permeability of the fluid-filled inclusions in porous media

Transp Porous Med (2010) 84:307–317DOI 10.1007/s11242-009-9503-1

Permeability of the Fluid-Filled Inclusions in PorousMedia

M. Markov · E. Kazatchenko · A. Mousatov ·E. Pervago

Received: 27 March 2008 / Accepted: 5 November 2009 / Published online: 27 November 2009© Springer Science+Business Media B.V. 2009

Abstract In this article, we propose an approach to obtain the equivalent permeability ofthe fluid-filled inclusions embedded into a porous host in which a fluid flow obeys Darcy’s law.The approach consists in the comparison of the solutions for one-particle problem describingthe flow inside the inclusion, firstly, by the Stokes equations and then by using Darcy’s law.The results obtained for spheres (3D) and circles (2D) demonstrate that the inclusion equiva-lent permeability is a function of its radius and, additionally, depends on the host permeability.Based on this definition of inclusion permeability and using effective medium method, wehave calculated the effective permeability of the double-porosity medium composed of thepermeable matrix (with small scale pores) and large scale secondary spherical pores.

Keywords Double porosity · Effective medium method · Effective permeability estimation ·Vugular porous media

1 Introduction

Rock permeability prediction is a crucial point for many areas of geological and geophysi-cal investigations such as hydrogeology, petroleum science, petrophysics, etc. (Renard andMarsily 1997). The relevant problem is the determination of hydraulic permeability fordouble-porosity carbonate formations characterized by complex pore microstructure. Thedetailed classification of the pore system of carbonate formations is given by Choquette andPray (1970), Bagrintseva (1999), and Lucia (1999). The concept of the double porosity wasintroduced by Barrenblatt et al. (1960) for solving transport problem in complex porousmedia. The double-porosity medium can be treated as heterogeneous materials composedof two components: (1) the homogeneous matrix with primary pores; where (2) inclusions(secondary pores) of different sizes and shapes are embedded. In carbonate formations, theprimary-pore system consists of small scale intercrystalline and intergranular pores, whereas

M. Markov (B) · E. Kazatchenko · A. Mousatov · E. PervagoInstituto Mexicano del Petróleo, Eje Central Lázaro Cárdenas 152, CP 07730, México DF, Mexicoe-mail: [email protected]

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308 M. Markov et al.

large scale cracks and cavities (vugs) correspond to the secondary pore systems. The studyof the pore size and shapes in carbonate rocks was performed by Song et al. (2002) andMoctezuma-Berthier et al. (2002).

There are various approaches for the permeability estimation of the heterogeneous mediathat include the effective media methods, numerical solution of differential equations thatdescribe fluid flow (Arbogast and Brunson 2007), and methods for stochastic modeling of sys-tems with elements of different permeabilities (Renard and Marsily 1997; Berkowitz 2002).At present, the effective medium method is widely used for the permeability assessment ofporous media (Koplic 1981; Schvidler 1985; Fokker 2001; Pozdniakov and Tsang 2004),because for various important cases the analytical solutions were obtained.

In order to apply the effective medium theory, the permeabilities of both matrix and inclu-sions have to be known. The permeability of the porous matrix may be calculated using thetraditional equations based on the relationships between the permeability and pore geome-try and concentration established for homogeneous well-sorted formations (Carman 1956;Timur 1968; Coates et al. 1991). However, the introduction of the secondary-pore permeabil-ity represented by inclusions of different form and size is a complicated question, because, atpresent, the permeability is only defined for bodies with, at least, one infinite size as cylinder,slab, and layer limited by two parallel planes (Gueguen and Palciauskas 1994). In this situa-tion, the simulation of the effective permeability of the double-porosity medium is performedby assigning same reasonable values of the permeability for the inclusions of finite size.

