The modeling and numerical simulation of fluid flow in naturally fractured carbonate karst reservoirs are extremely challenging due to non-Darcy flow in vugs and caves connected by fracture networks. The momentum balance of such flow has been shown to be better described by the Brinkman equation both physically and mathematically, and many methods have been proposed in the literature dealing with the steady-state Brinkman model. We carry Brinkman's idea one step further and propose a transient flow model which consists of the Brinkman equation and a generalized material balance equation, and the latter has proven to be exact in the fractured carbonate karst reservoir. Finite differences are implemented for the solution of the proposed transient flow model. This solution method is more straightforward, easier to derive and implement, and more apt to generalization from 2D to 3D cases than alternative techniques.
Numerical simulation of the transient Brinkman model requires explicit solution of not only pressure at the center of each grid block, but also velocities at the interfaces between the blocks, which exaggerates the computational cost and makes the computational process more difficult and less stable. In this paper, we propose a simplified finite difference formulation of the transient Brinkman model, which significantly reduces the computational time of the simulation process, and improves accuracy and stability of the simulation results. We update our reservoir simulator with this new formulation and illustrate it with a complex 3D naturally fractured carbonate karst reservoir model. The results of this study form the foundation for future 3D multi-phase reservoir cases.