effective stress law for the permeability of clay-rich sandstones

Effective stress law for the permeability of clay-rich sandstones

Widad Al-Wardy and Robert W. ZimmermanDepartment of Earth Science and Engineering, Imperial College of Science, Technology and Medicine, London, UK

Received 8 October 2003; revised 6 February 2004; accepted 2 March 2004; published 20 April 2004.

[1] Two models of clay-rich sandstones are analyzed to explain the relative sensitivityof permeability to pore pressure and confining pressure. In one model the clay lines theentire pore wall in a layer of uniform thickness, and in the second model the clay isdistributed in the form of particles that are only weakly coupled to the pore walls.Equations of elasticity and fluid flow are solved for both models, giving expressions forthe effective stress coefficients in terms of clay content and the elastic moduli of the rockand clay. Both models predict that the permeability will be much more sensitive tochanges in pore pressure than to changes in confining pressure. The clay particle modelgives somewhat better agreement with data from the literature and with new data on aStainton sandstone having a solid volume fraction of 8% clay. INDEX TERMS: 5114 Physical

Properties of Rocks: Permeability and porosity; 5139 Physical Properties of Rocks: Transport properties; 8168

Tectonophysics: Stresses—general; 3909 Mineral Physics: Elasticity and anelasticity; KEYWORDS:

permeability, effective stress, sandstones, clay

Citation: Al-Wardy, W., and R. W. Zimmerman (2004), Effective stress law for the permeability of clay-rich sandstones, J. Geophys.

Res., 109, B04203, doi:10.1029/2003JB002836.

1. Introduction

[2] Physical properties of porous sedimentary rocks, suchas permeability, k, depend on both the confining pressure,Pc, and pore pressure, Pp. If hysteresis is neglected, k canalways be expressed as some function k = f(Pc, Pp). If thepermeability can also be expressed as a function of thesingle parameter Pc � nkPp, i.e., k = f(Pc � nkPp), we thensay that it follows an effective stress law, where nk is theeffective stress coefficient, and Pc � nkPp is the effectivestress. Therefore nk is a measure of the sensitivity ofpermeability to pore pressure, relative to the sensitivity toconfining pressure [Bernabe et al., 1982].[3] The effective stress coefficient is of importance in, for

example, petroleum reservoir calculations. When oil or gasis produced from a reservoir, the pore pressure declines.Laboratory measurements of permeability, however, aremore easily made at nonelevated pore pressures. Changesin the effective stress are then simulated in the laboratory byvarying the confining pressure. The permeabilities thusmeasured would then be related to those that exist in situ,by applying the appropriate effective stress law.[4] Numerous studies have been conducted on the effect

of clay on permeability. These studies have tended to focuson issues such as clay particles clogging the pore throats,but only a few studies have considered the influence ofstress on this effect. McLatchie et al. [1957] found that thepercentage of reduction in permeability, due to an increasein confining stress, was much higher in shaly rocks of lowpermeability than in clay-free, high-permeability rocks.Berryman [1992] showed that nk for a rock consisting ofa single mineral, say quartz, should not exceed unity.

However, Zoback and Byerlee [1975] found experimentallythat the effective stress coefficient of some clay-rich sand-stones can in fact be as high as 3–4. Walls and Nur [1979]found that nk increased with clay fraction, and reachedvalues as high as 7 for sandstones with volumetric clayfractions of 20%.[5] To explain this behavior, Zoback and Byerlee [1975]

proposed a model in which the rock consists of quartz,permeated with cylindrical pores that are lined with anannular layer of clay. As the inner clay layer is morecompliant than the outer quartz layer, such a rock shouldbe more sensitive to changes in pore pressure than tochanges in confining pressure, and therefore have an effec-tive stress coefficient greater than unity. Although thismodel has frequently been invoked to explain experimentalresults, a quantitative discussion of this model does notseem to be available. A related model, proposed in thepresent paper, is one in which clay takes the form ofparticles that are only tangentially attached to the porewalls. The governing equations of elastic equilibrium andfluid flow will be solved for each model, yielding expres-sions for the effective stress coefficient in terms of claycontent and the elastic moduli of the rock and the clay.Using recently collected data on the elastic deformation ofclays [Farber et al., 2001], which shows clays to be about25 times more compliant than quartz, we find that bothmodels yield effective stress coefficients that increase withincreasing clay content. The second model gives highervalues of nk, which are in somewhat closer agreement withthose found in the literature. A value of nk was obtainedexperimentally in this study, for Stainton sandstone con-taining 8% clay by volume, which is also in a betteragreement with the second model.[6] The models considered in this paper attempt to

account for sandstone behavior only in the elastic region,

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 109, B04203, doi:10.1029/2003JB002836, 2004

Copyright 2004 by the American Geophysical Union.0148-0227/04/2003JB002836$09.00

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and only under hydrostatic stresses. The very complex andinteresting behavior exhibited by the permeability evolutionof sandstones and other rocks under deviatoric loads, and inthe inelastic range of behavior [Holt, 1990; Zhu and Wong,1999; David et al., 2001; Simpson et al., 2001; Ngwenya etal., 2003], is beyond the scope of the present work.

