Analysis of subdiffusion in disordered and fractured media using a Grünwald-Letnikov fractional calculus model

Abstract

The increasing applications of fractional calculus in simulating the anomalous transport behavior in disordered and fractured heterogeneous porous media has grown rapidly over the past decade. In the present study, a temporal fractional flux relationship is employed as a constitutive equation to relate the volumetric flow rate to the gradient of the pore pressure. The novelty of this paper entails interpreting the time fractional derivative operator in the flux relationship by the Grünwald-Letnikov (G-L) definition as opposed to the Caputo interpretation which has been widely considered. Subsequently, a numerical scheme based on the block-centered finite-difference discretization is formulated to handle the resulting non-linear fractional diffusion model. In addition, a linear stability analysis is successfully performed to establish the stability criterion of the developed numerical scheme. An expression for the modified incremental material balance index was derived to assess the effectiveness of the numerical discretization process. Finally, numerical experiments were performed to provide qualitative insights into the nature of pressure evolution in a hydrocarbon reservoir under the influence subdiffusion. In summary, the results establish that subdiffusion regime results in the development of higher pressure drop in the reservoir. This paper will provide a strong foundation for researchers interested in investigating anomalous diffusion phenomena in porous media.

Appendices

Appendix A: Definition of pseudo-diffusivity term

To solve the fractional flow model presented, we employ the Carman-Kozeny (1939) permeability-porosity relationship to describe the evolution of the pseudo-permeability. The pseudo-diffusivity is defined as shown below.

$$ \eta \,=\,\frac{K_{\gamma }}{\mu_{\text{ob}}\thinspace \exp\left[ c_{\mu }\left( p\,-\,p_{b} \right) \right]}\thinspace \text{with}\thinspace \thinspace K_{\gamma }\,=\,\frac{1}{72\tau }\frac{\phi^{3}{d_{p}^{2}}}{\left( 1\!-\phi \right)^{2}} $$

(1)

Please refer to nomenclature section for the definition of variables introduced above.

In addition, we assume the reservoir fluid to be typical of UAE crude oil, defined through the following empirical correlations presented by [74]:

$$ \mu =\mu_{ob}\thinspace \exp\left[ c_{\mu }\left( p-p_{b} \right)\right] $$

(2)

where μ_ob is the oil viscosity at bubble point pressure obtained from:

$$\begin{array}{@{}rcl@{}} \mu_{\text{ob}}&=&6.59927\times {10}^{5}R_{s}^{-0.597627}T^{-0.941624}\\&&\times \gamma_{g}^{-0.555208}{\text{API}}^{-1.487449} \end{array} $$

(3)

$$ p_{b}=-620.592 + 6.23087\thinspace \frac{R_{s}\gamma_{o}}{\gamma_{g}B_{o}^{1.38559}}+ 2.89868\thinspace T $$

(4)

$$ B_{\text{ob}}= 1.122018 + 1.410\times {10}^{-6}\frac{R_{s}T}{\gamma_{o}^{2}} $$

(5)

$$ c_{o}\,=\,\frac{\left( 1433 + 5R_{s}+ 17.2T-1180\gamma_{g}+ 12.61\thinspace \text{API} \right)}{\left( p\times {10}^{5} \right)} $$

(6)

Appendix B: Derivation of incremental balance check equation

In this section, the incremental material balance check at time level n + 1 including the effect of memory is derived by writing Eq. 19 for each grid block in the system (m = 1, 2, 3 ... M) and then summing up all m equations. The resulting equation is:

$$\begin{array}{@{}rcl@{}} &&{\sum}_{m = 1}^{M} \left\{ {\sum}_{l\epsilon \psi_{m}} T_{\mathrm{l,m}}^{n + 1} \left[ \left( p_{l}^{n + 1}+p_{m}^{n + 1} \right) \right] \right\} \\&&+{\sum}_{m = 1}^{M} \left( {\sum}_{l\epsilon \xi_{m}} q_{{\text{sc}}_{\mathrm{l,m,n}}}^{n + 1} +q_{\mathrm{sc,m}}^{n + 1}+q_{\mathrm{memory,m}}^{n + 1} \right) = \end{array} $$

$$ {\sum}_{m = 1}^{M} {\frac{V_{\text{bm}}}{{\Delta} t}\left[ \left( \frac{\phi }{B_{o}} \right)_{m}^{n + 1}-\left( \frac{\phi }{B_{o}} \right)_{m}^{n} \right]} $$

(7)

