Nonlinear quasi-static poroelasticity☆
Introduction
Poroelasticity refers to (Darcy) fluid flow within a deformable, porous medium. The development of this field has been inspired by geophysics and petroleum engineering problems, in particular, reservoir, environmental, and earthquake engineering. Mathematically, the subject was initiated in the 1D work of Terzaghi in the 1920s, and the groundbreaking consolidation theory developed by Biot in the 1940–50s [4]. It was Biot's work which instigated the rapid development and progress of this field. The relevant literature is now abundant, and we only list here representative fundamental treatments: [2], [3], [7], [11], [12], [18], [22], [27]. In all of the works motivated by geophysical applications, the poroelastic structures considered are soil and/or rock (for instance, in most of the aforementioned references). However, cartilages, bones, as well as brain, heart, and liver tissue etc., are also poroelastic structures. Therefore, the theory of poroelasticity can be used and applied to fluid flows inside cartilages, bones, and engineered tissue scaffolds, as well as in perfusion in the optic nerve head—see [5], [6], [26] and references and discussion therein.
From a mathematical point of view, poroelastic systems constitute a coupled system of a (possibly degenerate) parabolic fluid pressure and a hyperbolic (inertial) or elliptic (quasi-static) system of elasticity for the displacements of the porous matrix containing the fluid. The saturated elastic matrix is modeled through homogenization [2], [4], in the sense that the pressure and displacement are distributed quantities throughout the physical domain. In this treatment, we focus on poroelastic models with specific applications in biomechanics (in contrast to those tailored to geomechanical systems). Thus we work under the assumptions of full saturation, negligible inertia, small deformations, and (possibly) compressible mixture components. The applications of interest give rise to a permeability taken as a nonlinear function of the so called fluid content (a particular linear combination of pressure and dilation). This type of nonlinear coupling introduces a variety of complications detailed below, and, in particular, destroys the monotone nature of the problem.
Such a nonlinear poroelastic model was first considered—from a mathematical point of view—in [7], and shortly after in [5].1 The former reference [7] and sequel [8] focus on the compressible Biot model and construct weak solutions through a full spatio-temporal discretization, in the mathematically simplified framework of homogeneous Dirichlet boundary conditions for both fluid pressure and solid displacement. Those references take the earlier linear theory [2], [21], [22] as their primary motivation, and use Brouwer's fixed point theorem at the level of the fully discretized problem. The latter reference [5] focuses on Biot models with incompressible constituents and constructs weak solutions (also using discretizations in both time and space [27]) for both poroelastic and poro-viscoelastic systems with non-homogeneous, mixed boundary conditions that are physically relevant to opthalmological applications. A key theme in this latter work [5] is the careful analysis of the requisite boundary and source regularity for the construction of weak solutions, as this aspect is crucial in understanding the mechanisms leading to tissue damage in the optic nerve head, and consequent vision loss possibly associated with glaucoma. Both [5], [7] obtain a priori estimates in the fully discretized setting, and much of the challenge lies in adequately addressing the nonlinear and non-monotone coupling to obtain a weak solution in the limit. The reference [7] provides a straightforward regularity criterion for uniqueness of solutions, but does not actually consider smooth solutions, nor address the permissibility of multipliers used to obtain estimates.
In this treatment, we provide a careful mathematical construction of weak solutions using semi-discretization in space, in the setting of fully homogeneous boundary conditions. One primary goal is to clearly elucidate the challenges introduced into the Biot problem by the inclusion of non-monotone nonlinear coupling. We also include a novel, sharper uniqueness criterion for solutions of sufficient smoothness. Our approach is based on a priori estimates for the time-dependent linearization, from which we construct a fixed point correspondence. An interesting feature of this approach is that we cannot appeal to uniqueness of solutions for the aforementioned linear problem, as we do not satisfy requisite hypotheses for established theories (e.g., that in [21]). Indeed, weak solutions themselves are not permissible test functions, presenting a great hurdle in the analysis. To address this issue, we utilize a multi-valued fixed point approach, along with a careful construction of the correspondence between the permeability function and the resulting fluid content.
In summary: This paper addresses the existence of weak solutions to a quasilinear Biot system, based on a fixed point approach that circumvents the lack of monotonicity in the system's nonlinear coupling. We believe that the construction given here is quite natural, and illustrates the complexities in the analysis introduced by the presence of nonlinearity and its interaction with the boundary conditions.
Let Ω be an open, bounded subset of representing the spatial domain occupied by the (fully saturated) fluid-solid mixture, with smooth boundary . Let x be the position vector of each point in the body with respect to a fixed Cartesian reference frame. The symbol n will be used to denote the unit outward normal vector to Ω. Let be the volume occupied by the fluid component in a representative volume element centered at at time t. Then the porosity ϕ and the fluid content ζ are given by , where is a baseline (local) value for the porosity.
Under the assumptions of small deformations, full saturation of the mixture, and negligible inertia, we can write the balance of mass for the fluid component and the balance of linear momentum for the mixture as where T is the total stress, v is the discharge velocity (also commonly called the Darcy velocity [22]), F is a body force per unit of volume, and S is a net volumetric fluid production rate.
We complement the balance equations with the constitutive equations: The total stress of the mixture is given by where u is the solid displacement, the symmetrized gradient gives the strain tensor, α is the Biot-Willis constant, p is the Darcy fluid pressure, I is the identity tensor, and and are the elasticity parameters.
