Permeability prediction using stress sensitive petrophysical properties

C. Jones, J. M. Somerville, B. G. D. Smart, O. Kirstetter, S. A. Hamilton andK. P. Edlmann

Department of Petroleum Engineering, Heriot Watt University, Edinburgh EH14 4AS, UK

ABSTRACT: The correlation of stress sensitivity to various petrophysical parameterswas studied by analysis of experimental results from a range of sandstone core plugstested hydrostatically at room temperature. The parameters measured were: com-pressional wave velocity, porosity, permeability and electrical resistivity. Moredetailed information on the effects of sorting and grain size distributions wasobtained from experiments on artificial, unconsolidated sandstone cores. Themeasurements showed a high degree of stress sensitivity, which was different foreach core but, broadly, could be classified as either high or low stress sensitivity.Cores from the high permeability clean sand were less stress sensitive than the coresfrom the low permeability coarsening-upwards sequence and the petrophysicalvalues when combined into a synthetic log distinguished between the two lithologies.The results were compared to the predictions of a simple asperity deformationmodel. The experimental results and the model suggested a possible logging strategyto deduce permeability, by varying wellbore pressure.

KEYWORDS: stress sensitivity, acoustic property, permeability, grain sorting

INTRODUCTION

This paper reports the results of a study made on a set of sixcore plugs from a North Sea reservoir to investigate the stresssensitivity of a wide variety of petrophysical parameters. Thecore plugs studied were from a clean homogeneous sandstoneoverlying a coarsening-upwards unit. The core data indicatethat there is a large difference in permeability between the twounits with relatively little change in porosity (Table 1), making acorrelation between them practically impossible.

The aim of this study was to investigate the interrelationshipsof petrophysical measurements made on identical samples toimprove the understanding of the effect of lithology onpermeability. The parameters measured were ultrasonic com-pressional wave velocity (Vp), porosity, permeability and elec-trical conductivity. They were measured in a true triaxial cellwhich allowed independent control of each principal stress:vertical stress (�v), major horizontal stress (�H) and minorhorizontal stress (�h).The stresses were varied to encompassthe full range of isotropic stresses up to the reservoir in situstress. Ancillary experiments on synthetic cores were conducted

to investigate the effect of grain size and sorting on thepetrophysical parameters.

PRELIMINARY CHARACTERIZATION ANDDESCRIPTION OF THE UNIT STUDIED

The unit studied was of Cretaceous age and contains 5.8 m ofclean homogeneous upper shoreface clean sandstone (CS) thatoverlies a 1.5 m lower shoreface unit with a coarsening-upwards (CU) unit. The CS contains a 1.5 m bed of baritefractures, which show up on the bulk density and sonic logs.Visually, and from thin section measurements, the CS ishomogeneous with the exception of the barite fractures, with agrain size of 0.3–0.4 mm and with occasional large pores (Fig.1). The ambient porosity was measured during routine coreanalysis and found to be in the range 18–21%, with an ambientKlinkenberg permeability of 627–840 mD. The CU sequence isweakly laminated with a grain size gradually reducing to 0.1 mmat the base of the unit and with an ambient porosityand Klinkenberg permeability of 17–21% and 8–411 mD

Table 1. Properties of the core plugs tested

Core TVDSS (ft) Helium � (%)Klinkenberg corrected

permeability (mD)Mean grainsize (µm)

Grain sorting(span)

CS1 11536.3 18.2 737 340 1.09CS2 11536.5 18.0 742 341 1.11CS3 11537.4 18.0 829 363 1.03CU1 11543.5 21.8 58 292 1.40CU2 11544.2 21.6 11 247 1.48CU3 11545.0 20.7 13 216 1.48

For location of samples, see Figure 2.

Petroleum Geoscience, Vol. 7 2001, pp. 211–219 1354-0793/01/$15.00 � 2001 EAGE/Geological Society of London

respectively. Table 1 shows the large variation in permeabilityof the cores (and a relatively small variation in porosity).

