Effective Permeability of a Porous Medium with Spherical and Spheroidal Vug and Fracture Inclusions

Abstract

Vugs and fractures are common features of carbonate formations. The presence of vugs and fractures in porous media can significantly affect pressure and flow behavior of a fluid. A vug is a cavity (usually a void space, occasionally filled with sediments), and its pore volume is much larger than the intergranular pore volume. Fractures occur in almost all geological formations to some extent. The fluid flow in vugs and fractures at the microscopic level does not obey Darcy’s law; rather, it is governed by Stokes flow (sometimes is also called Stokes’ law). In this paper, analytical solutions are derived for the fluid flow in porous media with spherical- and spheroidal-shaped vug and/or fracture inclusions. The coupling of Stokes flow and Darcy’s law is implemented through a no-jump condition on normal velocities, a jump condition on pressures, and generalized Beavers–Joseph–Saffman condition on the interface of the matrix and vug or fracture. The spheroidal geometry is used because of its flexibility to represent many different geometrical shapes. A spheroid reduces to a sphere when the focal length of the spheroid approaches zero. A prolate spheroid degenerates to a long rod to represent the connected vug geometry (a tunnel geometry) when the focal length of the spheroid approaches infinity. An oblate spheroid degenerates to a flat spheroidal disk to represent the fracture geometry. Once the pressure field in a single vug or fracture and in the matrix domains is obtained, the equivalent permeability of the vug with the matrix or the fracture with matrix can be determined. Using the effective medium theory, the effective permeability of the vug–matrix or fracture–matrix ensemble domain can be determined. The effect of the volume fraction and geometrical properties of vugs, such as the aspect ratio and spatial distribution, in the matrix is also investigated. It is shown that the higher volume fraction of the vugs or fractures enhances the effective permeability of the system. For a fixed-volume fraction, highly elongated vugs or fractures significantly increase the effective permeability compared with shorter vugs or fractures. A set of disconnected vugs or fractures yields lower effective permeability compared with a single vug or fracture of the same volume fraction.

Appendices

Appendix 1: Gegenbauer Polynomials

The nth degree Gegenbauer polynomials of first and second kind and of degree $-\frac{1}{2}$ are defined as

$$\begin{aligned} G_n(x)= & {} \dfrac{1}{2n-1}\left( P_{n-2}(x)-P_n(x)\right) , \end{aligned}$$

(79)

$$\begin{aligned} G_0(x)= & {} 1,\end{aligned}$$

(80)

$$\begin{aligned} G_1(x)= & {} -x,\end{aligned}$$

(81)

$$\begin{aligned} H_n(x)= & {} \dfrac{1}{2n-1}\left( Q_{n-2}(x)-Q_n(x)\right) ,\end{aligned}$$

(82)

$$\begin{aligned} H_0(x)= & {} -x,\end{aligned}$$

(83)

$$\begin{aligned} H_1(x)= & {} -1, \end{aligned}$$

(84)

with the following properties

$$\begin{aligned} x^2G_0(x)= & {} G_0(x)-2G_2(x), \end{aligned}$$

(85a)

$$\begin{aligned} x^2G_1(x)= & {} G_1(x)+2G_3(x), \end{aligned}$$

(85b)

$$\begin{aligned} x^2G_2(x)= & {} \frac{1}{5}G_2(x)+\frac{4}{5}G_4(x), \end{aligned}$$

(85c)

$$\begin{aligned} x^2G_3(x)= & {} \frac{3}{7}G_3(x)+\frac{4}{7}G_5(x), \end{aligned}$$

(85d)

$$\begin{aligned} x^2G_n(x)= & {} \alpha _n G_{n-2}(x)+\gamma _nG_n(x)+ \beta _n G_{n+2}(x), \end{aligned}$$

(85e)

and

$$\begin{aligned} x^2H_0(x)= & {} x^2G_1(x), \end{aligned}$$

(86a)

$$\begin{aligned} x^2H_1(x)= & {} -x^2G_0(x), \end{aligned}$$

(86b)

$$\begin{aligned} x^2H_2(x)= & {} -\frac{1}{3}G_1(x)+\frac{1}{5}H_2(x)+\frac{4}{5}H_4(x), \end{aligned}$$

