Abstract
We consider a problem modeling a porous medium with a random perturbation. This model occurs in many applications such as biology, medical sciences, oil exploitation, and chemical engineering. Many authors focused their study mostly on the deterministic case. The more classical one was due to Biot in the 50s, where he suggested to ignore everything that happens at the microscopic level, to apply the principles of the continuum mechanics at the macroscopic level. Here we consider a stochastic problem, that is, a problem with a random perturbation. First we prove a result on the existence and uniqueness of the solution, by making use of the weak formulation. Furthermore, we use a numerical scheme based on finite differences to present numerical results.
1. Introduction
The theory of linear poroelasticity has been introduced and rigorously improved by many authors, among others, Biot [1–4], Bear and Bachmat [5], Bémer et al. [6], Barucq et al. [7], and Zenisek [8]. Nowadays, this theory leads to many applications in different disciplines, such as oil exploration, biological phenomena, medical sciences [9], and military applications [10].
In our work, we are interested in the study of fluid-saturated porous media, subject to a random disturbance when the phenomenon of consolidation is realized (see [11, 12]). The study of poroelastic properties or the problem of acoustic wave propagation in saturated porous media, for example, in the oil exploration, has been based on two approaches (see, e.g., [9, 10, 13]). The first focuses on microscopic laws; that is, the pore becomes an entire field of study and then derives the macroscopic laws that are involved across the porous media as a whole. This is where the homogenization techniques are used, by considering the fact that the microscopic structure is repeated periodically which leads to the periodicity of the solutions [14]. The second approach, the more classical, was due to Biot in the 50s, where he suggested to ignore everything that happens at the microscopic level, to apply the continuum mechanics principles at the macroscopic level.
2. The Model
Let be a porous medium and the density; we consider the porous matrix (porous or skeleton) to be filled with a relatively incompressible viscous fluid which diffuses through. The small movements of both solid and liquid phases are verified, neglecting the speed of filtration. We write the coupled system of hyperbolic-parabolic type as where , , and are the velocity of the solid matrix, the fluid pressure, and the external forces acting on the macroscopic element, for all in and all , respectively, with a source term. The parameters and denote the expansion and the shear modulus, respectively, whereas the coefficients , , and are positive such that is associated with the consolidation side effects and may cancel, is the coefficient that combines the porosity of the medium and the compressibility of the fluid-solid structure and is the one which takes into account the permeability of the medium and the viscosity of the fluid, since it is a measure of flow obeying the Darcy law for a given pressure gradient. The constant is a positive real number, representing the Biot-Willis constant which takes into account the effects of coupling the deformation and the pressure: it is actually a measure of the amount of fluid that can be placed in the porous matrix by increasing the pressure at constant volume.
3. One-Dimensional Biot Model
We are interested in the one-dimensional nonlinear model of the following form:
The nonlinear term in the first equation of (2) is due to the local geometry of the medium, such as sudden changes in contact areas or occlusions of cracks, with . Here and is a positive constant if and vanishes when . Besides the system physical parameters are assumed to be constant and independent of the space variable.
This system appears when we consider the particular case of a wave propagating in a single direction. In this case the displacement depends only on one variable denoted by and the scalar represents the component along the -direction and the same for the pressure .
Let us consider , , , to be the saturated porous medium which occurs in the propagation of the wave and let be the cylinder .
When , the system is transformed into a quasistatic system as follows: with the initial conditions In addition, when , we can write and the homogeneous Dirichlet boundary conditions Note that , and are strictly positive constants.
3.1. Galerkin Method
The Galerkin method is used to prove the existence of the discrete solution. We consider a family of vector spaces that approaches an infinite dimensional Hilbert space satisfying the following:(i);(ii) when in the sense that there exists a dense subspace of , such that, for all , we can find a sequence satisfying: for all , , and in when .
These approximation spaces are generated using a family for such that, for , . According to the choice of this family, we can construct solutions to problems that can be more or less regular.
In order to have sufficient regularity for the nonlinear term treatment, we make use of the eigenfunctions of the Laplace operator in . First let us set the following results.
3.2. Known Results for Laplace’s Operator
Proposition 1. There exists a sequence such that with . The set is a Hilbertian basis for and the space of the finite linear combinations of is dense in and in .
Proposition 2. Let where is defined by (7) and let be the projection onto defined on by where denotes the duality product between and its dual. Then the restriction of to on is with . In addition, the following properties are satisfied and the norms and are independent of .
4. One-Dimensional Nonlinear Stochastic Biot Model
Here we are interested in studying the Biot model in the presence of a stochastic perturbation. The idea is to develop a mathematical analysis of the above equations with a stochastic perturbation. This is based on analyzing the problem in a weak form by making use of appropriate functional spaces.
4.1. Weak Formulation
We study the model of consolidation in the case where and by using the weak formulation.