The gamma-ray data from the well (Fig. 2) showed a slightincrease in the CU sequence with depth from which an increasein clay content is inferred. This is confirmed by SEM measure-ments. The medium (ILM) and deep (ILD) induction logs showa separation in the CS unit from which invasion and, hence,permeability is inferred. The well was drilled with oil-based

mud so the ILM is more resistive than the ILD due to the oilfiltrate flushing brine away from the wellbore. Both inductionlog values decrease with depth in the CU unit and theseparation disappears. The sonic log shows very small changesbetween the two units, as does the bulk density. The baritefracture zone is visible on both the sonic and density log. Thinsection micrographs of material adjacent to each core tested inthis study show a clear increase in grain size moving up the CUunit into the CS unit (Fig. 3).

Samples of core adjacent to the core plug locations werelightly ground in a mortar and pessel to break the cement andliberate the grains. The material was seived to remove grainaggregates and the remaining isolated grains and cement wereanalysed using a Malvern particle size analyser (Roberts 1996).Repetition of the process produces consistent results. Compari-son of thin section grain size measurements allows a cut-off tobe determined for the upper grain size (material above the limitdeemed to be not dissagregated) and for the lower grain size(material deemed smaller than the interstitial material andtherefore contaminated). Output from the instrument (Fig. 4) isin the form of a graph of grain size distribution, a mean grainsize and the width of the grain size distribution, defined as thespan and given by Equation (1),

where Dx is the largest diameter present in the x% volumefraction. The values of mean grain size and span for eachsample are shown in Table 1. There is a clear contrast in grain

Fig. 1. Secondary electron (SEM) image of sample CS3 showing200–400 µm grains with quartz overgrowths. Some clay (kaolinite) isvisible to the left of the frame. The bright areas are small baritegrains.

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Fig. 2. Log and core data from thestudied units, the left-hand trace is thegamma-ray and the density log, thebarite fracture zone is visible in thedensity log, and the gamma-ray showsan increase in the CU. The centre righttrace is the ILM and ILD, the coreanalysis permeabilities and thepermeabilities of the tested cores. Thepermeability measured in the coreanalysis data and the laboratory samplescorrespond very well. The right-handtrace is the log-derived compressionalwave �T and the �T derived from themeasured velocity. Also the grain sizeand core analysis porosity are shown.

212 C. Jones et al.

sorting between the clean sand and the coarsening-upwardssequence.

Figure 5 shows a plot of the Klinkenberg corrected per-meability of each sample against the width of the grain sizedistribution as defined by the span. The data fall into twogroups. The CS samples which have a high permeability and alow span and an inferred narrow grain size distribution and the

CU samples which have a low permeability, high span and aninferred wider grain size distribution.

LABORATORY MEASUREMENT OFSTRESS-SENSITIVE PARAMETERS

Estimation of the in situ stress

The in situ effective stress field was estimated from the TVDSS,mud weight and the pore pressure; the effect of the sea depth isneglected. The TVDSS of the top of the sequence was 3579 m(11 742 ft) and, assuming a lithostatic stress gradient of0.0226 MPa m�1 (1 psi ft�1), the vertical stress was calculatedto be 81.0 MPa (11 742 psi). The hydrostatic mud pressuregradient was 0.0172 MPa m�1 (0.76 psi ft�1) which produceda mud pressure of 61.5 MPa (8915 psi). Stress measurements inthe North Sea (Edwards et al. 1998) suggest that the ratio of theminimum horizontal stress to the vertical stress (�h/�v) is inrange of 0.75-0.95. In calculating the horizontal stress a value of�h/�v=0.8 was used and the horizontal stresses were assumedto be equal, which produced a horizontal stress of 64.8 MPa(9394 psi) and a mean stress of 70.2 MPa (10 177 psi). The porepressure from a formation testing log was 35.9 MPa (5200 psi)which gave a mean effective stress of 34.3 MPa (4977 psi). Anupper stress limit of 34.5 MPa (5000 psi) was assumed for thesubsequent core tests; all the tests reported were conductedunder hydrostatic stress conditions.