(86c)

$$\begin{aligned} x^2H_3(x)= & {} \frac{1}{15}G_0(x)+\frac{3}{7}H_3(x)+\frac{4}{7}H_5(x), \end{aligned}$$

(86d)

$$\begin{aligned} x^2H_n(x)= & {} \alpha _n H_{n-2}(x)+\gamma _nH_n(x)+ \beta _nH_{n+2}(x), \end{aligned}$$

(86e)

with

$$\begin{aligned} \alpha _n= & {} \dfrac{(n-3)(n-2)}{(2n-3)(2n-1)},\quad n\ge 4 \end{aligned}$$

(87a)

$$\begin{aligned} \beta _n= & {} \dfrac{(n+1)(n+2)}{(2n-1)(2n+1)},\quad n\ge 4 \end{aligned}$$

(87b)

$$\begin{aligned} \gamma _n= & {} \dfrac{2n^2-2n-3}{(2n+1)(2n-3)},\quad n\ge 4. \end{aligned}$$

(87c)

Appendix 2: Series Expansion in Terms of the Legendre Polynomials

The orthogonality of the Legendre polynomials permits any function f(x) to be expressed in terms of a series in the basis of the Legendre polynomials $P_n(x)$ as

$$\begin{aligned} f(x)=\sum _{n=0}^\infty a_n P_n(x), \end{aligned}$$

(88)

with

$$\begin{aligned} a_n=\frac{2n+1}{2}\int _{-1}^{1}f(x)P_n(x)\mathrm{d}x. \end{aligned}$$

(89)

If f(x) is an even function, then the expansion includes only even terms; thus,

$$\begin{aligned} f(x)=\sum _{k=0}^\infty a_{2k} P_{2k}(x), \end{aligned}$$

(90)

with

$$\begin{aligned} a_k=(4k+1)\int _{0}^{1}f(x)P_{2k}(x)\mathrm{d}x. \end{aligned}$$

(91)

Fig. 14

Spheroidal coordinate and spherical coordinate systems

Appendix 3: Reduction to the Spherical Coordinate System

As it is shown in Fig. 14, when the semi-focal length c of the spheroidal coordinate system approaches zero, the coordinate system $(\xi , \eta , \phi )$ reduces to a spherical coordinate system $(\rho , \theta , \phi )$ where

$$\begin{aligned} \rho =\sqrt{x^2+y^2+z^2} =\sqrt{s^2+t^2-1}, \end{aligned}$$

(92)

when $c\rightarrow 0+$, noting that $s\ge 1$ and $-1\le t \le 1$; thus,

$$\begin{aligned} \lim _{c\rightarrow 0+}c~s= & {} \rho , \end{aligned}$$

(93)

$$\begin{aligned} \lim _{c\rightarrow 0+}t= & {} \cos \theta . \end{aligned}$$

(94)

As a result, the term in the spheroidal coordinate system representing the radius is proportional to the radius in the spherical coordinate system.

It is possible to show that the results obtained in the spheroidal coordinate system can be converted to the actual results in the spheroidal coordinate if one replaces s by s / c, and then takes the limit of $c^nG_n(s/c)$ and $c^{1-n}H_n(s/c)$ as $c\rightarrow 0+$ and then replaces s by r. We can show that

$$\begin{aligned} \lim _{c\rightarrow 0+}{~\dfrac{1}{2c}\ln {\dfrac{x+1}{x-1}}}=\dfrac{1}{\rho }, \end{aligned}$$

(95)

where we have used the expansion

$$\begin{aligned} \dfrac{1}{2}\ln {\dfrac{x+1}{x-1}}=\coth ^{-1}x=\dfrac{1}{x}+\dfrac{1}{3x^3}+\dfrac{1}{5x^5}+\cdots . \end{aligned}$$

(96)

The Gegenbauer polynomials have the equivalents when $c\rightarrow 0+$ given by

$$\begin{aligned}&c^0~G_0(s)\rightarrow 1, \quad c~G_1(s)\rightarrow -\rho ,\nonumber \\&\quad c^2 G_2(s)\rightarrow -\frac{1}{2}\rho ^2,\quad c^3 G_3(s)\rightarrow -\frac{1}{2}\rho ^3, \nonumber \\&\quad c^4 G_4(s)\rightarrow -\frac{5}{8}\rho ^4,\quad c^n G_n(s)\rightarrow (\text {const})\rho ^n,\quad n\ge 5, \end{aligned}$$

