,e] Sw = 50%, [c,f] Sw = 26%. The additional lines are as in 4.10.

T2-D 67

4.4 Conclusions

We presented a numerical technique to simulate and analyse T2-D diffusion

spectra on the basis of Xray-CT images, avoiding common approximations, which is

able to deal with non-linear internal gradients. The derived effective gradients are

in good agreement with literature. For the contrast of susceptibilities considered at

low fields, the internal gradients were found not to impact strongly on the diffusion

coefficient spectrum. It was assumed that the clay regions exhibit a homogeneous

susceptibility.

We showed that the internal gradients are much larger in the wetting phase

for typical susceptibility contrasts and can expect that at higher field the difference

in the internal gradient distributions would provide contrast to aid in the deriva-

tion of wetting properties and fluid configurations, as the variation of experimental

parameters allows to control this contrast mechanism. This is in addition to surface

relaxivites, which contribute to an incoherent T2 relaxation mechanism.

In this work we considered dodecane as non-wetting fluid, which has a well

defined diffusion coefficient. For many crude oils a distribution and correlation

of T2 and D values makes the fluids more distinguishable. We conclude that the

numerical simulations are in general agreement with experimental techniques and

give confidence in the interpretation of NMR experiments involving 2D inverse

Laplace transforms.

CHAPTER 5

RELATIVE PERMEABILITY CORRELATIONS FROM NMR T2-D

NMR is a non-destructive technique and also available as a downhole tool.

Fluids saturating the rock differ in their bulk diffusion constants, bulk relaxation

times, and surface relaxivities. They also because of wettability differences occupy

preferentially certain parts of the rock. These material properties and spatial con-

figurations give rise to contrast mechanisms used in fluid typing methods. Earlier

techniques introduced for fluid typing use combinations of spin-lattice T1 and spin-

spin T2 relaxation times [2, 18, 104]. Here, the main contrast is achieved by the

negligible surface relaxivity of the non-wetting fluid, which mainly relaxes by bulk

relaxation. NMR methods have long been used for the determination of fluid prop-

erties and the identification of fluids. More recently, the inclusion of a diffusion

dimension in the experiment led to the development of T2-D relaxation-diffusion

experiments [66, 132, 20] which generally allow a better separation of fluids. When

uncertainties arise in fluid typing in the presence of light and heavy fluids for in-

stance, varying the echo-spacing or waiting times may assist in fluid separation.

Additionally, accounting for other factors influencing the NMR response such as in-

ternal gradients and restricted diffusion is necessary for more enhanced fluid typing,

e.g. more accurate determination of diffusion coefficients. If fluids can be separated

in the T2-D maps, a standard analysis like the one presented in chapter 3 should be

feasible, allowing one to partition the response and eventually to predict effective

and relative permeabilities.

In this chapter, we utilize the relaxation-diffusion NMR response and predict

relative permeability similar to chapter 3. Here, we simulate low field (2 MHz)

NMR T2 − D responses on high resolution tomographic images of Bentheimer and

Berea sandstone samples. All material properties and the simulation technique for

the relaxation-diffusion measurements have been introduced in previous chapters.

In particular, the material properties can be found in Table 2.2 (page 29), Table

2.3 (page 29), and Table 2.4 (page 29). The pulse sequence is introduced in Fig 4.1

(page 54). This chapter is organized as follows: initially, low field (2 MHz) NMR

relaxation-diffusion responses for two-phase primary drainage are simulated with


short echo spacing time of tE = 0.4 msec and diffusional times of (Δ = 80 msec)

and (Δ = 480 msec). Following that, single fluid NMR diffusion-induced relaxation

responses are calculated for both wetting and non-wetting phases. This allows the

estimation of both absolute and effective permeabilities of each phase and thus two-

phase relative permeability using the modified SDR model (Eqn. 5.3 (page 82)).

Further, we consider the effects of internal field gradients on relative permeability

correlations. To examine such effects, echo-spacing time tE will be incremented

gradually and NMR responses for both phases will be simulated accordingly. After

that, medium field (12 MHz) and high field (400 MHz) NMR responses will be

carried out at the echo-spacing time tE which will be increased to include effects on

internal gradients. This will provide insight to the relationship between changing

magnetic field and relative permeabilities for both phases. With the presence of

some experimental results of wetting phase relative permeability for Bentheimer

and Berea sandstone, a comparison between our wetting phase relative permeability

with experimental results will be presented.

5.1 Introduction

In partially saturated media where hydrocarbon fluids and water co-exist,

significant information can be extracted via nuclear magnetic resonance (NMR)

measurements, which have the ability to distinguish and separate fluids. NMR ap-

plication for fluid typing have progressed significantly. NMR T2-D maps were not

available in early methods [97, 3] since practice was influenced by the contrast be-

tween spin-lattice and spin-spin relaxation times in variable hydrocarbon fluids and

water. Recently, majority of presented material on fluid typing [63, 134, 146, 20, 40]

provide visual relaxation-diffusion maps for distinguishing fluids by diffusion coeffi-

cients. The effects of restricted diffusion described in [64, 7, 63] infers considerable

information about pore size geometry and characteristics for fluid filled systems that

are found in restricted environments as in porous media. In water-wet systems, wa-

ter tends to relax dominantly by surface relaxation where as the non-wetting phase,

e.g. oil, would relax through bulk fluid relaxation time [2, 116]. The inclusion of

internal magnetic fields and restricted diffusion measurements allowed high accu-

racy in NMR relaxation-diffusion maps and as a result much clearer separation of

fluids [89, 40]. The relaxation-diffusion NMR response depends on some important

experimental parameters such as echo spacing time and the strength of magnetic

field gradient for fluid typing. The macroscopic definition [144] of T2D, which is the

diffusion component to relaxation (Eqn. 1.1) is expressed as the following:


T2D =D

12(γGtE)2 . (5.1)

Here (D) is the diffusion coefficient, (γ) is the gyromagnetic ratio of the

proton, (G) the magnetic field and (tE) is the echo-spacing which is the time spacing

for the echo-pulse in CPMG sequence [29, 75].

Internal field gradient effects are accounted for in our numerical simulations

as explained earlier. The Bloch-Torrey equation [143] is used to predict the magne-

tization evolution response when it is measured under a field gradients (G). Bloch-

Torrey model in NMR is adopted to calculate the relaxation response when ad-

ditional properties and material are included such as susceptibility contrast, and

surface relaxivity. Field inhomogeneities due to the presence of more than one field

gradient will cause dephasing [43] of molecules in the pore space and subsequently,

decrease in relaxation time. This dephasing takes place by the movement of fluid

molecules into regions where the strength of magnetic field is different leading into

different Larmor frequencies.

The effect of internal gradients in NMR diffusion measurements was analyzed

on clay minerals present in sandstone samples [157]. It was noted that stronger

diffusion effects are directly related to the magnitude of internal gradients. Dia-

magnetic mineralogy as in Kaolinite exhibited weaker internal gradients due to

absence of iron found in illite and chlorite. The magnitude of internal gradients in

porous media has been calculated [116] by increasing the echo-spacing time which

concludes that internal gradients is highly dependent on the echo-spacing time and

shortening such time might minimize and reduce the effects of this susceptibility

contrast. Also, a surprising observation that internal gradients effects were found

to correlate strongly more with oil base mud flushed to irreducible saturation than

water saturated samples. This indicates clearly that internal gradients are more

influential in non-wetting phase than in wetting phase. These remarks have been

supported and examined by [157, 5, 124], providing strong evidence that internal

gradients at a larger echo-spacing time would predominate the relaxation of non-

wetting phase. In contrast, evidence of weaker dependency of the NMR response of

the wetting phase on internal gradients is due to the stronger effects of surface relax-

ation. As we increase the echo-spacing time, transverse relaxation times decrease.

We will show how this effectively improved the relative permeability correlations

mainly for the non-wetting phase that was modeled with zero surface relaxivity,

particularly when we exert higher magnetic fields to simulate the NMR response

for both phases.


5.2 Methodology

Similar to our analysis in Chapters 3 and 4 for simulating 1-D and 2-D NMR

relaxation responses respectively on partitioned saturated images of Bentheimer and

Berea sandstone samples, we utilize the same workflow to obtain two-dimensional

NMR maps as function of saturation for two-phase and single phase simulations. In

the simulation code and tomographic images, we currently have a practical limit on

echo-spacing time starting from (tE = 50 μsec) up to (tE = 6 msec) at low field (2

MHz). Later, an increase of magnetic field should reflect use of shorter echo spacings

as a result. As we performed some initial analysis of T2-D maps in chapter 4, we will

initially simulate NMR relaxation-responses for Bentheimer and Berea sandstone

samples at low field and tE = 0.4msec. Initially, NMR responses of partial saturated

fluids in both samples will be simulated. Later, individual fluids T2-D will be also

simulated. Once we generate all relaxation-diffusion maps for all cases, we extract

the marginal transverse relaxation time distributions as function of saturations for

all phases. The Schlumberger-Doll-Research (SDR) NMR permeability correlations

(Eqn. 5.3 (page 82) , and 5.2 (page 82)) are used to estimate relative permeabilities.

