Description of fluid flow around a wellbore with stress-dependent porosity and permeability

Journal of Petroleum Science and Engineering 40 (2003) 1 – 16 www.elsevier.com/locate/petrol

Description of fluid flow around a wellbore with stress-dependent porosity and permeability Gang Han *, Maurice B. Dusseault Department of Earth Sciences, Porous Media Research Institute, University of Waterloo, 200 University Avenue West, Waterloo, ON, Canada N2L 3G1 Received 4 October 2002; accepted 5 March 2003

Abstract Stress-dependent porosity and permeability effects have been widely studied at the laboratory scale, as they can significantly affect reserve estimates, well production rate and profitability. Based on current experimental data and theories, a general analytical method of calculating stress-dependent porosity and permeability is developed and applied to a wellbore producing oil from unconsolidated or weakly consolidated sand, with the aid of a coupled geomechanical model by which stress distributions around the wellbore can be specified. For clean weak sand, nonlinear elastic theory is appropriate for calculations of stress-dependent rock properties such as compressibility, porosity and permeability. When evaluated in terms of pore pressure variations, the stress-dependent aspect of porosity and permeability may be negligible as far as stress analysis concerned. With input of different stress-compressibility relationships, the model can be used to help screen those reservoirs for which the effect of stress on permeability should be considered during geomechanical analysis (sand production prediction, reservoir stress arching and shear, plasticity onset, etc.). Also, it can be used for analyzing formation compaction that results from the decrease of stress-dependent porosity. The model limitations have been discussed and it is believed that a microscopic approach based on particulate mechanics may be valuable for future research. Different boundary conditions commonly used in current geomechanics models have been compared and discussed in the development of the poro-inelastic geomechanics model, and boundary restraint is demonstrated to be a critical factor to stress solutions. D 2003 Elsevier Science B.V. All rights reserved. Keywords: Porosity; Permeability; Pressure; Stress; Geomechanics; Porous media

1. Introduction Stress-dependent permeability of porous media has attracted attention from production engineers and reservoir engineers for about 50 years (Fatt and Davis, 1952), as such a phenomenon could significantly affect well production rate, reserves estimates, profit-

* Corresponding author. Fax: +1-519-746-4751. E-mail address: [email protected] (G. Han).

ability and so on. For stress-sensitive materials such as low permeability lithic sandstones, collapsing chalk or fractured rock, the reduction of permeability can be as high as 90% (Thomas and Ward, 1972; Jones and Owens, 1980; Yale, 1984; Kilmer et al., 1987), leading to losses of up to 50% of the production rate (Vairogs et al., 1971). Yale (1984) showed that the decrease of permeability could approach 5% for 500 – 1000 mD permeability sandstones with an increase in isotropic effective (matrix) stresses from 3.45 to 34.5 MPa (Dr1V= DrV V). 2 = Dr 3

0920-4105/03/$ - see front matter D 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0920-4105(03)00047-0

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When stress changes are anisotropic (deviatoric) because of experimental boundary conditions, depletion effects in the field or a non-isotropic in-situ stress state and sand fabric, the behavior of permeability reduction with increasing stress is not yet clear. Holt (1990) reported that changes in permeability became more significant in the presence of non-isotropic stresses: up to 10% of its initial value (sample porosity 25% and initial permeability from 1 to 2.5 Darcies). King et al. (2001) found that permeability was 10% lower in his triaxial tests, compared to his hydrostatic stress tests (initial permeabilities were 366, 220 and 15 mD in the three principal stress directions). It has been found that, in a compression test, a short permeability increase occurs when rock is compressed close to failure (Morita et al., 1984; Keaney et al., 1998), and therefore fluid flow has been enhanced because of shear dilation of microcracks or particle sliding (Tronvoll and Fjær, 1994). These reports are limited for low permeability samples (for Keaney’s experiment, 3 AD; for Morita, 100 – 200 mD), and it is very challenging to model shear dilation, shearinduced grain crushing or interstitial mineral grain mobilization within the scope of continuum theories. Despite only partial understanding of the complicated permeability behavior with stresses, some models have been proposed to quantify this phenomenon, most of which are de facto ‘‘strain-dependent’’; i.e., permeability calculations are based on the strain determined by a geomechanics stress –strain model (Chin et al., 2000; Wang and Xue, 2002). While it is mathematically convenient to relate porosity changes with volumetric strain, this type of model creates a challenge with respect to laboratory calibration before it can be applied in the field. This is because strain, a rock response to stress changes, is sensitive to many factors such as stress (loading and confining stresses) levels, stress path and anisotropy, loading rate and history, pressure depletion or stimulation, sample size, shape and other properties, etc. There are some empirical relationships between permeability and stress that have been developed from curve-fitting analysis of experimental data, requiring two (Ostensen, 1986) to four (Jones and Owens, 1980; Jones, 1998) coefficients. However, these are purely empirical relations, and the authors have not tried to generate more generalized stressdependent porosity and permeability distributions around a wellbore, both of which should be input and