(97)

$$\begin{aligned}&\frac{1}{c} H_2(s) \rightarrow \frac{1}{3\rho },\quad \frac{1}{c^2} H_3(s) \rightarrow \frac{1}{15\rho ^2}, \nonumber \\&\quad \frac{1}{c^3} H_4(s) \rightarrow \frac{2}{105\rho ^3},\quad \frac{1}{c^{n-1}} H_n(s)\rightarrow \frac{\text {const}}{\rho ^{n-1}},\quad n\ge 5. \end{aligned}$$

(98)

Prior to that, we have the following

$$\begin{aligned}&c^n P_n(s)\rightarrow (\text {const})\rho ^n,\quad n\ge 0, \nonumber \\&\quad c^{-(n+1)} Q_n(s)\rightarrow \dfrac{(\text {const})}{\rho ^{n+1}},\quad n\ge 0. \end{aligned}$$

(99)

Appendix 4: Fluid Flow in the Matrix Medium Including a Spherical Vug

1.1 Flow field in the matrix and in the spherical vug

For a spherical vug with a radius $\rho =\rho _o$ located in the center of the porous domain of permeability $k_\mathrm{m}$, the spherical coordinate system $(\rho , \theta , \phi )$ with unit vectors $(\mathbf {e}_{\rho },\mathbf {e}_{\theta },\mathbf {e}_{\phi })$ is used. The scale factors for the spherical coordinate system are given as

$$\begin{aligned} h_1=1, \quad h_2=\rho ,\quad h_3=\rho \sin \theta . \end{aligned}$$

(100)

The solution to Stokes stream function that satisfies $E^4\varPsi _\mathrm{s}=0$ in the spherical coordinate system together with the conditions in Eqs. 13 and 14 is given as

$$\begin{aligned} (\varPsi _\mathrm{v})_\mathrm{s}(\rho ,\theta )= & {} (B\rho ^2+C\rho ^4)\sin ^2\theta , \quad \text {in } (\varOmega _\mathrm{v})_\mathrm{s}, \end{aligned}$$

(101)

$$\begin{aligned} (u_\mathrm{v})_\mathrm{s}(\rho ,\theta )= & {} 2(B+C\rho ^2)\cos \theta , \quad \text {in } (\varOmega _\mathrm{v})_\mathrm{s},\end{aligned}$$

(102)

$$\begin{aligned} (v_\mathrm{v})_\mathrm{s}(\rho ,\theta )= & {} -2(B+2C\rho ^2)\sin \theta , \quad \text {in } (\varOmega _\mathrm{v})_\mathrm{s}, \end{aligned}$$

(103)

where we have used Eq. 9. The subscript “s” is used to point out that it is for a spherical vug. The pressure field can be determined by using Eq. 12 as

$$\begin{aligned} (p_\mathrm{v})_\mathrm{s}(\rho ,\theta )=20C~\mu ~\rho \cos \theta , \quad \text {in } (\varOmega _\mathrm{v})_\mathrm{s}. \end{aligned}$$

(104)

Outside the spherical vug, the pressure field satisfies the Laplacian equation previously given in Eq. 3. This equation together with the condition of regularity of the pressure gradient at infinity (Eq. 15) gives the following determination for the matrix pressure as

$$\begin{aligned} (p_\mathrm{m})_\mathrm{s}(\rho ,\theta )=(\dfrac{D}{\rho ^2}+|\nabla (p_\mathrm{m})_{\infty }|\rho )\cos \theta , \quad \text {in } (\varOmega _\mathrm{m})_\mathrm{s}. \end{aligned}$$

(105)