2. Review of Results for Clay-Free Rocks

[7] The permeability of a rock depends on the amount andstructure of the pore space. Hence it is to be expected that asthe pore volume changes, the permeability will also change.However, there is no simple one-to-one relationship betweenpore volume (porosity) and permeability [Scheidegger,1974]. In order to develop a model of how the permeabilityof a sandstone may vary with applied loads, we consider aporous rock in which the pores are an interconnectednetwork of cylinders of radius a. We start with theHagen-Poiseuille equation for fluid flow through a circulartube [White, 1974, p. 116]:

Q ¼ pa4

8hdP

dz; ð1Þ

where h is the viscosity of the flowing fluid and dP/dz is thepressure gradient along the pore axis. From equation (1), thepermeability of a rock containing a single pore follows as

k ¼ pa4

8A; ð2Þ

where A is the macroscopic area of the rock normal to theflow. According to this very simple model, the permeabilityis proportional to a4, which implies that permeability will besensitive to small changes in the pore radius.[8] This model can be generalized somewhat by includ-

ing a number density of pores in the numerator, and atortuosity factor in the denominator [Scheidegger, 1974].However, for our present purposes we are interested only inthe relative effects of pore pressure and confining pressure,which will not be affected by any multiplicative constantsappearing in equation (2). So, in this spirit we absorb allsuch factors into a constant G, and write k = Ga4. Theoverall permeability will also depend on the distribution ofpore sizes [Doyen, 1988; Lock et al., 2002], consideration ofwhich would have the effect of replacing the radius a bysome effective radius. Nevertheless, the main point is thatthe permeability will be controlled by the fourth power ofthe pore radius.[9] More sophisticated models might represent the pores

as elliptical in cross section, and thereby introduce an aspectratio into the analysis [Bernabe et al., 1982; Seeburger andNur, 1984; Sisavath et al., 2000]. This assumption in itselfcannot explain effective stress coefficients greater thanunity. Moreover, an elliptical pore model cannot readilybe extended to the case in which a layer of clay lines thepore wall, without resorting to extensive computations [Ru,1999], and so we will use the circular cross section as thebasis of our analysis.[10] From the definition of the effective stress law, the

effective stress coefficient for permeability, nk, can bedefined as the ratio of the sensitivity of permeability to

changes in pore pressure, to the sensitivity of the perme-ability to changes in confining pressure [Bernabe, 1987]:

nk ¼ � @k

@Pp

� �Pc

,@k

@Pc

� �Pp

; ð3Þ

where the subscripted variable outside the derivativeindicates a variable that is held constant. If the permeabilityvaries explicitly only with the pore radius, we can use thechain rule on equation (3) to find

nk ¼� @k

@Pp

� �Pc

@k

@Pc

� �Pp

¼� dk

da

� �@a

@Pp

� �Pc

dk

da

� �@a

@Pc

� �Pp

¼� @a

@Pp

� �Pc

@a

@Pc

� �Pp

: ð4Þ

According to this model, the effective stress coefficient for kis essentially the same as that for a (or for the pore volume).[11] To obtain the effective stress coefficient for the pore

radius a, we need to solve the elasticity equations for acylindrical pore in an elastic body. The equations of elasticequilibrium are, in general, three coupled partial differentialequations with three unknown components of the displace-ment vector. However, for a body with radial symmetry,subjected to hydrostatic loading, the equations simplifyconsiderably [Sokolnikoff, 1956, p. 184]. For a radiallysymmetric deformation under plain strain conditions, theonly nonzero component of the displacement vector is theradial component, ur, which can only vary with r. The onlynonzero strain components are

err ¼du

dr; eqq ¼

u

r; ð5Þ

where we write u for ur. Two of the three equations of stressequilibrium are identically satisfied. The third takes theform

dtrrdr

þ trr � tqqr

¼ 0: ð6Þ

The stress-strain equations take the usual form for anisotropic material:

trr ¼ l err þ eqqð Þ þ 2merr; ð7Þ

tqq ¼ l err þ eqqð Þ þ 2meqq; ð8Þ

where l and m are the Lame moduli. They are related to themore familiar bulk and shear moduli by G = m, and K = l +(2/3)m.[12] Combining equations (5) and (7) yields the stress-

