Modeling wavefields in saturated elastic porous media based on thermodynamically compatible system theory for two-phase solid-fluid mixtures

Highlights

• SHTC equations for poroelasticity are compared with Biot’s model of poroelasticity.

• It is demonstrated that fast and slow pressure waves are present in the SHTC model.

• The model is solved with a second-order staggered finite difference method.

• Physical consistency of the model is demonstrated on a series of 2D test cases.

Abstract

A two-phase model and its application to wavefields numerical simulation are discussed in the context of modeling of compressible fluid flows in elastic porous media. The derivation of the model is based on a theory of thermodynamically compatible systems and on a model of nonlinear elastoplasticity combined with a two-phase compressible fluid flow model. The governing equations of the model include phase mass conservation laws, a total momentum conservation law, an equation for the relative velocities of the phases, an equation for mixture distortion, and a balance equation for porosity. They form a hyperbolic system of conservation equations that satisfy the fundamental laws of thermodynamics. Two types of phase interaction are introduced in the model: phase pressure relaxation to a common value and interfacial friction. Inelastic deformations also can be accounted for by source terms in the equation for distortion. The thus formulated model can be used for studying general compressible fluid flows in a deformable elastoplastic porous medium, and for modeling wave propagation in a saturated porous medium. Governing equations for small-amplitude wave propagation in a uniform porous medium saturated with a single fluid are derived. They form a first-order hyperbolic PDE system written in terms of stress and velocities and, like in Biot’s model, predict three type of waves existing in real fluid-saturated porous media: fast and slow longitudinal waves and shear waves. For the numerical solution of these equations, an efficient numerical method based on a staggered-grid finite difference scheme is used. The results of solving some numerical test problems are presented and discussed.

Introduction

The modeling of fluid flows in porous media is of permanent interest in many geophysical and industrial applications. The starting point of research developments in this field was a series of pioneering works by Biot [1], [2], [3], in which a model of elastic wave propagation in saturated porous media was proposed. Some modifications and generalizations of the model have been made (see, for example [6], [21], [46] and references therein), and at present Biot’s approach is a commonly accepted and widely used one in geophysical community. Nevertheless, many actual technological and scientific problems, such as geothermal energy extraction, CO₂ storage, hydraulic fracturing, fuel cells, food production, etc. require new advanced models and methods.

To simulate the development of nonlinear, temperature dependent processes in porous media, methods of continuum mechanics and, in particular, multiphase theories can be successfully used. A two-phase approach to poroelasticity has been, perhaps, most consistently implemented by Wilmanski in [44], [45] (see also references therein). In particular, in a review paper, [44], the structure of Biot’s poroelastic model was analysed and its consistency with the fundamental principles of continuum mechanics was discussed.

In recent years, considerable attention has been paid to the modeling finite-strain saturated porous media and their applications in various fields, in particular in medicine, see, for example, [19], [29], [34]. Worthy of mention is a large-deformation model of a saturated porous medium proposed by Dorovsky [4], in which the key point of the model is thermodynamic consistency and hyperbolicity of the governing equations. Nevertheless, there is still no thermodynamically consistent formulation of a multiphase mixture flow model in a deforming porous medium with finite deformations.

In this paper, we apply a powerful method of designing new models of complex continuum media, which is based on a thermodynamically compatible system theory [12], [13], [14]. In [35], the mentioned theory was first time applied to designing a model for deformed porous medium saturated by a compressible fluid. In this paper the governing equations were formulated and small amplitude wave propagation has been studied showing a qualitative agreement with Biot’s theory. The present study contains further developments of ideas from [35] based on the unified model of continuum [9], [10] as well as a quantitative comparison with Biot’s theory and a series of numerical experiments proving the physical correctness of the model. The thermodynamically compatible system theory allows the development of well-posed models satisfying the fundamental laws of irreversible thermodynamics. In [14], [15], [32], [38], [40], a class of Symmetric Hyperbolic Thermodynamically Compatible (SHTC) systems was formulated to describe many known classical equations of continuum mechanics and electrodynamics (fluid mechanics, solid mechanics, electrodynamics, magnetohydrodynamics) including advective and dissipative processes. All SHTC systems have nice mathematical properties: symmetric hyperbolicity in the sense of Friedrichs [11] and a conservative form of the equations. The solutions to the governing PDE system satisfy fundamental laws of non-equilibrium irreversible thermodynamics: conservation of total energy (first law) and non-decreasing of physical entropy (second law).

The SHTC system theory is a first principle type theory. It allows the derivation of governing PDEs for a quite wide class of physical processes from a variational principle [32] (Hamilton's principle of stationary action). In particular, the SHTC approach has been successfully applied to the development of a hierarchy of compressible multi-phase flow models [30], [36], [37], [39]. Recently, a unified SHTC model of Newtonian continuum mechanics has been developed [9], [10]. It simultaneously describes the dynamics of elastoplastic solids, as well as of viscous and non-viscous fluids in the presence of electromagnetic fields. In the present paper, we extend this unified model to describe solid-fluid two-phase flows. The governing equations of the model also belong to the class of SHTC equations. The interfacial friction between the liquid and solid phases and shear stress relaxation are implemented in the model as relaxation-type source terms in accordance with the laws of thermodynamics. The latter allows taking into account the time dependence of elastic moduli on frequency and using numerical methods to study wave propagation problems.

The rest of the paper is organized as follows. Section 2 briefly describes the governing PDEs for a unified model of continuum mechanics and for a two-phase compressible fluid model. By combining the above two models, a governing master SHTC system for a two-phase solid-fluid medium is formulated. In Section 3, using this model governing equations are derived for small-amplitude wave propagation in a stationary saturated porous medium. In Section 4 we discuss the differences and similarities between the Biot and SHTC models. In Section 4.2, we derive a dispersion relation for the thus obtained acoustic equations and study the properties of the wavefields. In Section 5, an efficient finite difference method for small-amplitude wave propagation is presented and some numerical results are discussed.