Joint petrophysical inversion of electromagnetic and full ...

Joint petrophysical inversion of electromagneticand full-waveform seismic data

Guozhong Gao1, Aria Abubakar1, and Tarek M. Habashy1

ABSTRACT

Accurate determination of reservoir petrophysical parametersis of great importance for reservoir monitoring and characteri-zation. We developed a joint inversion approach for the directestimation of in situ reservoir petrophysical parameters such asporosity and fluid saturations by jointly inverting electromag-netic and full-waveform seismic measurements. Full-waveformseismic inversions allow the exploitation of the full content ofthe data so that a more accurate geophysical model can be in-ferred. Electromagnetic data are linked to porosity and fluid sat-urations through Archie’s equations, whereas seismic data arelinked to them through rock-physics fluid-substitution equa-tions. For seismic modeling, we used an acoustic approxima-tion. Sensitivity studies combined with inversion tests showthat seismic data are mainly sensitive to porosity distribution,whereas electromagnetic data are more sensitive to fluid-satura-tion distribution. The separate inversion of electromagnetic orseismic data is highly nonunique and thus leads to great ambi-guity in the determination of porosity and fluid saturations. Inour approach, we used a Gauss-Newton algorithm equipped

with the multiplicative regularization and proper data-weightingscheme. We tested the implemented joint petrophysical inver-sion method using various synthetic models for surface andcrosswell measurements. We found that the joint inversion ap-proach provides substantial advantage for an improved estima-tion of porosity and fluid-saturation distributions over the oneobtained from the separate inversion of electromagnetic andseismic data. This advantage is achieved by significantly redu-cing the ambiguity on the determination of porosity and fluidsaturations using multiphysics measurements. We also carriedout a study on the effects of using inaccurate petrophysicaltransform parameters on the inversion results. Our study demon-strated that up to 20% errors in the saturation and porosity ex-ponents in Archie’s equations do not cause significant errors inthe inversion results. On the other hand, if the bulk modulus anddensity of the rock matrix have a large percentage of errors (i.e.,more than 5%), the inversion results will be significantly de-graded. However, if the density of the rock matrix has an errorof less than 2%, the joint inversion can tolerate a large percen-tage of errors in the bulk modulus of the rock matrix.

INTRODUCTION

Seismic and electromagnetic (EM) methods have been playingimportant roles in oil and gas explorations as well as in reservoircharacterization and monitoring applications. One main goal of theformation evaluation is to provide accurate estimates of reservoirporosity and saturation (water, oil, and/or gas) distributions. Theseismic method has been used as an indispensable tool to providethe structural information of the subsurface. However, advances inrock physics have made it possible to link seismic data and reservoirproperties (Gassmann, 1951; Nur, 1992; Wang, 2001). On the otherhand, thanks to the pioneering work of Archie (1942) and others

(for example, Waxman and Smits [1968]), EM data have been usedas a routine quantitative tool to compute in situ fluid saturationsfrom resistivity logs. The differences in the fundamental physicsof these two types of methods make them sensitive to different re-servoir parameters. As a result, the joint inversion of these two typesof measurements enables the estimation of multiple reservoir prop-erties simultaneously, thus leading to improved reservoir character-ization. This paper summarizes our development of estimatingporosity and saturation distributions by using the joint inversionof full-waveform seismic and EM measurements.Inversion methods are widely used in geophysics to derive

earth’s geophysical properties from a variety of measurements in

Manuscript received by the Editor 2 May 2011; revised manuscript received 6 January 2012; published online 23 April 2012.1Schlumberger-Doll Research, Cambridge, Massachusetts, USA. E-mail: [email protected]; [email protected]; [email protected].

© 2012 Society of Exploration Geophysicists. All rights reserved.

WA3

GEOPHYSICS, VOL. 77, NO. 3 (MAY-JUNE 2012); P. WA3–WA18, 22 FIGS., 4 TABLES.10.1190/GEO2011-0157.1

an automated manner. Seismic inversion has been a powerful tool inreconstructing complex geologic structures of the subsurface(Tarantola, 1986; Pratt and Worthington, 1990; Abubakar et al.,2003; Hu et al., 2009b, among others). Especially in recent years,seismic full-waveform inversion (FWI) has become one of the pro-mising seismic imaging techniques. Virieux and Operto (2009) pro-vide a nice overview on the development of seismic FWIalgorithms. On the other hand, EM inversion has also been a usefultool to derive resistivity distribution around the borehole fromsingle-well induction measurements (Kriegshauser et al., 2001;Abubakar and Habashy, 2006; Abubakar et al., 2006), betweenboreholes from crosswell EM measurements (Spies and Habashy,1995; Abubakar et al., 2005), and below the subsurface (Constableet al., 1986; Abubakar et al., 2008; Commer and Newman, 2008;Weitemeyer et al., 2010). However, because the full-waveform seis-mic inversion reconstructs seismic velocity distributions whereasEM inversion produces resistivity distributions, the joint inversionof these two types of data is not straightforward (Harris et al., 2009).One way of performing joint inversion is through imposing struc-

tural similarity constraints between the resistivity and the seismicvelocities of targeted regions (Gallardo and Meju, 2003; Huet al., 2007, 2009a; Colombo et al., 2008; Moorkamp et al.,2011). The alternative approach is to use petrophysical linksbetween the resistivity and the seismic velocities (Hoverstenet al., 2006). The petrophysical relationships, such as Archie’sequation (Archie, 1942) and Waxman and Smits’ equation(Waxman and Smits, 1968), establish the link between resistivity,porosity, and water saturation, whereas the fluid-substitution equa-tions (Gassmann, 1951) link the seismic velocities to porosity andsaturations. These relationships are derived from core analyses,which are usually only meaningful in the region where core samplesare collected. Chen et al. (2007) and Chen and Dickens (2009) useda Bayesian inversion framework to quantify the uncertainty of thereservoir parameters’ inversion from marine seismic amplitudevariation with angle of incidence (AVA) and controlled-source elec-tromagnetics (CSEM) data. They show that uncertainty in the rock-physics model may have significant effects on inversion results.However, the statistical method can be impractical for solving alarge-scale multidimensional problem. Wang et al. (2007) proposea method to separate porosity and water saturation close to a well byusing borehole resistivity and sonic measurements. In this paper, weconsider the full nonlinear inversion of the full-waveform seismicdata and EM data at a reservoir-scale for obtaining 2D pixel porosityand saturation distributions using a massively parallel computingpower. We first perform a sensitivity analysis based on the petro-physical relationships of how seismic and EM measurements aresensitive to porosity and fluid saturations. Then, we present meth-ods and examples for jointly inverting full-waveform seismic andEM measurements for directly obtaining porosity and fluid (waterand oil) saturation distributions. We also carry out an inversionstudy where we introduce uncertainties in the parameters of thepetrophysical transforms.

PETROPHYSICAL RELATIONSHIPS

Archie’s equation

Archie (1942) discovered that formation conductivity σ can beexpressed as a function of porosity ϕ and water saturation Swaccording to the relationship

σ ¼ 1

aσwϕ

mSnw; (1)

where a is a tortuosity factor, m is the porosity/cementation expo-nent, n is the saturation exponent, and σw is the conductivity of theformation saline water. Archie’s equation is valid for clean sandformations. Since the development of Archie’s equation, other var-iants have been developed to account for the conduction of the claycontent in rocks, such as Waxman and Smits’ (1968) equation. Ar-chie’s equation and its variants have served as important tools toestimate the hydrocarbon saturation from EM measurements. Wenote that although in our numerical examples we only use Archie’sequation, the extension of the method for using other resistivity-transform equations is straightforward.For the inversion, we will also need the derivative of the conduc-

tivity with respect to porosity and water saturation, which can becomputed easily as follows:

∂σ∂ϕ

¼ maσwϕ

m−1Snw; (2)

∂σ∂Sw

¼ naσwϕ

mSn−1w : (3)

Gassmann’s equations

On the other hand, the fluid-substitution model (Gassmann,1951) links seismic velocities to reservoir parameters such as por-osity ϕ, water saturation Sw, and oil-saturation So or gas saturationSg. In fluid-saturated rocks, the compressional-wave (P-wave)velocity VP is expressed as follows:

VP ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiKsat þ 4

3μsat

ρsat

s; (4)

where

Ksat ¼ ð1 − βÞKma þ β2M; (5)

M ¼β − ϕ

Kma

þ ϕ

Kf

−1; (6)

Kf ¼Cw

SwKw

þ Co

SoKo

þ Cg

SgKg

−1; (7)

ρsat ¼ ð1 − ϕÞρma þ ϕðSwρw þ Soρo þ SgρgÞ: (8)

In equations 5 and 6, β is the Biot coefficient, which, in general, is afunction of porosity. In this study, we chose the critical porositymodel of Nur (1992) for β:

β ¼ϕ∕ϕc; 0 ≤ ϕ ≤ ϕc

1; ϕ > ϕc; (9)

WA4 Gao et al.