displacement relations

trr ¼ lþ 2mð Þ dudr

þ lu

rð9Þ

tqq ¼ ldu

drþ lþ 2mð Þ u

r: ð10Þ

B04203 AL-WARDY AND ZIMMERMAN: EFFECTIVE STRESS LAW FOR PERMEABILITY

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Substituting these stress expressions into the stress-equilibrium equation (6), and assuming that the moduliare locally uniform, yields

lþ 2mð Þ d2u

dr2þ 1

r

du

dr� u

r2

� �¼ 0: ð11Þ

The general solution of equation (11) is

u ¼ Ar þ B

r; ð12Þ

where A and B are constants which must be found by usingthe boundary conditions.[13] Many previous analyses of the effect of stress on

permeability have taken as their basic problem a single porein an infinite rock mass [Bernabe et al., 1982; Sisavath etal., 2000]. This model has the advantage of allowing avariety of pore shapes to be considered, but is appropriateonly for very low porosities, as it does not allow theporosity to enter into the analysis as a parameter. However,it is known from the voluminous literature on the effectivecompressibility of porous media that a body permeated withcylindrical pores, having total porosity f, can be quite wellrepresented [Hashin and Rosen, 1964; Warren, 1973] by asingle hollow cylinder with a finite outer radius b, chosen sothat f = (a/b)2. For the hollow cylinder model, the appro-priate boundary conditions are that the radial stress at theouter boundary must be �Pc, and the radial stress at theinner boundary must be �Pp, where we take compressivestresses to be negative. From equations (9) and (12), theseconditions take the forms

At r ¼ a 2A lþ mð Þ � 2mB

a2¼ �Pp; ð13Þ

At r ¼ b 2A lþ mð Þ � 2mB

b2¼ �Pc: ð14Þ

Solving for A and B gives [Jaeger and Cook, 1979, p. 135]

A ¼ Ppa2 � Pcb

2

2 lþ mð Þ b2 � a2ð Þ ; ð15Þ

B ¼Pp � Pc

� �a2b2

2m b2 � a2ð Þ : ð16Þ

Hence the radial displacement is given by

u rð Þ ¼� Pcb

2 � Ppa2

� �r

2 lþ mð Þ b2 � a2ð Þ �Pc � Pp

� �a2b2

2m b2 � a2ð Þr : ð17Þ

The change in the inner radius a, due to a change in the porepressure, is therefore given by

@u að Þ@Pp

� �Pc

¼ a3

2 lþ mð Þ b2 � a2ð Þ þab2

2m b2 � a2ð Þ ; ð18Þ

and the change in the inner radius due to a change in theconfining pressure is

@u að Þ@Pc

� �Pp

¼ �ab2

2 lþ mð Þ b2 � a2ð Þ �ab2

2m b2 � a2ð Þ : ð19Þ

Equations (18) and (19) can then be substituted back intoequation (4) to obtain an expression for nk, making use ofthe fact that (a/b)2 = f:

nk ¼lþ m 1þ fð Þ

lþ 2m¼ 1þ 1� 2nð Þf

2 1� nð Þ ; ð20Þ

where n = l/2(l + m) is the Poisson’s ratio. For example, arock with a Poisson’s ratio of 0.25 (i.e., l = m) has aneffective stress coefficient given by (2 + f)/3. The range ofvariation of n that is observed in rocks is relatively narrow,and variations in n have less influence on nk than dovariations in other parameters such as the clay fraction,rock/clay stiffness ratio, etc., so in the sequel we will oftentake n = 0.25, to simplify the presentation.[14] This effective stress coefficient depends on porosity,

and never exceeds unity. If the pores were assumed to beelliptical rather than cylindrical, the effective stress coeffi-cient would be higher, approaching unity in the limit of thincrack-like pores, but never exceeding unity [Bernabe et al.,1982; Zimmerman, 1991]. Hence it seems that values ofnk > 1 should only be expected to occur in a sandstone if itcontains components that have substantially differentelastic moduli, such as a sandstone containing some clay[Berryman, 1992]. (The moduli of most other rock-formingminerals, such as calcite, feldspar, etc., are not vastlydifferent from those of quartz, and rocks composed of amixture of these minerals can be treated as if they weremicroscopically homogeneous [Zimmerman, 1991], at leastunder isothermal conditions. The justification for assigninga relatively low elastic modulus to ‘‘clay’’ is discussedbelow.) Indeed, the data collected from various sources byKwon et al. [2001] show that nk increases almost linearlywith clay content, reaching values as high as 7 when claycontent is 20%.