In this article, we propose an approach for determining the permeability of inclusionswith finite size. This approach consists in solving the one-particle problem for the fluid-filled inclusion placed into the permeable porous matrix, where the fluid flow is governedby the Darcy law. The fluid flow into the inclusion is described, firstly, by the Stocks equa-tion and then by the Darcy law. By using these two solutions, the equivalent permeabil-ity of the inclusion is found. We have applied this approach for calculating the equivalentDarcy permeability for the inclusions of circular (2D case) and spherical (3D case) shapes.The obtained equivalent permeabilities for the circular and spherical inclusions depend onthe inclusion radii and matrix permeability. Knowing the equivalent inclusion permeability,the effective permeability of the double-porosity composite can be simulated by using theEffective Medium schemes such as Differential Effective Medium, self-consistent or Max-well methods. We present the examples of calculating the effective permeability for rockswith cavities (spheres) applying Maxwell theory.

2 Permeability of a Fluid-Filled Inclusion

We consider a porous matrix as a medium that obeys Darcy’s law (Fig. 1):

q = − 1

μk ∇ p, (1)

where the vector q is a volume flow rate per unit of a cross-sectional area, k, μ, and p are thepermeability, dynamic viscosity, and pressure in the matrix, respectively. Darcy’s flow rateq is a volume flux and can be related to the average velocity w of the fluid through pores byusing Dupuit–Forcheimer relation (Gueguen and Palciauskas 1994)

q = φw,

where φ is the matrix porosity.

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Permeability of the Fluid-Filled Inclusions in Porous Media 309

? P

Vn V?

Darcy area (porous matrix)

Stokes area (inclusion)

∇P

Vn Vτ

Darcy area (porous matrix)

Stokes area (inclusion)

Fig. 1 Fluid-filled inclusion in a porous medium. Vn and Vτ are normal and tangential components of velocityvector

The fluid pressure satisfies the Laplace equation:

�p = 0. (2)

The pressure field is homogeneous at infinity:

p → ∇ p∞ · Z , |Z | → ∞ (3)

where Z is the Cartesian coordinate.The flow field inside an inclusion contained viscous incompressible fluid is described

using the Stokes equations

�ν = − 1

μ∇ p, (4a)

∇ · ν = 0, (4b)

where ν is the fluid flow velocity.Two boundary conditions between a porous medium and a fluid-filled inclusion are the

continuity of the normal pressure and the normal velocity components

σnn = −p, (5a)

φwn = vn (5b)

using Dupuit–Forcheimer relation

qn = vn, (5c)

where σnn is the stress tensor in the fluid, qn is the normal component of the volume flow rateper unit of a cross-sectional area, wn and νn are the normal components of the fluid velocityin the matrix and in the inclusion, respectively.

The third boundary condition is not obvious. An empirical equation for tangential veloc-ity at the interface between a Newtonian fluid and a porous medium was first suggested by

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Beavers and Joseph (1967). Ross (1983) has obtained the boundary condition for a generalanisotropic porous medium in the form:

〈vi 〉 = − Ki j

μ

∂

∂x j〈p〉 + Ki j L jmk

∂

∂xk〈vm〉 + Ki j N jm∇2〈vm〉, (6)

where the tensors Ki j , L jmk, and N jm are mathematically defined in terms of the map fromthe average velocity to the point velocity.

Saffman (1971) proposed a different boundary condition that in the case of a flat boundarybetween two half spaces (y > 0 for the Stokes creep flow and y < 0 for the Darcy flow) hasthe form:

vτ = λ√

k∂vτ

∂y, (7)

where vτ is the tangential component of the fluid velocity in the inclusion and λ is a semi-empirically non-dimensional coefficient that varies in the interval of 0–5.

In the recently published article, Jäger and Mikelic (2000) studied the laminar viscouschannel flow over a porous surface. The authors obtained rigorously the Saffman interfacecondition that is valid when the pore size is smaller than the width of viscous boundarylayer. Based on this result, we have applied the Saffman condition (7) to solve the one-par-ticle problem for the fluid-filled inclusion. It should be noted that in the case of the doubleporosity medium, the use of the Saffman condition is only possible when the inclusion cur-vature radius is much more than

√k. The detailed review of the boundary conditions for the

fluid–porous medium interface can be consulted in the book of Nield and Bejan (2006).In the next section, we present the solutions for the fluid-filled inclusions of spherical (3D

case) and circular (2D case) shapes.