Medium field (12 MHz) and high field (400 MHz) NMR relaxation-responses will

be utilized to assess the marginal relaxation time induced by diffusion and relative

permeability correlations analyzed. The random walk method explained earlier

will facilitate these numerical computations. Around 100000 random walkers are

assigned for each simulation. Most of the observations remarked in [20] were based

on simulations at tE = 0.4 msec. We extend it here by allowing numerical NMR

responses at variable echo-spacing times.

5.3 Results and Analysis

5.3.1 T2-D NMR analysis for partially saturated samples

Capillary drainage of oil into water is simulated (see chapter 2) on the tomo-

graphic images of Bentheimer and Berea sandstone samples. For a series of water

saturations (Sw), NMR transverse relaxation responses are numerically calculated.

The drainage profiles in both samples illustrate a uniform shift of fluids starting from

the water line while oil invades the system until full oil saturations is reached (Fig-

5.1, and 5.2). NMR responses are simulated for different diffusional times (Eqn. 4.4

page 54) to assess the effect of diffusion coefficients. It is noted that much sharper

T2-D maps result at higher diffusional times; as a result, have more structure and

may lead to a more accurate prediction of the marginal relaxation-time distribu-

tions. From the profiles of the logarithmic mean relaxation time in both samples


(Fig-5.3), the logarithmic mean relaxation time profiles of both phases (combined)

exhibit increase in relaxation time distribution as system desaturates. The wetting

phase relaxation exhibits power law model compared to the one seen in our analysis

in chapter 3. The relaxation of oil from T2-D maps for short echo-spacing time (tE

= 0.4 msec) behaves linearly due to similar marginal relaxation time distribution

independent of saturation. The relaxation-diffusion NMR responses for partial sat-

urations will be analyzed for absolute permeabilities of both fluids. The individual

fluids will be examined separately to outline any significant findings on the behavior

of transverse relaxation measurements of fluids subjected to variable choice of echo-

spacing times. In two-phase, it was observed that at this short echo-spacing time,

the characteristics of fluid during drainage was much accurate, but much more at

higher diffusional time Δ.

5.3.2 T2-D NMR analysis of individual fluids

Single phase NMR relaxation-diffusion responses are simulated by switching

off the hydrogen index of the saturating fluids. Experimentally, this situation can

be realized using non-hydrogen fluids such as nitrogen for the drainage of water

or deuterium oxide (heavy water) to allow non-wetting phase protons to be only

captured. In the wetting phase scenario, surface relaxivity is non-zero and in com-

parison, the bulk relaxation of water is usually negligible, e.g. surface relaxivity

dominate the signal. The non-wetting phase is modeled with zero relaxivity and

bulk-fluid relaxation dominates the NMR transverse relaxation response. We vary

echo-spacing time to observe any changes in the NMR spin-spin relaxation time due

to internal gradients.

5.3.2.1 T2-D analysis of water phase

The log mean relaxation time responses for the defending wetting phase under

capillary drainage resembles a power law model. In Bentheimer and Berea sandstone

samples, the NMR relaxation-diffusion responses exhibit much recognized presenta-

tions of the wetting phase as drainage occurs (Fig-5.4, and 5.5). Clear observation

are seen regarding the decrease of logarithmic mean of the relaxation time as mov-

ing from fully saturation state toward partial saturations (Fig-5.6). This decrease

is dependent mainly on pore volume to surface area ratios that get smaller as pore

space desaturates and thus surface relaxation gets faster. To assess the importance

of internal gradients, we vary the echo spacing time from (tE = 0.4 msec) up to (tE

= 5 msec). Only a minor decrease in relaxation time is recorded (Fig-5.6). From

this we conclude that surface relaxation is dominant in the wetting phase relaxation


[a] [b]

[c] [d]

[e] [f]

Figure 5.1: Diffusion-relaxation correlation maps for Bentheimer sandstone for dif-fusion time of Δ = 480 ms (left) and Δ = 80 ms (right). [a,b] brine saturated(Sw = 100%), [c,d] Sw = 55%, and [e,f] oil saturated. The horizontal lines indicatethe diffusion coefficients of water (solid line) and dodecane (dashed line). The ver-tical line notes the bulk relaxation time for both fluids used in the simulation. Thesloped line indicates the relaxation-diffusion correlations for Alkanes.


[a] [b]

[c] [d]

[e] [f]

[g] [h]

Figure 5.2: Diffusion-relaxation correlation maps for Berea sandstone for diffusiontimes of Δ = 480 ms (left) and Δ = 80 ms (right). [a,b] brine saturated (Sw =100%), [c,d] Sw = 78%, [e,f] Sw = 40% and [g,h] oil saturated.


[a]0 0.2 0.4 0.6 0.8 1

Sw

0

0.2

0.4

0.6

0.8

1

T 2lm

(sec

)

T2lm (Water + Oil)T2lm (Oil)T2lm (Water)

[b]0 0.2 0.4 0.6 0.8 1

Sw

0

0.2

0.4

0.6

0.8

1

T 2lm

(sec

)

T2lm (Water + Oil)T2lm (Oil)T2lm (Water)

Figure 5.3: Extracted profiles of logarithmic mean relaxation times as function ofwater saturation for individual fluids and partial saturated fluids from NMR T2−Dresponses in [a] Bentheimer sandstone, and [b] Berea sandstone at low field (2 MHz)and echo spacing (tE = 0.4msec).


from T2-D measurements.

5.3.2.2 T2-D analysis of oil phase

The non-wetting phase is modeled with no surface relaxation, allowing bulk

fluid relaxation to dominate the response. The marginal relaxation responses from

diffusion in the non-wetting phase are acquired initially at lower echo-spacing time

(tE = 0.4 msec). For both Bentheimer and Berea samples, marginal increments

of logarithmic mean relaxation times profile (Fig-5.3) are noted during drainage

of oil phase. We conclude that majority of non-wetting phase NMR response is

dominated by fluid bulk relaxation time. At this stage, effects of internal gradients

are minimized due to short echo-spacing time. Following, the echo-spacing time

was increased at low field (2 MHz) up to (tE = 6 msec). In Bentheimer sandstone

sample, which exhibits weak internal gradients (Fig-4.4 (page 59)), T2-D maps

presented the response of oil phase at multiple saturations. Similar behavior of

oil phase relaxation-diffusion responses were examined in the fluid maps during

drainage (Fig-5.7). However, the increase in echo-spacing time resulted in a small

shift in the marginal relaxation responses to lower relaxation times. The magnitude

of internal gradients here was not sufficient to significantly change the relaxation

responses (Fig-5.10). In Berea sandstone, which exhibit stronger internal gradient

fields than Bentheimer, a similar trend of the NMR marginal relaxation response

was observed for short echo spacing (tE = 0.4 msec). Accordingly, echo-spacing

time was increased and enhancement in relaxation is seen and departure from linear

behavior to exponential was clearly visible (Fig-5.11, and 5.14). The internal field

effects here are much stronger than seen in Bentheimer.

Knowing that internal fields are important in our analysis, we applied higher

magnetic fields (12 MHz) and varied tE up to the practical limit of (tE = 3.5 msec).

In Bentheimer sample, T2-D fluid maps were changed with saturation (Fig-5.8) and

insignificant enhancement in relaxation time was noted (Fig-5.10). This suggests

that the magnitude of internal fields is important to provide an enhancement to

relaxation measurements induced by diffusion. In contrast, Berea sample at this

magnetic field and practical limit illustrated much more enhancement in relaxation

responses of oil phase during drainage (Fig-5.12, and 5.14). At low and medium

fields, the potential of internal field gradients showed that higher magnitude result in

enhancement in phase identification and thus enhanced relaxation time responses of

the non-wetting phase. At higher magnetic field (400 MHz) and tE = 0.4 msec, fluid

maps T2-D exhibit different behaviors of the non-wetting phase found at variable

relaxation times via different peaks. Internal gradients are too strong to allow a

reliable interpretation in term of permeability predictions (Fig-5.9 , and 5.13).


[a] [b]

[c] [d]

[e]

Figure 5.4: T2-D maps for water phase drainage response in Bentheimer sandstoneat low field (2 MHz) and echo-spacing time (tE = 0.4 msec) at a diffusional time of480 msec [a] Fully water saturated, [b] Sw = 55%, [c]Sw = 50%, [d] Sw = 25%, and[e] Sw = 14%.


[a] [b]

[c] [d]

[e] [f]

Figure 5.5: T2-D maps for water phase drainage response in Berea sandstone atlow field (2 MHz) and echo-spacing time (tE = 0.4 msec) at a diffusional time of480 msec [a] Fully water Saturated, [b] Sw = 78%, [c] Sw = 50%, [d]Sw = 40%,[e]Sw = 34%, and [f] Sw = 17%.


[a]0 0.2 0.4 0.6 0.8 1

Sw

0

0.2

0.4

0.6

0.8

1T 2l

m (s

ec)

T2lm (tE = 0.4 msec)T2lm (t=E = 5 msec)

[b]0 0.2 0.4 0.6 0.8 1

Sw

0

0.2

0.4

0.6

0.8

1

T 2lm

(sec

)

T2lm (tE = 0.4 ms)

Figure 5.6: Observed NMR relaxation-diffusion response of water phase at differentecho-spacing times and low field (2 MHz) in [a] Bentheimer sandstone, and [b] Bereasandstone.


