Multiscale dimension reduction for flow and transport problems in thin domain with reactive boundaries

Highlights

• Design of multiscale methods and concepts for thin domain problems.

• Dimension reduction using multiscale methods.

• Multiscale methods for problems with irregular boundaries.

• Coupled flow and transport problems in thin domains.

Abstract

In this paper, we consider flow and transport problems in thin domains. Modeling problems in thin domains occur in many applications, where thin domains lead to some type of reduced models. A typical example is one dimensional reduced-order model for flows in pipe-like geometries (e.g., blood vessels). In many reduced-order models, the model equations are described apriori by some analytical approaches. In this paper, we propose the use of multiscale methods, which are alternative to reduced-order models and can represent reduced-dimension modeling by using fewer basis functions (e.g., the use of one basis function corresponds to one dimensional approximation).

The mathematical model considered in the paper is described by a system of equations for velocity, pressure, and concentration, where the flow is described by the Stokes equations, and the transport is described by an unsteady convection-diffusion equation with non-homogeneous boundary conditions on walls (reactive boundaries). We start with the finite element approximation of the problem on unstructured grids and use it as a reference solution for two and three-dimensional model problems. Fine grid approximation resolves complex geometries on the grid level and leads to a large discrete system of equations that is computationally expensive to solve. To reduce the size of the discrete systems, we develop a multiscale model reduction technique, where we construct local multiscale basis functions to generate a lower-dimensional model on a coarse grid. The proposed multiscale model reduction is based on the Discontinuous Galerkin Generalized Multiscale Finite Element Method (DG-GMsGEM). In DG-GMsFEM for flow problems, we start with constructing the snapshot space for each interface between coarse grid cells to capture possible flows. For the reduction of the snapshot space size, we perform a dimension reduction via a solution of the local spectral problem and use eigenvectors corresponding to the smallest eigenvalues as multiscale basis functions for the approximation on the coarse grid. For the transport problem, we construct multiscale basis functions for each interface between coarse grid cells and present additional basis functions to capture non-homogeneous boundary conditions on walls. Finally, we will present some numerical simulations for three test geometries for two and three-dimensional problems to demonstrate the method's performance.

Introduction

Mathematical models in thin domains occur in many real-world applications, scientific and engineering problems. Fluid flow and transport in thin tube structures are widely used in biological applications, for example, to simulate blood flow in vessels [1], [2], [3]. In engineering problems, flow simulation is used to study fluid flow in complex pipe structures, for example in pipewise industrial installations, wells in oil and gas industry, heat exchangers, etc. In reservoir simulations, thin domains are related to the fractures that usually have complex geometries with very small thickness compared to typical reservoir sizes [4], [5], [6]. Such problems are often considered with complex interaction processes with surrounding media or walls. In many applications, these problems are transformed to reduced (e.g., one) dimensional problems via some type of apriori postulated models. Our goal is to present an alternative approach to analytical model reduction by using multiscale basis functions.

For the applicability of the convenient numerical methods for simulations of such problems, a very fine grid should be constructed to resolve the geometry's complex structure on the grid level. Moreover, a very small domain thickness provides an additional complexity in the grid construction for thin and long domains. For a fast and accurate solution of the presented problem, a homogenization (upscaling) technique or multiscale models are used, which are based on constructing the lower dimensional coarse grid models. The asymptotic behavior of the solutions in thin domains is intensively studied. In [7], [8], the authors propose a hierarchical model reduction approach, in which a coarse model featuring only the dominant direction dynamics is enriched locally by a fine model that accounts for the transverse variables via an appropriate modal expansion. In [9], the authors consider the problem of complete asymptotic expansion for the time-dependent Poiseuille flow in a thin tube. In [10], the method of asymptotic partial decomposition of a domain (MAPDD) was presented to reduce the computational complexity of the numerical solution of such problems. This method combines the three-dimensional description in some neighborhoods of bifurcations and the one-dimensional description out of these small subdomains, and it prescribes some special junction conditions at the interface between these 3D and 1D submodels. Numerical results were presented in [2] for the method of asymptotic partial domain decomposition for thin tube structures with Newtonian and non-Newtonian flows in large systems of vessels. Our goal is to provide an alternative systematic approach to model reduction for thin domain problems that can add complexity via additional multiscale basis functions.