3. Models for Clay-Rich Sandstones

[15] To explain the results mentioned above in aquantitative way, two different pore-clay models will beexamined. The first model is the Zoback and Byerlee[1975] model, in which the rock consists mainly of asingle mineral, say quartz, permeated with cylindricalpores that are lined with shell-like layer of clay(Figure 1a). In the second model, the rock again consistsmainly of quartz permeated with cylindrical pores, butwith the clay situated as particles that are touching, butonly weakly coupled to, the rock matrix (Figure 1b).These two models may be expected to represent extremecases with regard to the extent of coupling between theclay and rock. Although these models are highly ideal-ized, both pore-lining clays and discrete particles of clayare indeed often observed in sandstones [Neasham, 1977].Another possible structural model, in which quartz par-ticles are embedded in a clay matrix [Kwon et al., 2001],will not be expected to yield effective stress coefficients


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larger than unity, as in this rock the pores wouldessentially be embedded in an ‘‘effective’’ homogeneousmedium.

3.1. Clay Shell Model

[16] In this model the clay forms a thin layer that isuniformly distributed over the entire pore wall (Figure 1a).The clay occupies the region a < r < c, and the rockoccupies the region c < r < b. As in the clay-free case, thepermeability will depend only on the radius of the pore tube,a, and the effective stress coefficient will be given by theratio shown in equation (4). However, the dependence of aon the two applied stresses will be different in this case.This dependence can be found by solving the elasticityequations, subject to the appropriate boundary conditionsfor this geometry.[17] The governing equations of elastic equilibrium are

the same as in the clay-free case, except that the moduli willbe {lc, mc} in the clay region, a < r < c, and will take thevalues {lr, mr} in the rock region, c < r < b. The radialdeformation will therefore have the same form as inequation (12), but with constants {A1, B1} in the clayregion, and {A2, B2} in the rock region.[18] These boundary conditions are as follows. At the

pore wall,

trr að Þ ¼ �Pp: ð21Þ

At the outer boundary of the hollow cylinder,

trr bð Þ ¼ �Pc: ð22Þ

The radial stress must be continuous at the boundarybetween the clay and the rock, so

trr;clay cð Þ ¼ trr;rock cð Þ: ð23Þ

Finally, the displacement must also be continuous as therock/clay interface, so

uclay cð Þ ¼ urock cð Þ: ð24Þ

Using equations (21)–(24) in equations (9) and (12) yieldsfour equations for the constants {A1, A2, B1, B2}, which canbe solved to yield

A1 ¼2

Dmcmrc

2 þ lr þ mrð Þmcb2�

Ppa2 � Pcb

2� �

� lr þ mrð Þ c2 � b2� �

mcb2Pc � mra

2Pp

� ��; ð25Þ

A2 ¼2

Dmcmrc

2 þ lc þ mcð Þmra2�

Ppa2 � Pcb

2� �

� lc þ mcð Þ mcb2Pc � mra

2Pp

� �c2 � a2� ��

; ð26Þ

B1 ¼ 2 lc þ mcð ÞA1a2 þ Ppa

2�

=2mc; ð27Þ

B2 ¼ 2 lr þ mrð ÞA2b2 þ Pcb

2�

=2mr; ð28Þ

D ¼ 4 lc þ mcð Þ mcmrc2 þ lr þ mrð Þmcb2

� c2 � a2� �

� lr þ mrð Þ mcmrc2 þ lc þ mcð Þmra2

� c2 � b2� ��

: ð29Þ

These constants can be substituted back into the generalsolution (12) to find the displacement in each region, clayand rock, respectively [Al-Wardy, 2003]. For the presentpurposes we are interested only in the displacement at thepore wall. As the constants depend linearly on the twoapplied pressures, the displacement at the pore wall willhave the form

u að Þ ¼ A1aþB1

a¼ A1pPp þ A1cPc

� �aþ

B1pPp þ B1cPc

� �a

;

ð30Þ

Figure 1. Cross sections of pore-clay models: (a) clay shell model and (b) clay particle model.


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where for example, A1c is the coefficient that multipliesPc in the expression for A1. Using this notation, andequation (4), it follows that

nk ¼A1pa

2 þ B1p

A1ca2 þ B1c

: ð31Þ

Figure 2 shows the effective stress coefficient as a functionof clay content, Fc, for various values of the stiffness ratio g,defined as g = mrock/mclay. The clay fraction is defined as theratio of clay volume to total solid (clay + rock) volume. Theporosity is taken to be 20%, and the Poisson ratios of bothcomponents are taken to be 0.25.[19] At zero clay content, all curves begin at the value