[a] [b]

[c] [d]

[e] [f]

Figure 5.7: T2-D maps for capillary drainage of oil phase into water at differentsaturations for Bentheimer Sandstone at low field (2 MHz) and echo-spacing time(tE = 6 msec) at a diffusional time of (tD = 480 msec). [a] Fully oil saturated, [b]Sw = 55%, [c] Sw = 50%, [d] Sw = 33%, [e] Sw = 25%, and [f] Sw = 14%.


[a] [b]

[c] [d]

[e]

Figure 5.8: T2-D maps for capillary drainage of oil phase into water at differentsaturations for Bentheimer Sandstone at medium field (12 MHz) and echo-spacingtime (tE = 3.5 msec) at a diffusional time of (tD = 480 msec). [a] Fully oil saturated,[b] Sw = 55%, [c] Sw = 33%, and [e] Sw = 14%.


5.4 Relative permeability from NMR T2-D analysis

Using Schlumberger-Doll-research (SDR) NMR permeability model utilized

in chapter 3, relative permeability correlations for both phases are derived from

relaxation-diffusion responses. Relative permeabilities from this chapter are calcu-

lated using ratios of (Eqn.5.3, and 5.2).

kNMR = a ∗ T 22lm × φ4

1 . (5.2)

kNMR(Sw) = a ∗ T2lm(Sw)2 × φ42 . (5.3)

5.4.1 Relative permeability from NMR T2-D of wetting phase

The correlations between NMR relaxation-diffusion responses at partial sat-

urations for both samples were initially tested at low field (2 MHz) and at short

echo-spacing (tE = 0.4 ms). The wetting phase relative permeability from simulated

NMR response for both Bentheimer and Berea sandstone samples is in good agree-

ment to imaged based relative permeability predictions (Fig-5.15). Additionally,

there is an excellent match for the wetting phase relative permeability predictions

from NMR between numerical response simulation and published experimental re-

sults by Chen[41] (Fig-5.16). As we increase the echo-time spacing, transverse re-

laxation times decrease due to diffusion in internal gradients. However, the effect is

minor and the correlations remain strong. The wetting phase relative permeability

remains strong even at a medium field of (12 MHz) (Fig-5.16). At high field (400

MHz) and tE = 0.4 msec, relaxation times were much shortened due to internal

gradient effects and we were unable to find a correlation to permeability.

5.4.2 Relative permeability from NMR T2-D of non-wetting phase

In the non-wetting phase analysis from T2-D, relative permeability estimation

for both samples from NMR marginal relaxation responses at low field and initial

echo-spacing time of 0.4 msec result in a promising match with image based rela-

tive permeability correlations (Fig-5.15). The magnitude of internal gradient fields

were noted earlier to be much stronger in Berea than Bentheimer sandstone. Here,

we see whether a change in echo-spacing time (tE) and therefore variation in the

influence of internal gradients can improve the relative permeability. In Bentheimer

sandstone at low field (2 MHz), echo-spacing time was gradually increased and dif-

ferent relative permeability predictions were observed, but no significant change in

the correlations was detected. However, progressive enhancement in the relative


permeability correlations was visible when incrementing echo spacing time tE (Fig-

5.17). In the Berea sample, the same approach was used to assess the strength of

non-wetting phase relative permeability. In the low field measurements and variable

echo-spacing times, the correlation is stronger at longest echo-spacing (tE = 6 msec)

due to the magnitude of influence coming from internal gradients on relaxation re-

sponses and thus better relative permeability correlations (Fig-5.18). At medium

field (12 MHz) and long echo-spacing (tE = 3.5 MHz), relative permeability corre-

lations from the NMR transverse relaxation responses are much stronger. For both

samples, high magnetic field (400 MHz) NMR relaxation-diffusion responses result

in the destruction of relative permeability correlations due to the fast transverse

relaxation magnetization decay in the presence of internal gradients effects.

5.4.2.1 Non-wetting phase correlations with known bulk relaxation

The bulk relaxation in our simulations of the non-wetting phase is set (T2b

=1). To establish a connection between the observed increase in relaxation with

saturations while incrementing the magnitude of internal gradients, internal gradi-

ents are inversely proportional to pore size. The simulation of oil relaxation in the

presence of internal gradients will then be subjective to subtract the bulk relax-

ation rate from the measured relaxation time where the difference will be the direct

relaxation induced by diffusion. By looking at (Eq.5.1, the diffusion induced decay

rate is proportional to the square of internal gradients. This will deviate the SDR

model to utilize different power law models for wetting phase which is depedant

on surface relaxation (Power 2), while the non-wetting phase will use (Power 1) as

shown in (Eq.5.4 and Eq.5.5). In Bentheimer and Berea sandstone sample, relative

permeability predictions of non-wetting phase after using the new power resulted in

progressive enhancements in Bentheimer due to weak internal fields but noticeable

improvements in Berea because of stronger internal fields in both low and medium

fields (Fig.5.19 and Fig.5.20) compared to previously established in (Fig-5.17) and

(Fig-5.18).

kNMR = a ∗ T2lm × φ41 . (5.4)

kNMR(Sw) = a ∗ T2lm(Sw) × φ42 . (5.5)


[a] [b]

[c] [d]

Figure 5.9: T2-D maps for capillary drainage of oil phase into water at differentsaturations for Bentheimer sandstone at high field (400 MHz) and echo-spacingtime (tE = 3 msec) at a diffusional time of (tD = 480 msec). [a] Fully oil saturated,[b] Sw = 55%, [c] Sw = 33%, and [d] Sw = 25%.


[a]0 0.2 0.4 0.6 0.8 1

Sw

0.7

0.72

0.74

0.76

0.78T 2l

m (s

ec)

(tE = 0.4 msec)(tE = 1 msec)(tE = 6 msec)

[b]0 0.2 0.4 0.6 0.8 1

Sw

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

T 2lm

(sec

)

(tE = 0.4 msec )(tE = 2 msec)(tE = 3.5 msec)

Figure 5.10: Sensitivity analysis of observed NMR relaxation-diffusion response ofoil phase in Bentheimer sandstone at different echo-spacings. [a] Low filed (2 MHz),and [b] medium field (12 MHz).


[a] [b]

[c] [d]

[e]

Figure 5.11: T2-D maps for capillary drainage of oil phase into water at differentsaturations for Berea sandstone at low field (2 MHz) and echo-spacing time (tE =6 msec) at a diffusional time of (tD = 480 msec). [a] Sw = 17%, [b] Sw = 34%, [c]Sw = 40%, [d] Sw = 50%, and [e] Sw = 78%.


[a] [b]

[c] [d]

[e]

Figure 5.12: T2-D maps for capillary drainage of oil phase into water at differentsaturations for Berea sandstone at medium field (12 MHz) and echo-spacing time(tE = 3 msec) at a diffusional time of (tD = 480 msec). [a] Fully oil saturated, [b]Sw = 17%, [c] Sw = 40%, [d] Sw = 50%, and [e] Sw = 78%.


[a] [b]

[c] [d]

Figure 5.13: T2-D maps for capillary drainage of oil phase into water at differentsaturations for Berea sandstone at high field (400 MHz) and echo-spacing time (tE= 3 msec) at a diffusional time of (tD = 480 msec). [a] Sw = 17%, [b] Sw = 40%,[c] Sw = 50%, and [d] Sw = 78%.


[a]0 0.2 0.4 0.6 0.8 1

Sw

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

T 2lm

(sec

)

tE = 0.4 msec (2 MHz)tE = 6 msec (2 MHz)tE = 3 msec (12 MHz)

Figure 5.14: Sensitivity analysis of observed NMR relaxation-diffusion response ofoil phase in Berea sandstone at different echo-spacing times and magnetic fields.


[a]0 0.2 0.4 0.6 0.8 1

Sw

0

0.2

0.4

0.6

0.8

1R

elat

ive

Perm

eabi

lity

- Kr

Krw_LBMKrw_NMRKro_LBMKro_NMR

[b]0 0.2 0.4 0.6 0.8 1

Sw

0

0.2

0.4

0.6

0.8

1

Rel

ativ

e Pe

rmea

bilit

y - K

r

Krw_LBMKrw_NMRKro_LBMKro_NMR

Figure 5.15: Relative permeability predictions from NMR relaxation-diffusion mea-surements at low field (2 MHz) and short echo spacing (tE = 0.4 msec) for [a]Bentheimer sandstone, and [b] Berea sandstone.


[a]

0 0.2 0.4 0.6 0.8 1Sw

0

0.2

0.4

0.6

0.8

1R

elat

ive

Perm

eabi

lity

- Kr

Krw_NMR (tE = 0.4 msec "2 MHz")Krw_NMR (tE = 5 msec "2 MHz")Krw_NMR_Exp (Chen 1994)

[b]0 0.2 0.4 0.6 0.8 1

Sw

0

0.2

0.4

0.6

0.8

1

Rel

ativ

e Pe

rmea

bilit

y - K

r

Krw_NMR (tE = 0.4 msec "2 MHz" )Krw_NMR_EXP (Chen 1994)

Figure 5.16: Relative permeability of the wetting phase from relaxation-diffusionmeasurements at different echo-spacings and low magnetic field (2 MHz) for [a]Bentheimer sandstone, and [b] Berea sandstone.