The discharge (Darcy) velocity has the following formula via Darcy's law [22]: where the viscosity of the fluid was assumed to be equal to 1. The particular form of the relationship between the permeability k and the porosity ϕ depends on the geometrical architecture of the pores in the elastic matrix and the properties of the fluid. We allow for k to be a general continuous function, assuming only that it is bounded above and below (as discussed below, in Assumption 1.1, and consistent with [5], [7]).
The fluid content is given by where is the constrained specific storage coefficient [2], [12], [22]. Using the relation between porosity and fluid content, as well as the definition of permeability, we can see that permeability in the system depends nonlinearly on the fluid content. In the special case of incompressible constituents, due to the fact that the constrained storage coefficient and , the permeability becomes a nonlinear function of dilation alone. This is the scenario that is specifically addressed in [5].
We consider homogeneous Dirichlet boundary conditions for both the structural displacement u (and hence , when defined) and the fluid pressure p This choice is in line with the model considered in [7], [16]. In our previous work [5], we considered complex physical configurations, incorporating both nonhomogeneous Dirichlet and Neumann boundary conditions for the elastic displacement and fluid pressure. These physically motivated mixed boundary conditions could be incorporated here, and this is the subject of future work.
Initial conditions are to be specified for the fluid content, ζ, as it is the only term which appears under temporal differentiation in the mass-balance equation (1.1):
.
Remark 1.1 In discussing various notions of solutions (as in Section 6), one can find the requirement that for some specified independently, taken in an appropriate space (see [27], as well as [5]). In these works, a different construction for solutions is utilized. We do note that for the linear case, in the most general “weak” setting [22], only should be needed. In this weak situation, the construction is done independent of a priori estimates obtained in standard Hilbert spaces such as and . In [16], solutions are also constructed in the linear case, but initial data is taken to be smoother than “finite energy” considerations require.
To summarize, below is the nonlinear system under consideration:
Note that (1.6) can be written equivalently as where the Laplacian above is interpreted component-wise.
Assumption 1.1 Bounds on the permeability function We assume that the permeability function is continuous and that there exist constants and s.t. Remark 1.2 With a slight abuse of notation, we denote by the Nemytskii operator associated with k. Using our assumptions on the function k, and the theory of superposition operators [19], [25], we have that the operator k is bounded and continuous from into . In order to obtain uniqueness of solution in Section 6, we will further assume that k is a globally Lipschitz function, i.e., .
Assumption 1.2 Other assumptions In what follows, for simplicity, we set to unity non-essential (from the mathematical point of view) parameters. This is to say, we take . The parameter is retained as is, with no dependence on other parameters, as we will take in the construction of weak solutions for the case of incompressible constituents.
Section snippets
Notation, function spaces, and conventions
We make the following conventions for the rest of the paper. Norms are taken to be for a domain D. Inner products in are written as , where the subscript will be omitted when the context is clear. The standard Sobolev space of order s defined on a domain D [15] will be denoted by , with denoting the closure of in the norm (which we denote by or ). Vector valued spaces will be denoted as and . We make use
Fundamental operators and translation of momentum source
In this section we introduce the principal operators that are used in the proofs of the main theorems, along with their properties. We follow the abstract framework provided in [22]. In the last part of the section we provide formal “translations” of the linear and nonlinear problems that allow us to consider the problem with null distributed force in the balance of linear momentum, upon translating the initial data and the pressure source S.
In what follows, it will be necessary to invoke the
Existence of solutions for the linear problem
In this section we recapitulate the relevant linear Theorem 2.1 to be invoked in our fixed point construction. As we noted in the Introduction and Remark 2.5, the abstract theory of time-dependent linear evolutions from [20], [21] produces an equivalent result. For self-containedness, and to tailor the analysis to the specific estimates utilized in later sections, we provide a brief discussion of the setup here, and also a traditional Galerkin construction of solutions in Appendix B.
Recall that
Nonlinear problem - existence of solutions
This section contains the proof of Theorem 2.2. We divide the proof into two parts. First, we focus on the case of compressible constituents, i.e., . Our strategy in this scenario is to show that the map provided by Theorem 2.1 has a fixed point. In part two of the proof, we obtain existence of solutions for the case of incompressible mixture constituents using a limiting process .
Uniqueness - proof of Theorem 2.3
In this section we divide our considerations into two approaches, depending on which terms have point-wise-in-t control: (i) the first one considers a weak solution to the full dynamical system in both dependent variables ; (ii) the second approach considers the reduced system phrased in terms of p and Bp. There are subtle differences in the approaches and in the requisite hypotheses to obtain unique solutions. We point out that—since we do not construct strong solutions in this paper—we
Appendix A: Multivalued fixed point
We begin with a handful of definitions and straightforward theorems that will be relevant to the fixed point we are using in the construction of weak solutions. All of these considerations are taken from [1].
The basic setting considers as a correspondence, where, for each , represents a subset of Y. (We do not use the equivalent point of view that .) The ↠ notation indicates that ϕ need not be a function, but is thought of as a “multi-valued function.”
Definition 6 Notions of closedness and compactness A correspondence
Appendix B: Galerkin construction for linear problem
Proof of Lemma 4.1: Proof Due to Assumption 1.1 on the permeability operator k, the following Proposition is immediate. Proposition 8.1 The bilinear form satisfies the following properties: Continuity: s.t. , a.e. in . Coercivity: , for all .
We use Galerkin approximations. Let be an orthogonal basis of V, and an orthonormal basis in . (For example, we can take to be the
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