where ϕc is the critical porosity above which solids become suspen-sions. In equations 4–9, Ksat, μsat, and ρsat are the bulk modulus, theshear modulus, and the bulk density of the fluid-saturated rock, Kf

is the bulk modulus of the pore fluid; Kma and ρma are the bulkmodulus and the density of the matrix (solid or grain); Kw, Ko,and Kg are the bulk modulus of water, oil, and gas, respectively;ρw, ρo, and ρg are the density of water, oil, and gas, respectively;and Cw, Co, and Cg are correction terms for water, oil, and gas,respectively. Note that Cg is necessary in the existence of gas(Hoversten et al., 2006), whereas Cw and Co are usually equalto unity. In this paper, we use the acoustic approximation, whichimplies that the shear modulus is neglected, although we haveextended it to full-waveform elastic data (Gao et al., 2011).Similar to EM measurements, for the inversion, we will need the

derivatives of Ksat and ρsat with respect to porosity and water satura-tion (assuming we are dealing with the two-phase flow with oil andwater), which are given by

∂Ksat

∂ϕ¼

−β2M2

Kma

þ 2βM − Kma

∂β∂ϕ

− β2M2

1

Kf

−1

Kma

; (10)

∂ρsat∂ϕ

¼ Swρw þ ð1 − SwÞρo − ρma; (11)

∂Ksat

∂Sw¼ −ϕβ2M2

Cw

Kw

−Co

Ko

; (12)

∂ρsat∂Sw

¼ ϕðρw − ρoÞ: (13)

For three-phase flow, the derivatives can be computed similarly.Note that different models for the Biot coefficient can be used.For the model shown in equation 9, the derivative is given by

∂β∂ϕ

¼1∕ϕc; 0 ≤ ϕ ≤ ϕc

0; ϕ > ϕc. (14)

SENSITIVITY ANALYSIS

To understand the inversion results better, it is necessary to under-stand how seismic and EM measurements are sensitive to changesin porosity or water-saturation distributions. From Archie’s equa-tion given in equation 1, it is straightforward to observe that, math-ematically, ϕ and Sw are nearly interchangeable because m and nusually have similar values, which indicates that EM data are effec-tively a function of the product of ϕ and Sw, i.e., a function of thewater-filled porous space. Typically ϕ varies from 0.1 to 0.3,whereas Sw can change from zero to one. As a result, the changesin the formation conductivity σ are more influenced by water-saturation changes. Such observation can be seen more clearly fromFigure 1, which plots a 2D map of σ for ϕ between zero and 0.4 andfor Sw between zero and one. To generate this map, we use a typicalset of Archie’s parameters: σw ¼ 5.5 S∕m, a ¼ 1, m ¼ 1.6, and

n ¼ 2. It is obvious that neither ϕ nor Sw can be inferred uniquelyfrom EM data only.Gassmann’s equations, given by equations 4–8, show a more

complicated dependency on porosity and fluid-saturation distribu-tions. For typical rock parameters ϕc ¼ 0.4, Kma ¼ 32 GPa,Kw ¼ 2.81 GPa, Ko ¼ 0.75 GPa, ρma ¼ 2560 kg∕m3, ρw ¼1050 kg∕m3, and ρo ¼ 750 kg∕m3; we show in Figure 2 the 2Dmaps of P-wave velocity VP and bulk density ρsat for ϕ betweenzero and 0.4 and Sw between zero and one. In these plots, the hor-izontal axis is ϕ, and the vertical axis is Sw. Figure 2 clearly demon-strates that VP and ρsat are very sensitive to changes in ϕ; however,they have little sensitivity to Sw. This suggests that the inversion ofseismic measurements is more likely to provide more reliable infor-mation on ϕ, but not as robustly on Sw. Note that this sensitivityanalysis is for a two-phase reservoir model: water and oil. Thepresence of gas could lead to a different conclusion. From theabove analysis, one can easily observe that it is difficultfor either EM or seismic measurements to determine ϕ and Sw.

Figure 1. Sensitivity analysis: σ versus (ϕ, Sw) using σw ¼5.5 S∕m, a ¼ 1, m ¼ 1.6, and n ¼ 2 in Archie’s equation.

Figure 2. Sensitivity analysis: (a) VP versus (ϕ, Sw) and (b) ρsatversus (ϕ, Sw) using ϕc ¼ 0.4, Kma ¼ 32 GPa, Kw ¼ 2.81 GPa,Ko ¼ 0.75 GPa, ρma ¼ 2560 kg∕m3, ρw ¼ 1050 kg∕m3, and ρo ¼750 kg∕m3 in Gassmann’s equations.

Joint petrophysical EM and seismic inversion WA5

However, the joint inversion of EM and seismic data could help indetermining ϕ and Sw. In the following sections, we present a jointpetrophysical inversion algorithm to determine ϕ and Sw simulta-neously.It is also worth to point out that if the porosity values are ex-

tremely low, it will be difficult to estimate porosity and watersaturation because the sensitivity of the data with respect to porositywill significantly decrease. Our numerical study indicates that theinversion is still effective for porosity as low as 5%. Also note thatit is rare to encounter conventional reservoirs with a porositybelow 5%.

INVERSION ALGORITHMS

The petrophysical relationships make it possible for us to invertporosity and fluid-saturation distributions from seismic and EMmeasurements separately or jointly. In this section, we summarizethe inversion methodologies for separate/single-physics and jointinversions.

Single-physics inversion

The inversion domain is discretized into Nx × Nz uniform grids,where Nx and Nz are the number of cells in the x-direction and thez-direction, respectively. Assume thatmϕ andmSw are the unknownvectors representing the porosity and water-saturation distributions,respectively. The total unknown vector ism ¼ ½mϕ mSw T, where Tdenotes the transpose of a matrix. As a result, the size of m is2N ¼ 2NxNz. The cost function for the single-physics petrophysi-cal inversion to be minimized is written as follows:

ΦðmÞ ¼ ΦdðmÞ þ λΦmðmÞ; (15)

which is, for the Gauss-Newton method, equivalent to the multipli-cative cost function (as introduced by van den Berg et al. [1999])when the regularization parameter λ at each iteration n is set to be

λn ¼ΦdðmnÞΦmðmnÞ

: (16)

The normalized data misfit Φd is given by

ΦdðmÞ ¼ 1

2

XNF

k¼1

ðηkÞ2PNS

i¼1

PNR

j¼1 jwd;i;j;k½di;j;k − si;j;kðmÞj2PNS

i¼1

PNR

j¼1 jwd;i;j;kdi;j;kj2

¼ 1

2kWd½d − sðmÞk2; (17)

where NF, NS, and NR are the number of frequencies, sources, andreceivers, respectively. The vector d is the measured data, and s isthe simulated response vector for a given model parameter m. Thesimulated response is calculated using the forward algorithms de-scribed in Appendix A combined with the petrophysical relation-ships. The matrix wd;i;j;k is the data-weighting matrix whosediagonal elements are the inverse of the estimates of the standarddeviation of the measurement noise. For surface EM and seismicinversions, wd;i;j;k includes the weighting based on the Jacobianof the initial model to safeguard against measurements with relativehigh amplitudes that tend to dominate the inversion (seeAbubakar et al., 2009). The frequency weighting factor ηk is used

for the seismic inversion as a way to prevent the high-frequencycomponents from dominating the inversion process, which isgiven by

ηk ¼ω−2kPNF

i¼1 ω−2i

: (18)

The regularization cost function Φm is defined as follows:

Φmn ðmÞ ¼ 1

2

Zd

b2nðrÞfj∇mðrÞj2 þ δ2gdV; (19)

where r is the spatial coordinate and where

b2nðrÞ ¼1R

d fj∇mnðrÞj2 þ δ2ngdV(20)

for the l2-norm regularizer and

b2nðrÞ ¼1

V1

j∇mnðrÞj2 þ δ2n(21)

for the weighted l2-norm regularizer as introduced in van den Bergand Abubakar (2001). The function δ2n is chosen to be equal toΦdðmnÞ∕ðΔxΔzÞ, whereΔx andΔz are the size of the discretizationcell in the x- and z-directions, respectively. We use the Gauss-Newton minimization approach to perform the inversion, whichis described in the next section.

Joint inversion

The cost function for the simultaneous petrophysical joint inver-sion is written as

ΦðmÞ ¼ ΦSMd ðmÞ þ γΦEM

d ðmÞ þ λΦmðmÞ; (22)

whereΦSMd is the seismic data misfit andΦEM

d is the EM data misfit.They are defined in equation 17. The model regularization Φm isdefined in equation 19. The symbol γ is the relative weighting be-tween seismic data misfitΦSM

d and EM data misfitΦEMd to safeguard

against any data set from dominating the inversion process. Intui-tively, the choice of γ could be critical for obtaining robust inversionresults. In our study, we choose γ as follows:

γ ¼ ΦSMd ðm0Þ

ΦEMd ðm0Þ

; (23)

where m0 is the initial model. As in Abubakar et al. (2011), we canalso use a different γ value at each iteration n.

