Abstract Fractured geologic media can yield anisotropies in solute and heat diffusion due to the formation of changing fluid network connectivity in a rock matrix. In this paper we model Steady-state anisotropic heat diffusion as an elliptic partial differential equation with a symmetric positive definite second rank thermal conductivity tensor. We model diffusive flux as a non-diagonal symmetric tensor, which can potentially have jump discontinuities that are not aligned with the coordinate axis. The presence of jump discontinuities due to joints and faults in a rock matrix impose difficulties on existing, well-established numerical schemes that model diffusive transport. In our scheme, we model diffusive flux using mimetic finite difference operators, which are discrete analogs of the classical continuous differential operators. We introduce a 2nd- and 4th-order mimetic formulation for computing anisotropic fluxes. Numerical results demonstrate our formulation yields a substantial improvement compared to similar mimetic schemes.

中文翻译：

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各向异性椭圆方程的高阶拟态有限差分

摘要 由于岩石基质中不断变化的流体网络连通性的形成，破裂的地质介质可以产生溶质和热扩散的各向异性。在本文中，我们将稳态各向异性热扩散建模为具有对称正定二阶热导率张量的椭圆偏微分方程。我们将漫射通量建模为非对角对称张量，它可能具有与坐标轴不对齐的跳跃不连续性。由于岩石基质中的节理和断层而导致跳跃不连续性的存在给现有的、完善的扩散输运建模数值方案带来了困难。在我们的方案中，我们使用模拟有限差分算子对扩散通量进行建模，这些算子是经典连续微分算子的离散模拟。我们引入了用于计算各向异性通量的二阶和四阶模拟公式。数值结果表明，与类似的模拟方案相比，我们的配方产生了实质性的改进。