Sample preparation

The core was taken during normal coring operations with anon-orientated core barrel using an oil-based mud. Core plugswere taken according to an agreed schedule from the core usinga kerosene flushed core barrel. The ends were cut parallel andnormal to the sides to the specifications of Hawkes & Mellor(1970). The samples were cleaned in a Soxhlet Refluxer withdichloromethane and dried in an oven at 70�C. The dimensions,weight, porosity and permeability of all samples were measured.The samples were then vacuum saturated with simulatedformation brine and pressure saturated at 2000 psi for 2 days.One hundred percent saturation was confirmed by sampleweight. The brine had a measured resistivity of 0.093 �m.

The saturation of the cores was representative of the waterleg in the reservoir. The oil zone would contain a reduced brinesaturation at appoximately 20% brine (80% oil). However, even

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Fig. 3. Thin section photographs ofmaterial adjacent to each core testedshowing a constant grain size in the CSunit and a decrease in grain sizethroughout the CU sequence.

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Stress sensitive permeability prediction 213

for the core from the water leg, it would be flushed to a certainextent with oil-based mud filtrate, which may explain the slightdiscrepancy between the laboratory and log sonic travel timesshown on Fig. 2.

TRUE TRIAXIAL CELL AND MEASUREMENTTECHNIQUES

North Sea cores

The true triaxial cell (Smart 1995) used in this study canindependently vary the major horizontal stress (�H) and theminor horizontal stress (�h), so replicating the true triaxialstress field as found in the reservoir. The cell was installed in aservo-controlled stiff testing machine which applied an axialload (vertical stress, �v) to the ends of the core via a ram andplatens. The radial stresses were generated by 24 plastic tubeslocated in the aluminium body of the cell which appliedpressure to the sides of the core via a nitrile rubber sleeve. Inthese experiments the true triaxial cell was used to apply onlyhydrostatic pressure to the samples. The stresses were appliedat a constant rate of 1.0 MPa min�1 up to a maximum of34.5 MPa (5000 psi). After an increment in stress, 15 min wereallowed for the sample to stabilize before any measurementswere taken. This loading rate and dwell period were chosen inthe light of previous experience and the literature (Longeron etal. 1989) that has shown that sandstones react virtually instan-taneously to changes in stress. This is confirmed in this studywhere, following an increament in stress from 4.1–5.5 MPa(600–800 psi), the axial displacement achieves a constant valueafter 15 min (Fig. 6).

Each sample was cycled once up to 34.5 MPa (5000 psi) in500 psi increments prior to taking the measurements to closeany pre-existing microcracks which may have opened duringthe core recovery process. The stress was increased to34.5 MPa (5000 psi) in increments of 500 psi and held until thestrain became constant. The sample was then unloaded to2.8 MPa (400 psi), the first experimental point. This pointwas chosen from the stress displacement curve – the majorityof microcracks in previously tested sandstones open belowthis stress – therefore, to hold the stress at this level re-duces anomalous pore volume reduction and compressibilityartefacts.

The end platens used in these tests were equipped withtransducers to simultaneously measure a number of petrophysi-cal parameters: permeability, pore volume reduction, electricalresistance and Vp.

A graduated pipette connected to the core via porous platensis used to measure pore volume reduction. Changes in fluidvolume in the pipette reflected the changes in the pore volumeas the effective stress was increased. This was then used todetermine the porosity of the sample at different stresses withan accuracy of 0.2%, e.g. for a porosity of 20% the accuracy ofthe measurement is 20.0�0.2%.

Resistance was measured axially through the core usingelectrical contacts in the end caps of the platens at 1 kHzfrequency by an impedance meter. This was converted to aresistivity using the dimensions of the sample; the error in themeasurement was �7%.

Vp was measured axially through the core by pulse transmis-sion with the source and receiver in the end caps of the platens.For brine-saturated samples the Vp can be obtained with anaccuracy of �1% above 10 MPa applied stress. Below 10 MPathe signals are increasingly poor, with �3% a more realisticaccuracy.

Single-phase steady state permeability was determined axiallythrough the sample by flowing brine at a constant rate andmeasuring the differential pressure across the core. The per-meability was calculated from Darcy’s equation, the error in themeasurement was �7%.