[a]0 0.2 0.4 0.6 0.8 1

Sw

0

0.2

0.4

0.6

0.8

1R

elat

ive

Perm

eabi

lity

- Kr

Kro_LBM(tE = 0.4 msec)(tE = 1 msec )(tE = 6 msec)

[b]0 0.2 0.4 0.6 0.8 1

Sw

0

0.2

0.4

0.6

0.8

1

Rel

ativ

e Pe

rmea

bilit

y - K

r

Kro_LBM(tE = 0.4 msec)(tE = 2 mesc)(tE = 3.5 msec)

Figure 5.17: Relative permeability of non-wetting phase from NMR relaxation-diffusion measurements for Bentheimer sandstone at different echo-spacings [a] lowfield (2 MHz), and [b] medium field (12 MHz).


[a]0 0.2 0.4 0.6 0.8 1

Sw

0

0.2

0.4

0.6

0.8

1R

elat

ive

Perm

eabi

lity

- Kr

Kro_NMR (tE = 0.4 msec)Kro_NMR (tE = 6 msec)Kro_LBM

[b]0 0.2 0.4 0.6 0.8 1

Sw

0

0.2

0.4

0.6

0.8

1

Rel

ativ

e Pe

rmea

bilit

y - K

r

Kro_NMR (tE = 0.4 msec)Kro_NMR (tE = 3 msec)Kro_LBM

Figure 5.18: Relative permeability of non-wetting phase from relaxation-diffusionmeasurements for Berea sandstone at different echo-spacings [a] low field (2 MHz),and [b] medium field (12 MHz)


[a]0 0.2 0.4 0.6 0.8 1

Sw

0

0.2

0.4

0.6

0.8

1

Rel

ativ

e Pe

rmea

bilit

y - K

rKro_LBM(tE = 6 msec) "Power 1"

[b]0 0.2 0.4 0.6 0.8 1

Sw

0

0.2

0.4

0.6

0.8

1

Rel

ativ

e Pe

rmea

bilit

y - K

r

Kro_LBM(tE = 3.5 msec) "Power 1"

Figure 5.19: Relative permeability of non-wetting phase from NMR relaxation-diffusion measurements for Bentheimer sandstone at longest echo-spacing allowedand utilizing SDR model with power of 1 at [a] low field (2 MHz), and [b] mediumfield (12 MHz).


[a]0 0.2 0.4 0.6 0.8 1

Sw

0

0.2

0.4

0.6

0.8

1R

elat

ive

Perm

eabi

lity

- Kr

Kro_LBMKro_NMR (tE = 6 msec) "Power 1"

[b]0 0.2 0.4 0.6 0.8 1

Sw

0

0.2

0.4

0.6

0.8

1

Rel

ativ

e Pe

rmea

bilit

y - K

r

Kro_LBMKro_NMR (tE = 3 msec) "Power 1"

Figure 5.20: Relative permeability of non-wetting phase from relaxation-diffusionmeasurements for Berea sandstone at longest echo-spacing allowed and utilizingSDR model with power 1 at [a] low field (2 MHz), and [b] medium field (12 MHz)


5.5 Conclusions

From the relaxation-diffusion NMR responses, we simulate T2-D fluid maps

for Bentheimer and Berea sandstone samples at different magnetic fields and echo-

spacing times. In partial saturations, absolute permeabilities were calculated and

effective permeabilities eventually were also calculated but via simulating individ-

ual phases NMR responses. Relative permeability correlations at low field and

short-echo spacing time were much stronger in wetting phase than seen in non-

wetting phase. Even when increasing echo-spacing time and introduce influence

from internal gradient fields, correlations remain strong due to domination of sur-

face relaxation on the wetting phase. Wetting phase relative permeability from our

analysis here for both samples matched published NMR experimental results.

In the non-wetting phase, both samples initially at low field and short echo

spacing time observe promising agreement to image based relative permeability.

As we increased echo-spacing in both samples, internal field gradient effects were

present but much stronger in Berea than Bentheimer. In Bentheimer sample, no

significant change was observed in the permeability correlation at both low field and

medium field. Relative permeability correlations were stronger in Berea sandstone

at the practical limit of both low and and much stronger in medium fields. The

magnitude of internal gradients in Berea resulted in assisted/weighted enhanced

relaxation for the oil phase much recognized at medium field.

At known oil bulk relaxation, the analysis of oil phase in the presence of

internal gradients was re-analyzed by subtracting the bulk relaxation rate from the

total relaxation. As a result, the relaxation rate induced by diffusion is linked to

the square of internal gradients and thus modified the power model of relaxation

time in SDR model (power 1). The observed correlations retained much stronger

with LBM at both low and medium fields but much recognized in Berea due to

higher internal fields than Bentheimer.

Last, high magnetic field of [400 MHz] in the both phases NMR relaxation-

diffusion responses was subjected to shorter relaxation times due to effects of inter-

nal gradients and restricted diffusion. Subsequently, this has resulted in destroying

the relative permeability correlations for both phases.

CHAPTER 6

RECOMMENDATIONS AND FUTURE RESEARCH

The cross-correlations observed from NMR responses on partial saturated

samples and relative permeability from tomographic images in this thesis are based

under water-wet wettability conditions and relatively clean and homogeneous reser-

voir samples. Additionally, modeling the NMR transverse relaxation response re-

lied here on a fixed surface relaxivity for both the solid phase and the clay-fraction

intermediate phase. The samples were imaged at ambient conditions and no stress-

related NMR simulations are included in this work. As such, the generality or appli-

cability limits of the NMR-permeability and relative permeability cross-correlations

has not yet been established. Recommendations and future topics are itemize below.

• Experimental validations of the NMR responses on partial saturated sam-

ples, considering the internal gradients where a specification of a long-

enough echo spacing allows the inclusion of such effects. First, measur-

ing the NMR T2 response for the water phase in a nitrogen experiment to

construct the profile of water saturations versus transverse relaxation time.

For the oil phase measurements, eliminate the hydrogen signal from water

phase by soaking the samples in heavy water D2O until fully saturated,

then inject pore volumes of oil until the sample is fully saturated. This

would allow one to record the NMR responses of both phases separately,

leading to relative permeability predictions.

• Experimental validation of the results obtained by NMR relaxation-diffusion

T2-D simulations, including for each fluid, to develop partitioning of fluids

on the basis of T2-D maps. This can be performed via using non-hydrogen

fluids stated earlier.

• Examining wettability effect on NMR relaxation or relaxation-diffusion re-

sponses and thus their effect on relative permeability estimation. In oil-wet

conditions, oil will exhibit shorter relaxation time and the investigation of


such effect on relative permeability correlations would be interesting. Addi-

tionally, simulating the NMR relaxation response on fluid saturated images

rather than numerically simulating saturations would confirm that wetta-

bility is not altered.

• Construction of a spatial model for clay fraction and type leading to distri-

bution of relaxivity and susceptibility value to account for clay heterogene-

ity. This would predict more realistic NMR relaxation responses. The bene-

fit of such model could ultimately lead to much more accuracy in predicting

NMR response numerically and thus more realistic match to conventional

NMR experiments.

• Predict NMR response on tomographic images from the samples at different

stages of applied stress using a pressure-cell vessel, this would impact the

NMR response from clay-fractions fluids and provide accurate modeling for

clay fluids and thus refine cut-off analysis.This experiments can also benefit

from using variable temperature ranges to observe the transition of NMR

experimental responses accordingly.

• The use of different non-wetting reservoir fluids and accounting for temper-

ature and reservoir fluid properties subjectively. The diffusion coefficients

will vary as temperature changes and as a result different NMR responses,

especially in the relaxation-diffusion T2-D maps. This will be interesting to

test whether permeability correlations can be established.

• Test thesis approach on heterogeneous carbonates via complex modeling

of micro-porosity and susceptibility. It is demonstrated in this work how

internal gradients are significant contributor to the NMR response and thus

relative permeability in particular the non-wetting phase that was modeled

at zero relaxation value. For carbonates, this might require either Ferroan

dolomites or working with higher fields.

• In this work, we have utilized three NMR frequencies which are 2 MHz,

12 MHz and 400 MHz. As observed in the results, Samples which bear

high susceptibility contrast accumulate excellent agreement with respect

to relative permeability NMR based predictions against both numerical

LBM and experimental wetting phase. The results from 2 MHz and 12

MHz are amazing, but when at 400 MHz the correlations were destroyed.

Future study could benefit from utilizing a broader range of medium to high

frequencies and simulate the NMR response to define what the upper limit

would be at which those correlations will no longer be valid.

BIBLIOGRAPHY

[1] Ridvan Akkurt. Practical NMR Logging. Lecture Notes in NMR Applications.

[2] Ridvan Akkurt. NMR logging of natural gas reservoirs. In SPWLA 36thAnnual Logging Symposium. SPWLA, June 1995.

[3] Ridvan Akkurt, Duncan Mardon, John S. Gardner, Dave M. Marschall,and Fernando Solanet. Enhance diffusion: Expanding the range of nmrdirect hydrocarbon-typing applications. In SPWLA 39th Annual LoggingSymposium. SPWLA, May 1998.

