Development of Coupled SPH-FDM Numerical Procedures for Simulating the Flow of Fluid Through Deformable Porous Media
Authors: Shwetank Kumar , Deb Debasis , Pramanik Ranjan 
Published: September 2023

ABSTRACT
A coupled numerical framework using smoothed particle hydrodynamics (SPH) and finite difference method (FDM) has been developed to evaluate the fracturing phenomenon and fluid flow in a deformable porous medium. The key aspect of the proposed coupled model is the simulation of flow of fluid in deformable porous medium using two distinct layers to characterize the solid and fluid phase. The respective governing equations of both phases are solved simultaneously with mutual transfer of variables from one to the other. SPH being a meshless technique has an advantage over grid-based methods in modeling large deformations and fracture propagation, and is used to simulate the solid phase, whereas FDM has been utilized for studying the variation of pore pressure in the fluid phase. Rate of change of porosity, rate of change of velocity, strain rate and stress rate are calculated in the solid phase by SPH. These rates are used to calculate porosity, velocity, strain, stress and volumetric strain of all the particles at any given time step. Porosity rate and volumetric strain calculated in SPH solver are then transferred to FDM solver to calculate rate of change of pressure to find out pressure at every FDM node. This research has provided a detailed mathematical framework and process describing the reciprocal flow of information between SPH and FDM. The coupling mechanism of the proposed coupled model has been validated with two standard benchmark problem in order to establish its accuracy and versatility by comparing the numerical results with the analytical solutions.

INTRODUCTION
The poroelasticity theory studies the mechanical behaviour of the solid phase, or solid skeleton, and the fluid phase in the pore space of a saturated porous medium. The coupled process of fluid phase interaction within the deformable porous solid phase is crucial in many industrial processes, including oil and gas extraction, hydrofracking, coal bed methane extraction, CO2 sequestration, ground subsidence due to excessive fluid extraction, and other engineering applications.
Based on the effective stress concept and an incompressible solid and fluid for a one-dimensional consolidation investigation, Terzaghi (1943) developed the mechanics of solid and fluid phase interaction in deformable porous media. By Biot (1941, 1955), Hsu and Jeng (1994) and Pastor et al. (2009), this work was expanded to three dimensions and took into account the impact of pore fluid diffusion on the kinematics and deformation of porous media. The mechanics and description of porous media are either based on mixture theory, which was developed on the concept of volume fraction provided by Morland (1972), Bowen (1980, 1982), and Muraleetharan and Wei (1999), or on averaging theory, which was developed by Whitaker (1977), Hassanizadeh and Gray (1979) and Scherer (1990). The mechanics involved are described in detail in the works of de Boer (1996), Lewis and Schrefler (1998), de Boir (2000) and Verruijt (2013).