Find ; such that ; , ; , , and ; , satisfying for almost every in and . Here is real and is defined by the relation . It is assumed that the initial conditions and (when ) are in and , respectively, and the source terms belong to . We are also supposed to have a major regularity for the disturbance , so that the resolution of the stochastic equation is reduced for each element of the probability space , to a deterministic equation. Hence is a continuous stochastic process with values in (i.e., continuous trajectory of the disturbance) and defined on a probability space . Any solution of the variational formulation (10) is called the solution of the nonlinear stochastic consolidation Biot model. This solution is obtained by solving the equations in (10) for each .
Theorem 3. Let be real such that , , , , and and . Let be a stochastic process defined in . There is a single pair of random variables such that and , satisfying the system of (10).
4.2. Existence of the Solution
In this section, we propose to formulate the equations whose solutions constitute the Faedo-Galerkin approximation type of our problem, in (for more details on the Faedo-Galerkin approximations the reader is referred to [15, 17] and references therein). A solution is constructed as the limit of a sequence of approximate solutions denoted by . This sequence is defined from in by where is the sequence defined in Proposition 1 of Section 3.2. To each integer , we associate a new discrete unknown by the use of the sequence that is defined by solving the differential system for all , with initial conditions where , .
is a continuous stochastic process with values in and defined on a probability space . The solution of (12) is obtained by solving the equations for each fixed . Therefore we consider the following deterministic equations: for all , with initial conditions with , , and is a continuous function with values in .
Theorem 4. The existence of sequences , , and satisfying the properties is a consequence of Propositions 1 and 2 in Section 3.2. Problem (-) satisfies the Cauchy-Lipschitz conditions and, from the nonlinear differential equations theory, ( ) admits a unique maximal solution in , , such that .
The idea is to show that there exists that converges to problem (-) solution . It is sufficient to extract a converging sequence , where its existence is satisfied by a priori estimates which prove that the sequence is bounded in suitable functional spaces from the bounded differential equation solution’s principle. The time for which solutions exist is equal to the initially given time .
Lemma 5. The sequence of the solutions of problem (-) satisfies the following properties:(i) is bounded in ,(ii) is bounded in ,(iii) is bounded in ,(iv) is bounded in ,(v) is bounded in ,(vi) is bounded in .
Each estimate is independent of the physical parameter but depends only on and from , , and .
Lemma 6. Let be the sequence of the solutions of (-). Then(i) is bounded in ,(ii) is bounded in .
For the proof of Lemma 6, we use the same as arguments as [7].
For the proof of Theorem 3, we proceed as follows.
From Theorem 4 we have, for almost every , problem (, ) with having one and only one solution: , satisfying ((12), (13)). It is sufficient to prove that defined by the unique solution in Theorem 4 is measurable. Let and be two elements of and let be the solution of (, ) corresponding to and , respectively. Using the uniqueness of the solution we have That is to say in and in .
This proves that the mapping: is continuous and, from the compacity, is measurable in the considered topology.
5. Numerical Results
5.1. Discretisation and Stability of the Scheme
We consider (2), taking and : We use the Euler scheme for the time discretisation, the central finite differences for the space variable, and to obtain with ; ; ; ; ; , , ; , , ; ; ; ; ; ; ; ; .
Using the Fourier stability analysis, we write
This is the so-called CFL condition, which is the stability condition leading us to the right choice of the discretisation time and space steps.
5.2. Numerical Results and Comments
In Figures 1–3, we present the Brownian motion, the associated velocities, and pressures for different space and time steps as solutions of (15) using the finite difference scheme considered in Section 5.1. The numerical simulations were carried out using MATLAB with the parameters set as , , , , , , , and . The data are set as and .
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5.2.1. Comments on the Numerical Results
According to the numerical experimentations, in Figure 1, we present results for the deterministic case where the Brownian motion is equal to . As shown in Figures 2 and 3, it is clear that the velocity and the pressure of the fluid behave randomly because of the stochastic part in (10). We also noticed that we experimented no unstability behavior as long as we respect the CFL condition (the stability condition) that is deducted from the stability analysis of the finite difference scheme considered (see Figure 3). To show this numerical instability behavior, we present numerical results for a condition equal to twice the CFL condition (see Figure 2).
5.3. Conclusion
We solved a system of two partial differential equations (PDEs) in a bounded domain, modeling the velocity and pressure; the first equation has a nonlinear part and another random part. A weak solution has been found in the space by the principle of the resolution of a stochastic differential equation (SDE) trajectory path. The velocity and the pressure were approached numerically by the Euler method for the time and the central finite difference for the space variable, where the time and space discretisation steps were chosen according to the numerical stability condition (CFL). The numerical results are resumed in Figures 1–3 and commented on above. It has to be pointed out that this model is quite important and can be used in many applications, for example, in image processing, and for this purpose a study on a comparison of this model with the one in [16] is under consideration.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by the Algerian Research National Project PNR977/23/2011–2013. The author is grateful to Professors H. F. Yashima and F. Z. Nouri for fruitful discussions, recommendations, and suggestions that made the accomplishment of this work possible.