Discontinuous finite volume element method for Darcy flows in fractured porous media☆
Introduction
The modeling of fluid flow and transport in fractured porous media has become a significant problem due to its application in many environmental and energy problems, such as ground water contamination, oil migration, nuclear waste and carbon dioxide storage. Since the fracture is supposed to have a small thickness with respect to the width of the porous medium, the dimension of the fracture is reduced to be in a dimensional domain. In practical application, the fracture permeability may vary by several orders of magnitude, thus the flow in fractured porous medium is strongly influenced by the presence of the fracture. In general, there are two kinds of fractures. On one hand, the fractures with higher permeability than the surrounding domains act as fast pathways, the fluid tends to flow into fractures and then along them, so the velocity is not supposed to be continuous across fractures. And on the other hand that those with lower permeability act as geological barriers, in this case, the fluid has a tendency to avoid fractures, so the pressure is not identical on both sides of fractures. For both types of fractures, the reduced coupled bulk fracture models have been presented for single phase Darcy flow developed in [1], where an averaging process is applied, and the fracture is treated as an interface that crosses the bulk region. The fracture is additionally assumed to be filled of debris, so that the flow therein can still be modeled by the Darcy law. To close the problem, interface conditions are enforced that to relate the average and jump of the bulk pressure to the normal flux of the fracture pressure.
Various numerical methods have been proposed to solve this reduced coupled bulk fracture model, such as methods to keep robust on polygonal meshes, including mimetic finite difference method [2], [3], hybrid high-order method [4], weak Galerkin method [5], discontinuous Galerkin method [6], mixed finite element method [7], [8], [9], [10], finite volume method[11], [12], [13], [14], [15], two-grid algorithm [16] and methods where the fractures are allowed to arbitrarily cut the bulk grid, including eXtended finite element method [17], [18], [19], [20]. We also assign the following literatures in which the problem of discrete fracture networks have been considered by various novel approaches, for example, virtual element method [21], a PDE-constrained optimization method [22], [23], and so on.
Our focus is here on the discontinuous finite volume element (DFVE) method which is originally introduced in [24] to the second-order elliptic problem, and later applied to solve the Stokes problem [25]. The DFVE method uses discontinuous piecewise polynomial for the trial functions as in discontinuous Galerkin method, and uses piecewise constant for test functions as in finite volume method, so it has the flexibility, high order of accuracy of discontinuous Galerkin method, and the simplicity, conservative property of finite volume element method. The fascinating nature of DFVE method reflects on the smaller conservation control volume which is less than half the size of control volume applied in the existed finite volume method. The localizability of the discontinuous element and its dual partition in DFVE method provides another advantage for parallel computing. There are many works on the discontinuous finite volume element method for both the second order elliptic problem [26], [27], [28], [29] and the Stokes equation [30], [31], [32]. DFVE method has been used to solve elliptic problem with adaptive technique and other applications [33], [34]. Recently, there are a few papers applying DFVE method to nonlinear problem [35], [36] and also for the coupled system [37], [38], [39], [40], [41].
The DFVE method proposed here hinges on a primal formulation in the bulk coupled with a primal formulation inside the fracture. To keep the exposition as simple as possible while retaining all the key difficulties, we focus on the two-dimensional case. More specifically, considering the discontinuity of solutions at the bulk fracture interface, it is quite natural to choose DFVE method to solve the flow problem in fractured porous media. However, this is not the only motivation to use DFVE method. Indeed, the coupling conditions at the interfaces between bulk and fracture flow be formulated using jump and average operators, which can be easily embedded in the variational formulation, so that DFVE method turns out to be a powerful tool for efficiently handling the coupling of bulk fracture model.
In this paper we propose a discretization which combines a DFVE approximation for the problem in the bulk with a conforming finite element approximation in the fracture. Compared with these existing methods, DFVM has the property of local mass conservation at a so-called diamond control volume level. And because of using low-order polynomial as the trial function, DFVM is easy to code, but also, a high order of accuracy can be achieved. Moreover, we obtained the optimal order error estimates corresponding variables in -norm and broken -norm. Some experiments are presented to illustrate the theory.
The paper is organized as follows. In Section 2, we recall and state the reduced model. In Section 3, the discrete formulation coupled the discontinuous finite volume element method for the bulk problem with the conforming finite element method for the fracture problem is introduced. Section 4 is used to derive some convergence analysis. Section 5 presents numerical experiments on some examples with exact analytical solutions and a series of benchmark problems. Finally we draw a conclusion in Section 6.
Section snippets
Model statement
We consider a bounded, open, convex domain , composed of a fractured porous medium, and let , denote the Dirichlet boundary and Neumann boundary, respectively, as depicted in Fig. 1. We denote a network formed by the union of fractures , for . Each is a dimensional manifold and we have
In this paper, we suppose that the fractures can intersect only at their endpoints, i.e.,
Then we can subdivide the boundary of fractures in
Discontinuous finite volume element method
We consider a family of meshes made of triangular elements in two dimension which are aligned with the fracture , is the diameter of element . Every triangle is divided into three subtriangles by connecting the barycenter of the triangle to its corner nodes, then we define the dual partition of the primal partition to be the union of the triangles as shown in Fig. 2. We define the mesh parameter . Let denote the length of edge in element , be the set
Error estimates
In this section, we use the equivalence relationship between the DFVE method and IPDG approximations for the coupled problem (2.1)–(2.12). This relationship plays a key role in the error estimation. We treat the DFVE method as a perturbation of the interior penalty method, as the error estimates of the interior penalty method are developed in [6], we can easily develop the error estimates for the DFVE method by means of the estimation of the difference between the solution of the DFVE method
Numerical experiments
In this section, we present some numerical examples to test the problem (2.1)–(2.12) in two-dimensional. All numerical results demonstrate that the proposed scheme (3.5) is robust, correct and easy to implement. The first three numerical examples are provided to show the convergence and accuracy. Moreover, in the last four examples we investigate the robustness of our method by contrasts between the permeability in the bulk and fracture. In all numerical examples, we take penalty parameters
Conclusion
In this paper, we develop a discrete scheme for the reduced model which describes the flow in the fractured porous media. This scheme couples DFVE methods for the bulk flow with a conforming finite element approximation for the fracture flow. The optimal convergence rate is obtained under the suitable norms. Numerical results in last section are not only consistent with our theory, but also demonstrate that our scheme is accurate and robust for fractures with high or low permeability. In future
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This work is partially supported by the National Natural Science Foundation of China (Nos. 11901372, 11771262,11771350), and Natural Science Foundation of Shaanxi Province, China (Nos. 2019JQ-077,2020JQ-403), and the Fundamental Research Fund for the Central Universities of China (Nos. GK201903007, GK201901008), and Foundation CMG.