The errors in each quantity (Ø, resistivity, Vp, and per-meability) are calculated from the measurement errors ineach parameter and combined using standard methods (Taylor1997). For example, to calculate the error in the per-meability involves a combination of the errors in the sampledimensions the pressure drop across the sample and the fluidviscosity.

The sequence of measurements during each experiment wasas follows: the stress was incremented to the next value whileexpelled fluid volume was noted. After 15 min an acousticwaveform was recorded, and the electrical resistance and phaseangle recorded. Brine was then pumped across the sample at1 cm3 min�1 and the differential pressure noted once equilib-rium was reached after approximately 15 min. Pumping wasthen stopped and when the volume of fluid expelled from thesample became constant and the differential pressure went tozero, the next increment in stress was then applied.

Synthetic unconsolidated sandstone cores

To investigate the apparent lack of correlation between porosityand permeability in the North Sea reservoir sandstone, controlsamples made from unconsolidated sand with specific sortingattributes were prepared. These were made from LochalineSandstone (Lowden et al. 1991), a very clean, mineralogicallysupermature, quartz-rich (>99.6%) rock which has very lowapparent cohesion, but with a high compressive strength at lowconfining pressure. Samples of this rock were disaggregated,seived and two sets of samples formed to represent well-sortedrock and poorly sorted rock.

The well-sorted sample consisted of grains from a singlesieve with diameters between 0.212–0.250 mm, the poorlysorted sample had equal weights from 10 different sieves withgrain diameters from 0.063–0.5 mm (Fig. 7).

The samples for testing were prepared by carefully placingthe sand into a shrinkfit plastic jacket and compacting the sand(by hand) to produce the desired porosity. The ends of thejacket were sealed with stainless steel mesh, which allowedaccess to fluid but retained the grains. Samples were saturatedin oil prior to testing. The P-wave velocity and the permeabilitywere measured at increasing hydrostatic stress from ambient to48 MPa (7000 psi).

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214 C. Jones et al.

RESULTS AND DISCUSSION

North Sea core samples

The results of the Vp measurements for the six cores revealtheir sensitivity to stress (Fig. 8). For the CS cores there was aninitial rapid increase in Vp for all cores due to the closure oflow aspect ratio cracks and pores (Bourbie et al. 1987). This wasfollowed by a slower increase as the influence of more equantpores becomes important. The response of the CU cores wasdifferent to the CS cores. All CU samples have lower velocity atlow stress but they increased to values similar to those of theCS cores at high stress. The stress sensitivity of Vp of the CUcores was greater than the CS cores. The changes in gradient inthe low stress region were less pronounced than in the CS unit,however, the gradient in the higher stress region of the CU unit

was greater than in the corresponding clean reservoir section.The values of the gradients at 35 MPa are listed in Table 2. Theabsolute values of Vp at high stresses correspond well to thedownhole logged values in this section, the average sonic logVp is 4162 m s�1 (Fig. 2). The difference between downholeand lab velocities may be due to saturation differences betweenin situ and laboratory measurements. There may also bedispersion effects related to the different frequencies used in thelaboratory compared to downhole conditions

The variation in porosity in response to effective stress wasvery similar for all cores (Fig. 9). The curves fall into twogroups: three slightly higher porosity cores from the CU unitand three lower porosity cores from the CS unit. The absolutevalues agree well with core analysis values (Fig. 2). The generalform of the stress sensitivity was as expected from similarmeasurements (Fatt 1953). There was a variation between thecores, with the CU cores being more stress sensitive thanthe CS cores. The CU cores have an average porosity reductionof 1.15%, the CS cores have an average porosity reduction of0.82%.

