[Paper Review] Homogenization of a Biot-Stokes system modeling deformable vuggy porous media

Zhaoqin Huang, Xu Zhou+4 more|arXiv (Cornell University)|Apr 12, 2018

Advanced Mathematical Modeling in Engineering14 references3 citations

TL;DR

This paper develops a homogenized macroscopic poroelastic model for deformable vuggy porous media by coupling Biot's equations in the porous matrix with Stokes flow in vugs, using an extended Beavers-Joseph-Saffman condition at interfaces. The key contribution is a systematic derivation of effective coefficients—such as permeability, Young’s modulus, and Biot coefficient—via three cell problems, enabling accurate macro-scale simulation of fluid-structure interactions in complex rock systems.

ABSTRACT

Vugs are small to medium-sized cavities inside rock, which have significant effects on the fluid flow in rock. Moreover, the presence of vugs may have non-trivial impacts on the geomechanical behavior of rock. How to quantify and analyze such effects is still an opening problem. To this end, we derive a macroscopic poroelastic model for a single-phase viscous fluid flow through a deformable vuggy porous medium. At first, a vuggy porous medium is divided into two parts: the porous matrix and vugs. Then, we model the hydro-mechanical coupling process on the fine scale using Biot's equations within porous matrix, Stokes equations within the vugs, and an extended Beavers-Joseph-Saffman boundary condition on the porous-fluid interface. Next, based on the homogenization theory, we obtain a macroscopic Biot's equations governing the hydro-mechanical coupling behavior of vuggy porous media on larger scale. Subsequently, the macroscopic poroelastic coefficients, such as the effective Darcy permeability, effective Young's modulus and effective Biot coefficient, can be computed from three cell problems. Finally, several numerical examples are designed to demonstrate the computational procedure of evaluating the geomechanical behavior of vuggy porous media.

Motivation & Objective

• To model hydro-mechanical coupling in deformable vuggy porous media where vugs significantly influence fluid flow and mechanical response.
• To address the lack of a systematic macro-scale model that captures the coupled behavior of fluid flow and solid deformation in heterogeneous media with vugs.
• To derive effective poroelastic coefficients (permeability, Young’s modulus, Biot coefficient) from fine-scale physics using homogenization theory.
• To provide a computationally tractable framework for simulating geomechanical behavior in vuggy reservoirs.

Proposed method

• Decompose the vuggy porous medium into a porous matrix and vuggy regions.
• Model fluid flow in the matrix using Biot’s equations for poroelasticity and in vugs using Stokes equations for viscous flow.
• Apply an extended Beavers-Joseph-Saffman condition at the matrix-vug interface to account for slip effects.
• Use asymptotic homogenization theory to derive a macroscopic Biot-like system governing the effective behavior.
• Formulate three independent cell problems to compute the effective poroelastic coefficients.
• Validate the approach through numerical examples demonstrating the computational procedure and accuracy of the effective parameters.

Experimental results

Research questions

• RQ1 How can the hydro-mechanical coupling in vuggy porous media be modeled at a macroscopic scale when the microstructure includes both porous matrix and vuggy cavities?
• RQ2 What are the effective poroelastic coefficients (e.g., permeability, Young’s modulus, Biot coefficient) that emerge from the fine-scale heterogeneity of vuggy media?
• RQ3 How does the extended Beavers-Joseph-Saffman condition influence the effective behavior at the macroscopic level?
• RQ4 Can the homogenization framework accurately predict the geomechanical response of vuggy porous media without resolving all microscale features?
• RQ5 What is the computational procedure for evaluating the effective coefficients from the solution of cell problems?

Key findings

• The homogenized macroscopic model is a Biot-type system that captures the effective poroelastic behavior of vuggy porous media.
• Effective Darcy permeability, effective Young’s modulus, and effective Biot coefficient are derived from three distinct cell problems.
• The extended Beavers-Joseph-Saffman condition plays a critical role in accurately modeling the slip at the matrix-vug interface.
• Numerical examples confirm the feasibility and computational efficiency of the proposed homogenization framework.
• The method enables macro-scale simulation of fluid-structure interaction in vuggy rocks without resolving the full microscale geometry.
• The derived effective coefficients are consistent with physical expectations and provide a basis for predictive modeling in reservoir engineering.