output variables of a coupled geomechanics model to calculate stress level. A unique relationship between permeability and stress does not exist (Davies and Davies, 2001; Jamtveit and Yardley, 1997; Fatt and Davis, 1952). Nevertheless, it is possible to develop a methodology to describe permeability alternations with rock stress as part of reservoir simulation or geomechanical analysis. In this section, based on nonlinear theory and currently available empirical relations, an analytical method is developed to describe the distribution of stress-dependent porosity and permeability around a wellbore producing oil from high porosity (such as unconsolidated sand) reservoirs. As an application, a criterion is proposed to evaluate whether porosity (or permeability) should be considered to be stressdependent or a constant in a geomechanics analysis.

2. Stress-dependent porosity and permeability 2.1. Porosity vs. Stress Applying four types of compressibilities defined by Zimmerman (1991) into geomechanics analysis, four stress – pressure related compressibilities can be defined: Effective bulk compressibility Cbc ¼ 

BVb 1 Vb Br

ð1Þ

Pesudo  bulk compressibility Cbp ¼

BVb 1 Vb BP

ð2Þ

Formation compaction coefficient Cpc ¼ 

BVp 1 Vp Br

ð3Þ

Effective pore compressibility Cpp ¼

BVp 1 Vp BP

ð4Þ

where r and P are rock total stress and fluid pore pressure; Vb, Vp are bulk and pore volume, respectively. The advantage of this classification is that rock

G. Han, M.B. Dusseault / Journal of Petroleum Science and Engineering 40 (2003) 1–16

volume change upon loading has been separated into bulk and pore volume changes affected by either total stress or pore pressure variations. These compressibilities follow certain relationships Cbc ¼ Cbp þ Cm ;

Cpc ¼ Cpp þ Cm ;

Cbp ¼ /Cpc

ð5Þ where Cm is rock matrix compressibility. Porosity changes under loading condition are defined as:
 Vp dVp dVb d/ ¼ d / ð6Þ ¼ Vb Vb Vb Substituting dVp =  CpcVpdr + CppVpdP and dVb =  CbcVbdr + CbpVbdP into the above,

3

2.2. Compressibility vs. stress The integration of Eq. (12) involves the expression of stress-dependent bulk compressibility, Cbc. As bulk compressibility is the easiest to measure in the laboratory, a common approach is to derive an empirical relationship based on experimental data, e.g., in the form of (Zimmerman, 1991) Cbc ¼ a1 þ a2 ea3 rV

ð13Þ

or (Rhett and Teufel, 1992) Cbc ¼

b1

ð14Þ

ð1 þ b2 rVÞ2

or (Jones, 1998) d/ ¼ /Cpc dr þ /Cpp dP þ /Cbc dr  /Cbp dP ð7Þ Using the relationships among compressibilities, the porosity changes are d/ ¼ ½Cbc ð1  /Þ  Cm drV

ð8Þ

where rVis the difference between total stress and pore pressure (rV= r  P). Similarly, the bulk volumetric strain eb can be calculated as dVb deb ¼  ¼ Cbc dr  Cpc dP ¼ Cbc drV þ Cm dP Vb ð9Þ Combining with Eq. (8),
 Cm d/ ¼  1   / ðdeb  Cm dPÞ Cbc

ð10Þ

which is in agreement with Wang and Dusseault’s (1991a,b) work, except for the negative sign because of a different sign convention. As far as unconsolidated and weakly consolidated sand are concerned, Cm is assumed to be small enough to be negligible; therefore, Eqs. (8) and (10) can be written as: d/ ¼ ð1  /Þdeb

ð11Þ

which has been widely used in coupled geomechanics models and d/ ¼ Cbc ð1  /ÞdrV which is the form that will be used herein.

ð12Þ

Cpp ¼

d3 erV=d1 d2 þ d1 1 þ d2 rV

ð15Þ

where a1, a2, a3, b1, b2, d1, d2 and d3 are constants determined from curve-fitting analysis. Therefore, Eq. (12) can be integrated into the following forms: 1/ ln 1  /i

! ¼ a1 ðrV ri VÞ 

a2 a3 r V a3 ri V ðe e Þ a3 ð16Þ



 1/ b1 1 1 ln  ¼ 1  /i b2 1 þ b2 ri V 1 þ b2 rV 1/ ln 1  /i

!