[4] Ridvan Akkurt, Manfred G. Prammer, and M. Andrei Moore. Selection ofoptimal acquisition parameters for mril logs. In SPWLA 37th Annual LoggingSymposium. SPWLA, June 1996.

[5] Vivek Anand and George Hirasaki. Paramagnetic relaxation in sandstone:Distringuishing t1 and t2 dependence on surface relaxation, internal gradientsand dependence on echo spacing. J. Magn. Res., 190:68–85, 2007.

[6] Matthias Appel, J. Justin Freeman, John S. Gardner, George H. Hirasaki,Q. Gigi Zhang, and John L. Shafer. Interpretation of restricted diffusionin sandstones with internal field gradients. Magnetic Resonance Imaging,19:535–537, 2001.

[7] Matthias Appel, J. Justin Freeman, Rod B. Perkins, and Jan P. Hofman. Re-stricted diffusion and internal gradients. In 40th Annual Logging Symposium,page paper FF, Oslo, Norway, May-June 1999. Society of Petrophysicists &Well Log Analyst.

[8] G. E. Archie. The electrical resistivity log as an aid in determining somereservoir characteristics. Trans. AIME, 146:54, 1942.

[9] C. H. Arns, Tariq AlGhamdi, and Ji-Youn Arns. Numerical analysis of NMRrelaxation-diffusion responses of sedimentary rock. New J. of Physics, ac-cepted, 2010.

[10] C. H. Arns, Tariq AlGhamdi, Ji-Youn Arns, Lauren Burcaw, and K. E. Wash-burn. Analysis of T2-D relaxation-diffusion NMR measurements for partiallysaturated media at different field strength. Number SCA2010-17 in The 24rdInternational Symposium of the Society of Core Analysts, pages 1–12, Halifax,October 4-7 2010. Society of Core Analysts.

Bibliography 100

[11] C. H. Arns, H. Averdunk, F. Bauget, A. Sakellariou, T. J. Senden, A. P.Sheppard, R. M. Sok, W. V. Pinczewski, and M. A. Knackstedt. Digital corelaboratory: Analysis of reservoir core fragments from 3D images. In 45thAnnual Logging Symposium, page paper EEE, Noordwijk, The Netherlands,June 2004. Society of Petrophysicists & Well Log Analyst.

[12] C. H. Arns, H. Averdunk, F. Bauget, A. Sakellariou, T. J. Senden, A. P.Sheppard, R. M. Sok, W. V. Pinczewski, and M. A. Knackstedt. Digital corelaboratory: Reservoir core analysis from 3D images. In Gulf Rocks 2004:Rock Mechanics across Borders & Disciplines, The 6th North American RockMechanics Symposium, page paper 498, Houston, June 2004.

[13] C. H. Arns, Y. Melean, L. Burcaw, P. T. Callaghan, and K. E. Washburn.Comparison of experimental NMR measurements with simulated responses ondigitized images of mono-mineralic rocks using xray-CT. In Improved CoreAnalysis for Unconventional Fields, number SCA2009-44 in The 23rd Inter-national Symposium of the Society of Core Analysts, pages 1–6, Noordwijk,September 27-30 2009. Society of Core Analysts.

[14] C. H. Arns, A. Sakellariou, T. J. Senden, A. P. Sheppard, R. Sok, W. V.Pinczewski, and M. A. Knackstedt. Digital core laboratory: Reservoir coreanalysis from 3D images. Petrophysics, 46(4):260–277, 2005.

[15] C. H. Arns, A. P. Sheppard, M. Saadatfar, and M. A. Knackstedt. Predictionof permeability from NMR response: surface relaxivity heterogeneity. In 47thAnnual Logging Symposium, pages paper GG:1–13, Veracruz, Mexico, June2006. Society of Petrophysicists & Well Log Analyst.

[16] C. H. Arns, A. P. Sheppard, R. M. Sok, and M. A. Knackstedt. NMRpetrophysical predictions on digitized core images. In 46th Annual LoggingSymposium, pages paper MMM:1–16, New Orleans, Texas, June 2005. Societyof Petrophysicists & Well Log Analyst.

[17] C. H. Arns, A. P. Sheppard, R. M. Sok, and M. A. Knackstedt. NMR petro-physical predictions on digitized core images. Petrophysics, 48(3):202–221,2007.

[18] C. H. Arns, K. E. Washburn, and P. T. Callaghan. Multidimensional NMRinverse Laplace spectroscopy in petrophysics. Petrophysics, 48(5):380–392,2007.

[19] Christoph H. Arns. A comparison of pore size distributions derived by NMRand Xray-CT techniques. Physica A, 339(1-2):159–165, 2004.

[20] Christoph H. Arns, Tariq AlGhamdi, and Ji-Youn Arns. Numerical analysisof nuclear magnetic resonance relaxationdiffusion responses of sedimentaryrock. New Journal of Physics, 13:015004, 2011.

[21] Christoph H. Arns, Mark A. Knackstedt, and Nicos Martys. Cross-propertycorrelations and permeability estimation in sandstone. Phys. Rev. E,72:046304, 2005.

Bibliography 101

[22] Ji-Youn Arns, Christoph H. Arns, Adrian P. Sheppard, Rob M. Sok, Mark A.Knackstedt, and W. Val Pinczewski. Relative permeability from tomographicimages; effect of correlated heterogeneity. J. Petr. Sc. Eng., 39(3-4):247–259,2003.

[23] Ji-Youn Arns, Vanessa Robins, Adrian P. Sheppard, Robert M. Sok, W. V.Pinczewski, and Mark A. Knackstedt. Effect of network topology on relativepermeability. Transport in Porous Media, 55:21–46, 2004.

[24] S. Bakke and P. E. Øren. 3-D pore-scale modelling of sandstones and flowsimulations in the pore networks. SPE Journal, 2:136–149, 1997.

[25] B. A. Baldwin and W. S. Yamanashi. Nmr imaging of fluid dynamics inreservoir core. Magnetic Resonance Imaging, 6(5):493–500, 1998.

[26] Jayanth R. Banavar and Lawrence M. Schwartz. Magnetic resonance as aprobe of permeability in porous media. Phys. Rev. Lett., 58:1411–1414, 1987.

[27] David J. Bergman, Keh-Jim Dunn, Lawrence M. Schwartz, and Partha P.Mitra. Self-diffusion in a periodic porous medium: A comparison of differentapproaches. Phys. Rev. E, 51:3393–3400, 1995.

[28] Djomice Beugre, Sbastien Calvo, Grard Dethier, Michel Crine, and DominiqueToye. Lattice Boltzmann 3D flow simulations on a metallic foam. J. ofComputational and Applied Mathematics, 2009.

[29] N. Bloembergen, E. M. Purcell, and R. V. Pound. Relaxation effects in nuclearmagnetic resonance absorption. Phys. Rev., 7:679–712, 1948.

[30] S. Bobroff, G. Guillot, C. Riviere, L. Cuiec, and J. C. Roussel. Quantitativeporosity profiles and wettability contrast visualisation in sandstone by CPMGimaging. Magnetic Resonance Imaging, 14(7/8):907–909, 1996.

[31] Robert J. S. Brown and Irving Fatt. Measurements of fractional wettabil-ity of oilfieldrocks by the nuclear magnetic resonance method. PetroleumTransactions (AIME), 207(262), 1956.

[32] K. R. Brownstein and C. E. Tarr. Importance of classical diffusion in NMRstudies of water in biological cells. Phys. Rev. A, 19:2446–2453, 1979.

[33] Lauren Burcaw. Multi-dimensional Nuclear Magnetic Resonance Methodsin the Inhomogeneous Magnetic Field. PhD thesis, Victoria University ofWellington, 2011.

[34] Paul T. Callaghan. Principles of Magnetic Resonance Microscopy. ClarendonPress, Oxford, 1991.

[35] H. Y. Carr and E. M. Purcell. Effects of diffusion on free precession in nuclearmagnetic resonance problems. Phys. Rev., 94:630, 1954.

[36] Quan Chen, Andrew E. Marble, Bruce G. Colpitts, and Bruce J. Balcom.The internal magnetic field distribution, and single exponential magnetic res-onance free induction decay, in rocks. J. Magn. Res., 175:300–308, 2005.

Bibliography 102

[37] S. Chen, S. P. Dawson, G. D. Doolen, D. R. Janecky, and A. Lawniczak.Lattice methods and their applications to reacting systems. Computers inChemical Engineering, 19(6-7):617–646, 1995.

[38] Songhua Chen, Hsie-Keng, and A. Ted Watson. Fluid saturation-dependentnuclear magnetic resonance spin-lattice relaxation in porous media and porestructure analysis. J. Appl. Phys., 74(3):1473–1479, 1993.

[39] Songhua Chen, F. Kim, K.-H.and Qin, and A.T. Watson. Quantitative nmrimaging of multiphase flow in porous media. Magnetic Resonance Imaging,10(5):815–826, 1992.

[40] Songhua Chen, Lilong Li, Gigi Zhang, and Jason Chen. Magnetic resonancefor down-hole complex lithology earth formation evaluation. New Journal ofPhysics, 13:085015, 2011.

