Fluid flow through anisotropic and deformable double porosity media with ultra-low matrix permeability: A continuum framework

TL;DR

The paper develops a thermodynamically consistent continuum framework for anisotropic, deformable double porosity media with ultra-low matrix permeability, coupling two pore networks through pressures $p_1$ and $p_2$ and an energy-based effective stress $\bar{\boldsymbol{\sigma}}$. It introduces a nonlinear, non-Darcy flow law for the matrix porosity, a Darcy flow for the fracture porosity with an upscaled ${\boldsymbol{k}}_2$, and a leakage mechanism governed by $\gamma$, all embedded in a VTI poroelastic host described by ${\mathbb{C}}^e$ and Biot-like coupling tensors. An upscaling procedure based on volume integration yields physically meaningful ${\boldsymbol{k}}_2$, and model applications demonstrate rich hydro-mechanical behavior in consolidation and 3D loading scenarios, including anisotropy effects, Mandel-Cryer-type transients, and double-shell dynamics. The framework provides a path toward hybrid models combining continuum and discrete fracture descriptions for shale gas and other heterogeneous reservoirs, with significant implications for predicting pressure and displacement fields in complex geologic media.

Abstract

Fractured porous media or double porosity media are common in nature. At the same time, accurate modeling remains a significant challenge due to bi-modal pore size distribution, anisotropy, multi-field coupling, and various flow patterns. This study aims to formulate a comprehensive coupled continuum framework that could adequately consider these critical characteristics. In our framework, fluid flow in the micro-fracture network is modeled with the generalized Darcy's law, in which the equivalent fracture permeability is upscaled from the detailed geological characterizations. The liquid in the much less permeable matrix follows a low-velocity non-Darcy flow characterized by threshold values and non-linearity. The fluid mass transfer is assumed to be a function of the shape factor, pressure difference, and (variable) interface permeability. The solid deformation relies on a thermodynamically consistent effective stress derived from the energy balance equation, and it is modeled following anisotropic poroelastic theory. The discussion revolves around generic double porosity media. Model applications reveal the capability of our framework to capture the crucial roles of coupling, poroelastic coefficients, anisotropy, and ultra-low matrix permeability in dictating the pressure and displacement fields.

Figures (17)

• Figure 1: An arbitrary control volume $P$ within the computational domain where $\rho$ is a density term, $\vec{\varrho}$ is the corresponding flux term of $\rho$, and $\vec{n}$ is the unit outward normal vector of the infinitesimal area $\mathrm{d} A$. The $\rho$ and $\vec{\varrho}$ are specified in Figure \ref{['Fig:02']}.
• Figure 2: Phase diagram of the double porosity media.
• Figure 3: The interactions and relationships among different components of mathematical theories.
• Figure 4: Schematic of a rectangular local solution domain $\Omega$ with explicit micro-fractures, the blue triangle represents the pressure boundary for the first solution, and the green triangle represents the pressure boundary for the second solution.
• Figure 5: An actual computational domain ($1$$\rm m$$\times$$1$$\rm m$) whose scale is much smaller than the typical reservoir Yan2018Yan2020, and that is why the fractures drawn here can be regarded as the micro-fractures. The twelve micro-fractures could be divided into three groups based on the value of the aperture. The red group has an aperture of $5 \times 10^{-5}$$\rm m$, the green group has an aperture of $1 \times 10^{-5}$$\rm m$, and the blue group has an aperture of $5 \times 10^{-6}$$\rm m$. The intrinsic fracture permeability is obtained using the cubic law. The matrix permeability $k_1$ is $10^{-18}$$\rm m^2$, the fluid viscosity $\mu_f$ is 0.001 $\rm Pa\cdot s$, $p_{\rm in} = 1$$\rm MPa$, and $p_{\rm out} = 0$$\rm MPa$.