0.733, given by equation (20). In the limiting case inwhich the stiffness ratio is 1 (i.e., when mrock = mclay), thesystem is mechanically equivalent to a rock without clay,and the effective stress coefficient is consequently insen-sitive to clay fraction. For higher stiffness ratios, theeffective stress coefficient increases with clay content, ata rate that increases with increasing stiffness ratio. How-ever, no realistic combination of parameters seems to becapable of yielding values of nk that are greater that about,say, 3 or 4.[20] At clay fractions greater than about 35%, the effec-

tive stress coefficient for the clay-shell model actuallybegins to decrease with increasing clay fraction. This isunderstandable, since, for example, if the clay fraction in themodel shown in Figure 1a reaches 100%, the rock againconsists of a single mineral component, in this case ‘‘clay.’’Although the sensitivity of the permeability to both the porepressure and confining pressure would be greater than for arock consisting only of quartz, the relative sensitivities tothe two pressures would be the same as that of the all-quartzcase. This is apparent from equation (20), which shows thatnk for a single-mineral rock does not depend on the absolute

magnitude of the elastic moduli, although it does have aweak dependence on the Poisson ratio. Hence all the curvesin Figure 2 must eventually return to the value 0.733 whenthe clay fraction reaches 100%. Nevertheless, rocks withclay fractions greater than 35% would probably not bemodeled by the clay-shell model of Figure 1a but morelikely should be represented by the ‘‘quartz-dispersed-in-clay’’ model shown in Figure 9f of Kwon et al. [2001].

3.2. Clay Particle Model

[21] In this model, the clay exists in the form of particlesthat are tangentially connected to the pore walls (Figure 1b).In this configuration, the clays will have essentially noinfluence on the effect that the confining pressure has on thepore geometry. An increase in confining stress will causethe pore channel to deform in the same manner as if the claywere not present, and, moreover, will have essentially noinfluence on the geometry of the clay particles. The porepressure will cause the pore wall at r = a to expand radially,exactly as in the clay-free case. However, the pore pressure,which acts over essentially the entire outer boundary of theclay particle, will cause a uniform hydrostatic compressionof the clay particle. If we assume that the clay is isotropic,as we do in order to avoid introducing too many parametersinto the model, then the clay will contract uniformly,regardless of the specific shape of the clay particle.[22] The permeability will depend on the geometry of the

region of the pore that is not occupied by the clay particles.In order to yield a tractable, two-dimensional flow problem,we assume that the clay exists as a solid cylinder of radius c,touching the pore wall. The region available for fluid flow isthen the region between eccentric cylinders of radii a and c(Figure 3), in the limiting case in which the inner cylinder istouching the outer one.[23] We first consider the general case in which the

centers of the two cylinders are separated by a distance

Figure 2. Effective stress coefficient, nk, as a function ofclay fraction Fc, for different values of the rock:claystiffness ratio g, according to the clay shell model. Theporosity is taken to be 20%.

Figure 3. Two eccentric cylinders, the larger one of radiusa representing the pore, and the smaller one of radius crepresenting the clay particle.


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l (Figure 3). The volumetric flow Q through this eccentricannulus is given by [White, 1974, p. 121]

Q ¼ p8h

dP

dz

� �(a4 � c4 � 4l2M2

b� a:�8l2M2

X1n¼1

ne�n bþað Þ

sinh n b� að Þ½ �

);

ð32Þ

where

M ¼ F2 � a2� �1=2

; F ¼ a2 � c2 þ l2

2l; ð33Þ

a ¼ 1

2lnF þM

F �M; b ¼ 1

2lnF � l þM

F � l �M: ð34Þ

The expression for Q is quite complicated, and the infiniteseries converges very slowly in the case where the claycylinder is tangent to the pore wall, i.e., l = a � c, which isthe case of interest. Moreover, this expression does not lenditself to being differentiated to find the effective stresscoefficient. Fortunately, if we evaluate equation (32) for thecase l = a � c, we see (Figure 4) that the expression for Qcan be approximated quite well by a cubic polynomial ofthe form

Q ¼ pa4

8hdP

dz

� �h1:0 � 0:57752 c=að Þ � 2:6126 c=að Þ2

þ 2:1928 c=að Þ3i: ð35Þ

This approximation holds for the range 0.1 < (c/a) < 0.7,which covers the range in which the clay fills between 0 and

50% of the pore. In this range, therefore, the modelpermeability becomes

k ¼ Ga4h1:0� 0:57752 c=að Þ � 2:6126 c=að Þ2 þ 2:1928 c=að Þ3

i;

ð36Þ

where we again absorb all numerical prefactors (tortuosity,etc.) into the constant G.[24] To find the sensitivities of the permeability to pore

and confining pressure, the derivatives appearing in equa-tion (3) must be calculated using chain rule as

@k

@Pp

¼ @k

@a

@a

@Pp

þ @k

@c

@c

@Pp

; ð37Þ

since the pore pressure affects both the pore and the clayvolumes. However, the confining pressure has no effect onthe clay volume, so the sensitivity to confining pressure canbe calculated as