Consequently, the velocity in the matrix domain is given as

$$\begin{aligned} (u_\mathrm{m})_\mathrm{s}(\rho ,\theta )= & {} \dfrac{k_\mathrm{m} }{\mu }\left( 2\dfrac{D}{\rho ^3}-|\nabla (p_\mathrm{m})_{\infty }|\right) \cos \theta , \quad \text {in } (\varOmega _\mathrm{m})_\mathrm{s},\nonumber \\ (v_\mathrm{m})_\mathrm{s}(\rho ,\theta )= & {} \dfrac{k_\mathrm{m} }{\mu }\left( \dfrac{D}{\rho ^3}+|\nabla (p_\mathrm{m})_{\infty }|\right) \sin \theta , \quad \text {in } (\varOmega _\mathrm{m})_\mathrm{s}. \end{aligned}$$

(106)

Using Eq. 6, the stream function exterior to the vug can be written as

$$\begin{aligned} (\varPsi _\mathrm{m})_\mathrm{s} (\rho ,\theta )=\left( \dfrac{k_\mathrm{m} }{\mu }\dfrac{D}{\rho }+\dfrac{1}{2}U_\infty \rho ^2\right) \sin ^2\theta , \quad \text {in } (\varOmega _\mathrm{m})_\mathrm{s}, \end{aligned}$$

(107)

where $U_\infty =-\frac{k_\mathrm{m} }{\mu }|\nabla (p_\mathrm{m})_{\infty }|.$ The three sets of boundary conditions given in Eqs. 16, 17, and 20 are applied on the interface of the matrix and Stokes domain as

$$\begin{aligned} (u_\mathrm{v})_\mathrm{s}= & {} (u_\mathrm{m})_\mathrm{s},\quad \text {on } \rho =\rho _o,\nonumber \\ -(p_\mathrm{v})_\mathrm{s}+2\mu ~\dfrac{\partial (u_\mathrm{v})_\mathrm{s}}{\partial \rho }= & {} -(p_\mathrm{m})_\mathrm{s}, \quad \text {on } \rho =\rho _o,\nonumber \\ \dfrac{\lambda }{\sqrt{k_\mathrm{m} }}~(v_\mathrm{v})_\mathrm{s}= & {} \rho \dfrac{\partial }{\partial \rho }\left( \dfrac{(v_\mathrm{v})_\mathrm{s}}{\rho }\right) +\dfrac{1}{\rho }\dfrac{\partial (u_\mathrm{v})_\mathrm{s}}{\partial \theta },\quad \text {on } \rho =\rho _o. \end{aligned}$$

(108)

Applying the boundary conditions of Eq. 108 to the velocity and pressure fields already obtained for the interior and exterior of the vug, which results in a system of three equations and three unknowns B, C, D, solvable as

$$\begin{aligned} C= & {} \dfrac{\dfrac{3}{2}\dfrac{k_\mathrm{m}}{\mu }\lambda |\nabla (p_\mathrm{m})_{\infty }|}{\lambda \rho _o^2 +12\lambda k_\mathrm{m}-3\sqrt{k_\mathrm{m}}\rho _o}, \end{aligned}$$

(109)

$$\begin{aligned} B= & {} C(3\sqrt{k_\mathrm{m}}-2\lambda \rho _o)\rho _o,\end{aligned}$$

(110)

$$\begin{aligned} D= & {} -|\nabla (p_\mathrm{m})_{\infty }|\rho _o^3+\dfrac{18k_\mathrm{m}\lambda |\nabla (p_\mathrm{m})_{\infty }|\rho _o^3}{\lambda \rho _o^2 +12\lambda k_\mathrm{m}-3\sqrt{k_\mathrm{m}}\rho _o}. \end{aligned}$$

(111)

Instead of the generalized form of the Beavers–Joseph–Saffman boundary condition, Markov et al. (2010) have used the simplified form for the boundary condition expressed as

$$\begin{aligned} \dfrac{\lambda }{\sqrt{k_\mathrm{m}}}~v_\mathrm{s}=\dfrac{\partial v_\mathrm{s}}{\partial \rho },\quad \text {on } \rho =\rho _o. \end{aligned}$$

(112)

If one uses the simplified form of the Beavers–Joseph–Saffman boundary condition as in Eq. 112, then the matrix pressure field is different and is given as

$$\begin{aligned} D=-|\nabla (p_\mathrm{m})_{\infty }|\rho _o^3+\dfrac{18k_\mathrm{m}\lambda |\nabla (p_\mathrm{m})_{\infty }|\rho _o^3}{\lambda \rho _o^2 +12\lambda k_\mathrm{m}-4\sqrt{k_\mathrm{m} }\rho _o}. \end{aligned}$$