Gauss-Newton minimization

For the single-physics inversion and the simultaneous joint inver-sion, we use the Gauss-Newton minimization framework describedin Habashy and Abubakar (2004). At the nth iteration, we obtain aset of linear equations for the search vector pn that identifies theminimum of the approximated quadratic cost function, namely,

Hnpn ¼ −gn; (24)

WA6 Gao et al.

where the gradient of the cost function gn is given by

gn ¼ JTnWTdWd½d − sðmnÞ þ λnLnmn (25)

and the Gauss-Newton Hessian matrix Hn is given by

Hn ≈ JTnWTdWdJn þ λnLn: (26)

The derivative of the regularization cost function is given by

Lnmn ¼ ∇ · ½bnðrÞ∇mnðrÞ: (27)

The Jacobian matrix J is defined as the derivative of the simulateddata with respect to the model parameter given by the followingexpression:

Ji;j;k;τ ¼∂si;j;kðmÞ

∂mτ: (28)

For EM and seismic data, the Jacobian matrices with respect to geo-physical parameters (conductivity, bulk modulus or inverse of com-pressibility, and mass density) are computed using an adjointformulation (see, for example, McGillivray and Oldenburg,1990; Abubakar et al., 2008). After the Jacobian matrices, with re-spect to the geophysical parameters, are computed, we use the chainrule to compute the Jacobian matrices with respect to the petrophy-sical parameters. The elements of the EM Jacobian matrixJEMðmÞ ¼ ½JEMðmϕÞ JEMðmSwÞ are calculated as

JEMi;j;k;τ ¼∂sEMi;j;kðmÞ

∂στ∂στ∂mτ

; (29)

where the derivatives of conductivity with respect to porosity andwater saturation are given in equations 2 and 3. The elements of theseismic Jacobian matrix JSMðmÞ ¼ ½JSMðmϕÞ JSMðmSwÞ are calcu-lated as

JSMi;j;k;τ ¼∂sSMi;j;kðmÞ∂Ksat;τ

∂Ksat;τ

∂mτþ ∂sSMi;j;kðmÞ

∂ρsat;τ∂ρsat;τ∂mτ

; (30)

where the derivatives of the bulk modulus and mass densitywith respect to porosity and water saturation are given in equa-tions 10–13.Because we are dealing with a large number of model parameters,

we solve equation 24 using the conjugate-gradient least-squares(CGLS) method (Golub and van Loan, 1996). After we obtainthe search vector pn, the model parameters are updated as follows:

mnþ1 ¼ mn þ νnpn; (31)

where νn is determined by a line-search algorithm as described inHabashy and Abubakar (2004). This line-search procedure is usedto guarantee the cost-function reduction as the iteration proceeds. Inaddition, we also used a nonlinear transform to force the invertedmodel parameters to lie within their physical bounds: mmin ≤m ≤ mmax . The detailed implementation of this constrained mini-mization can be found in Abubakar (2004). In our implementation,the forward response, the Jacobian construction, and the CGLSiterative process are parallelized using the message passinginterface (MPI) library, and their memory use is distributed nearlylinearly among all processors. The parallel implementation of ouralgorithm has good scalability, which is nearly linear. The later iscrucial for the application to an extremely large-scale problem.

INVERSION EXAMPLES

As demonstrations, we show a time-lapse crosswell fluid mon-itoring example and a surface exploration example. All inversionresults are obtained using the l2-norm regularization if it is notexplicitly specified. In addition, to investigate how errors in labora-tory-measured petrophysical parameters in Archie’s and Gass-mann’s equations affect the inversion results, we also carried outinversions using petrophysical transform parameters with errorsof up to 20%.

Time-lapse crosswell monitoring

Model setup

Figure 3 shows the true porosity, water-saturation distributionsfor the baseline crosswell model, and water-saturation distributions

Distance (m)

Dep

th (

m)

0 50 100

720

a) b) c)

740

760

780

800

820

840

860

φ 0

0.1

0.2

0.3

Distance (m)

Dep

th (

m)

0 50 100

720

740

760

780

800

820

840

860

Sw

0

0.2

0.4

0.6

0.8

1

Distance (m)

Dep

th (

m)

0 50 100

720

740

760

780

800

820

840

860

Sw

0

0.2

0.4

0.6

0.8

1

Figure 3. (a) True porosity, (b) water-saturation distributions of the baseline model, and (c) water-saturation distributions for the time-lapsecrosswell model.


for the time-lapse crosswell model. Figure 4 shows the correspond-ing resistivity, P-wave velocity, and mass-density distributions.These models are 2D. The resistivity model shown in Figure 4ais constructed from an actual field located in Devine, Texas. Theobservation wells are located at x ¼ 0m and x ¼ 104 m. Bothwells are nearly vertical. Between 784 and 822 m is a porousand permeable sandstone layer that is oil saturated, with shale layersabove and below. A hard formation with extremely low porosity (aslow as a few percentage points) and permeability is locatedbelow 846 m.We assume that water is injected using a vertical well located

around x ¼ 80 m. After the water-injection process, the residualoil saturation is about 0.2. This scenario is reasonable becausethe water normally does not completely replace the oil. Figure 3cshows the true water-saturation distribution after a certain time ofsaline water injection, which clearly indicates a waterflooded zonethat has a typical shape due to gravity effect and the special poregeometry. We refer to this as the time-lapse model. In our inversion,we do not take into account the salinity variation. This assumptionwill only be valid if one injects water that has also been producedfrom the same area. An example of an inversion approach that takesinto account the salinity variation for reservoir-scale measurementscan be found in Liang et al. (2010). Figure 5 shows the correspond-

ing resistivity, P-wave velocity, and mass-density distributions ofthe time-lapse model. The true Archie’s parameters are a ¼ 1,m ¼ 1.6, n ¼ 2, and σw ¼ 5.5 S∕m. The true rock-physicsparameters are ϕc ¼ 0.4, Kma ¼ 32 GPa, Kw ¼ 2.81 GPa,Ko ¼ 0.75 GPa, ρma ¼ 2560 kg∕m3, ρw ¼ 1050 kg∕m3, andρo ¼ 750 kg∕m3.

Measurement setup

For the EM survey, we use 57 vertical magnetic dipole sources inone well and 57 receivers in the other well. The receivers measureonly the magnetic-field component along the wellbore axis. Sourcesand receivers are distributed uniformly with a spacing of 2.5 m fromz ¼ 700 m to z ¼ 880 m. The operating frequencies for EMsources are 500 Hz and 1 kHz.For the seismic survey, we use 57 monopole sources and 57 re-

ceivers. The receivers measure scalar pressure fields. The distancebetween two sources or receivers is 2 m. The seismic data are time-domain data. However, in the inversion, we use only data thatcorrespond to operating frequencies of 25, 75, and 125 Hz.Synthetic EM and seismic data are generated for the baseline and

time-lapse models. We then added 2% Gaussian random noise toEM and seismic data. As initial models, we use homogeneous

Distance (m)

Dep

th (

m)

0 50 100

720

740

760

780

800

820

840

860

R (Ω⋅m)

3.16

10.00

31.62

100.00

316.23

Distance (m)

Dep

th (

m)

0 50 100

720

740

760

780

800

820

840

860

VP

(m/s)2000

2500

3000

3500

Distance (m)

Dep

th (

m)

0 50 100

720

740

760

780

800

820

840

860

ρ (kg/m3)1800

2200

2600a) b) c)

Figure 5. (a) True resistivity, (b) P-wave velocity, and (c) mass-density distributions of the time-lapse model calculated from the true porositydistribution shown in Figure 3a and the true water-saturation distribution shown in Figure 3c.

Distance (m)

Dep

th (

m)

0 50 100

720

a) b) c)

740

760

780

800

820

840

860

R (Ω⋅m)

3.16

10.00

31.62

100.00

316.23

Distance (m)

Dep

th (

m)

0 50 100

720

740

760

780

800

820

840

860

VP

(m/s)2000

2500

3000

3500

Distance (m)

Dep

th (

m)

0 50 100

720

740

760

780

800

820

840

860

ρ (kg/m3)1800

2200

2600

Figure 4. (a) True resistivity, (b) P-wave velocity, and (c) mass-density distributions of the baseline model calculated from the true porosity andwater-saturation distributions shown in Figure 3a and 3b.

WA8 Gao et al.

models with 0.2 for porosity and 0.5 for water saturation. The in-version domain is discretized into 78 cells in the x-direction and 88cells in the z-direction. The cell size in the x- and z-directions is 2 m.


First we carried out an EM-only petrophysical inversion for base-line and time-lapse models. Figure 6 shows inverted porosity andwater-saturation distributions for the baseline model and the time-lapse model, respectively.We clearly observe that the EM-only petrophysical inversion

could not accurately recover porosity and water-saturation distribu-tions simultaneously. Furthermore, the recovered water-saturationdistributions are much better than porosity distributions. It is worth-while to point out that because the porosity and water saturationappear as a product in Archie’s equation, theoretically one canresolve only the product of the two. One possible reason that theinverted water-saturation distributions are better than the invertedporosity distributions is the choice of the Archie parameters mand n. Furthermore, the variation of water saturation is larger thanthat of porosity distribution.Next, we perform the seismic-only inversion on the baseline and

time-lapse data. Figure 7 shows the inverted porosity and water-saturation distributions for the baseline model and the time-lapsemodel, respectively.Similar to the EM-only inversion, the seismic-only inversion

could not accurately recover porosity and water-saturation distribu-tions simultaneously. The porosity distribution was recovered much

better than the water-saturation distribution, as indicated by thesensitivity analysis. However, we observe that the inverted porositydistributions for baseline and time-lapse models are very close to thetrue ones (hence, also P-wave velocity and mass-density distribu-tions [not shown]). This is because we use a full-waveform inver-sion approach instead of a migration or a traveltime tomographyapproach, which also shows that the seismic data have higherresolution than the EM data.

