Modeling wavefields in saturated elastic porous media based on thermodynamically compatible system theory for two-phase solid-fluid mixtures
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.
Section snippets
System of governing PDEs for poroelastic media
To develop a poroelastic model whose governing equations form a symmetric hyperbolic thermodynamically compatible (SHTC) system, a unified thermodynamically compatible continuum model formulated in [9], [33] is coupled to a two-phase compressible fluid model [36].
System of governing PDEs for small-amplitude wave propagation in saturated porous media
In this section, we derive equations for small-amplitude wave propagation in a saturated porous medium at equilibrium. For the derivation, we first transform Eq. (22) to a more convenient form. Instead of the total momentum equation and the equation for the relative velocity, consider momentum equations for each of the constituents. These are derived from Eq.(22a) and Eq.(22e) and read as
Theoretical comparison
The most comprehensive comparative study of Biot’s model and models developed by a classical two-phase approach based on continuum thermodynamics has been made by Wilmanski [44]. The conclusion of this paper is that the two-phase model contains all information in Biot’s model about the features of wave propagation in a saturated porous medium, and, in particular, it predicts the slow pressure waves and gives a qualitatively correct description of the dependence of the phase velocities on
Finite difference implementation
To discretize the governing equations, the velocity-stress formulation, which was proposed for elastic-wave equations in [20], [43], is used. The use of a numerical scheme on staggered grids is quite natural due to the fact that the equations form a symmetric first order system of evolution equations for the mixture components, relative velocities, pressure, and shear stress.
Following the notation of [43] and [16], we introduce a time-space grid with integer nodes and
Conclusions
An extension of the unified model of continuum fluid and solid mechanics [9] for compressible fluid flows in elastoplastic porous media has been proposed. The derivation is based on the Symmetric Hyperbolic Thermodynamically Compatible (SHTC) theory [32], and the resulting model represents a combination of the unified continuum model from [9] with the SHTC model for two-phase compressible flows from [36]. The governing equations satisfy two laws of thermodynamics (energy conservation and
In memoriam
This paper is dedicated to the memory of Dr. Douglas Nelson Woods (*January 11th 1985 - †September 11th 2019), promising young scientist and post-doctoral research fellow at Los Alamos National Laboratory. Our thoughts and wishes go to his wife Jessica, to his parents Susan and Tom, to his sister Rebecca and to his brother Chris, whom he left behind.
CRediT authorship contribution statement
Evgeniy Romenski: Writing - original draft. Galina Reshetova: Writing - original draft. Ilya Peshkov: Writing - original draft. Michael Dumbser: Writing - original draft.
Declaration of Competing Interest
None.
Acknowledgements
The authors are grateful to M. Yudin for valuable help in the manuscript preparation. The research of E.R. and G.R. in Sects.2-4 was supported by the Russian Science Foundation under grant 19-77-20004, the research in Sect.5 was supported by the Russian Foundation for Basic Research under grant 19-01-00347. I.P. gratefully acknowledges the support of Agence Nationale de la Recherche (FR) (grant ANR-11-LABX-0040-CIMI) under program ANR-11-IDEX-0002-02. The work of M.D. and I.P. was partially
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The work by I.P. has been started while being at Paul Sabatier University, Institut de Mathématiques de Toulouse, Toulouse, France