(113)

1.2 Equivalent Permeability of a Single Spherical Vug

A porous spherical inclusion $(\varOmega _\mathrm{m})_{\mathrm{in},\mathrm{s}}$ of radius $\rho =\rho _o$ is located in the center of porous medium. The effective permeability of this porous inclusion $(k_\mathrm{m} )_{\mathrm{in},\mathrm{s}}$ is chosen so that the pressure and flow field out of the inclusion stay the same as in the case of a fluid-filled vug. The pressure field inside the porous inclusion should satisfy the following equation

$$\begin{aligned} \nabla ^2(p_\mathrm{m})_{\mathrm{in},\mathrm{s}}=0,\quad \text { in }(\varOmega _\mathrm{m})_{\mathrm{in},\mathrm{s}}, \end{aligned}$$

(114)

where $(p_\mathrm{m})_{\mathrm{in},\mathrm{s}}$ represents the pressure in the porous inclusion. When considering the bounded pressure at the center of inclusion, the following applies

$$\begin{aligned} (p_\mathrm{m})_{\mathrm{in},\mathrm{s}}=D_\mathrm{in}\rho \cos \theta ,\quad \text { in }(\varOmega _\mathrm{m})_{\mathrm{in},\mathrm{s}}. \end{aligned}$$

(115)

This means that the first approximation of stream function inside the porous inclusion is a straight line which is a result of the use of Darcy’s law to describe the flow inside the inclusion. The boundary condition for coupling two porous domains inside and outside the inclusion is the no-jump boundary condition on the pressure and the normal fluxes on the interface as

$$\begin{aligned} (p_\mathrm{m})_\mathrm{s}= & {} (p_\mathrm{m})_{\mathrm{in},\mathrm{s}},\quad \text {on } \rho =\rho _o, \end{aligned}$$

(116)

$$\begin{aligned} \dfrac{k_\mathrm{m} }{\mu }\dfrac{\partial (p_\mathrm{m})_\mathrm{s}}{\partial \rho }= & {} \dfrac{(k_\mathrm{m})_{\mathrm{in},\mathrm{s}}}{\mu }\dfrac{\partial (p_\mathrm{m})_{\mathrm{in},\mathrm{s}}}{\partial \rho },\quad \text {on } \rho =\rho _o, \end{aligned}$$

(117)

where $(p_\mathrm{m})_\mathrm{s}$ is given in Eq. 105 and $(k_\mathrm{m})_{\mathrm{in},\mathrm{s}}$ is the effective permeability of equivalent spherical porous inclusion to be found. Solving the system of Eqs. 116 and 117 for $\mathbb {D}$ and $(k_\mathrm{m} )_{\mathrm{in},\mathrm{s}}$, we obtain

$$\begin{aligned} (k_\mathrm{m} )_{\mathrm{in},\mathrm{s}}=k_\mathrm{m} \bigg (1-\dfrac{3 D}{(D+|\nabla (p_\mathrm{m})_{\infty }|\rho _o^3)}\bigg ), \end{aligned}$$

(118)

or

$$\begin{aligned} (k_\mathrm{m} )_{\mathrm{in},\mathrm{s}}=\dfrac{\rho _o^2}{6}\left( 1- \dfrac{3\sqrt{k_\mathrm{m} }}{\lambda \rho _o}\right) . \end{aligned}$$

(119)

Note that the simplified form of the Beavers–Joseph–Saffman boundary condition in Eq. 112 yields the effective permeability of the porous inclusion as

$$\begin{aligned} (k_\mathrm{m} )_{\mathrm{in},\mathrm{s}}=\dfrac{\rho _o^2}{6}\left( 1- \dfrac{4\sqrt{k_\mathrm{m} }}{\lambda \rho _o}\right) . \end{aligned}$$

(120)

Appendix 5: Reduction in a Spheroidal Vug to Spherical Vug

In this section, we investigate the behavior of the flow for the coupled Stokes flow and Darcy’s law in a vuggy porous medium with a spheroidal vug in the limit when the semi-focal length of the spheroid approaches zero. This behavior is expected to comply with the results of the solution for the vuggy porous medium with a spherical vug presented in Appendix D.1.