Joint inversion

We simultaneously invert EM and seismic data for the baselineand time-lapse models. Figure 8 shows the inverted porosity distri-butions for the baseline and time-lapse models as well as their dif-ferences. The inversion recovered porosity distribution very welland, as expected, the differences between the baseline and time-lapse models are very small.Figure 9 shows the inverted water-saturation distributions for the

baseline and time-lapse models and their differences. Both inver-sions recovered water-saturation distributions quite well. However,the resolution of the inverted water-saturation distributions is lowerthan that of the inverted porosity distributions. Nevertheless, thedifference in water saturation clearly indicates the location of thewaterflooded region. We also observe that the inverted water-satura-tion distributions obtained by the joint inversion algorithm are sig-nificantly better than those obtained by the EM-only petrophysicalinversion as shown in Figure 6b and 6d. This shows an addedbenefit of the joint inversion approach.

Distance (m)

Dep

th (

m)

0 50 100

720

a) b) c) d)

740

760

780

800

820

840

860

φ 0

0.1

0.2

0.3

Distance (m)

Dep

th (

m)

0 50 100

720

740

760

780

800

820

840

860

Sw0

0.2

0.4

0.6

0.8

1

Distance (m)

Dep

th (

m)

0 50 100

720

740

760

780

800

820

840

860

φ 0

0.1

0.2

0.3

Distance (m)

Dep

th (

m)

0 50 100

720

740

760

780

800

820

840

860

Sw0

0.2

0.4

0.6

0.8

1

Figure 6. (a) Inverted porosity and (b) water-saturation distributions of the baseline model, and (c) inverted porosity and (d) water-saturationdistributions for the time-lapse model using the EM-only inversion.

Distance (m)

Dep

th (

m)

0 50 100

720

740

760

780

800

820

840

860

φ 0

0.1

0.2

0.3

Distance (m)

Dep

th (

m)

0 50 100

720

740

760

780

800

820

840

860

Sw

0

0.2

0.4

0.6

0.8

1

Distance (m)

Dep

th (

m)

0 50 100

720

740

760

780

800

820

840

860

Sw

0

0.2

0.4

0.6

0.8

1

Distance (m)

Dep

th (

m)

0 50 100

720

740

760

780

800

820

840

860

φ0

0.1

0.2

0.3a) b) c) d)

Figure 7. (a) Inverted porosity and (b) water-saturation distributions of the baseline model. (c) Inverted porosity and (d) water-saturationdistributions for the time-lapse model using the seismic-only inversion.


Figure 10 shows the resistivity distribution for the baseline andtime-lapse models and their differences calculated based on the in-verted porosity shown in Figure 8 and the inverted water saturationshown in Figure 9. The differences are plotted on a logarithmic

scale. Both inversions confirm that we can reconstruct the resistivitydistributions reasonably because porosity and water-saturation dis-tributions are also well recovered. The resistivity differences clearlyindicate the location of the waterflooded zone.

Distance (m)

Dep

th (

m)

0 50 100

720

a) b) c)

740

760

780

800

820

840

860

φ 0

0.1

0.2

0.3

Distance (m)

Dep

th (

m)

0 50 100

720

740

760

780

800

820

840

860

φ 0

0.1

0.2

0.3

Distance (m)

Dep

th (

m)

0 50 100

720

740

760

780

800

820

840

860

∆φ−1

−0.5

0

0.5

1

Figure 8. Inverted porosity distributions for (a) baseline modeland (b) the time-lapse model. (c) Their differences using the joint inversion.

Distance (m)

Dep

th (

m)

0 50 100

a) b) c)

740

760

780

800

820

840

860

Sw

0

0.2

0.4

0.6

0.8

1

Distance (m)

Dep

th (

m)

0 50 100

720

740

760

780

800

820

840

860

Sw

0

0.2

0.4

0.6

0.8

1

Distance (m)

Dep

th (

m)

0 50 100

720

740

760

780

800

820

840

860

∆Sw

−1

−0.5

0

0.5

1

Figure 9. Inverted water-saturation distributions for (a) baseline model and (b) the time-lapse model. (c) Their differences using the jointinversion.

Distance (m)

Dep

th (

m)

0 50 100

720

a) b) c)

740

760

780

800

820

840

860

R (Ω⋅m)

3.16

10.00

31.62

100.00

316.23

Distance (m)

Dep

th (

m)

0 50 100

720

740

760

780

800

820

840

860

R (Ω⋅m)

3.16

10.00

31.62

100.00

316.23

Distance (m)

Dep

th (

m)

0 50 100

720

740

760

780

800

820

840

860

∆R−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Figure 10. Resistivity distribution of (a) the baseline model and (b) the time-lapse model. (c) Their differences calculated from the invertedporosity distribution given in Figure 8 and water-saturation distribution given in Figure 9. The resistivity (and difference) is plotted onlog scale.

WA10 Gao et al.

Figure 11 shows P-wave velocity distributions for the baselineand time-lapse models and their relative differences, which areshown as a percentage. We observe that both P-wave velocity dis-tributions are well recovered. Note that in Figure 11c, we observethe effect of the water-injection process. However, the shape and thelocation of the waterflooded zone are not as obvious as those shownby the resistivity changes. This is what we expected because theresistivity is more sensitive to the water-saturation variations (theresistivity variations are larger than the P-wave velocity variationswhen one replaces oil with water). For completeness, Figure 12shows mass-density distributions for baseline and time-lapse mod-els and their relative differences. The relative differences are givenas a percentage. Again, the results confirm that both inversionscould recover mass-density distributions, although the observedwaterflooding zone is not as obvious as the one shown in the re-sistivity changes.To quantify the improvement that the joint inversion approach

brings for the data interpretation, we show in Table 1 the valuesof the model misfit between the inverted model and the true modelfor the EM-only inversion, the seismic-only inversion, and the joint

inversion, for all of the preceding baseline and the time-lapse mod-els. The model misfit is defined as follows:

Em ¼ 100%

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPNxNzτ¼1 ðminv

τ −mtrueτ Þ2PNxNz

β¼1 ðmtrueβ Þ2

vuut ; (32)

where minv and mtrue are the inverted model and the true model,respectively.The model misfits in Table 1 clearly show the advantage of the

joint inversion approach. For the baseline and the time-lapse mod-els, the water-saturation distribution from the EM-only inversionhas a model error of roughly 27%, whereas that from the joint in-version is only 8%; the porosity distribution from the seismic-onlyinversion has a model error of roughly 10.4%, whereas that from thejoint inversion is only 1.3%.The inversion results demonstrate that (1) simultaneous joint in-

version significantly reduces ambiguity in recovering porosity andwater-saturation distributions and (2) joint inversion provides an

Distance (m)

Dep

th (

m)

0 50 100

720

740

760

780

800

820

840

860

ρ (kg/m3)1800

2200

2600

Distance (m)

Dep

th (

m)

0 50 100

720

740

760

780

800

820

840

860

∆ρ (%)−10

−8

−6

−4

−2

0

2

4

6

8

10

Distance (m)

Dep

th (

m)

0 50 100

720

740

760

780

800

820

840

860

ρ (kg/m3)1800

2200

2600a) b) c)

Figure 12. The mass-density distributions for (a) the baseline model and (b) the time-lapse model. (c) Their relative differences in percentagecalculated from the inverted porosity as shown in Figure 8 and the inverted water saturation as shown in Figure 9.

Distance (m)

Dep

th (

m)

0 50 100

720

740

760

780

800

820

840

860

VP

(m/s)2000

2500

3000

3500

Distance (m)

Dep

th (

m)

0 50 100

720

740

760

780

800

820

840

860

∆VP

(%)−10

−8

−6

−4

−2

0

2

4

6

8

10

Distance (m)

Dep

th (

m)

0 50 100

720

740

760

780

800

820

840

860

VP

(m/s)2000

2500

3000

3500a) b) c)

Figure 11. The inverted P-wave velocity distributions for (a) the baseline modeland (b) the time-lapse model (c) Their relative differences inpercentage calculated from the inverted porosity given in Figure 8 and the inverted water saturation given in Figure 9.


effective tool to monitor fluid movements in the reservoir-monitoring process.

Marine surface exploration example

In recent years, marine CSEM technology has become a promis-ing tool to identify resistive targets for offshore reservoir explora-tion. In this section, we provide an example to invert for porosityand water-saturation distributions from EM and full-waveformseismic (using the acoustic approximation) measurements in marineenvironments.

Model setup

Figure 13 shows porosity and water-saturation distributions ofthe true model and the corresponding resistivity, P-wave velocity,and mass-density distributions calculated from the petrophysical

relationships. The seawater is assumed to have porosity and watersaturation of one. The bathymetry is flat and water depth is 1 km.The subsurface has two porosity layers: the top layer (z ¼ 1 to2 km) has a porosity of 0.15, whereas the bottom layer (belowz ¼ 2 km) has a porosity of 0.1. There are two 4-km-long hydro-carbon-bearing reservoirs with a horizontal separation of 4 kmlocated above 2 km depth. The left reservoir has a thickness of300 m, and the right one is 200 m thick. The porosity values ofthe reservoirs are 0.25, and their water saturation values are 0.2.Figure 13c shows the true resistivity model calculated using the Ar-chie parameters: a ¼ 1, m ¼ 1.6, n ¼ 2.0, and σw ¼ 20:55 S∕m.The formation below the seabed has a resistivity of roughly1.25 ohm-m, whereas the reservoir resistivities are about20 ohm-m. Figure 13d and 13e shows the P-wave velocityand mass-density distributions assuming the following rockparameters: Kma ¼ 32 GPa, Kw ¼ 2.56 GPa, Ko ¼ 0.75 GPa,

ρma ¼ 2600 kg∕m3, ρw ¼ 1050 kg∕m3, ρo ¼750 kg∕m3, and ϕc ¼ 0.4. All fluid-correctionterms are set to one. The P-wave velocity andmass density for the formation below the seabedare roughly 3211 m∕s and 2442 kg∕m3, whereasthose of the reservoirs are about 2487 m∕sand 2152 kg∕m3.