The pore volume reductions can be used to calculate thepore compressibilities (Zimmerman 1991) from:

where Cpc is the pore compressibility, Vi the initial porevolume, and the term in brackets the change in pore volumewith confining pressure at constant pore pressure. The com-pressibility is greatest at low stress and approaches a constantvalue at high stress (Fig. 10); however, the values are still stresssensitive even at high stress. The compressibility values (Table

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Table 2. Stress sensitivity of the petrophysical properties of Vp, Ø, CP, FR and k, calculated at reservoir conditions of 35 MPa

Clean sandCoarsening

upwardsSynthetic

well sorted poorly sorted

�VP/�P (ms�1/MPa) 7.27 18.01 7.27 10.68�Ø/�P (%/MPa) �0.008 �0.011

CP (1/psi*10�6) 4.16 4.84�CP/�P(1/psi*10�6/MPa) �0.048 �0.082

FR† 35.8/30.6 34.8/21.9�FR/�P (/MPa) 0.18 0.22

�k (k0�k35 MPa)/k0 0.22 0.53 0.11 0.52

†2.8 MPa measured/calculated.

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Stress sensitive permeability prediction 215

2) are similar to those of other workers (Zimmerman 1991;Tiab 1996). Although there was some variation, the CS and CUcores have different values, with the CS core having, onaverage, lower values of pore compressibility, from which thepresence of more equant pores is inferred, which are moredifficult to close.

Resistivity increases with stress for all cores (Fig. 11). SampleCU1 suffered slight desaturation at the platens at 9 MPa and14 MPa. This was rectified by reflooding the core with brineand the measurements then returned to the normal trend. Theincrease in resistivity is due to the deformation of the porestructure causing the closure of conductive, brine-filled path-ways and this corresponds to the reduction in pore volumemeasured. On average, the samples from the CU unit wereslightly more sensitive to stress than those in the clean sand(Table 2).

The resistivities can be expressed in terms of the formationresistivity factor (FR), which is the ratio of the resistivity of a100% brine-saturated sample to that of the brine. The FR isrelated to the porosity by the Archie equation (Archie 1942).

where is the porosity, a and m are constants for a particularrock, and m is known as the cementation exponent. Thesimplest expression for FR is FR=1/ 2. This expression is usedto calcluate the formation factors of the samples based on theinitial porosity. The calculated FR of samples CS1, CS3, CU1and CU2 agree very well with the measured values. The othertwo samples have higher measured FR values than the calcu-

lated FR values (Table 2). The fit for CS1 is particularly good atthe lowest stress. When the stress is increased the porositychanges and the resistivity can be recalculated using the Archieequation. The fit between the measured and calculated relation-ship FR=1/ 2 degrades as the stress is increased (Fig. 12). Oneexplanation for this is that the cementation exponent m is afunction of stress. This is physically reasonable since m is afunction of the shape and distribution of the pores (Tiab &Donaldson 1996) and there is ample evidence of a changingpore network from the Vp, Ø, resistivity and permeabilitychanges. The m value to achieve a perfect fit is calculated fromFR=1/ m (Fig. 12) and the type of reduction in m is in linewith observations by Longeron et al. (1989).

The variations in permeability with applied stress are shownon Fig. 13. The initial permeabilities of the samples vary bythree orders of magnitude, therefore the results were normal-ized to the 400 psi values to show the stress sensitivity trendsmore clearly. Decreases in permeability were again due todeformation of the pore network. The samples from the CUunit were, on average, more stress sensitive than the clean sand(Table 2).

Synthetic unconsolidated sandstone cores

Two synthetic samples were prepared from disaggregatedLochaline Sandstone to directly test the influence of grain sizeand sorting. The well-sorted sample is much less stress sensitivethan the poorly sorted sample (Fig. 14). The Vp increased lessin the well-sorted sample than in the poorly sorted sample. The

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216 C. Jones et al.

permeability decline of the well-sorted sample was much lessthan that of the poorly sorted sample (Table 2), and thepermeabilities for well-sorted and poorly sorted samples were2200 mD and 680 mD, respectively, at 35 MPa.

‘BED OF NAILS’ MODEL

The Vp data were analysed using the ‘bed of nails’ model ofCarlson & Gangi (1985). This is a simple theoretical asperitydeformation model, which models the stress sensitivity of poredeformations. The model treats the pore structure of the rockas an equivalent rough-walled crack (Fig. 15). The crack wallsare held open by asperities, which are modelled as springs witha power law distribution in lengths. The model can be used todevelop a relationship (Equation 4) between Vp and the appliedstress such that,

where V0 is the velocity at zero stress, P0 is a factor accountingfor the number of asperities in contact at zero stress, D is aparameter describing the asperity height function. Equation 4contains three adjustable parameters, V0, P0 and D. These arefitted to the data using a non-linear least-squares fit.