The sum of all inter-block flow terms in the reservoir, which are expressed by the first term on the LHS of Eq. (B.1), add up to zero, while the second term on the LHS represents the M algebraic sum of all production rates through wells ${\sum }_{m = 1}^{M} q_{\mathrm {sc,m}}^{n + 1}$, those across reservoir $\left ({\sum }_{m = 1}^{M} {\sum }_{l\epsilon \xi _{m}} q_{{\text {sc}}_{\mathrm {l,m,n}}}^{n + 1} \right )$, and those resulting from the effect of memory ${\sum }_{m = 1}^{M} q_{\mathrm {memory,m}}^{n + 1}$.The RHS of this equation represents the sum of the accumulation terms in all blocks in the reservoir. Therefore, Eq. B.1 becomes

$$\begin{array}{@{}rcl@{}} &&{\sum}_{m = 1}^{M} \left( {\sum}_{l\epsilon \xi_{m}} q_{{\text{sc}}_{\mathrm{l,m,n}}}^{n + 1} +q_{\mathrm{sc,m}}^{n + 1}+q_{\mathrm{memory,m}}^{n + 1} \right) \\&&={\sum}_{m = 1}^{M} {\frac{V_{bm}}{{\Delta} t}\left[ \left( \frac{\phi }{B_{o}} \right)_{m}^{n + 1}-\left( \frac{\phi }{B_{o}} \right)_{m}^{n} \right]} \end{array} $$

(8)

Dividing (B.2) by the term on the LHS yields

$$ 1=\frac{{\sum}_{m = 1}^{M} {\frac{V_{bm}}{{\Delta} t}\left[ \left( \frac{\phi }{B_{o}} \right)_{m}^{n + 1}-\left( \frac{\phi }{B_{o}} \right)_{m}^{n} \right]} }{{\sum}_{m = 1}^{M} \left( {\sum}_{\mathrm{l\epsilon }\mathrm{\xi }_{\mathrm{m}}} q_{{\text{sc}}_{\mathrm{l,m,n}}}^{n + 1} +q_{\mathrm{sc,m}}^{n + 1}+q_{\mathrm{memory,m}}^{n + 1} \right) } $$

(9)

where

$$\begin{array}{@{}rcl@{}} q_{\mathrm{memory,m}}^{n + 1}&=&{\sum}_{k = 1}^{\frac{t}{{\Delta} t}} \psi \left( \gamma ,k \right){\sum}_{\mathrm{l\epsilon }\mathrm{\psi }_{\mathrm{m}}} T_{l,m}^{n + 1}\\&&\times \left[ \left( P_{l}^{n + 1-k}+P_{m}^{n + 1-k} \right) \right] \end{array} $$

(10)

Appendix C: Analytical solution to simplified fractional diffusion model

Applying to both sides of Eq. 11, the operation of fractional integration leads to the following fractional diffusion equation:

$$ \frac{\partial }{\partial x}\left( \thinspace \thinspace \frac{\beta_{c}K_{\gamma }}{\mu_{o}}\frac{A_{x}}{B_{o}}\frac{\partial p}{\partial x} \right){\Delta} x=\left( \frac{V_{b}\phi c_{t}}{\alpha_{c}B_{\text{ob}}} \right)\frac{\partial^{\gamma }p}{{\partial t}^{\gamma }}\thinspace $$

(11)

Initial condition:

$$ p\left( x,0 \right)=p_{i} $$

(12)

Boundary conditions:

$$ Q_{x0}=-\left( \frac{\beta_{c}K_{\gamma }A}{\mu_{o}B_{o}} \right)\frac{\partial^{1-\gamma }}{{\partial t}^{1-\gamma }}\left( \frac{\partial p}{\partial x} \right)_{x = 0}=\text{constant} $$

(13)

$$ \left( p \right)_{x=L}=p_{i} $$

(14)

For convenience, the rock and fluid properties are considered as constant.

Taking the Laplace transform of Eq. C.1 with respect to time leads to:

$$ \sigma_{1}\hat{P}_{xx}=\sigma_{2}\left[ s^{\gamma }\hat{P}-s^{\gamma -1}p_{i} \right] $$

(15)

With the coefficients defined below are all constants.

$$ \sigma_{1}=\frac{\beta_{c}K_{\gamma }}{\mu_{o}}\frac{A_{x}{\Delta} x}{B_{o}}\thinspace ,\thinspace \sigma_{2}=\frac{V_{b}\phi c_{t}}{\alpha _{c}B_{o}}\thinspace \thinspace ,\mathrm{{\Theta} }=\thinspace \frac{\sigma_{2}}{\sigma_{1}}=\frac{\phi \mu_{o}c_{t}}{\beta_{c}\alpha_{c}K_{\gamma }}. $$

(16)

Therefore (C.5) can be re-written as:

$$ \hat{P}_{\text{xx}}-s^{\gamma }\mathrm{{\Theta} }\hat{P}=-\mathrm{{\Theta} }s^{\gamma -1}p_{i} $$

(17)

Equation C.7 is a non-homogeneous second order differential equation hence the solution would consist of the complimentary and the particular-solution.