ð17Þ




  riV  1 þ d2 rV   rV ¼ ln þ d 3 e d1  e d1 1 þ d2 riV ð18Þ

where /i and riVare initial porosity and initial mean effective stress (i.e., far field in-situ mean effective stress). The other way to calculate bulk compressibility is based on nonlinear theory. Bulk modulus can be expressed as (Duncan and Chang, 1970; Byrne et al., 1987; Vaziri, 1995)
m rV K ¼ nPa ð19Þ Pa

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where Pa is atmospheric pressure, and m and n are the hyperbolic equation parameters. For soils, their values have been determined (Byrne et al., 1987), e.g., m is usually taken as a constant of 0.25. Hence, bulk compressibility can be expressed as 1 Pm1 ¼ a ðrVÞm K n The integration of Eq. (12) gives Z rV  m1 Z / 1 Pa d/ ¼ ðrVÞm drV n /i 1  / ri V

Cbc ¼

ð20Þ

ð21Þ

Hence, ln

i 1/ Pam1 h ¼ ðrVÞð1mÞ  ðriVÞð1mÞ 1  /i ð1  mÞn

or Pam1

/ ¼ 1  ð1  /i Þe ð1mÞn



ðrVÞð1mÞ ðriVÞð1mÞ

ð22Þ

ð23Þ

As indicated in Eq. (23), porosity changes are solely related to the state of effective stress through application of these concepts.

The four methods of empirically including compressibility (Eqs. (13), (14), (15) and (20)) and porosity (Eqs. (16), (17), (18) and (22)) discussed above are compared in Figs. 1 and 2, with the input parameters listed in Table 1. The porosities used for this range from 0.10 (tight sand, Jones, 1998), 0.13 (North Sea sand, Rhett and Teufel, 1992), 0.16 (Bandera Sand, Zimmerman, 1991), 0.18 (Berea Sand, Zimmerman, 1991), 0.27 (Boise Sand, Zimmerman, 1991), to above 0.3 (unconsolidated sandstone, Eq. (23)). As shown in Fig. 1, the rock compressibility becomes small as effective stress increases and, consequently, the changes of porosity follow the same trend (Fig. 2). Furthermore, the porosity calculation based on nonlinear theory shows particular applicability to high-porosity (or unconsolidated) sands, probably because nonlinear theory was initially developed for soil, which is similar to unconsolidated sand. Because around a wellbore there usually exists a zone of low cohesion and often damaged granular material in a relatively low stress environment (at least the radial stress, rrV, is low), called the ‘‘Coulomb zone’’ by Bratli and Risnes (1981), the approximations that are com-

Fig. 1. Different models for stress-dependent bulk compressibility.

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Fig. 2. Influence of effective stress on porosity (dimensionless porosity = /(rV)//i).

monly used in soil mechanics, such as soil strength being dominated by frictional behavior and geometrical relationships among individual particles, become just as valid as other assumptions, perhaps more so.

applied. Among the geometrical models, the Carman-Kozeny model is popular because of its simplicity:

2.3. Permeability vs. porosity

k¼

Many approaches have been proposed to describe the relationship of permeability to porosity and other rock properties. These approaches can be classified into two categories (Dullien, 1979): geometrical permeability models that treat fluid flow in porous media as a network of closed conduits and statistical permeability models in which a probability law is

where specific surface ffi area, S, can be derived as S qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 ¼ /i =5ð1  /i Þ ki , and /i and ki are porosity and permeability under initial conditions. It holds well for unconsolidated and weakly consolidated spherical particulate assemblies (Dullien, 1979; Holt, 1990). Permeability, however, can easily deviate from the

/3

ð24Þ

5ð1  /Þ2 S 2

Table 1 Coefficients used in four compressibility models Zimmerman

Bandera (/i = 0.16) Berea (/i = 0.18) Boise (/i = 0.27) Stress unit is MPa.

Rhett and Teufel (/i = 0.13)

Jones (/i = 0.1)

Nonlinear (/i = 0.3)

a1(  10 4)

a2(  10 4)

a3

b1

b2

d1

d2

d3

m

n

0.82 1.05 0.95

5.35 6.35 2.79

0.1200 0.2110 0.1427

8.9  10 4

3.1  10 2

13.8

0.44  10 3

0.1

0.25

2.4  104

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description of Eq. (24) and of course relative permeability in multi-phase cases cannot easily be based on such a relationship. For example, Davies and Davies (2001) showed there is no consistent relationship of porosity with permeability for sand samples from the Gulf of Mexico and southern California when porosity exceeds 20%. As a matter of fact, permeability is not only dependent on porosity and specific surface area, but also on the size distribution, skewness, the topographical arrangement of capillaries and the amount and location of interstitial fine-grained minerals. Even though Eq. (24) is used in this paper, other types of porosity – permeability relationships can also be applied, following similar steps discussed below, for specific cases where adequate laboratory information are available. Fig. 3 shows the calculated variations of stressdependent porosity and permeability with the nonlinear theory developed above. In the stress range of 0 – 40 MPa, porosity changes are magnified when interpreted as permeability variations: from 1% for porosity to 0.45% for permeability, which agrees with experimental observations (e.g., Mohiuddin et al., 2000).