Model reduction techniques usually depend on a coarse grid approximation, which can be obtained by discretizing the problem on a coarse grid and choosing a suitable coarse-grid formulation of the problem. In the literature, several approaches have been developed to obtain the coarse-grid formulation for the problems in heterogeneous domains, including multiscale finite element method [11], [12], [13], mixed multiscale finite element method [14], [15], generalized multiscale finite element method [16], [17], [18], [19], [20], multiscale mortar mixed finite element method [21], multiscale finite volume method [22], [23], variational multiscale methods [24], [25], [26], and heterogeneous multiscale methods [27] etc. The non - conforming multiscale method is considered for the solution of the Stokes flow problem in a heterogeneous domain in [28], [29]. In [30], [31], we presented the Generalized Multiscale Finite Element Method (GMsFEM) for the solution of the flow problems in perforated domains with continuous multiscale basis functions. GMsFEM shows a good accuracy for solving the nonlinear (non - Newtonian) fluid flow problems [32]. In [33], we presented the Discontinuous Galerkin Generalized Multiscale Finite Element Method (DG-GMsFEM) for the solving the two - dimensional problems in perforated domains.

In this paper, we consider flow and transport processes in thin structures with reactive boundaries. The mathematical model is described using Stokes equations and the unsteady convection-diffusion equation with non - homogeneous boundary conditions. Non-homogeneous boundary conditions occur in many applications. For example, the pore-scale modeling and simulation of reactive flow in porous media have many applications in many branches of science such as biology, physics, chemistry, geomechanics, and geology [34], [35], [36]. In order to handle the complex geometry of the walls and non-homogeneous boundary conditions on them, we present additional spectral multiscale basis functions. In this work, we continue developing the multiscale model reduction techniques for problems with multiscale features and the generalization of the techniques for DG-GMsFEM. In our previous work [37], [38], we considered elliptic problems in perforated domains and constructed additional basis functions to capture non-homogeneous boundary conditions on perforations. In [37], we proposed a non-local multi-continua (NLMC) method for Laplace, elasticity, and parabolic equations with non-homogeneous boundary conditions on perforations. In [38], we considered the Continuous Galerkin Generalized Multiscale Finite Element Method (CG-GMsFEM) for problems in perforated domains with non-homogeneous boundary conditions, where we constructed one additional basis function for local domains with perforations. Recently, we extended this technique for solving unsaturated flow problems in heterogeneous domains with rough boundary [39]. In this work, we consider DG-GMsFEM and present additional spectral basis functions for rough non-homogeneous boundary conditions in transport problems. The concept is based on the separation of the snapshots for each feature and shares a lot of similarities with multiscale methods for fractured media presented in our previous works [40], [41]. Our work is also motivated by a recently developed Edge GMsFEM, where multiscale basis functions are constructed for each coarse grid interface [42], [43].

In this work, we use the DG-GMsFEM for constructing multiscale basis functions for problems in thin domains. The goals of this paper are in constructing the general approach for problems in complex thin geometries with an accurate approximation of the velocity space and transport processes. We construct local multiscale basis functions to generate a lower-dimensional model on a coarse grid. In DG-GMsFEM for flow problem, we start with constructing the snapshot space that captures possible flows between coarse cell interfaces. After constructing the snapshot space, we perform a dimension reduction by a solution of the local spectral problems. For the pressure approximation, we use piecewise constant functions on the coarse grid. For the transport problem, we construct multiscale basis functions for each interface between coarse grid cells and present additional basis functions to capture non-homogeneous boundary conditions on walls. The presented snapshot spaces can accurately capture processes on the rough wall boundaries with non-homogeneous boundary conditions on them. We present the results of the numerical simulations for three test geometries for two and three-dimensional problems.

As we mentioned earlier, our goal is to investigate the use of multiscale and generalized multiscale methods for dimension reduction for problems in this domains. Many existing approaches propose analytical or semi-analytical reduced-dimensional problems, where the dimensions are determined apriori. Our idea is to use generalized multiscale method and identify the dimension. The proposed approach combined with a posteriori error estimate can further identify the dimension across thin layers and allow obtaining more accurate solution.

The paper is organized as follows. In Section 2, we describe the problem formulation and the fine-scale approximation. In Section 3, we present the multiscale method for flow and transport processes in thin structures with rough reactive boundaries. In Section 4, we present numerical results. The paper ends with a conclusion.