Measurement setup

For the EM survey, there are 41 inline electricsources located at a depth of 950 m and spaced500 m apart. There are 40 inline electric receivers

Table 1. Model misfits for the EM-only inversion, the seismic-only inversion,and the joint inversion for the baseline model and the time-lapse model.

Baseline model Time-lapse model

ϕ misfit Sw misfit ϕ misfit Sw misfit

Inversion (%) (%) (%) (%)

EM 5.20 27.11 4.82 27.17

Seismic 10.42 74.25 10.46 68.10

Joint 1.32 8.10 1.31 8.54

T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T TR R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R

Distance (km)

Dep

th (

km)

0 2 4 6 8 10 12 14 16 18 20

0.51

1.52

2.5φ 0

0.1

0.2

0.3


Distance (km)

Dep

th (

km)

0 2 4 6 8 10 12 14 16 18 20

0.51

1.52

2.5S

w

00.20.40.60.81


Distance (km)

Dep

th (

km)

0 2 4 6 8 10 12 14 16 18 20

0.51

1.52

2.5R (Ω⋅m)

0.32 1.00 3.1610.0031.62


Distance (km)

Dep

th (

km)

0 2 4 6 8 10 12 14 16 18 20

0.51

1.52

2.5V

P(m/s)

15002000250030003500


Distance (km)

Dep

th (

km)

0 2 4 6 8 10 12 14 16 18 20

0.51

1.52

2.5

ρ (kg/m3)10001400180022002600

a)

b)

c)

d)

e)

Figure 13. (a) True porosity and (b) water-saturation distributionsfor the marine model, as well as corresponding (c) true resistivity,(d) P-wave velocity, and (e) density distributions. Resistivity is inlog scale.


Distance (km)

Dep

th (

km)

0 2 4 6 8 10 12 14 16 18 20

0.51

1.52

2.5φ 0

0.1

0.2

0.3


Distance (km)

Dep

th (

km)

0 2 4 6 8 10 12 14 16 18 20

0.51

1.52

2.5S

w

00.20.40.60.81


Distance (km)

Dep

th (

km)

0 2 4 6 8 10 12 14 16 18 20

0.51

1.52

2.5R (Ω⋅m)

0.32 1.00 3.1610.0031.62


Distance (km)

Dep

th (

km)

0 2 4 6 8 10 12 14 16 18 20

0.51

1.52

2.5

ρ (kg/m3)10001400180022002600


Distance (km)

Dep

th (

km)

0 2 4 6 8 10 12 14 16 18 20

0.51

1.52

2.5

VP

(m/s)15002000250030003500

a)

b)

c)

d)

e)

Figure 14. (a) Inverted porosity and (b) water-saturation distribu-tions using the EM data only, as well as the (c) corresponding re-sistivity, (d) P-wave velocity, and (e) density distributions.Resistivity is in log scale.

WA12 Gao et al.

located at the sea bottom, spaced at 500 m. The EM survey fre-quency is 0.25 Hz.For the seismic survey, we use 41 explosive seismic sources

and 40 receivers located at the sea bottom and spaced 500 m apart.The seismic survey frequencies are 1, 2, 3, 4, and 5 Hz. For full-waveform inversion, the use of low-frequency data (1 and 2 Hz) isnecessary for inversion algorithm stability. This constraint might berelaxed to some extent when we have a better initial model. We add2% Gaussian random noise to EM and seismic data. We also use thesame grid setup for seismic and EM inversions. The cell size in thex-direction is 100 m, and that in the z-direction is 50 m. As initialmodels, we use homogeneous models, with 0.1 for porosity and 0.8for water saturation.


Figure 14 shows the inverted porosity and water-saturation dis-tributions using EM data only, as well as the corresponding resis-tivity, P-wave velocity, and density distributions. The final EM datamisfit is 1.27%. We observe that the EM-only inversion recoverswater-saturation distribution much better than porosity distribution.However, the resistivity distribution is recovered quite well, whichindicates the high nonuniqueness of the solution. The well-recovered resistivity indicates that the inverted water-saturation dis-tribution may not be reliable because porosity distribution is notcorrect. Neither the P-wave velocity distribution nor the mass-density distribution is correct because porosity distribution is notrecovered correctly.Figure 15 shows the inverted porosity and water-saturation dis-

tributions using the seismic data only, as well as the corresponding

resistivity, P-wave velocity, and density distributions. The final seis-mic data misfit is 1%. Unlike the EM-only petrophysical inversionresults, seismic inversion recovers porosity distribution very well.Furthermore, the inverted porosity distribution resolution is signif-icantly higher than the inverted water-saturation distribution fromthe EM-only inversion. This is consistent with the sensitivity ana-lysis presented earlier. The P-wave velocity and density distribu-tions are recovered correctly: however, the resistivity distributionis not correct because the inversion was unable to recover the watersaturation. Notice that the two porosity layers were also identifiedproperly. It is worthwhile to note that because the P-wave velocityand mass density have some sensitivity to water saturation; whenthe inverted water-saturation distribution is incorrect, the invertedporosity distribution may not be reliable.

Joint inversion

Figure 16 shows the inverted porosity and water-saturation dis-tributions jointly using EM and seismic data, as well as the corre-sponding resistivity, P-wave velocity and density distributions. Thefinal seismic data misfit is 1.31%, whereas the final EM data misfitis 1.37%. These data misfit values are similar as those of the sepa-rate inversions. Figure 16a and 16b clearly shows that porosity andwater-saturation distributions are properly recovered. This is alsoconfirmed by Figure 16c, 16d, and 16e, in which all derived geo-physic properties (resistivity, P-wave velocity, and mass density)were recovered correctly, which cannot be achieved either by onlyEM inversion or only seismic inversion.Table 2 shows the model misfits for the EM-only inversion, the

seismic-only inversion, and joint inversion, respectively. This model


Distance (km)

Dep

th (

km)

0 2 4 6 8 10 12 14 16 18 20

0

1

2

3 φ 0

0.1

0.2

0.3


Distance (km)

Dep

th (

km)

0 2 4 6 8 10 12 14 16 18 20

0

1

2

3S

w

00.20.40.60.81


Distance (km)

Dep

th (

km)

0 2 4 6 8 10 12 14 16 18 20

0

1

2

3R (Ω⋅m)

0.32 1.00 3.1610.0031.62


Distance (km)

Dep

th (

km)

0 2 4 6 8 10 12 14 16 18 20

0

1

2

3V

P(m/s)

15002000250030003500

a)

b)

c)

d)

e)T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T TR R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R

Distance (km)

Dep

th (

km)

0 2 4 6 8 10 12 14 16 18 20

0

1

2

3ρ (kg/m3)

10001400180022002600

Figure 15. (a) Inverted porosity and (b) water-saturation distribu-tions using the seismic data only, as well as the (c) correspondingresistivity, (d) P-wave velocity, and (e) density distributions. Resis-tivity is in log scale.


Distance (km)

Dep

th (

km)

0 2 4 6 8 10 12 14 16 18 20

0.51

1.52

2.5φ 0

0.1

0.2

0.3


Distance (km)

Dep

th (

km)

0 2 4 6 8 10 12 14 16 18 20

0.51

1.52

2.5S

w

00.20.40.60.81


Distance (km)

Dep

th (

km)

0 2 4 6 8 10 12 14 16 18 20

0.51

1.52

2.5R (Ω⋅m)

0.32 1.00 3.1610.0031.62


Distance (km)

Dep

th (

km)

0 2 4 6 8 10 12 14 16 18 20

0.51

1.52

2.5

VP

(m/s)15002000250030003500


Distance (km)

Dep

th (

km)

0 2 4 6 8 10 12 14 16 18 20

0.51

1.52

2.5

ρ (kg/m3 )10001400180022002600

a)

b)

c)

d)

e)

Figure 16. (a) Inverted porosity and (b) water-saturation distribu-tions using EM and seismic data jointly, as well as the (c) corre-sponding resistivity, (d) P-wave velocity and (e) densitydistributions. Resistivity is in log scale.


misfit comparison clearly shows that joint inversion significantlyimproves the estimates of porosity and water-saturation distribution.The joint inversion reduces the model error of water saturationto 11.18%, whereas it reduces the model error of theporosity to 10.57%. Note that both model errors for the EM-onlyinversion are significantly larger than those of the joint inversion,whereas the model error of the porosity for the seismic-only inver-sion is also larger than that of the joint inversion, which indicatesthat errors in water saturation can cause errors in inverted porosityvalues.

Sensitivity analysis for parametersin petrophysical relationships

Because some of the parameters in Archie’s equation and Gass-mann’s equations are derived empirically, they inevitably includeerrors. As a result, it is necessary to study how errors in theseparameters affect the inversion results. We focus on four param-eters: m, n, Kma, and ρma for random errors of 5%, 10%, and20%. In our study, when there are errors in m and n, Kma andρma will be kept accurate, and vice versa. When we compute thegeophysical models from inverted petrophysical models, the cor-rupted petrophysical parameters are used.