2.1 Spherical Inclusion

In order to simplify the solution for the spherical inclusion, we have used the spherical coor-dinate system (r, θ, ϕ) with origin placed in the sphere center. The solution of Eq. 2 for thepressure outside of the sphere has the following form

p = cos(θ)

(XS1

r2 + |∇ p∞|r)

, (8a)

where XS1 is an unknown constant, r and θ are the radial coordinate (radius) and elevationangle, respectively. According to Levich (1962) and Happel and Brenner (1981), the radialand tangential components of the fluid velocity inside of the sphere governed by Eqs. 4a and4b can be presented as

vr = 2(XS2 + XS3r2) cos(θ), (8b)

vθ = −2(2XS2 + 4XS3r2) sin(θ), (8c)

where XS2 and XS3 are unknown constants. Since the pressure is symmetric with respectto its gradient direction, Eqs. 8b–8c do not depend on the azimuth angle ϕ. The boundaryconditions (5a, 5c, and 7) in the spherical coordinates (radial stress component in the inclu-sion is equal to the pressure in the matrix (1), continuity of the radial component of the fluidvelocity (2), and (3) the Saffman condition for the tangential velocity component) can berewritten as follows

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σrr = −p, (9a)

vr = − k

μ

∂p

∂r, (9b)

vθ = λ√

k∂vθ

∂r. (9c)

Using these boundary conditions, the unknown constant XS1, XS2, and XS3 can be found.Taking into account that the permeability determination requires only knowing the expres-sion for the pressure outside of the spherical inclusion (8a), it is enough to find the coefficientXS1 which is a function of the sphere radius a and the matrix permeability k

XS1 = −|∇ p∞|a3 + 18|∇ p∞|a3k

a2 + 12k − 4aλ√

k. (10)

2.2 Circular Inclusion

In the case of the circular inclusion, the solution of Eq. 2 for the pressure outside of thecircle and (4a and 4b) for the radial and tangential components of the fluid velocity insidethe circular inclusion was given by Raja Sekhar and Sano (2000, 2003),

p = cos(ϕ)

(XC1

ρ+ |∇ p∞|ρ

), (11a)

vρ = 2(XC2 + XC3ρ

2/8)

cos(ϕ), (11b)

vϕ = − (XC2 + 3XC3ρ

2/8)

sin(ϕ), (11c)

where ρ and ϕ are the radial and polar cylindrical coordinates with origin in the circle center,XC1, XC2, and XC3 are unknown constants. Using the corresponding boundary conditionsfor the fluid-filled circular inclusion

σρρ = −p (12a)

vρ = − k

μ

∂p

∂ρ(12b)

vϕ = λ√

k∂vϕ

∂ρ, (12c)

the constant XC1 required for determining the pressure can be obtained

XC1 = −|∇ p∞|b2 + 4|∇ p∞|b2kλ

λb2 + 4kλ − 3b√

k, (13)

where b is the circle radius.

2.3 Equivalent Permeability of Fluid-Filled Inclusion

In order to introduce formally the permeability of the fluid-filled inclusion, we have solvedthe one-particle problem for the same geometry of inclusions, but in this case inclusions weretreated as permeable components with the inside fluid-flow governed by Darcy’s law. Thematrix, as well in the solved above problems, was considered as the Darcy-type medium.

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The pressure outside of the spherical inclusion pout presented in the spherical coordinateshas the same structure as in Eq. 8a

pout = cos(θ)

(X S1

r2 + |∇ p∞|r)

, (14a)

where XS1 is an unknown constant. Inside the inclusions, the pressure is

pin = cos(θ)r XS2. (14b)

Here XS1 and XS2 are unknown constants. The boundary conditions for the Darcy-type inclu-sions in the permeable matrix also described by the Darcy law are continuity of the pressureand normal component of the fluid flow rate

pout(r = a) = pin(r = a) (15a)k

μ

∂pout(r = a)

∂r= ks

μ

∂pin(r = a)

∂r. (15b)

Therefore, from Eqs. 14a, 14b, 15a and 15b, the coefficient XS1 for the outside pressure canbe obtained

XS1 = −|∇ p∞|a3 k − ks

2k + ks. (16)