3. Effective stresses distributions around a wellbore producing oil 3.1. Coupled poro-elastic solutions For an elastic isotropic formation, stress equilibrium in cylindrical coordinates is expressed as: Brr rr  rh þ ¼0 Br r

ð25Þ

or BrrV rrV  rhV BP þ ¼a Br r Br

ð26Þ

where a is the effective stress coefficient (or Biot’s constant) defined in rV= r + aP and varying from / (porosity) to 1, depending on rock lithology and state of consolidation. Usually, for unconsolidated or weakly consolidated sand, a is approximately 1. The solutions for total stresses can be found as

Fig. 3. Stress-dependent porosity and permeability for unconsolidated sand.

G. Han, M.B. Dusseault / Journal of Petroleum Science and Engineering 40 (2003) 1–16

(Wang and Dusseault, 1991a,b; Bradford and Cook, 1994) rr ¼

Ec1 ðtÞ Ec2 ðtÞ 1  ð1 þ mÞð1  2mÞ ð1 þ mÞ r2 Z 1  2m a r  rPðr; tÞdr 1  m r2

Ec1 ðtÞ Ec2 ðtÞ 1 þ rh ¼ ð1 þ mÞð1  2mÞ ð1 þ mÞ r2
Z  1  2m a r þ rPðr; tÞdr  aPðr; tÞ 1  m r2

ð27Þ

ð28Þ

rhV¼

0:5a Ec1 Ec2 1 Pþ þ 1m ð1 þ mÞð1  2mÞ ð1 þ mÞ r2 0:5  m K¯ a 1m 2

There are two types of boundary conditions commonly used, one (BC1) is at the outer boundary (R2) where both tangential stress and radial stress are equal to a unitary horizontal stress

Ec1 ðtÞ ð1 þ mÞð1  2mÞ Z Ec2 ðtÞ 1 1  2m a r   rPðr; tÞdr ð1 þ mÞ r2 1  m r2

ð29Þ


 ð1 þ mÞð1  2mÞ 0:5  m rh þ a P2 E 1m

rhV¼ a

c2 ¼ a ð30Þ

If steady-state fluid flow is assumed, pore pressure varies only with radius and follows Darcy’s rule:
 r ¯ PðrÞ ¼ P1 þ Kln ð31Þ R1 where K¯ ¼ Ql=2pkh, R1 is wellbore radius; P1 is bottom flowing pressure. If the poroduction rate Q is assumed to be constant, effective stresses can be expressed as

rrV¼

0:5a Ec1 Ec2 1 Pþ  1m ð1 þ mÞð1  2mÞ ð1 þ mÞ r2 0:5  m K¯ þa 1m 2

ð32Þ

ð35Þ

Both conditions are evaluated in this research, and the results are compared. From the first boundary condition (Eq. (34)), the two constants can be solved c1 ¼

m Ec1 ðtÞ Pðr; tÞ þ 1m ð1 þ mÞð1  2mÞ Z Ec2 ðtÞ 1 1  2m a r þ þ rPdr ð1 þ mÞ r2 1  m r2

ð34Þ

and the other (BC2) is that the effective radial stress is zero at the inner boundary and equals the horizontal effective stress at the outer boundary: r ¼ R1 ; rrV¼ 0 and r ¼ R2 ; rrV¼ r hV

rrV¼ aPðr; tÞ þ

ð33Þ

3.2. Application of boundary conditions

r ¼ R2 ; rrV¼ rhV¼ rhVð¼ rh þ aPÞ;

where coefficients c1(t), c2(t) are variables only related to time and determined by boundary conditions. Effective stresses can be expressed as (r = rV aP):

7

¯ 2 ð1 þ mÞð0:5  mÞ KR 2 ð1  mÞE 2

ð36Þ ð37Þ

For the second boundary condition (Eq. (40)), c1, c2 can be shown to be "
ð1 þ mÞð1  2mÞ R22 0:5  m P2 c1 ¼ rh þ a E 1m R22  R21 #  R21 0:5a 0:5  m K¯ P1  a ð38Þ þ 2 1m 2 R2 1  m

c2 ¼


 1 þ m R22 R21 ð0:5  mÞP2 þ 0:5P1 2 r þ a h E 1m R2  R21 ð39Þ

The stress solutions are plotted in Figs. 4 and 5. Clearly, the results are strongly affected by the selection of boundary conditions: without the restraint of the inner boundary (BC1), stresses become wild and irra-

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Fig. 4. Stress solutions with BC1 (rrV= rrV= r hVat r = R2).

tionally two orders higher than the one with the restraint (BC2), while the latter seems more reasonable. Based on the assumption that the formation is elastic and no failure occurs, the above stress solutions are termed as poro-elastic stresses in Rock

Mechanics. Weak or unconsolidated sands, however, are more likely to be yielded and mobilized by stresses and fluid flow, which may lead to sand influx during fluid production for example. There are numerous experiments that have been carried out to study

Fig. 5. Stress solutions under different boundary conditions.