Taking the auxiliary solution of $\hat {P}_{\text {xx}}-s^{\gamma }\mathrm {{\Theta } }\hat {P}= 0$, gives:

$$ m^{2}-s^{\gamma }{\Theta} = 0 $$

(18)

and then, $m=\pm \sqrt {s^{\gamma }{\Theta } }$. Hence, the complimentary solution is:

$$ \hat{P}_{c}=c_{1}e^{\left( \sqrt {s^{\gamma }{\Theta} } \right)x}+c_{2}e^{-\left( \sqrt {s^{\gamma }{\Theta} } \right)x} $$

(19)

where c₁ and c₂ are two constants to be determined from the boundary conditions. Noting that the right-hand side (RHS) of Eq. C.7 is independent of the space variable x, so $\hat {P}_{p}=\frac {p_{i}}{s}$ forms a particular-solution of Eq. C.7.

$$ \hat{P}(x)=c_{1}e^{\left( \sqrt {s^{\gamma }{\Theta} } \right)x}+c_{2}e^{-\left( \sqrt {s^{\gamma }{\Theta} } \right)x}+\frac{p_{i}}{s}\thinspace $$

(20)

Therefore, Eq. C.10 is a general solution of Eq. C.7. To determine c₁ and c₂, we take the Laplace transform of the boundary conditions represented by Eqs. C.3 and C.4. From Eq. C.3, we observe:

$$ {s^{1-\gamma }\hat{P}}_{x}\left( \mathrm{0} \right)=-\frac{Q_{\mathrm{x0}}\mu_{o}B_{o}}{{\beta_{c}K}_{\gamma }A\thinspace s}=:\frac{\sigma_{4}}{s} $$

(21)

Simplifying (C.11) leads to:

$$ \hat{P}_{x}\left( \mathrm{0} \right)=\frac{\sigma_{4}}{s^{2-\gamma }} $$

(22)

Likewise, at right boundary condition, Eq. C.4 we observe:

$$ \thinspace \hat{P}_{x=L}=\frac{p_{i}}{s} $$

(23)

Thus, applying the boundary condition (C.12) into Eq. C.10 leads to:

$$ \hat{P}_{x}\left( \mathrm{0} \right)=\frac{\sigma_{4}}{s^{2-\gamma }}\thinspace =c_{1}\left( \sqrt {s^{\gamma }{\Theta} } \right)-c_{2}\left( \sqrt{s^{\gamma }{\Theta} } \right) $$

(24)

Therefore, it follows that:

$$ c_{1}=\frac{\sigma_{4}}{s^{2-\gamma }\sqrt {s^{\gamma }{\Theta} } }+c_{2} $$

(25)

Likewise substituting (C.13) into Eq. C.10 leads to:

$$ \frac{p_{i}}{s}=c_{1}e^{\left( \sqrt {s^{\gamma }{\Theta} } \right)L}+c_{2}e^{-\left( \sqrt {s^{\gamma }{\Theta} } \right)L}+\frac{p_{i}}{s} $$

(26)

Substituting (C.15) into (C.16) returns:

$$ \left[ \frac{\sigma_{4}}{s^{2-\gamma }\sqrt {s^{\gamma }{\Theta} } }+c_{2} \right]\thinspace e^{\left( \sqrt {s^{\gamma }{\Theta} } \right)L}+c_{2}e^{-\left( \sqrt {s^{\gamma }{\Theta} } \right)L}= 0 $$

(27)

Simplifying (C.17) results in:

$$ c_{2}\left[ e^{\left( \sqrt {s^{\gamma }{\Theta} } \right)L}+e^{-\left( \sqrt {s^{\gamma }{\Theta} } \right)L} \right]+\frac{\sigma_{4}}{s^{2-\gamma }\sqrt {s^{\gamma }{\Theta} } }e^{\left( \sqrt {s^{\gamma }{\Theta} } \right)L}= 0\thinspace $$

(28)

Therefore,

$$ c_{2}=-\frac{\sigma_{4}}{s^{2-\gamma }\sqrt {s^{\gamma }{\Theta} } \left[ 1+e^{-\left( 2\sqrt {s^{\gamma }{\Theta} } \right)L} \right]}\thinspace $$

(29)

Consequently,

$$ c_{1}=\frac{\sigma_{4}}{s^{2-\gamma }\sqrt {s^{\gamma }{\Theta} } }\left[ \frac{e^{-2\left( \sqrt {s^{\gamma }{\Theta} } \right)L}}{1+e^{-2\left( \sqrt {s^{\gamma }{\Theta} } \right)L}} \right] $$

(30)

Finally, the Pressure profile in the reservoir in Laplace space is expressed by:

$$ \hat{P}=\frac{\sigma_{4}}{s^{2-\gamma }\sqrt {s^{\gamma }{\Theta} } }\left[ \frac{e^{\left( \sqrt {s^{\gamma }{\Theta} } \right)\left( x-2L \right)}-e^{-\left( \sqrt {s^{\gamma }{\Theta} } \right)x}}{1+e^{-2\left( \sqrt {s^{\gamma }{\Theta} } \right)L}} \right]+\frac{p_{i}}{s} $$

(31)