Scenario 1: Effects of errors in m and n

Figure 17 shows the inverted porosity distributions for 0%, 5%,10%, and 20% errors in m and n, respectively, and Figure 18 is thecorresponding inverted water-saturation distributions. Table 3 liststhe model misfits after inversions for different error levels inArchie’s parametersm and n. The comparisons shown in Figures 17and 18 as well as in Table 3 clearly demonstrate that, for this ex-ample, the inversion results are not affected significantly by errorsin m and n, even when the error is as high as 20%. Furthermore, themodel misfits in Table 3 show that those errors mainly caused someerrors only in water-saturation distribution.

Scenario 2: Errors in Kma and ρma

Figure 19 shows inverted porosity distributions for 0%, 5%, and10% errors in Kma and ρma, and Figure 20 is the corresponding in-verted water-saturation distributions. The first part of Table 4 liststhe model misfits for equal error levels in rock-physics parametersKma and ρma. The comparisons shown in Figures 19 and 20 as wellas in Table 4 clearly suggest that, for this example, the inversion

Table 2. Final model misfits for the EM inversion, theseismic inversion, and the joint inversion for the marinesurface example.

ϕ misfit Sw misfit

Inversion (%) (%)

EM 26.89 17.91

Seismic 12.02 63.51

Joint 10.57 11.18


Distance (km)

Dep

th (

km)

0 2 4 6 8 10 12 14 16 18 20

0.51

1.52

2.5φ 0

0.1

0.2

0.3


Distance (km)

Dep

th (

km)

0 2 4 6 8 10 12 14 16 18 20

0.51

1.52

2.5φ 0

0.1

0.2

0.3


Distance (km)

Dep

th (

km)

0 2 4 6 8 10 12 14 16 18 20

0.51

1.52

2.5φ 0

0.1

0.2

0.3


Distance (km)

Dep

th (

km)

0 2 4 6 8 10 12 14 16 18 20

0.51

1.52

2.5φ 0

0.1

0.2

0.3

a)

b)

c)

d)

Figure 17. Inverted porosity distributions for (a) 0%, (b) 5%, (c)10%, and (d) 20% errors in m and n.


Distance (km)

Dep

th (

km)

0 2 4 6 8 10 12 14 16 18 20

0.51

1.52

2.5S

w

00.20.40.60.81


Distance (km)

Dep

th (

km)

0 2 4 6 8 10 12 14 16 18 20

0.51

1.52

2.5S

w

00.20.40.60.81


Distance (km)

Dep

th (

km)

0 2 4 6 8 10 12 14 16 18 20

0.51

1.52

2.5S

w

00.20.40.60.81


Distance (km)

Dep

th (

km)

0 2 4 6 8 10 12 14 16 18 20

0.51

1.52

2.5S

w

00.20.40.60.81

a)

b)

c)

d)

Figure 18. Inverted water-saturation distributions for (a) 0%, (b)5%, (c) 10%, and (d) 20% errors in m and n.

Table 3. Model misfits for different error levels in Archie’sparameters m and n.

Model misfit

Porosity ϕ Water saturation SwErrors in m and n (%) (%)

0 (%) 10.57 11.18

5 (%) 10.16 13.10

10 (%) 10.27 17.14

20 (%) 10.50 24.89

WA14 Gao et al.

results are more affected by errors in Kma and ρma than those in mand n, especially when Kma and ρma have a large percentage oferrors (i.e., more than 10%) the inversion results will be severelydegraded.

Scenario 3: Either Kma or ρma has a relatively low error level

When a large percentage of errors exist in Kma and ρma, the in-version results are significantly affected. To further understand theeffects of these errors on the inversion results, we fix one parameterwith a relatively low error level (i.e., 2%) and we test the effects of


Distance (km)

Dep

th (

km)

0 2 4 6 8 10 12 14 16 18 20

0.51

1.52

2.5φ 0

0.1

0.2

0.3


Distance (km)

Dep

th (

km)

0 2 4 6 8 10 12 14 16 18 20

0.51

1.52

2.5φ 0

0.1

0.2

0.3


Distance (km)

Dep

th (

km)

0 2 4 6 8 10 12 14 16 18 20

0.51

1.52

2.5φ 0

0.1

0.2

0.3

a)

b)

c)

Figure 19. Inverted porosity distributions for (a) 0%, (b) 5%, and(c) 10% errors in Kma and ρma.


Distance (km)

Dep

th (

km)

0 2 4 6 8 10 12 14 16 18 20

0.51

1.52

2.5S

w

00.20.40.60.81


Distance (km)

Dep

th (

km)

0 2 4 6 8 10 12 14 16 18 20

0.51

1.52

2.5S

w

00.20.40.60.81


Distance (km)

Dep

th (

km)

0 2 4 6 8 10 12 14 16 18 20

0.51

1.52

2.5S

w

00.20.40.60.81

a)

b)

c)

Figure 20. Inverted water-saturation distributions for (a) 0%, (b)5%, and (c) 10% errors in Kma and ρma.

Table 4. Model misfits for different error levels inrock-physics parameters Kma and ρma.

Model misfit

ϕ SwErrors in Kma Errors in ρma (%) (%)

0 (%) 0 (%) 10.57 11.18

5 (%) 5 (%) 15.82 14.61

10 (%) 10 (%) 26.34 20.76

5 (%) 2 (%) 13.34 12.91

10 (%) 2 (%) 22.13 19.27

20 (%) 2 (%) 39.88 29.23

2 (%) 5 (%) 20.99 17.48

2 (%) 10 (%) 19.84 18.45

2 (%) 20 (%) 28.15 22.17


Distance (km)

Dep

th (

km)

0 2 4 6 8 10 12 14 16 18 20

0.51

1.52

2.5S

w

00.20.40.60.81


Distance (km)

Dep

th (

km)

0 2 4 6 8 10 12 14 16 18 20

0.51

1.52

2.5S

w

00.20.40.60.81


Distance (km)

Dep

th (

km)

0 2 4 6 8 10 12 14 16 18 20

0.51

1.52

2.5S

w

00.20.40.60.81


Distance (km)

Dep

th (

km)

0 2 4 6 8 10 12 14 16 18 20

0.51

1.52

2.5S

w

00.20.40.60.81

a)

b)

c)

d)

Figure 22. Inverted water-saturation distributions for 0% (a), 5%(b), 10% (c), and (d) 20% errors in Kma and for 2% errors in ρma.


Distance (km)

Dep

th (

km)

0 2 4 6 8 10 12 14 16 18 20

0.51

1.52

2.5φ 0

0.1

0.2

0.3


Distance (km)

Dep

th (

km)

0 2 4 6 8 10 12 14 16 18 20

0.51

1.52

2.5φ 0

0.1

0.2

0.3


Distance (km)

Dep

th (

km)

0 2 4 6 8 10 12 14 16 18 20

0.51

1.52

2.5φ 0

0.1

0.2

0.3


Distance (km)D

epth

(km

)

0 2 4 6 8 10 12 14 16 18 20

0.51

1.52

2.5φ 0

0.1

0.2

0.3

a)

b)

c)

d)

Figure 21. Inverted porosity distributions for (a) 0%, (b) 5%, (c)10%, and (d) 20% errors in Kma and for 2% errors in ρma.


different error levels in the other parameter on the inversion results.Figure 21 shows inverted porosity distributions for 0%, 5%, 10%,and 20% errors inKma and for 2% errors in ρma. Figure 22 shows thecorresponding inverted water-saturation distributions. The modelmisfits after inversions are also listed in Table 4. Figures 21 and22 indicate that with a relatively low error level in ρma, the inversioncan cope with a large percentage of errors in Kma without signifi-cantly affecting inverted petrophysical parameters. We observe thatthe inverted porosity and water-saturation distributions becomeworse with increasing errors in Kma. However, the structure of por-osity and water-saturation distributions is preserved quite well.On the other hand, when Kma has a relatively low error level,

errors in ρma will significantly affect the inversion results. For thosecases, the inversion results are similar to Figure 19c, where thestructure of the inverted image is destroyed. In Table 4, we alsoshow the model errors for 0%, 5%, 10%, and 20% errors in ρma

and for 2% errors in Kma. Note that the model errors shown inTable 4 are misleading for this purpose because the definition ofthe model error only reflects the overall error in the inversion resultwithout accounting for the structure of the inverted model. Whenρma has a relatively low error level, the errors in Kma do not causesignificant errors in the model structure. However, a large percen-tage of errors in ρma severely destroys the model structure. This ob-servation suggests that to make our inversion approach practical, weneed to have good a priori information of ρma. Fortunately, thismight be achievable because for most geologic environments,the lithology is known rather well. The parameter might also beobtained from wireline or core measurements.