[41] Songhua Chen, Hsie-Keng Liaw, and A.Ted Watson. Measurements and anal-ysis of fluid saturation-dependent nmr relaxation and linebroadening in porousmedia. Magnetic Resonance Imaging, 12(2):201–202, 1994.

[42] Songhua Chen, F. Qin, and A.T. Watson. Determination of fluid saturationsduring multiphase flow experiments using nmr imaging techniques. ChemicalEngineering Journal, 40(7):1238–1245, 1994.

[43] George R. Coates, Lizhi Xiao, and Manfred G. Prammer. NMR LoggingPrinciples and Applications, chapter Fundamentals of NMR HydrocarbonTyping. Halliburton Energy Services, 1999.

[44] R. M. Cotts, M. J. R. Hoch, T. Sun, and J. T. Markert. Pulsed field gradi-ent stimulated echo methods for improved NMR diffusion measurements inheterogeneous systems. J. Magn. Res., 83:252–266, 1989.

[45] In Chan Kim Dinko Cule and Salvatore Torquato. Comment on ”Walker dif-fusion method for calculation of transport properties of composite materials”.Phys. Rev. E, 61(4):4659–4660, 2000.

[46] D. W. Curwen and C. Molaro. Permeability from magnetic resonance imaginglogs. In SPWLA 36th Annual Logging Symposium. SPWLA, June 1995.

[47] K-J Dunn, M.Appel, J.J. Freeman, J.S. Gardner, G.J. Hirasaki, J.L. Shafer,and G. Zhang. Interpretation of restricted diffusion and internal field gradi-ents in rock data. In SPWLA 42th Annual Logging Symposium. Society ofPetrophysicists & Well Log Analyst, June 2001.

[48] Keh-Jim Dunn. Magnetic susceptibility contrast induced field gradients inporous media. Magnetic Resonance Imaging, 19:439–442, 2001.

[49] Keh-Jim Dunn, Gerald A. LaTorraca, and David J. Bergman. Permeabil-ity relation with other petrophysical parameters for periodic porous media.Geophysics, 64(2):470–478, 1999.

Bibliography 103

[50] L. Elden and I. Skoglund. Algorithms for the regularization of ill-conditionedleast squares problems with tensor product structure, and application tospace-variant image restoration. Technical Report LiTH-MAT-R-1982-48,Dept. Math., Linkoping Univ., Linkoping, Sweden, 1982.

[51] B. Ferreol and D. H. Rothman. Lattice-Boltzmann simulations of flow throughFontainebleau sandstone. Transport in Porous Media, 20:3–20, 1995.

[52] R. Freedman, N. Heaton, M. Flaum, G. J. Hirasaki, C. Flaum, andM. Hurlimann. Wettability, saturation, and viscosity from NMR measure-ments. Soc. Pet. Eng. J., SPE 87340:317–327, 2003.

[53] Robert Freedman. Advances in nmr logging. J. Petr. Tech., pages 60–66,2006.

[54] A. Ghous, F. Bauget, C. H. Arns, A. Sakellariou, T. J. Senden, A. P. Shep-pard, R. M. Sok, W. V. Pinczewski, R. G. Harris, G. F. Beck, and M. A.Knackstedt. Resistivity and permeability anisotropy measured in laminatedsands via digital core analysis. SPWLA 46th Annual Logging Symposium,New Orleans, Texas, June 2005. Society of Petrophysicists & Well Log Ana-lyst.

[55] Denis S. Grebenkov. Use, misuse, and abuse of apparent diffusion coefficients.Concepts in Magnetic Resonance, 36A(1):24–35, 2010.

[56] H. Guan, D. Brougham, K. S. Sorbie, and K. J. Packer. Wettability effectsin a sandstone reservoir and outcrop cores from NMR relaxation time distri-butions. J. Petr. Sc. Eng., 34:35–54, 2002.

[57] Andrew K. Gunstensen and Daniel H. Rothman. Lattice Boltzmann modelof immiscible fluids. Phys. Rev. A, 43(8):4320–4327, 1991.

[58] Karl G. Helmer, Martin D. Hurlimann, Thomas M. De Swiet, Pabitra N. Sen,and C. H. Sotak. Determination of ratio of surface area to pore volume fromrestricted diffusion in a constant field gradient. J. Magn. Res. A, 115:257–259,1995.

[59] Irwan Hidajat, Mohit Singh, Josh Cooper, and Kishore K. Mohanty. Perme-ability of porous media from simulated NMR response. Transport in PorousMedia, 48:225–247, 2002.

[60] M. Hilpert and C. T. Miller. Pore-morphology based simulation of drainagein totally wetting porous media. Advances in Water Resources, 24:243–255,2001.

[61] G. J. Hirasaki, Sho-Wei Lo, and Ying Zhang. Nmr properties of petroleumreservoir fluids. Magnetic Resonance Imaging, 21:269–277, 2003.

[62] Christoph P. Hunt, B.M. Moskowtiz, and S. K. Banerjee. Practical Handbookof Physical Constants, chapter Magnetic Properties of Rocks and Minerals,pages 189–204. American Geophysical Union, Minnesota, 1995.

Bibliography 104

[63] M. D. Hurlimann, M. Flaum, L. Venkataramanan, C. Flaum, R. Freedman,and G. J. Hirasaki. Diffusion-relaxation distribution functions of sedimentaryrocks in different saturation states. Magnetic Resonance Imaging, 21:305–310,2003.

[64] M. D. Hurlimann, K. G. Helmer, L. L. Latour, and C. H. Sotak. Restricteddiffusion in sedimentary rocks. determination of surface-area-to-volume ratioand surface relaxivity. J. Magn. Res. A, 111(2):169–178, 1994.

[65] M. D. Hurlimann and L. Venkataramanan. Quantitative measurement oftwo-dimensional distribution functions of diffusion and relaxation in grosslyinhomogeneous fields. J. Magn. Res., 157:31–42, 2002.

[66] M. D. Hurlimann, L. Venkataramanan, and C. Flaum. The diffusion-spinrelaxation time distribution function as an experimental probe to characterizefluid mixtures in porous media. J. Chem. Phys., 117(22):10223–10232, 2002.

[67] Martin D. Hurlimann. Effective gradients in porous media due to susceptibil-ity differences. J. Magn. Res., 131:232–240, 1998.

[68] Furqan Hussain. Experimental validation of image based drainage relativepermeability. PhD thesis, University of New South Wales, 2011.

[69] M. A. Ioannidis, I. Ghatzis, and C. Lemaire an R. Perunarkilli. Unsaturatedhydraulic conductivity from nuclear magnetic resonance measurements. WaterResources Research, 42:6 PP,W07201, 2006.

[70] Oleksandr P. Ivankhnenko and David K. Potter. Magnetic susceptibility ofpetroleum reservoir fluids. Physics and Chemistry of the Earth, 29:899–907,2004.

[71] Thomas L. James. Fundamentals of NMR. Indian Academyof Sceinces, Lecture notes, 1998, 1998. available online:http://www.ias.ac.in/initiat/scied/resources/chemistry/James.T.pdf.

[72] A. J. Katz and A. H. Thompson. Quantitative prediction of permeability in porousrock. Phys. Rev. B, 34(11):R8179–R8181, 1986.

[73] A. J. Katz and A. H. Thompson. Prediction of rock electrical conductivity frommercury injection experiments. J. Geophys. Res., 92:599–607, 1987.

[74] Kristina Keating and Rosemary Knight. A laboratory study of the effect of fe(ii)-bearing minerals on nuclear magnetic resonance (nmr) relaxation measurements.Geophysics, 75(3):F72–F82, 2010.

[75] W. E. Kenyon. Nuclear magnetic resonance as a petrophysical measurement. Nucl.Geophys., 6:153–171, 1992.

[76] W. E. Kenyon. Petrophysical principles of applications of NMR logging. The LogAnalyst, March-April:21–43, 1997.

http://www.ias.ac.in/initiat/scied/resources/chemistry/James.T.pdf

Bibliography 105

[77] W. E. Kenyon, P. Day, C. Straley, and J. Willemsen. Compact and consistent rep-resentation of rock NMR data from permeability estimation. In Proc. 61st AnnualTechnical Conference and Exhibition, New Orleans, 1986. SPE 15643.

[78] W. E. Kenyon, P. Day, C. Straley, and J. Willemsen. A three part study of NMRlongitudinal relaxation properties of water saturated sandstones. SPE formationevaluation, 3(3):626–636, SPE 15643, 1988.

[79] R. L. Kleinberg. Utility of NMR t2 distributions, connection with capillary pressure,clay effect, and determination of the surface relaxivity parameter ρ2. MagneticResonance Imaging, 14(7/8):761–767, 1996.

[80] R. L. Kleinberg, Christian Straley, Ridvan Akkurt, and S.A. Farooqui. Nuclearmagnetic resonance of rock: T1 vs. t2. SPE, SPE 26470, 1993.

[81] M. A. Knackstedt, C. H. Arns, A. Limaye, A. Sakellariou, T. J. Senden, A. P.Sheppard, R. M. Sok, W. V. Pinczewski, and G. F. Bunn. Digital core laboratory:Properties of reservoir core derived from 3D images. In Asia Pacific Conferenceon Integrated Modelling for Asset Management, page SPE 87009, Kuala Lumpur,March 2004. SPE 87009.

