Modeling single-phase fluid flow in porous media through non-local fractal continuum equation

Abstract

Modeling fluid flow in highly heterogeneous porous media is an open research topic due to the degree of complexities and uncertainties attributable mainly to the spatial variations of media properties. Mathematical models capable of effectively capturing such complexities are valuable tools to obtain insights into the underlying phenomenology and characterize the system behavior under different boundary conditions or model parameters. In this context, we present a non-local model complemented by the fractal continuum theory to describe single-phase flow in highly heterogeneous porous media. We use the generalized Laplacian operator and the non-local (subdiffusive) Darcy’s law to formulate a fluid flow model suitable for fractal porous media with non-local effects. We analyze the dynamics of radially symmetric flow geometry according to radially convergent flow appearing in reservoirs and aquifers. Pressure-transient drop and rate-transient flow scenarios were analyzed for different fractal dimensions and anomalous parameters. The results show that the model parameters associated with the anomalous flow have clear and distinctive effects in field tests, providing a theoretical tool to explore different scenarios under anomalous flow that could be useful for improving our understanding of fluid flow in porous media with complex geometries.

Appendix A: Analytical solution in the Laplace domain

The Laplace transforms of (28) is

$$\begin{aligned}{} & {} \frac{1}{r_{D}^{d-1} }\frac{\mathrm{{d}}}{\mathrm{{d}}r_{D} }\left( {\,r_{D}^{\alpha } \frac{\mathrm{{d}}\hat{{p}}_{D} }{\mathrm{{d}}r_{D} }} \right) -\gamma ^{-1}s^{\beta }\,\hat{{p}}_{D} =-\gamma ^{-1}s^{\beta -1}p_{D} \left( 0 \right) \nonumber \\{} & {} \quad +\frac{s^{\beta -1}}{r_{D}^{d-1} }\left[ {{ }_{0}^{RL} I_{t}^{\beta } \frac{\partial }{\partial r_{D} }\left( {\,r_{D}^{\alpha } \frac{\partial p_{D} }{\partial r_{D} }} \right) } \right] _{t=0}. \end{aligned}$$

(37)

The right-hand side in both equations contain the initial conditions which, according to the definition of the dimensionless variables, vanish. Therefore, the model can be written in terms of the modified Bessel equation in the following form:

$$\begin{aligned} r_{D} \frac{\mathrm{{d}}^{2}\hat{{p}}_{D} }{\mathrm{{d}}r_{D}^{2} }+\alpha \frac{\mathrm{{d}}\hat{{p}}_{D} }{\mathrm{{d}}r_{D} }-\Omega ^{2}r^{d-\alpha }\hat{{p}}_{D} =0. \end{aligned}$$

(38)

The general analytical solution to the modified Bessel equation (38) in the Laplace domain is given by [45, 46]

$$\begin{aligned} \hat{{p}}_{D} \left( {r,s} \right) =r^{\frac{1-\alpha }{2}}\left[ {C_{1} \,I_{\nu } \left( {\Psi \,\,r^{1/\theta }} \right) +C_{2} \,K_{\nu } \left( {\Psi \,\,r^{1/\theta }} \right) } \right] . \end{aligned}$$

(39)

where $\theta =2/\left( {\alpha +1} \right)$, $\Psi =\theta \left| \Omega \right|$, $\nu =\theta \left( {1-\alpha } \right) /2$, and $I_{\nu } \left( \cdot \right)$ and $K_{\nu } \left( \cdot \right) K_{\nu } \left( \cdot \right)$ are, respectively, the modified Bessel functions of first and second kind of order $\nu$.