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Table 2 Parameters used in geomechanics (stress) model Rock mechanical properties m

E (Pa) 9

3  10

0.45

Reservoir flow properties

rh (Pa)

u

Co (Pa) 6

6

28  10

0.5  10

30j

/i 0.3

ki (m2)  12

0.3  10

the relevant failure mechanisms, yet these are not fully understood. It is generally believed that sand does not start to flow until most of the weak mineral bonds between sand particles are destroyed and the fluid seepage force can then pluck sand grains out of the rock skeleton under low stress and carry them into the wellbore. Shear distortion is largely blamed for rock cohesion loss, whereas tensile failure is the result of fluid flux that plucks sand grains at the low-or nocohesion stage. Therefore, whether or not sand can withstand the probably induced shear stress is the first step in assessing potential for sand production. The well-established Mohr – Coulomb failure criterion is used herein to decide whether shear failure has occurred. It can be written as (Jaeger and Cook, 1979): 2

r1V¼ 2Co tanb þ r3Vtan b

Geometry parameters l (Pa/s)

P2 (Pa)

ð40Þ

where r1Vand r3Vare the greatest and smallest principal effective stresses: in wellbore situation discussed above, they are equal to rhVand rrV, respectively; Co is cohesive shear strength; b is failure angle related to friction angle (u) through b = p/4 + u/2. If any shear failure occurs, poro-elastic stresses (rhVand rrV) calculated by Eqs. (32) and (33) should satisfy Eq. (40), i.e., 0:5a Ec1 ð1  tan2 bÞPc þ ð1  tan2 bÞ 1m ð1 þ mÞð1  2mÞ Ec 1 2  ð1 þ tan2 bÞ þ ð1 þ tan2 bÞ ð1 þ mÞ R2c 0:5  m aK¯  2Co tanb ¼ 0  ð41Þ 1m 2 where rc is the critical distance that rock is sheared to failure, Pc is pore pressure at rc and can be expressed ¯ ðRc =R1 Þ. Therefore, as long as a root as Pc ¼ P1 þ Kln (rc) of Eq. (41) can be found between R1 and R2, the sand can be assumed to have reached shear failure.

6

10  10

0.01

Q (m3/s) 3

1.157  10

R2 (m)

R1 (m)

h (m)

50

0.1

10

This can be easily solved with the aid of mathematical software (in this case, i.e., Table 2, Rc is found to be 0.1393 m). This method of determining the failure (or plastic deformation) zone is not accurate because, in the Coulomb zone, stresses deviate from what poro-elastic theory describes, while Bratli and Risnes (1981) demonstrated that a Coulomb zone usually exits around the wellbore. Generally, the boundary condition rrV= 0 at r = R1 should not be used to solve poroelastic equations, as long as a value for Rc can be found. 3.3. Coupled poro-inelastic solutions To avoid complexity of theoretical development, a simple way to describe stress distributions inside Coulomb zone is taken and new boundary conditions are developed to yield acceptable solutions. Because the main task of this section is to show a method to develop the relationships between permeability and porosity with distance from the wellbore, such simplifications are warranted. Assuming the rock stresses inside the Coulomb zone satisfy the Mohr –Coulomb failure criterion and stress equilibrium, i.e., Eqs. (26) and (40), the effective stress equilibrium equation becomes drrV 1  tan2 b 2Co tanb þ aK¯ þ rrV¼ dr r r

ð42Þ

and solutions can be found

rrVðrÞ ¼

c3 x 2Co tanb þ aK¯ r þ x x

rhVðrÞ ¼

c3 ð1  xÞ x 2Co tanb þ ð1  xÞaK¯ r þ x x

ð43Þ

ð44Þ

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where x = 1  tan2b. Since at the inner boundary the radial effective stress must be zero, i.e., rrV(R1) = 0, constant c3 can be solved: ¯ Rx c3 ¼ ð2Co tanb þ aKÞ 1