[82] Mark A. Knackstedt, Christoph H. Arns, Mohammad Saadatfar, Tim J. Senden,Ajay Limaye, Arthur Sakellariou, Adrian P. Sheppard, Rob M. Sok, WolfgangSchrof, and H. Steininger. Elastic and transport properties of cellular solids de-rived from three-dimensional tomographic images. Proc. Royal. Soc. A, 2006.

[83] Raghavendra Kulkarni, A. Ted Watson, and Jan-Erik Nordtvedt. Estimationof porous media flow functions using NMR imaging data. Magnetic ResonanceImaging, 16:707–709, 1998.

[84] Munish Kumar, Tim J. Senden, Adrian P. Sheppard, Jill P. Middleton, and Mark A.Knackstedt. Visualizing and quantifying the residual phase distribution in corematerial. In Improved Core Analysis for Unconventional Fields, number SCA2009-16 in The 23rd International Symposium of the Society of Core Analysts, pages1–12, Noordwijk, September 27-30 2009. Society of Core Analysts.

[85] Shane Latham, Trond Varslot, and Adrian Sheppard. Image registration: enhancingand calibrating X-ray micro-CT imaging. In Challenges in Carbonates and Solutionsfor Field Development, number 35:1-12 in The 22rd International Symposium of theSociety of Core Analysts, Abu Dhabi, UAE, 29 October-2 November 2008. Societyof Core Analysts.

[86] G.A. LaTorraca and K. J. Dunn. Magnetic susceptibility contrast effect on NMRT2 logging. In SPWLA 36th Annual Logging Symposium. SPWLA, June 1995.

[87] C. L. Lawson and R. J. Hansen. Solving Least Squares Problems. Prentice-Hall,1974.

[88] Pierre Le Doussal and Pabitra N. Sen. Decay of nuclear magnetization by diffusionin a parabolic magnetic field: An exactly solvable model. Phys. Rev. B, 46(6):3465–3485, 1992.

Bibliography 106

[89] Lilong Li and Songhua Chen. Incorporating internal gradients and restricted diffu-sion effects in nuclear magnetic resonance log interpretations. Basic Principles ofDiffusion Theory, Experiment and Application, 14(4):1–5, 2010.

[90] Hsie-Keng Liaw, R.Kulkarni, S.Chen, , and A. Ted Watson. Characterization of fluiddistributions in porous media by nmr techniques. Chemical Engineering Journal,42(2):538–546, 1996.

[91] S.-W. Lo, G. J. Hirasaki, W. V. House, and R. Kabayashi. Mixing rules andcorrelations of NMR relaxation times with viscosity, diffusivity, and gas/oil ratio ofmethane/hydrocarbon mixtures. Soc. Pet. Eng. J., 7:24–34, 2002.

[92] D. Lu, M. Zhou, J. H. Dunsmuir, and H. Thomann. NMR T2 distributions andtwo phase flow simulations from x-ray micro-tomography images of sandstones.Magnetic Resonance Imaging, 19:443–448, 2001.

[93] N. S. Martys, J. G. Hagedorn, D. Goujon, and J. E. Devaney. Large scale simulationsof single and multi component flow in porous media. SPIE, 309:205–213, 1999.

[94] Nicos S. Martys and Hudong Chen. Simulation of multicomponent fluids in complexthree-dimensional geometries by the lattice Boltzmann method. Phys. Rev. E,53(1):743–750, 1996.

[95] K. S. Mendelson. Percolation model of nuclear magnetic relaxation in porous media.Phys. Rev. B, 41:562–567, 1990.

[96] Oliver Mohnke, Benjamin Ahrenholz, and Norbert Klitzsh. Joint numerical mi-croscale simulations of muti-phase flow and nmr relaxation behavior in porous me-dia. Geophysical Research Abstracts, 13, 2011.

[97] M. A. Moore and R. Akkurt. Nuclear magnetic resonance applied to gas detectionin a highly laminated gulf of mexico turbidite invaded with synthetic oil filtrate. InPresented at the 1996 Annual Technical Conference and Exhibition of the Societyof Petroleum Engineers, Denver, October 1998. Soc. Pet. Eng.

[98] C. Neuman. Spin echo of spins diffusing in a bounded medium. J. Chem. Phys.,60(11):4508–4511, 1974.

[99] V. H. Nguyen, A. P. Sheppard, M. A. Knackstedt, and W. V. Pinczewski. A dynamicnetwork model for film-flow and snap-off in imbibition displacements. J. Petr. Sc.Eng., 52:54–70, 2006.

[100] P. E. Øren and W. V. Pinczewski. Pore-scale network modelling of waterfloodresidual oil recovery by immiscible gas flooding. In SPE/DOE Ninth Symposiumon Improved Oil Recovery, number 27814, pages 345–359. Soc. Pet. Eng., 1994.

[101] D. K. Potter, T. M. AlGhamdi, and O. P. Ivakhnenko. Sensitive carbonate reservoirrock characterization from magnetic susceptibility: mineral quantification, correla-tion with petrophysical properties, and anisotropy. In International Symposium ofthe Society of Core Analysts, pages SCA2008–10, Abu Dhabi, 2008. Society of CoreAnalysts.

Bibliography 107

[102] M. G. Prammer. NMR pore size distributions and permeability at the well site.In Presented at the 1994 SPE Annual Technical Conference and Exhibition, pages55–64, New Orleans, 1994. Soc. Pet. Eng.

[103] Y.-H. Qian and Y. Zhou. Complete Galilean-invariant lattice BGK models for theNavier-Stokes equation. Europhys. Lett., 42(4):359–364, 1998.

[104] Xie Ranhong and Xiao Lizhi. Advanced fluid-typing methods for nmr logging.Journal of Petroleum Science, 8:163–169, 2011.

[105] Baldwin Robertson. Spin-echo decay of spins diffusing in a bounded region. Phys.Rev., 151(1):273–277, 1966.

[106] William P. Rothwell and H. J. Vinegar. Petrophysical applications of NMR imaging.Applied Optics, 24(23):3969–3972, 1985.

[107] Hakon Rueslatten, Terje Eidesmo, Carsten Slot-Petersen, and Jim White. Nmrstudies of an iron-rich sandstone oil reservoir. In International Symposium of theSociety of Core Analysts, pages SCA1998–10, The Hague, 1998. Society of CoreAnalysts.

[108] A. Sakellariou, T. J. Senden, T. J. Sawkins, M. A. Knackstedt, M. L. Turner, A. C.Jones, M. Saadatfar, R. J. Roberts, A. Limaye, C. H. Arns, A. P. Sheppard, andR. M. Sok. An x-ray tomography facility for quantitative prediction of mechanicaland transport properties in geological, biological and synthetic systems. In UlrichBonse, editor, Developments in X-Ray Tomography IV, volume 5535 of Proceedingsof SPIE, pages 473–484, Denver, August 2004. SPIE.

[109] Arthur Sakellariou, Christoph H. Arns, Adrian P. Sheppard, Rob M. Sok, HolgerAverdunk, Ajay Limaye, Anthony C. Jones, Tim J. Senden, and Mark A. Knack-stedt. Developing a virtual materials laboratory. Materials Today, 10(12):44–51,2007.

[110] Arthur Sakellariou, Tim J. Sawkins, Tim J. Senden, Christoph H. Arns, AjayLimaye, Adrian P. Sheppard, Robert M. Sok, and Mark A. Knackstedt. Micro-CT facility for imaging reservoir rocks at pore scales. In SEG Technical ProgramExpanded Abstracts, pages 1664–1667, Dallas, October 2003. SEG.

[111] Lawrence M. Schwartz and Stephen Kimminau. Analysis of electrical conductionin the grain consolidation model. Geophysics, 52:1402–1411, 1987.

[112] D. 0. Seevers. A nuclear magnetic method for determining the permeability ofsandstones. In SPWLA 7th Annual Logging Symposium. SPWLA, 1966.

[113] John Georg Seland, Geir Humborstad Sørland, Klaus Zick, and Bjørn Hafskjold.Diffusion measurements at long observation times in the presence of spatially vari-able internal magnetic field gradients. J. Magn. Res., 146:14–19, 2000.

[114] John Georg Seland, Kathryn E. Washburn, Henrik W. Anthonsen, and JosteinKrane. Correlations between diffusion, internal magnetic field gradients, andtransvers relaxation in porous systems containing oil and water. Phys. Rev. E,70:051305, 2004.

Bibliography 108

[115] P. N. Sen, C. Straley, W. E. Kenyon, and M. S. Whittingham. Surface-to-volumeratio, charge density, nuclear magnetic relaxation, and permeability in clay-bearingsandstones. Geophysics, 55(1):61–69, 1990.

[116] J. L. Shafer, D. Mardon, and J. Gardner. Diffusion effects on nmr response of oil &water in rock: Impact of internal gradients. In Reservoir engineering applicationsof Core Analysis Data, number SCA-9916 in The 13rd International Symposium ofthe Society of Core Analysts, Golden, Colorado, 1999. Society of Core Analysts.

[117] Xiaowen Shan and Hudong Chen. Lattice Boltzmann model for simulating flowswith multiple phases and components. Phys. Rev. E, 47(3):1815–1819, 1993.