The ‘bed of nails’ model represents the pores in the rockstatistically by a mechanically equivalent crack. The parametersgenerated do not correspond to any one set of pores only a

volume average. With regard to the parameter, D, low valuescorrespond to crack walls, which fit together very well and areheld apart by a few asperities. This type of pore will close underlow stress (Fig. 16). Conversely, high values of D imply a poorfit between the crack walls and more equant pore space. Higherstresses are required to close these pores. This could be relatedto grain sorting as a poorly sorted material will have a greatertendency to produce well-fitting crack walls with a low D value.Well-sorted materials will produce less well-fitting crack wallsand a higher D. The model fits are plotted on Figure 8 for theCS and CU cores and on Figure 14 for the synthetic cores andlisted in Table 3.

DISCUSSION

The six cores from the two lithological units and the syntheticsamples showed, on average, two distinct stress-sensitive

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Fig. 15. Bed of nails representation of the pore in a rock: (A)idealized set of pores; (B) the mechanically equivalent bed of nails.

Fig. 16. Schematic of (A) poorly and (B) well-sortedrocks showingthe difference in aspect ratio of pores between the two. The stressesacting on the rock are also shown – the vertical stress �v andhorizontal stress �h. The poorly sorted rock has the less equant thatclose more easily under the action of stress.

Table 3. ‘Bed of nails’ fit to data from this study, with unconsolidated sand for acomparison

Sample D value P0 value V0 value

CS1 0.753 0.490 2443CS2 0.788 0.277 2766CS3 0.903 0.582 3552CU1 0.531 2.833 2327CU2 0.580 2.174 2174CU3 0.601 2.533 2419Well sorted 0.729 8.494 1651Poorly sorted 0.618 14.121 1708

Stress sensitive permeability prediction 217

behaviours. For Vp, Ø, FR and k the CS cores were less stresssensitive than the CU cores, the CS core had a measurablynarrower grain size distribution and larger average grain size.The synthetic samples mirrored the two types of stress sensi-tivity with the well-sorted sample less stress sensitive than thepoorly sorted sample and the two samples had similar averagegrain sizes.

Stress sensitivity appears to discriminate between the twounits, since the sensitivity of the petrophysical parameters ofthe CU unit is greater than that of the CS unit. The two unitshave very similar mineralogy, the only difference is that the CUunit has a small amount of clay. Grain size and sorting aredifferent in each unit in that the CS is better sorted and coarsergrained than the CU sequence. It is unlikely that grain size alonecauses the change in stress sensitivity, especially for Vp andporosity. The ultrasonic waves used in the Vp measurementshave a frequency of 0.6 MHz. The velocity is approximately4000 m s�1 producing a wavelength of 6 mm, equivalent to atleast 18 grain diameters for the maximum mean grain size ofthe samples (0.34 mm). Propagation of these waves will becontrolled by the effective medium theory not ray theory(Marion et al. 1994). Since the effective medium bulk propertiesof the sample control the propagation of the waves, at thisfrequency the waves are not affected by the individual grains,only the bulk elastic properties.

Grain sorting could account for the change in stress sensi-tivity. The CS could be represented by a collection of perfectlywell-sorted quartz grains that produce high aspect ratio porespaces, which will be difficult to deform by the application ofstress. A sample from the CU sequence could be represented bya collection of grains of different diameters. This model couldproduce lower aspect ratio pores, which may be deformed andclosed by the application of stress (Fig. 16).

The experiments on the controlled grain size distributionunconsolidated samples were designed to examine the influenceof sorting on stress sensitivity. The experiments show that, forthe unconsolidated sand samples the samples with a narrowdistribution of grain sizes were less stress sensitive than thesamples with a wide distribution of grain sizes. This wouldindicate that stress sensitivity is a function of grain sizedistribution, at least for this material.