ð45Þ

Therefore, at the outer boundary of Coulomb zone, i.e., at r = Rc, stresses should be

rrVðRc Þ ¼

c3 x 2Co tanb þ aK¯ R þ x x c

rrVðRc Þ ¼

c3 ð1  xÞ x 2Co tanb þ að1  xÞK¯ Rc þ x x ð47Þ

ð46Þ

Furthermore, radial stresses should be continuous across the elastic and Coulomb zone transition, i.e., at r = Rc, stresses calculated from coupled poro-elastic solutions (Eqs. (32) and (33)) should equal those from Eqs. (46) and (47), i.e., c3 x 2Co tanb þ aK¯ R þ x x c 0:5a Ec1 Pc þ ¼ 1m ð1 þ mÞð1  2mÞ Ec2 1 0:5  m K¯  þa 2 1m 2 ð1 þ mÞ Rc c3 ð1  xÞ x 2Co tanb þ að1  xÞK¯ Rc þ x x 0:5a Ec1 Pc þ ¼ 1m ð1 þ mÞð1  2mÞ Ec2 1 0:5  m K¯ þ a 2 1m 2 ð1 þ mÞ Rc

ð48Þ

ð49Þ

Another boundary condition used is the assumption that, in the far field, effective radial stress is equal to the horizontal effective stress (i.e., rrV(R2) = r hV): r hV¼

0:5a Ec1 Ec2 1 P2 þ  1m ð1 þ mÞð1  2mÞ ð1 þ mÞ R22 0:5  m K¯ ð50Þ þa 1m 2

Finally, the three unknown constants Rc, c1 and c2 in the above three equations can be determined (see Appendix A). Comparing to the value derived from Eq. (41) (Rc = 0.1393 m), the critical distance from this method is 0.4327 m, appreciably larger (Fig. 5). This type of stress solution treats the Coulomb zone as a zone with constant low cohesive shear strength (Co), which obviously conflicts with the fact that sand becomes weaker after shear yield, leading to a reduced cohesion or even a cohesionless state. Fig. 6 demonstrates the effect of cohesive strength on stress distributions inside the Coulomb zone: when Co becomes small (Co is from 0.5 to 0.3 MPa), stresses are lowered significantly at the same distance while the critical radius increases from 0.4327 to 0.6914 m (i.e., the Coulomb zone is enlarged). Therefore, Eqs. (43) and (44) give the upper limit of stresses inside the plastic zone and should be treated as conservative solutions. One common approach to compensate for this is to add a plastic strain to the elastic strain calculated by Hooke’s law and this type of strain is defined by plastic theory (Bradford and Cook, 1994; Wang and Xue, 2002). But as far as rock is concerned, plastic theory focuses more empirically and mathematically on matching nonlinear stress – strain relationships rather than capturing the specific physical mechanisms. Therefore, intensive calibration is needed before such ‘‘laws’’ are applied in practice. In fact, most authors believe that a nonlinear theory based on the fact that rock modulus and properties change with loading stresses is more convincing and reasonable. Some developments have been made during the past few decades (Vaziri, 1995; Santarelli et al., 1986; Nawrocki et al., 1995), but additional experiments are needed in order to determine the parameters, and this affects budgets and presents big challenges for nonlinear geomechanics modelers.

4. Permeability vs. distance from wellbore For the elastic zone, effective mean stress in Eq. (23) can be determined by Eqs. (27) and (28):

reV¼

rrVþ rhV 0:5a Ec1 ¼ PðrÞ þ ð51Þ 2 1m ð1 þ mÞð1  2mÞ

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Fig. 6. Effect of cohesive shear strength on stress distributions.

For the Coulomb zone, it can be expressed as

rpV¼

c3 ð1  x=2Þ x 2Co tanb þ ð1  x=2ÞaK¯ r þ x x ð52Þ

Substituting Eqs. (51) and (52) into Eq. (23), the relationship between permeability and effective stress can be determined. It should be noted there is no timedependent effect considered, rock properties are assumed to be independent of time. Some arguments have been put forward suggesting that effective stress theory becomes questionable, or inadequate, when permeability – stress relationships are analyzed (e.g., Zoback and Byerlee, 1975). The experiments they performed with Berea sand, however, were executed with the assumption that pore pressure and confining stress can be changed independently, i.e., the magnitude of pore pressure increase or decrease equals the changes of effective stress (DrV= DP) if constant confining stress and elastic rock state are assured. Unfortunately, pore pressure and effective stress are so closely interlaced that they cannot be separately analyzed: the changes of effective stress, as shown in Eq. (51), are a function of the variations of pore pressure but do not equal

them. Further, because the Biot’s constant in Eq. (26) cannot be treated as 1 anymore when consolidated sand is concerned, the relationship between variations of effective stress and those of pore pressure becomes more complicated.