Stream Function Inside the Vug Equation 47 gives the first approximation for the stream function inside the spheroidal vug. In the limit $c\rightarrow 0+$, and using Eqs. 93, 94, and 97, the stream function reduces to the same form as the stream function inside the spherical vug given in Eq. 101 as

$$\begin{aligned} \varPsi _\mathrm{v}(s,t)=\bigg ( \underbrace{c^2 a_{2}G_2(s)}_{\equiv -\dfrac{a_2}{2} \rho ^2}+\underbrace{c^4~\bigg (\dfrac{\beta _2 C_{2} }{10} \bigg )G_{4}(s)}_{\equiv \dfrac{-C_2}{20}\rho ^4} \bigg )\underbrace{G_{2}(t)}_{\equiv \dfrac{1}{2}\sin ^2\theta }. \end{aligned}$$

(121)

which is

$$\begin{aligned} \varPsi _\mathrm{v}(\rho ,\theta )=\left( \mathbb {B}\rho ^2+\mathbb {C}\rho ^4 \right) \sin ^2\theta , \end{aligned}$$

(122)

with

$$\begin{aligned} \mathbb {B}=-\dfrac{a_2}{4} ,\quad \mathbb {C}=\dfrac{-C_2}{40}. \end{aligned}$$

(123)

Pressure Field Inside the Vug The first approximation for the pressure field inside the spheroidal vug is given in Eq. 49 as

$$\begin{aligned} p_\mathrm{v}(s,t)=-\dfrac{1}{2}\mu ~c~C_{2}st, \end{aligned}$$

(124)

and when $c\rightarrow 0+$, using Eqs. 93 and 94, Eq. 124 reduces to

$$\begin{aligned} p_\mathrm{v}(\rho ,\theta )=20\mu ~\mathbb {C}\rho \cos \theta , \end{aligned}$$

(125)

which is in accordance with the results obtained for the pressure inside the spherical vug given in Eq. 104.

Pressure Field Outside the Vug The first approximation for the pressure field outside the spheroidal vug in the matrix medium is given in Eq. 52 as

$$\begin{aligned} p_\mathrm{m}(s,t)=c~A_{2}\left( \frac{1}{2}s\log \left( \dfrac{s+1}{s-1}\right) -1\right) t+c|\nabla (p_\mathrm{m})_{\infty }|st, \end{aligned}$$

(126)

when $c\rightarrow 0+$, and using Eqs. 93, 94, and 96, Eq. 126 reduces to the same form as the pressure outside the spherical vug given in Eq. 105, i.e.,

$$\begin{aligned} p_\mathrm{m}(\rho ,\theta )=\left( \frac{\mathbb {D}}{\rho ^2} +|\nabla (p_\mathrm{m})_{\infty }|\rho \right) \cos \theta , \end{aligned}$$

(127)

where

$$\begin{aligned} \mathbb {D}\equiv \frac{c^3 A_{2}}{3}. \end{aligned}$$

(128)

Stream Function Outside the Vug The first approximation of the stream function outside the spheroidal vug is given in Eq. 56, and when $c\rightarrow 0+$, this equation reduces to

$$\begin{aligned} \varPsi _\mathrm{m}(s,t)=-c^2\dfrac{k_\mathrm{m} }{\mu }\dfrac{A_{2}}{2}\sin ^2\theta \left( -\frac{2c}{3\rho } -\frac{c^3}{3\rho ^3} \right) - \dfrac{1}{2} \dfrac{k_\mathrm{m} }{\mu } c^2|\nabla (p_\mathrm{m})_{\infty }|\dfrac{\rho ^2-c^2}{c^2}\sin ^2\theta . \end{aligned}$$

(129)

This equation is equivalent to the stream function external to the spherical vug given in Eq. 107, i.e.,

$$\begin{aligned} \varPsi _\mathrm{m}(s,t)= \dfrac{k_\mathrm{m} }{\mu } \left( \frac{\mathbb {D}}{\rho } -\dfrac{1}{2}|\nabla (p_\mathrm{m})_{\infty }|\rho ^2\right) \sin ^2\theta . \end{aligned}$$

(130)