@k

@Pc

¼ @k

@a

@a

@Pc

: ð38Þ

The derivatives of k with respect to a and c are found bydifferentiating equation (36), and the derivative of a withrespect to the two pressures are given by equations (18) and(19) for the hollow cylinder. The derivative @c/@Pp, whichis the change in clay particle radius due to a change in porepressure, is given by �c/2(lc + mc), where (l + m) is theplane strain ‘‘areal’’ bulk modulus.[25] Substituting all the derivatives back into equation (3),

the effective stress coefficient for permeability of the clayparticle model is found to be

nk ¼2þ fþ 1� fð ÞFc

3þ 0:0481g 1� fð Þ 1� Fcð Þg c=að Þ;

ð39Þ

g c=að Þ ¼ c=að Þ þ 9:048 c=að Þ2�11:39 c=að Þ3

1� 0:4331 c=að Þ � 1:306 c=að Þ2 þ 0:5482 c=að Þ3; ð40Þ

where Fc is the clay fraction by solid volume, g is thestiffness ratio, and we again take, for simplicity, bothPoisson ratios to be 0.25. The ratio c/a can be related to theporosity and clay fraction through the relation c/a =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� fð ÞFc= Fc þ f 1� Fcð Þ½ �p

. The first term on the right-hand side of equation (39) is the effective stress coefficientfor the single-component rock (equation (20)) modified sothat the ‘‘effective’’ porosity now reflects the volume ofboth the pore space and the clay particle, which in this casedoes not bear any of the external load. The second term isdue to the compression of the clay particle, and itsconsequent effect on the permeability. Recalling theapproximation used in equation (35), we must bear in mindthat equation (39) is intended only for the range 0.1 <(c/a) < 0.7, which is to say, when the fraction of the porethat is obstructed by clay is no more than 50%.[26] Figure 5 shows the predicted effective stress coeffi-

cient nk as a function of clay fraction, Fc, for differentstiffness ratios and a porosity of 20%. The results arequalitatively quite similar to those of the clay shell model,

Figure 4. Flow in a tube obstructed by a clay particleattached to its wall (see Figures 1b and 3), normalized withrespect to flow in an unobstructed tube, i.e., Q(c,l = a � c)/Q(c = 0), as a function of (c/a), according to equation (32).


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in that the coefficient nk increases as clay fraction increases,with the effect more pronounced for higher values of thestiffness ratio. However, the numerical values of nk arelarger for this model than for the clay shell model; note thedifferent scales in Figures 2 and 5. For example, for astiffness ratio of 25:1 and a clay fraction of 0.1, the clayshell model yields an effective stress coefficient of 1.83,whereas the clay particle model predicts a value of 3.25.[27] The model used in the above calculation, in which

the clay is assumed to exist as a circular cylinder sittingalongside the pore wall, is obviously highly idealized. Itwas chosen in large part so as to yield a solvable flowproblem. Alternative models could include, for example,spherical clay particles attached to the pore walls at randomlocations. Different geometric models would almost surelyyield relationships between the effective stress coefficientand clay fraction that differ from equation (39). All such‘‘clay particle’’ models, however, would predict a fairlylarge effective stress coefficient, as the permeability willagain be very sensitive to the size of the flow-obstructingclay particle, and this size would always be much moresensitive to pore pressure than to confining pressure.

4. Permeability Measurements

4.1. Specimen Description

[28] To provide some data against which to test our twomodels, permeability measurements were carried out on acore of Stainton sandstone having 16% porosity and 8%volumetric clay content. A cylindrical sample with a diam-eter of 38 mm and a length of 77 mm was used, with 5%NaCl brine as the pore fluid. This rock comes from the CoalMeasures of Carboniferous age. It is a fine-medium, sub-angular grained rock, yellow to grey in color. The grainsconsist mainly of quartz (85%), with some feldspar(7%). The clay (8%) is of kaolinite type. As seen inFigure 6, some of the pores are almost completely filled

with clay, some are nearly clay-free, and others are partiallyfilled. Clay that completely lines the pore in a continuousshell-like structure, as in Figure 1a, does not seem to beobserved in this rock. The large pore in the center of theimage is partially filled with kaolinite, encircled in black.Although this kaolinite clump is in contact with a largeportion of the bottom edge of the pore, its location vis-a-visnearby quartz grains seems to indicate that it is not loadbearing and is, in fact, mechanically uncoupled from theeffects of confining pressure, as in our ‘‘clay particle’’model. Several such ‘‘uncoupled’’ clumps of clay wereobserved in various thin section images of the Stainton,although in general, it could not be said that they formedcylindrical particles.