CONCLUSIONS

We have developed an inversion framework to jointly invert full-waveform seismic (using an acoustic approximation) and EM mea-surements for improving estimates of in situ formation porosity andfluid-saturation distributions. The inversion of porosity and fluidsaturations is made possible by using petrophysical relationshipsthat link geophysic properties to petrophysical properties. ForEM data, Archie’s equation is used; whereas for seismic data, Gass-mann’s equations coupled with the critical porosity model are used.Other petrophysical transforms may be used straightforwardly. Inour inversion algorithm, EM and seismic full-waveform data areinverted using a regularized constrained Gauss-Newton approach,in which the regularization is performed in an automatic mannerusing the multiplicative technique. The derivatives of the EM dataand seismic data with respect to the petrophysical parameters arecomputed using the adjoint approach and the proper chain rule.To handle a large-scale problem, the algorithm is implementedon a parallel computer infrastructure using an MPI library.We tested our inversion algorithm using a time-lapse crosswell

monitoring example and a marine surface exploration example.Joint-inversion results are compared with those from separateEM and seismic inversions. Sensitivity analysis and inversion ex-amples show that the joint-inversion approach can correctly recoverporosity and water-saturation distributions simultaneously, whereasneither the EM-only inversion nor the seismic-only inversion can.Morerover, the joint-inversion approach is especially useful to iden-tify the waterflooded zone in the monitoring process. Inversion ex-ercises are also performed to test how the errors in petrophysicalparameters of the petrophysical relationships affect the inversionresults. We find that the inversion can tolerate a relative high

percentage of errors in Archie’s parameters; however, the resultsare quite sensitive to errors in the rock-physics parameters. Furthertests also show that when we have only a relatively low error level inthe density of the rock matrix ρma, the joint inversion approachcan tolerate a large error level in the bulk modulus of the rockmatrix Kma.

ACKNOWLEDGMENTS

The authors thank V. Druskin from Schlumberger-Doll Researchfor providing the 2.5D EM forward modeling code and G. Pan fromSchlumberger-Doll Research for providing the 2D acoustic forwardand inversion codes. We also acknowledge much help provided byJ. Liu from Schlumberger-Doll Research on the 2.5D EM inversion.We thank Luis Gallardo, Max A. Meju, the associate editor, andthree anonymous reviewers for many constructive commentsthat improved the quality of our paper.

APPENDIX A

FORWARD SIMULATIONS

In our inversion algorithm, we use 2D and 2.5D finite-differencefrequency-domain (FDFD) forward solvers for seismic and electro-magnetic simulations, respectively. Forward solvers are used tocompute responses for a given model as well as the sensitivitymatrices (derivatives with respect to model parameters).

Seismic modeling

We simulate seismic wave propagation in an isotropic and inho-mogeneous medium using the following acoustic wave equation(Fokkema and van den Berg, 1993):

∇ ·

1

ρ∇p

þ κω2p ¼ −s; (A-1)

where p is the pressure field (Pa); s is the frequency-dependentsource term, which is either a monopole or dipole source; κ isthe compressibility (inverse of the bulk modulus K) in 1∕Pa, ρis the mass density (kg∕m3); ∇ is the spatial differential operatorwith respect to x and z; ω is the angular frequency (rad/s); andj2 ¼ −1. The forward simulator is an FDFD approach withfourth-order accuracy. To reduce the numerical reflections causedby truncating the computational domain, an absorbing boundarycondition with a PML (Berenger, 1994) is used. The details ofour PML implementation can be found in Pan et al. (2012). Afterdiscretization, a linear system of equations (corresponding to thediscretized equation A-1) is solved using a multifrontal LU decom-position method (Duff and Reid, 1983). For 2D problems, the use ofa direct solver is very efficient for multiple-source simulation.

EM modeling

The 2.5D EM forward simulation is carried out by solving thefollowing electric-field equation:

∇ × ∇ × E − jωμσE ¼ ∇ ×Kþ jωμJ; (A-2)

WA16 Gao et al.

where E ¼ Eðx; y; zÞ is the electric-field vector; J ¼ Jðx; y; zÞ andK ¼ Kðx; y; zÞ are the electric and magnetic current sources, re-spectively; μ is constant magnetic permeability; and σ ¼ σðx; zÞis electrical conductivity. The time convention is expð−jωtÞ, whereω is the angular frequency and t is time.EM waves are diffusive fields in rock formations; hence, instead

of using the PML boundary condition, we use a zero Dirichletboundary condition at infinity and efficiently extended the compu-tational domain to infinity using the optimal grid technique de-scribed in Ingerman et al. (2000). To solve equation A-2, first aFourier transform in the y-direction is applied; then the equationis solved in the Fourier domain for each ky, after which the electricfield is computed via an inverse Fourier transform, i.e.,

Eðx; y; zÞ ¼ 1

2π

Z∞

−∞Eðx; ky; zÞe−jkyydky; (A-3)

where E is the electric-field vector in the Fourier domain. The mag-netic field can be computed from

Hðx; y; zÞ ¼ 1

jωμ∇ × Eðx; y; zÞ: (A-4)

Similar to seismic forward modeling, we use a multifrontal LU de-composition to solve the forward problem because it is efficient formultisource simulations. The Jacobian matrix is computed using theadjoint method, which only requires one additional forwardcomputation for each inversion iteration. Details can be found inAbubakar et al. (2008).

REFERENCES

Abubakar, A., and T. M. Habashy, 2006, Three-dimensional single-wellimaging of the multi-array triaxial induction logging data: 76th AnnualInternational Meeting, SEG, Expanded Abstracts, 411–415.

Abubakar, A., T. M. Habashy, V. Druskin, D. Alumbaugh, and L. Knizhner-man, 2008, 2.5D forward and inversion modeling for interpreting low-frequency electromagnetic measurements: Geophysics, 73, no. 4,F165–F177, doi: 10.1190/1.2937466.

Abubakar, A., T. M. Habashy, V. Druskin, D. Alumbaugh, P. Zhang, M.Wilt, H. Denaclara, E. Nichols, and L. Knizhnerman, 2005, A fast andrigorous 2.5D inversion algorithm for cross-well electromagnetic data:75th Annual International Meeting, SEG, Expanded Abstracts,534–537.

Abubakar, A., T. M. Habashy, V. Druskin, S. Davydycheva, and L. Knizh-nerman, 2006, A 3D parametric inversion algorithm for triaxial inductiondata: Geophysics, 71, no. 1, G1–G9, doi: 10.1190/1.2168009.

Abubakar, A., T. M. Habashy, M. Li, and J. Liu, 2009, Inversion algorithmsfor large-scale geophysical electromagnetic measurements: InverseProblems, 25, no. 12, doi: 10.1088/0266-5611/25/12/123012.

Abubakar, A., M. Li, G. Pan, J. Liu, and T. M. Habashy, 2011, Joint MTandCSEM data inversion using a multiplicative cost function approach: Geo-physics, 76, no. 3, F203–F214, doi: 10.1190/1.3560898.

Abubakar, A., P. M. van den Berg, and J. T. Fokkema, 2003, Towards non-linear inversion for characterization of time-lapse phenomena through nu-merical modelling: Geophysical Prospecting, 51, 285–293, doi: 10.1046/j.1365-2478.2003.00369.x.

Archie, G. E., 1942, The electrical resistivity log as an aid in determiningsome reservoir characteristics: Petroleum Transactions of AIME, 146, 54–62.

Berenger, J. P., 1994, A perfectly matched layer for the absorption of elec-tromagnetic waves: Journal of Computational Physics, 114, 185–200, doi:10.1006/jcph.1994.1159.

Chen, J., and T. Dickens, 2009, Effects of uncertainty in rock-physics mod-els on reservoir parameter estimation using seismic amplitude variationwith angle and controlled-source electromagnetics data: Geophysical Pro-specting, 57, 61–74, doi: 10.1111/gpr.2008.57.issue-1.

Chen, J., M. Hoversten, D. Vasco, Y. Rubin, and Z. Zhou, 2007, A Bayesianmodel for gas saturation estimation using marine seismic AVA and CSEMdata: Geophysics, 72, no. 2, WA85–WA95, doi: 10.1190/1.2435082.

Colombo, D., M. Mantovani, S. Hallinan, and M. Virgilio, 2008, Sub-basaltdepth imaging using simultaneous joint inversion of seismic and electro-magnetic MT data: A CRB field study: 78th Annual International Meet-ing, SEG, Expanded Abstracts, 2674–2678.

Commer, M., and G. Newman, 2008, New advances in three-dimensionalcontrolled-source electromagnetic inversion: Geophysical Journal Inter-national, 172, 513–535, doi: 10.1111/gji.2008.172.issue-2.

Constable, S. C., C. S. Cox, and A. D. Chave, 1986, Offshore electromag-netic surveying techniques: 46th Annual International Meeting, SEG,Expanded Abstracts, 81–82.

Duff, I. S., and J. K. Reid, 1983, The multifrontal solution of indefinitesparse symmetric linear systems: ACM Transactions on MathematicalSoftware, 9, 302–325.

Fokkema, J. T., and P. M. van den Berg, 1993, Seismic applications of acous-tic reciprocity: Elsevier Science.

Gallardo, L. A., and M. A. Meju, 2003, Characterization of heterogeneousnear-surface materials by joint 2D inversion of DC resistivity and seismicdata: Geophysical Research Letters, 30, 1658, doi: 10.1029/2003GL017370.

Gao, G., A. Abubakar, and T. M. Habashy, 2011, Inversion of porosity andfluid saturations from joint electromagnetic and elastic full-waveformdata: 81st Annual International Meeting, SEG, Expanded Abstracts,30, 660–665.

Gassmann, F., 1951, Über die Elastizität poröser Medien: Vierteljahrsschriftder Naturforschenden Gesellschaft in Zürich, 96, 1–23.

