A new numerical continuum \textit{one-domain} approach (ODA) solver is presented for the simulation of the transfer processes between a free fluid and a porous medium. The solver is developed in the \textit{mesoscopic} scale framework, where a continuous variation of the physical parameters of the porous medium (e.g., porosity and permeability) is assumed. The Navier-Stokes-Brinkman equations are solved along with the continuity equation, under the hypothesis of incompressible fluid. The porous medium is assumed to be fully saturated and can potentially be anisotropic. The domain is discretized with unstructured meshes allowing local refinements. A fractional time step procedure is applied, where one predictor and two corrector steps are solved within each time iteration. The predictor step is solved in the framework of a marching in space and time procedure, with some important numerical advantages. The two corrector steps require the solution of large linear systems, whose matrices are sparse, symmetric and positive definite, with $\mathcal{M}$-matrix property over Delaunay-meshes. A fast and efficient solution is obtained using a preconditioned conjugate gradient method. The discretization adopted for the two corrector steps can be regarded as a Two-Point-Flux-Approximation (TPFA) scheme, which, unlike the standard TPFA schemes, does not require the grid mesh to be $\mathbf{K}$-orthogonal, (with $\mathbf{K}$ the anisotropy tensor). As demonstrated with the provided test cases, the proposed scheme correctly retains the anisotropy effects within the porous medium. Furthermore, it overcomes the restrictions of existing mesoscopic scale one-domain approachs proposed in the literature.

翻译：本文提出了一种新的数值连续介质单域方法求解器，用于模拟自由流体与多孔介质之间的传递过程。该求解器在介观尺度框架下开发，假设多孔介质的物理参数（如孔隙度和渗透率）连续变化。在不可压缩流体假设下，求解纳维-斯托克斯-布林克曼方程和连续性方程。假设多孔介质完全饱和且可能具有各向异性。计算域采用非结构化网格离散化，允许局部加密。采用分步时间推进过程，每个时间迭代步中求解一个预测步和两个校正步。预测步在时空推进框架下求解，具有若干重要的数值优势。两个校正步需要求解大型线性系统，其矩阵为稀疏对称正定矩阵，且在Delaunay网格上具有$\mathcal{M}$-矩阵特性。通过预处理共轭梯度法实现快速高效的求解。两个校正步采用的离散化方法可视为两点通量近似格式，与标准两点通量近似格式不同，该方法不要求网格是$\mathbf{K}$-正交的（其中$\mathbf{K}$为各向异性张量）。通过提供的算例验证表明，本文提出的格式正确保留了多孔介质内的各向异性效应。此外，该方法克服了现有文献中介观尺度单域方法的限制。