4.2. Experimental Apparatus and Procedure

[29] The experimental system essentially consists of foursubsystems: a system to control the confining pressure, asystem to control the flow and pore pressure, a triaxialloading cell in which the cylindrical sample sits, and acomputer for the data collection (Figure 7). The confiningpressure system itself consists of an axial pressure controlsystem and a radial pressure control system. The axialpressure is applied using the loading equipment thatincludes a closed loop, servocontrolled, rock mechanics testsystem incorporating a 2000 kN capacity loading frame(ESH Testing Ltd.). The radial pressure is applied bycompressing the oil (Shell Tellus 68) surrounding thesample radially by mean of an air compressor. This pressureis monitored by an electronic pressure transducer. For thepurpose of this research, the samples were compressedhydrostatically only, by keeping the axial and the radialpressures equal.[30] The pore pressure system consists mainly of the

GDS advanced digital pressure/volume controller of high-pressure range (64 MPa), and is connected at the inlet of thesample. This apparatus is used to control the upstream flow

Figure 5. Effective stress coefficient nk, as a function ofclay fraction Fc, for different values of the rock:claystiffness ratio g, according to the clay particle model. Theporosity is taken to be 20%.

Figure 6. SEM image of a thin section of Staintonsandstone. Large gray regions are quartz grains, the lighter,brighter grain toward the lower right is feldspar, the darkestregions are epoxy (used to impregnate the pores), and thelight gray, mottled regions are microporous regions ofkaolinite clay. The total width of this image is 715 mm.


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and the pore pressure through the sample. The pressure ismeasured by means of an internal pressure transducer, andthe pressure/volume readings are displayed on the digitaldisplay/control unit. The target pressure and the rate of thepumping can be controlled by entering the appropriateparameters. The downstream pressure is controlled by aback pressure regulator, of 40 MPa maximum capacity,which is connected at the outlet of the sample. A pressuretransducer is connected at the sample outlet to measure thedownstream pressure. By controlling the upstream and thedownstream pressures, a differential pressure is introducedacross the sample, and the fluid that flows through thesample is collected in a beaker.[31] The experimental procedure can be described as

follows:[32] 1. The sample is placed in the triaxial cell, and a

confining pressure of 5 MPa is applied to it. The porepressure is set at 1 MPa as an initial condition, while thevalve at the outlet is closed and the back pressure regulatorset at maximum pressure (i.e., closed). The sample is leftunder these conditions for several minutes to equilibrate,before proceeding with the flow process.[33] 2. The confining pressure is increased to 10 MPa,

and the pore pressure is increased to 2 MPa. The outletvalve is then opened.[34] 3. The backpressure regulator is then adjusted to give

a pressure difference of around 0.2 MPa across the sample.At this stage, the pore pressure at the inlet will drop becauseof pressure drop at the outlet. This is adjusted by increasingthe inlet pressure such that the mean pressure in the sampleis 2 MPa.[35] 4. After reaching the required pressure difference at

the intended pore pressure, the sample is left under theseconditions for few minutes until the pressure drops and theflow rates stabilize.[36] 5. The same procedure is repeated at 5 and 8 MPa

pore pressure.

[37] 6. The pore pressure is then reduced to 5 MPa, andthe outlet valve is closed to build up the pore pressure. Theconfining pressure is increased to 20 MPa and the outletvalve is opened, with the backpressure regulator adjusted tothe desired pressure. The same procedure of permeabilitymeasurements is repeated at pore pressures of 5, 10, and15 MPa.

4.3. Results and Discussion

[38] Figure 8 shows the two sets of permeabilities, mea-sured at confining pressures of 10 and 20 MPa. The perme-

Figure 7. Schematic diagram of the system used for permeability measurements under stress.

Figure 8. Permeability of Stainton sandstone as a functionof pore pressure, at two different confining pressures.Curves essentially coincide if shifted by DPp = 1.84 MPa.


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ability increases with increasing pore pressure, whereas itdecreases with increasing confining pressure, with valuesranging between 10.8 to 11.8 mdarcy. To calculate theeffective stress coefficient, we note from equation (3) thatnk can also be expressed as [Bernabe, 1987]

nk ¼@Pc

@Pp

� �k

: ð41Þ

The effective stress coefficient can therefore be calculatedfrom the horizontal offset between the two curves. The bestagreement between the two curves occurs if DPp =1.84 MPa. Substituting this value in equation (41), withDPp = 10 MPa, gives an effective stress coefficient of 5.4.[39] To check the ability of the two pore-clay models to

match real rock data, in at least a qualitative way, we plottheir predictions against some values of nk found in theliterature, as collected by Kwon et al. [2001], and our valuemeasured for Stainton sandstone (Figure 9). The experi-mental value from this study is somewhat higher than thosefound in literature, for this clay fraction. Note, however, thatthe various experimental data points correspond to rockshaving different porosities; see Table 1. In order to plotthem together and compare them to the model predictions,we choose a porosity of 20% for the model calculations.Similarly, the stiffness ratio will probably vary for thedifferent rocks, but we use a value of 25:1 for the plottedmodel predictions, consistent with the recent measurementsby Farber et al. [2001] that showed clay to be about25 times less stiff than quartz. Despite these unavoidablesimplifications, we see that both models give the same trendas observed in the data, with the clay particle model giving asomewhat better fit.