Golub, G. H., and C. F. van Loan, 1996, Matrix computations: 3rd ed., TheJohns Hopkins University Press.

Habashy, T. M., and A. Abubakar, 2004, A general framework for constraintminimization for the inversion of electromagnetic measurements:Progress in Electromagnetic Research, 46, 265–312, doi: 10.2528/PIER03100702.

Harris, P., Z. Du, H. H. Soleng, and L. MacGregor, 2009, Rock physics in-tegration of CSEM and seismic data: A case study based on the Luva gasfield: 79th Annual International Meeting, SEG, Expanded Abstracts,1741–1745.

Hoversten, G. M., F. Cassassuce, E. Gasperikova, G. A. Newman,J. Chen, Y. Rubin, Z. Hou, and D. Vasco, 2006, Direct reservoirparameter estimation using joint inversion of marine seismic AVAand CSEM data: Geophysics, 71, no. 3, C1–C13, doi: 10.1190/1.2194510.

Hu, W., A. Abubakar, and T. M. Habashy, 2007, Joint inversion algorithmfor electromagnetic and seismic data: 77th Annual International Meeting,SEG, Expanded Abstracts, 1745–1749.

Hu, W., A. Abubakar, and T. M. Habashy, 2009a, Joint electromagnetic andseismic inversion using structural constraints: Geophysics, 74, no. 6,R99–R109, doi: 10.1190/1.3246586.

Hu, W., A. Abubakar, and T. M. Habashy, 2009b, Simultaneous multi-frequency inversion of full-waveform seismic data: Geophysics, 74,no. 2, R1–R14, doi: 10.1190/1.3073002.

Ingerman, D., V. L. Druskin, and L. Knizhnerman, 2000, Optimal finite dif-ference grids and rational approximations of the square root, I: Ellipticproblems: Communications on Pure and Applied Mathematics, 53,1039–1066, doi: 10.1002/(ISSN)1097-0312.

Kriegshauser, B., S. McWilliams, O. Fanini, and L. Yu, 2001, An efficientand accurate pseudo 2-D inversion scheme for multicomponent inductionlog data: 71st Annual International Meeting, SEG, Expanded Abstracts,369–372, doi: 10.1190/1.1816619.

Liang, L., A. Abubakar, and T. M. Habashy, 2010, Crosswell electromag-netic inversion constrained by the fluid flow simulator: Presented at theAnnual Technical Conference and Exhibition, SPE, 135069.

McGillivray, P. R., and D. W. Oldenburg, 1990, Methods for calculatingFréchet derivatives and sensitivities for the non-linear inverseproblem: A comparative study: Geophysical Prospecting, 38, 499–524,doi: 10.1111/j.1365-2478.1990.tb01859.x.

Moorkamp, M., B. Heincke, M. Jegen, A. W. Roberts, and R. W. Hobbs,2011, A framework for 3-D joint inversion of MT, gravity and seismicrefraction data: Geophysical Journal International, 184, 477–493, doi:10.1111/gji.2010.184.issue-1.

Nur, A., 1992, Critical porosity and the seismic velocities in rocks: EOS,Transactions, American Geophysical Union, 73, 43–66.

Pan, G., A. Abubakar, and T. M. Habashy, 2012, An effective perfectlymatched layer design for acoustic fourth-order frequency-domain fi-nite-difference scheme: Geophysical Journal International, 188, 211–222, doi: 10.1111/gji.2012.188.issue-1.

Pratt, R. G., and M. H. Worthington, 1990, Inverse theory applied tomulti-source cross-hole tomography. Part 1: Acoustic wave-equationmethod: Geophysical Prospecting, 38, 287–310, doi: 10.1111/gpr.1990.38.issue-3.

Spies, B. R., and T. M. Habashy, 1995, Sensitivity analysis of crosswellelectromagnetics: Geophysics, 60, 834–845, doi: 10.1190/1.1443821.

Tarantola, A., 1986, A strategy for nonlinear elastic inversion of seismicreflection data: Geophysics, 51, 1893–1903, doi: 10.1190/1.1442046.


http://dx.doi.org/10.1190/1.2937466

http://dx.doi.org/10.1190/1.2937466

http://dx.doi.org/10.1190/1.2937466

http://dx.doi.org/10.1190/1.2168009

http://dx.doi.org/10.1190/1.2168009

http://dx.doi.org/10.1190/1.2168009

http://dx.doi.org/10.1088/0266-5611/25/12/123012

http://dx.doi.org/10.1088/0266-5611/25/12/123012

http://dx.doi.org/10.1190/1.3560898

http://dx.doi.org/10.1190/1.3560898

http://dx.doi.org/10.1190/1.3560898

http://dx.doi.org/10.1046/j.1365-2478.2003.00369.x

http://dx.doi.org/10.1046/j.1365-2478.2003.00369.x

http://dx.doi.org/10.1046/j.1365-2478.2003.00369.x

http://dx.doi.org/10.1046/j.1365-2478.2003.00369.x

http://dx.doi.org/10.1046/j.1365-2478.2003.00369.x

http://dx.doi.org/10.1046/j.1365-2478.2003.00369.x

http://dx.doi.org/10.1046/j.1365-2478.2003.00369.x

http://dx.doi.org/10.1006/jcph.1994.1159




http://dx.doi.org/10.1111/gpr.2008.57.issue-1





http://dx.doi.org/10.1190/1.2435082

http://dx.doi.org/10.1190/1.2435082

http://dx.doi.org/10.1190/1.2435082

http://dx.doi.org/10.1111/gji.2008.172.issue-2





http://dx.doi.org/10.1029/2003GL017370

http://dx.doi.org/10.1029/2003GL017370

http://dx.doi.org/10.1029/2003GL017370

http://dx.doi.org/10.2528/PIER03100702



http://dx.doi.org/10.1190/1.2194510

http://dx.doi.org/10.1190/1.2194510

http://dx.doi.org/10.1190/1.2194510

http://dx.doi.org/10.1190/1.3246586

http://dx.doi.org/10.1190/1.3246586

http://dx.doi.org/10.1190/1.3246586

http://dx.doi.org/10.1190/1.3073002

http://dx.doi.org/10.1190/1.3073002

http://dx.doi.org/10.1190/1.3073002

http://dx.doi.org/10.1002/(ISSN)1097-0312

http://dx.doi.org/10.1002/(ISSN)1097-0312

http://dx.doi.org/10.1190/1.1816619

http://dx.doi.org/10.1190/1.1816619

http://dx.doi.org/10.1190/1.1816619

http://dx.doi.org/10.1111/j.1365-2478.1990.tb01859.x






















http://dx.doi.org/10.1190/1.1443821

http://dx.doi.org/10.1190/1.1443821

http://dx.doi.org/10.1190/1.1443821

http://dx.doi.org/10.1190/1.1442046

http://dx.doi.org/10.1190/1.1442046

http://dx.doi.org/10.1190/1.1442046

van den Berg, P. M., and A. Abubakar, 2001, Contrast source inversionmethod: State of the art: Progress in Electromagnetic Research, 34,189–218, doi: 10.2528/PIER01061103.

van den Berg, P. M., A. L. Broekhoven, and A. Abubakar, 1999,Extended contrast source inversion: Inverse Problems, 15, 1325–1344,doi: 10.1088/0266-5611/15/5/315.

Virieux, J., and S. Operto, 2009, An overview of full-waveform inversionin exploration geophysics: Geophysics, 74, no. 6, WCC1–WCC26,doi: 10.1190/1.3238367.

Wang, Z., 2001, Fundamentals of seismic rock physics: Geophysics, 66,398–412, doi: 10.1190/1.1444931.

Wang, G. L., C. Torres-Verdin, J. Ma, and T. B. Odumosu, 2007, Combinedinversion of borehole resistivity and sonic measurements to estimate watersaturation, porosity, and dry-rock elastic moduli in the presence ofinvasion: 48th Annual Logging Symposium, Society of Petrophysicistsand Well Log Analysts, Expanded Abstract, Paper LLL.

Waxman, M. H., and M. J. L. Smits, 1968, Electrical conductivities in oil-bearing shaly sands: SPE Journal, 8, no. 2, 107–122, doi: 10.2118/1863-A.

Weitemeyer, K., G. Gao, S. Constable, and D. Alumbaugh, 2010, Thepractical application of 2D inversion to marine controlled-source electro-magnetic data: Geophysics, 75, no. 6, F199–F211, doi: 10.1190/1.3506004.

WA18 Gao et al.



http://dx.doi.org/10.1088/0266-5611/15/5/315

http://dx.doi.org/10.1088/0266-5611/15/5/315

http://dx.doi.org/10.1190/1.3238367

http://dx.doi.org/10.1190/1.3238367

http://dx.doi.org/10.1190/1.3238367

http://dx.doi.org/10.1190/1.1444931

http://dx.doi.org/10.1190/1.1444931

http://dx.doi.org/10.1190/1.1444931

http://dx.doi.org/10.2118/1863-A

http://dx.doi.org/10.2118/1863-A

http://dx.doi.org/10.1190/1.3506004

http://dx.doi.org/10.1190/1.3506004

http://dx.doi.org/10.1190/1.3506004

http://dx.doi.org/10.1190/1.3506004

Joint petrophysical inversion of electromagnetic and full ...

Documents

Transcript of Joint petrophysical inversion of electromagnetic and full ...