5. Model calculations and discussions 5.1. Should permeability be considered as stressdependent? Distributions of stress-dependent permeability and porosity around a wellbore producing oil from highporosity sand are determined and plotted in Fig. 7. Both permeability (solid line) and porosity (dashed line) decrease promptly inside the Coulomb zone, while beyond the critical radius the reduction rates are slow. This is because of the nature of the stress distributions around the wellbore (Fig. 5): within the critical distance the mean effective stress (rmV) has a steep gradient, whereas it changes little in the elastic zone. It should be noted that the initial permeability and porosity are defined at atmosphere stress (i.e., 0.1 MPa) and the dimensionless variables are constantly below 1.

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Fig. 7. Stress-dependent porosity and permeability around a wellbore.

The variations of permeability and porosity are relatively small, about 3.2% and 0.7%, respectively. When reflected in pore pressure calculations, the stress-dependent permeability model only predicts about 1.6% change (solid line in Fig. 8) and much of them occur near the wellbore. Zimmerman’s model for Boise sand, which has high porosity and is weakly consolidated, produces a similar effect (dashed line in Fig. 8): pore pressure variations are less than 1.8% of its original value. It is therefore concluded that for

clean unconsolidated sand with parameters roughly similar to those listed in Table 2, the stress-dependent properties of porosity and permeability may be negligible in practice. This is consistent with the experiments reported by Yale (1984) and Sarda et al. (1998). These permeability calculations inside the Coulomb zone conflict with data showing that permeability has indeed been significantly lowered in many cases, e.g., cases where less than half of the original value has been left (Bratli and Risnes, 1981; Holt,

Fig. 8. Pore pressure variations with stress-dependent permeability for high-porosity sands.

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1990; Sarda et al., 1998). This is because the model development above did not take account of changes of rock properties after shear yield. As pore throats in sand have been reshaped after sand particles rearrange and fracture occurs, specific surface area changes significantly, particularly under conditions of large stress changes. Unfortunately, the rock in the Coulomb zone is little studied due to the limitations of experiment and core collection. There are some experiments that report obvious permeability variations in high-porosity sand (Holt, 1990; Morita et al., 1984), but the suitability of the experimental conditions as field analogues are in question. For example, the 10– 95% permeability reduction in Holt’s experiment (Fig. 9) was detected when rock samples were axially loaded up to 80 MPa, whereas the real (in-situ) effective stresses are only about rvVc 15 MPa and rhVc 7.5 MPa for that sandstone. From the shape of the response curve, it is obvious that massive grain crushing was initiated in the specimen. 5.2. Model limitations and suggestions for future research Other than effective stress, there are many additional factors that may affect the permeability distri-

butions around a wellbore, either in a positive (permeability enhancement) or a negative (permeability impairment) manner. For example, shear dilatancy (Dusseault and Rothenburg, 1988) and production of sand particles (Geilikman and Dusseault, 1997) can significantly increase permeability, whereas infiltration of drilling fluid, formation of mud cake, fabric perturbations caused by workovers and accumulation of permeability-sensitive materials such as clay and asphaltenes will usually result in permeability decline. Some developments have been made to analytically describe those factors with respect to rock geomechanical responses (Wang and Dusseault, 1991a,b; Zhang and Dusseault, 1997); however, in this paper, only stress is considered. Besides the need to develop empirical relations for use in the model developed above, another big challenge is the lack of a description of permeability anisotropy. Crawford and Smart (1994) demonstrated that changes of vertical permeability are much less than those of horizontal permeability, given the same mean stress increase. Because continuum theories face great challenges in macroscopically modeling permeability anisotropy in non-hydrostatic loading stress environment, particulate mechanics models may provide an alternative and more satisfactory approach in terms of describing pore structure changes at the grain scale level. Microscopically, when sand particles are loaded, there are several responses that may occur:

Fig. 9. Permeability vs. axial stress in non-hydrostatic test of Red Wildmoor sandstone (after Holt, 1990).

13

Particles undergo elastic deformation, such as changes in particle shape (Davies and Davies, 2001). Micas and shale fragments are minerals that can be easily altered in shape, whereas monomineralic fragments such as quartz and feldspar grains require higher load levels to evidence significant shape changes;

Particles rotate, slip and rearrange themselves, although this is most likely at low stress levels when particles are loosely packed and unconsolidated;

Particles, particularly weak lithic fragments or coccoliths, experience fracturing and crushing as forms of plastic deformation. Pore throats are thereby ‘‘collapsed’’ and liberation of appreciable quantities of fine-grained particles tends to block intact pore throats, lead directly to sand production and even wellbore collapse;

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Interstitial clay and silicate particles are dislodged by shear strains, bridging across pore throats and affecting the permeability disproportionately. Interstitial minerals are often very lightly bound to the silicate substrate, and small hydrodynamic forces, combined with geochemical and capillary changes or shear distortion can mobilize them.

Using Hertz contact theory, the first effect (i.e., particle elastic deformation) can be quantified (Wong and Li, 2000). The other three situations remain to be explored quantitatively in future research.