[40] Appropriate values of the elastic moduli for ‘‘clay’’have been a matter of controversy for some time, particu-larly in the context of seismic modeling [Tosaya and Nur,1982; Vanorio et al., 2003]. One the one hand, pure,nonporous clay minerals should have elastic moduli thatare on the same order of magnitude as other rock-formingminerals such as quartz or feldspar. However, individualclay particles in the pores of a sandstone are often not veryclosely packed, and form a microporous material, as can beobserved in Figure 6. Hence the appropriate elastic modulivalues of clay, for our modeling purposes, would actually bethe moduli of a highly porous clay assemblage. However,Coyner [1984] pointed out, in an unpublished Ph.D. thesis,that if we assume that the pore fluid permeates the micro-pores of the clay, then the deformation of the clay will begoverned by the so-called ‘‘unjacketed’’ bulk modulus[Zimmerman, 1991], which is essentially equal to that ofthe nonporous mineral. According to Coyner, then, a highcompressibility for the ‘‘clay’’ can only be explained by thepossible presence of very compressible organic material, airbubbles, etc., within the clay assemblages.[41] Vanorio et al. [2003] measured the bulk modulus of

pure clay to be about 3–6 times less than that of quartz.According to our two micromodels, such moderate stiffnessratios are insufficient to explain effective stress coefficientsgreater than about 2. Indeed, in order to explain the higheffective stress coefficients, our models require that the clayassemblages be at least twenty times more compressiblethan quartz. Mavko et al. [1998] list bulk moduli of 37 GPafor quartz and 1.5 GPa for kaolinite, which is consistentwith our assumed stiffness ratio of 25:1, as are the valuesmeasured by Farber et al. [2001]. We conclude that it maywell be the case that the effective compressibility of clayassemblages in the pores of a sandstone is much greater thanthat of pure clay minerals, although the explanation of thislatter fact is as yet unknown.

5. Conclusions

[42] Two conceptual models for clay-rich sandstoneshave been analyzed: a clay-shell model in which the claylines the pore walls, and a clay-particle model in which theclay exists as particles tangentially attached to the porewalls. Solutions of the elasticity and fluid flow equationswere presented for both models. Both models yield effectivestress coefficients that are typically higher than 1, and whichincrease with clay content. The rate of increase depends onthe rock:clay stiffness ratio, g. Using realistic values for this

Figure 9. Effective stress coefficient as a function of clayfraction. The points are data from the present work and fromthe literature (see Table 1), and the lines refer to the twomodels, with the Poisson ratio of rock and clay taken as 0.25,a porosity of 20%, and a rock:clay stiffness ratio of 25:1.

Table 1. Description of Samples Whose Effective Stress Coeffi-

cients Are Plotted in Figure 9

Rock TypeClay

Fraction Porosity nk Source

St. Peter 0.01 0.20 1.2 Walls and Nur [1979]Massillon 0.05 0.23 3.2 Walls and Nur [1979]Massillon 0.06 0.24 3.5 Zoback [1975]Berea 0.08 0.20 3.3 Walls and Nur [1979]Berea (k bedding) 0.08 0.20 2.2 Zoback and Byerlee [1975]Berea (? bedding) 0.08 0.20 4.0 Zoback and Byerlee [1975]Stainton 0.08 0.16 5.4 present workBandera 0.20 0.16 7.1 Walls and Nur [1979]


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ratio, the clay shell model does not yield values of nk largerthan 3 or 4. The clay particle model, on the other hand, cangive higher values, and gives a somewhat better fit to thedata set collected by Kwon et al. [2001]. Comparison ofFigures 5 and 9 shows that the clay particle model can fitthe data very well if the stiffness ratio is taken to be about50:1, although such a low value of the elastic modulus ofclay may be difficult to justify. Stainton sandstone, with 8%clay content, showed a value at the high end of those foundin the literature.[43] Our two models are highly idealized, and in principle

could be made more sophisticated, by, for example, con-sidering elliptical pores, or considering a combination of thetwo idealized clay geometries. Nevertheless, the modelsdo explain, at least qualitatively, the observed variationswith clay content of the effective stress coefficient forpermeability.

[44] Acknowledgments. The authors thank Petroleum DevelopmentOman (PDO) for financial support, John Dennis for assistance with thepermeability measurements, Harry Shaw for assistance with interpreting theSEM images of the Stainton core, and Yves Bernabe and Christian Davidfor their thoughtful and constructive reviews.

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��W. Al-Wardy and R. W. Zimmerman, Department of Earth Science and

Engineering, Imperial College of Science, Technology and Medicine,London, SW7 2AZ, UK. ([email protected])


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effective stress law for the permeability of clay-rich sandstones

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