Assuming that the pressure p in the heterogeneous medium when the fluid flow in the inclu-sion is described by Stocks Eq. 8a is equal to the pressure pout in the same heterogeneousmedium when the flow rate in the inclusion obeys the Darcy law (14a), we can define theequivalent permeability of sphere ks . By comparing Eqs. 10 and 16 for the coefficients XS1

and XS1, the equation for the permeability ks is found in explicit form

ks = a2

6

(1 − 4

λ√

k

a

). (17)

In the same way, we can introduce the equivalent permeability for the circular inclusion in2D case. Using the expressions for the pressure outside and inside of the circle

pout = cos(ϕ)

(XC1

ρ+ |∇ p∞|ρ

), (18a)

pin = cos(ϕ)ρ XC2 (18b)

and corresponding boundary conditions

pout(ρ = b) = pin(ρ = b) (19a)k

μ

∂pout(ρ = b)

∂ρ= kc

μ

∂pin(ρ = b)

∂ρ(19b)

the coefficient XC1 for the outside pressure is

XC1 = −|∇ p∞|b2 k − kc

k + kc. (20)

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Fig. 2 Dependence of thenormalized inclusionpermeability ks

a2/6and kc

a2/2on

the jump coefficient λ and radiusof inclusions a. Matrixpermeability k = 5 · 10−12 m2. aSpherical inclusions, b circularinclusions

0 1 2 3 4 50.0

0.2

0.4

0.6

0.8

1.0

1.2

k ef/(a

2 /6)

λ

a=50μm a=100μm a=200μm a=300μm a=500μm

A

0 1 2 3 4 50.2

0.4

0.6

0.8

1.0

1.2

kef

/(a2 /2

)

λ

a=50μm a=100μm a=200μm a=300μm a=500μm

B

From Eqs. 13 and 20, the equivalent permeability of the circular inclusion can bepresented as

kc = b2

2

(1 − 3

λ√

k

b

). (21)

It should be noted that Eqs. 17 and 21 are valid for the relations a � √k and b � √

kthat correspond to the applicability of Saffman’s interface condition. The variation of thenon-dimensional permeability with λ is shown in Fig. 2 for different inclusion radiuses. Inthis example, the matrix permeability k = 5 ·10−12 m2. Our calculations have shown that theinfluence of the tangential velocity jump (even for this high value of the matrix permeability)is significant only for small inclusions with radiuses a < 300 µm. In carbonate formationsthe real matrix does not frequently exceed 10−14 to 10−15 m2 and the obtained formula forthe sphere equivalent permeability can be applied for the secondary pores such as vugs and

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cavities with radii large than 30 µm. Taking into account that the coefficient λ varies in thesmall range of 0−5, satisfying the requirement of Saffman’s conditions means, at the sametime, that the sphere permeability is weakly related to the matrix permeability. Therefore, forthe above-mentioned conditions, the permeabilities of spherical and circular inclusions canbe approximated with high accuracy degree by simple equations

ks = a2

6(22a)

kc = b2

2. (22b)

3 Effective Permeability of a Medium with Spherical Inclusions

Knowing the equivalent inclusion permeability, one can simulate the effective permeabilityof the double-porosity medium composed of the permeable porous matrix and fluid-filledinclusions of finite size, by using the Effective Medium schemes such as Differential Effec-tive Medium, self-consistent or Maxwell methods. All these methods are well known andwidely used in micromechanics of porous media. Here, we apply the Maxwell theory devel-oped especially for a microhomogeneous composite material with the inclusions of sphericalshapes (Schvidler 1985; Berryman and Berge 1996). Following the Maxwell formula, theeffective permeability kef of the double-porosity formations with spherical inclusions is

kef = k2k + ks + 2C (k − ks)

2k + ks − C (k − ks)(23)

where C is the inclusion concentration (the secondary porosity). Substituting the sphericalinclusion permeability from Eq. 17 in the formula (23), one can obtain the expression for theeffective permeability of the media containing spherical fluid-filled inclusions

ke f =k

(a2(1 + 2C) − 12k(C − 1) − 4a(1 + 2C)λ

√k)

a2(1 − C) + 6k(2 + C) + 4a(C − 1)λ√

k. (24)

When the spheres equivalent permeability is much more than the matrix permeability (inclu-sion radius a goes to infinity), the effective permeability ceases to depend on the jumpcoefficient λ

kef = k1 + 2C

1 − C, (25)

and coincides with the asymptotic expressions for the standard Maxwell’s theory (Schvidler1985).