For the North Sea core there is the additional complicationthat the average grain size changes as well as the sortingbetween the CS and CU units. As noted above, Vp is unlikelyto be detectably influenced by grain size. The same is true

for porosity and porosity changes, since for purely elasticdeformation the porosity change with stress will be largelyindependent of grain size.

The results of the ‘bed of nails’ model fit to the Vp data aresummarized in Table 3. They show that the less stress sensitiveCS samples have a consistently higher values of D than thesamples from the CU unit. The high values of D imply a moreequant pore space that is more difficult to close. With thesynthetic unconsolidated sand samples, again the well-sortedsample has a higher D value than the poorly sorted sample.

POTENTIAL APPLICATIONS

A logging strategy could be developed that exploits the influ-ence of grain sorting on stress sensitivity to derive estimates ofpermeability. Sonic travel time would be measured on twoindependent logging runs, the first logging run would be takenat normal wellbore pressure conditions and the second atelevated wellbore pressure conditions. An increase in wellborepressure can be achieved by increasing the wellhead pressure.This would be transmitted via the mud column to the for-mation, thus causing a reduction of the effective stress. In thesandstone units in this study the sonic travel time will changemore in the stress sensitive CU unit than in the CS. Byoverlaying the logs, differentiation of the two units should bepossible. Figure 17 shows an example of this type of syntheticlog calculated from the study well sonic log and the stresssensitivity data from the experimental results using a 7 MPa(1000 psi) change in stress. The stress sensitivity used was thegradient calculated between 27.5 and 34.5 MPa, the valueclosest to reservoir conditions.

Having determined the locations of the different sequences,estimates of the permeabilities may be possible if the grain sizedistributions are known. If the permeability–stress relationshipscan be normalized for a variety of grain size distributions,reference to a series of characteristic curves could yield esti-mates of the permeability. It is unlikely that simple relationshipswill encapsulate precisely the flow/porosity/acoustic relation-ships; however, they should provide a first pass estimation ofthe permeability variation present.

The wellhead pressure would have to be relatively low toavoid fracturing the formation and producing significantleak-off. The permeability of the mud filter cake is assumedto be zero and the pore pressure is not affected by filtrateleak-off.

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218 C. Jones et al.

Since the stress effects will be confined to the near-wellboreregion, then the penetration depth of the sonic wave will bevery important. If the wave penetrates too far then it willsample an undisturbed volume of rock. For sonic tools thefrequency of operation is in the range of 20–50 kHz and thepenetration of a sonic wave is approximately three wavelengths(Bassiouni 1994) which equates to a penetration of around30 cm. For a typical openhole diameter of 20 cm, the sonic logsamples between one and two diameters into the formation.Finally, the stress near the wellbore are modified by the stressconcentrations of the wellbore (Fjear et al. 1992), which maycomplicate the analysis.

CONCLUSIONS

(1) The petrophysical properties of Vp, Ø, FR and per-meability showed different levels of stress sensitivity in aclean sand and coarsening-upwards sequence.

(2) There was a relationship between permeability and grainsize distribution for all the samples tested: the well-sortedsamples had a high permeability and the poorly sortedsamples a low permeability.

(3) The permeability stress sensitivity and the Vp stresssensitivity of laboratory produced unconsolidated sampleswas related to the particle size distribution of the samples.The samples with the narrow particle size distributionshows stress sensitivity similar to the clean sand cores andthe response of the unconsolidated sample with a wideparticle size distribution is similar to the coarseningupwards section cores.

(4) Modelling of the deformation of the rock pore space usingthe ‘bed of nails’ model produced theoretical indices thatdifferentiate the clean sand from the coarsening-up sec-tion. The model related the deformation to the sizedistribution of asperities between the rock grains.

(5) The results indicated the nature of a field test to determinevariations in permeability in different formation facies bymeasuring changes in the logged Vp for changes in theapplied wellbore pressure.

Conversion factors

1000 psi=6.8947 MPa

Thanks to the sponsors of the PEGASUS project; Amerada Hess,Amoco, BHP, BG plc, Chevron, Fina, Saga, Schlumberger, Shell,Statoil, and EPSRC.

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Received 29 November 1999; revised typescript accepted 31 October 2000

Stress sensitive permeability prediction 219