[118] Adrian P. Sheppard, Rob M. Sok, and Holger Averdunk. Techniques for imageenhancement and segmentation of tomographic images of porous materials. PhysicaA, 339(1-2):145–151, 2004.

[119] Clinton DeW Van Siclen. Anomalous walker diffusion through composite systems.J. Phys. A: Math. Gen., 32:5763–5771, 1999.

[120] Yi-Qiao Song. Novel NMR techniques for porous media research. MagneticResonance Imaging, 21:207–211, 2003.

[121] Yi-Qiao Song, Seungoh Ryu, and Pabitra N. Sen. Determining multiple lengthscales in rocks. Nature, 406(13):178–181, 2000.

[122] WM. Scott Dodge Sr, John L. Shafer, and Angel G. Guzman-Garcia. Core and logNMR measurements of an iron-rich, glauconitic sandstone reservoir. In SPWLA36th Annual Logging Symposium. SPWLA, June 1995.

[123] E. O. Stejskal and J. E. Tanner. Spin diffusion measurements: Spin echoes in thepresence of a time-dependent field gradient. J. Chem. Phys., 42(1):288–292, 1965.

[124] Simon W. Stonard, Gerald A. LaTorraca, and Keh-Jim Dunn. Effects of magneticsusceptibility contrast on oil and water saturation determinations by nmr t2 labora-tory and well log measurements. Number SCA1996-19 in International Symposiumof the Society of Core Analysts, pages 1–12, Montpellier, France, September 8-101996. Society of Core Analysts.

[125] C. Straley, C. E. Morris, W. E. Kenyon, and J. J. Howard. NMR in partially sat-urated rocks: laboratory insights on free fluid index and comparison with boreholelogs. In SPWLA 32th Annual Logging Symposium. SPWLA, June 1991.

[126] Christian Straley, Abigail Matteson, Sechao Feng, and Lawrence M. Schwartz. Mag-netic resonance, digital image analysis, and permeability of porous media. Appl.Phys. Lett., 51(15):1146–1148, 1987.

[127] Christian Straley, Harold Vinegar, and Chris Morriss. Core analysis by low fieldnmr. pages SCA1994–9404, 1994.

[128] J. H. Strange, J. B. W. Webber, and S. D. Schmidt. Pore size distribution mapping.Magnetic Resonance Imaging, 14(7/8):803–805, 1996.

Bibliography 109

[129] Alexander L. Sukstanskii and Dmitriy A. Yablonskiy. Effects of restricted diffusionon MR signal formation. J. Magn. Res., 157:92–105, 2002.

[130] Boqin Sun and Keh-Jim Dun. Core analysis with two dimensional NMR. InInternational Symposium of the Society of Core Analysts, pages SCA2002–38:1–12,MontereySome limitations to such approach might occur in the presence of heavyoil and irreducible water saturations as they would be challenging to distinguish[132]., Canada, September 2002.

[131] Boqin Sun and Keh-Jim Dunn. Probing the internal field gradients of porous media.Phys. Rev. E, 65:051309, 2002.

[132] Boqin Sun and Keh-Jim Dunn. Methods and limitations of NMR data inversion forfluid typing. J. Magn. Res., 169:118–128, 2004.

[133] Boqin Sun and Keh-Jim Dunn. A global inversion method for multi-dimensionalNMR logging. J. Magn. Res., 172:152–160, 2005.

[134] Boqin Sun and Keh-Jim Dunn. Two-dimensional nuclear magnetic resonance petro-physics. Magnetic Resonance Imaging, 23:259–262, 2005.

[135] Olumide Talabi. Pore Scale simulation of NMR response in porous media. PhDthesis, London Imperial College, 2008.

[136] J. E. Tanner. Use of the stimulated echo in NMR diffusion studies. J. Chem. Phys.,52(5):2523–2526, 1970.

[137] H. Taud, R. Martinez-Angeles, J. F. Parrot, and L. Hernandez-Escobedo. Porosityestimation method by X-ray computed tomography. J. Petr. Sc. Eng., 47:209–217,2005.

[138] J. J. Tessier and K. J. Packer. NMR measurements and numerical simulation of fluidtransport in porous solids. Fluid Mechanics and Transport Phenomena, 43(7):1653–1661, 1997.

[139] Jean J. Tessier and Ken J. Packer. The characterization of multiphase fluid trans-port in a porous solid by pulsed gradient stimulated echo nuclear magnetic reso-nance. Phys. Fluids, 10(1):75–85, 1998.

[140] A.H. Thompson, S. W. Sinton, S. L. Huff, A. J. Katz, and R. A. Raschke. Deuteriummagnetic resonance and permeability in porous media. Applied Physics, 65(8):3259–3263, 1989.

[141] J.-F. Thovert, F. Yousefian, P. Spanne, C. G. Jacquin, and P. M. Adler. Grainreconstruction of porous media: Application to a low-porosity Fontainebleau sand-stone. Phys. Rev. E, 63:061307, 2001.

[142] A. Timur. Pulsed nuclear magnetic resonance studies of porosity, movable fluid,and permeability of sandstone. J. Petr. Tech., 21:775–786, 1969.

[143] H. C. Torrey. Bloch equations with diffusion terms. Phys. Rev., 104:563–565, 1956.

Bibliography 110

[144] E. Toumelin and C. Torres-Verdın. Modeling of multiple echo-time NMR mea-surements for complex pore geometries and multiphase saturations. SPE ReservoirEvaluation & Engineering, SPE 85635, 2003.

[145] E. Toumelin, C. Torres-Verdın, S. Chen, and D. M. Fischer. Reconciling NMRmeasurements and numerical simulations: Assessment of temperature and diffusivecoupling effects on two-phase carbonate samples. Petrophysics, 44(2):91–107, 2003.

[146] Emmanual Toumelin, Carlos Torres-Verdin, Boqin Sun, and Keh-Jim Dunn.Random-walk technique for simulating nmr measurements and 2D NMR maps ofporous media with relaxing and permeable boundaries. J. Magn. Res., 188:83–96,2007.

[147] Emmanuel Toumelin, Carlos Torres-Verdin, Boqin Sun, and Keh-Jim Dunn. Limitsof 2d nmr interpretation techniques to quatify pore size, wettability, and fluid type:A numerical sensitivity study. SPE Journal, pages 354–363, September 2006.

[148] M. Turner, L. Knufing, C. H. Arns, A. Sakellariou, T. J. Senden, A. P. Sheppard,R. M. Sok, A. Limaye, W. V. Pinczewski, and M. A. Knackstedt. Three-dimensionalimaging of multiphase flow in porous media. Physica A, 339(1-2):166–172, 2004.

[149] R. M. E. Valckenborg, H. P. Huinink, J. J. v. d. Sande, and K. Kopinga. Random-walk simulations of NMR dephasing effects due to uniform magnetic-field gradientsin a pore. Phys. Rev. E, 65:21306, 2002.

[150] Lalitha Venkataramanan, Yi-Qiao Song, and Martin D. Hurliman. Solving Fredholmintegrals of the first kind with tensor product structure in 2 and 2.5 dimensions.IEEE Trans. on Signal Process., 50(5):1017–1026, 2002.

[151] R. C. Wayne and R. M. Cotts. Nuclear-magnetic-resonance study of self-diffusionin a bounded medium. Phys. Rev., 151(1):264–272, 1966.

[152] Hildegard Westphal, Iris Surhold, Christinan Kiesl, Holger F. Thern, and ThomasKruspe. NMR measurements in carbonate rocks: Problems and an approach to asolution. 162(2005):549–570, 2005.

[153] David J. Wilkinson, David Linton Johnson, and Lawrence M. Schwartz. Nuclearmagnetic relaxation in porous media: The role of the mean lifetime τ(ρ, d). Phys.Rev. B, 44:4960–4973, 1991.

[154] Robert C. Wilson and Martin D. Hurlimann. Relationship between susceptibilityinduced field inhomogeneities, restricted diffusion, and relaxation in sedimentaryrocks. J. Magn. Res., 183:1–12, 2006.

[155] Song Xue. Towards improved methods for determining porous media multiphaseflow functions. PhD thesis, Texas AM University, 2004.

[156] Gigi Q. Zhang and George J. Hirasaki. CPMG relaxation by diffusion with con-stant magnetic field gradient in a restricted geometry: numerical simulation andapplication. J. Magn. Res., 163:81–91, 2003.

Bibliography 111

[157] Gigi Q. Zhang, George J. Hirasaki, and Waylon V. House. Effect of internal fieldgradients on NMR measurements. Petrophysics, 42(1):37–47, 2001.

[158] Gigi Q. Zhang, George J. Hirasaki, and Waylon V. House. Internal field gradientsin porous media. Petrophysics, 44(6):422–434, 2003.

[159] Lukasz Zielinski, Raghu Ramamoorthy, Chanh Cao Minh, Khadija Ahmed AlDaghar, Raza Hassan Sayed, and Atef Abdelaal. Restricted diffusion effects in sat-uration estimates from 2D diffusion-relaxation NMR maps. Number SPE 134841,Florence, Italy, 19-22 September 2010. Society of Petroleum Engineers.

NMR petrophysical cross correlations for partially saturated reservoir ...

Documents

Transcript of NMR petrophysical cross correlations for partially saturated reservoir ...