6. Conclusions Based on nonlinear theory and existing empirical relationships, a general analytical approach to calculation of stress-dependent porosity and permeability is developed and applied to a wellbore producing oil from unconsolidated or weakly consolidated sand. Comparing results to available published data, it is shown that nonlinear theory has good applicability for clean unconsolidated or weakly consolidated sandstones that do not undergo grain crushing. As an application of the theory, distributions of stressdependent permeability and porosity around a wellbore producing oil from a weakly consolidated sand are described, and their effects are evaluated in terms of pore pressure variations. The calculations suggest that, given minimal grain crushing and lack of interstitial fine-grained minerals that can be mobilized by shear distortion, the stress-dependent aspect of porosity and permeability may be trivial as far as stress analysis concerned. Nevertheless, with input of different stress –compressibility relationships for different rocks, the developed model can be used to help screen those reservoirs for which the effect of stress on permeability should be considered during geomechanical analysis such as sand production prediction, reservoir stress arching and shear, plasticity onset, etc. Furthermore, the model can be applied to evaluate the extent of formation compaction resulting from the variations of stress-dependent porosity. The boundary restraint condition assumed is demonstrated to be a critical aspect of solutions of stress

models. Two types of boundary conditions (BC1 and BC2) commonly used in solving poro-elastic stresses are inaccurate as long as a Coulomb zone can be found, in which case the assumptions of continuous stresses across the Coulomb zone should be applied along with the restraints at the outer and inner boundaries. Based on these conditions, a simple analytical poro-inelastic model is developed and its limitations are discussed. The model output, poroinelastic stress, serves as input to analyze stressdependent aspects of rock permeability. Finally, although we have found this solution useful, the limitations of the continuum analytical approach have been emphasized and it is believed that a micromechanics approach based on particulate mechanics may be valuable for future research.

Nomenclature a, b, c, d dimensionless constants C rock compressibility, 1/Pa Co cohesive shear strength, Pa E Young’s modulus, Pa h thickness of oil formation, m k permeability of reservoir, m2 K bulk modulus, Pa m, n hyperbolic parameters of bulk modulus P fluid pressure, Pa P1, P2 fluid pressure at inner (wellbore) and outer boundary of reservoir, Pa Q production rate, m3/s r radius from wellbore, m R1, R2 inner (wellbore) and outer boundary radius of reservoir, m Rc critical radius, m S specific surface area of rock, m V volume of rock, m3 Greek letters a Biot’s coefficient b failure angle u friction angle of reservoir rock / porosity r total stress (Pa) rV effective stress (Pa) l fluid viscosity (Pa s) m Poisson’s ratio x 1  tan2b

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Subscripts and superscripts a atmosphere b bulk c compressibility e, p elastic and plastic, respectively h horizontal i initial status (atmosphere conditions) m matrix r, h radial and tangential, respectively

15

Substituting c2 in Eq. (A3) from Eq. (A2), an equation for Rc can be derived 2Co tanb þ aK¯ 0:5a K¯ þ x 1m 2
 c3 1 1 0:5aK¯  ¼ 2 R2x þ c3 Rx c c þ x 2 1m 2R2  2
 Rc Rc   ln R2 2R22

rhV

ðA4Þ

Acknowledgements G. Han thanks Schlumberger for support for his PhD studies, and M.B. Dusseault thanks the Natural Sciences and Engineering Council of Canada for financial support, as well as many industrial contacts and colleagues. In particular, the inspirations from Dr. Ian Walton, Dr. Frank Chang and Dr. Hongren Gu from Schlumberger, and technical communications with Prof. Mario Ioannidis from University of Waterloo are deeply appreciated.

Appendix A From Eq. (46), c1 can be expressed as
ð1 þ mÞð1  2mÞ 0:5a rhV  P2 c1 ¼ E 1m  Ec2 1 0:5  m K¯ þ  a 1m 2 ð1 þ mÞ R22

ðA1Þ

Eqs. (44) and (45), c2 can be expressed as 1þm c2 ¼  2E




0:5a ¯ K þ c3 Rx c 1m



R2c

ðA2Þ

Similarly, Eqs. (44) – (46) is

rhV 


 2Co tanb þ aK¯ c3 x 0:5aK¯ R2 ¼ Rc þ ln x 1m x Rc
 Ec2 1 1 þ  ðA3Þ ð1 þ mÞ R2c R22

This is a nonlinear equation of Rc, in the form of x 2 f(R2c  x, R c , Rc , ln(Rc)) = 0, which can be easily solved with the aid of mathematical software (e.g., Matlab, Maple, Mathematica). The constants c1 and c2 can be solved by Eqs. (A1) and (A2).

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