In the case of 2D problem, the effective permeability of the medium with circular inclu-sions is

kef =k

(b2(1 + C) − 2(C − 1)k − 3b(1 + C)λ

√k)

b2(1 − C) + 2(C + 1)k + 3b(C − 1)λ√

k(26)

and when the cylinder radius tends to infinity it becomes

kef = k1 + C

1 − C. (27)

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Fig. 3 The influence of theconcentration C and radius ofspherical and circular inclusionsa on the effective permeability.λ = 1, k = 5 · 10−12 m2. aSpherical inclusions, b circularinclusions

1E-4 1E-3

1.1

1.2

1.3

1.4

1.5

1.6

k ef//k

a, m

C=0.05 C=0.1 C=0.15

A

1E-4 1E-31.05

1.10

1.15

1.20

1.25

1.30

1.35

1.40

k ef/k

a

C=0.05 C=0.1 C=0.15

B

The results of calculations for the double-porosity medium containing spherical (vugs andcavities) or circular inclusions are presented in the Figs. 3 and 4. For these examples, thecoefficient λ accepted was equal to 1. The calculations demonstrated that the effective perme-ability significantly depends on the inclusion concentration. The influence of the inclusionradius takes place for small inclusions with a ≤ 300 µm and especially in the case of highinclusion concentrations, while for the spheres with a > 300 µm the effective permeabilityis practically constant. In this case, we can calculate the permeability using Eqs. 25 and27. The radius value from which it is possible to use these asymptotic formulae dependson the matrix permeability and it can be considered as the critical radius. This behavior ofthe effective permeability is related to Saffman’s boundary condition. In Eqs. 24 and 26,the terms containing

√k correspond to a viscous boundary layer at the interface between a

porous medium and fluid (Saffman 1971). When the inclusion size is much more than thethickness of this boundary layer, one can neglect the influence of these terms and considerthe inclusion permeability equal to infinity.

Expressions for the equivalent (17, 21) and effective (24, 26) permeabilities can be appliedfor the practical permeability evaluation of double-porosity rocks using information about

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0.00005 0.00010 0.00015 0.00020 0.000251.46

1.48

1.50

1.52

1.54

a3 cr

a2 cra

1 cr

k ef/k

a, m

k1=10-12m2

k2=5*10-12m2

k3=10-11m2

Fig. 4 Relative effective permeability of double porosity medium as a function of the inclusion radius andmatrix permeability. λ = 1, C = 0.15, a1cr, a2cr, a3cr are the critical radiuses for matrix permeabilitiesk1 = 10−12 m2, k2 = 5 ∗ 10−12 m2, k3 = 10−11 m2, respectively

the pore size obtained from the nuclear magnetic resonance log (Prammer 1994) and theconcentration of the matrix and secondary vugular pores determined by the joint inversionof the conventional well logs (Kazatchenko et al. 2007).

4 Conclusions

Using fundamental hydrodynamic equations, we have calculated the equivalent permeabilityof circular and spherical fluid-filled inclusions embedded in Darcy’s medium. The resultsobtained were applied in explicit Maxwell’s scheme to simulate the effective permeability ofporous medium containing isolated spherical (3D) and circular (2D) inclusions. This schemeunderestimates the inclusion interaction for high porosities, but it is applicable for real dou-ble-porosity media such as carbonate rocks. Our calculations of fluid inclusion permeabilitycan be used in the other effective medium approximations schemes such as the DifferentialEffective Medium method and the self-consistent Effective Medium Approximation.

We have shown that in the case of small inclusions with the radius less than the critical,the permeability of effective media depends on both the size and concentration of inclusions.The critical radius, for which the influence of inclusion size is negligible, is determined bythe matrix permeability.

The approach proposed can be extended for determining the equivalent permeability ofinclusions with more complicated geometry.

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permeability of the fluid-filled inclusions in porous media

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