A unified multi-physics model for flow through naturally fractured carbonate karst reservoirs

Naturally fractured carbonate karst reservoirs are composed of porous materials and at the same time contain relatively large void spaces in the forms of fractures, small cavities, and caves. These features are typically interconnected via fractures on multiple scales. While the flow through the porous regions can be modeled by Darcy's law, the Stokes equation has to be used to describe the flow through the void spaces. This presents a major challenge in modeling the fluid flow through such formations because of the co-existence of porous and free-flow regions on multiple scales.;The Stokes-Brinkman equation was proposed as a fine-scale model for flow through carbonate karst reservoirs. This equation is a unified description of multi-physics and multi-scale flow in porous media. It is the first time that this equation is used to solve the flow through carbonate karst reservoirs though it was developed in the 1940s. We will carry an in-depth discussion of the Stokes-Brinkman equation and compare the differences and relation of this model to the coupled Darcy-Stokes models. The homogenization process of the Stokes-Brinkman equation is also discussed to show the multi-scale nature of the flow problem in carbonate karst reservoirs.;Taylor-Hood mixed finite element method is used to discretize the Stokes-Brinkman equation. We use the Schur complement with preconditioned conjugate gradient method to solve the resulting system of linear algebraic equations.;Numerical experiments were performed to validate this model. It is shown that the Stokes-Brinkman equation has almost identical accuracy compared to coupled Darcy-Stokes equations with greatly simplified numerical treatment. We apply the Stokes-Brinkman equation in typical fracture-cave configurations to study the sensitivity of fracture permeability on the effective permeability. The wide range of permeability values used in this study also shows the flexibility of the Stokes-Brinkman equation compared to the coupled Darcy-Stokes model, which is not capable of dealing with intermediate flow regimes.;Finally, the Stokes-Brinkman equation is used as a fine-scale flow model to solve local flow problems in flow-based scale-up analysis. The effective permeability is calculated and then used to solve Darcy's law on a coarse-scale grid. We developed an efficient numerical scheme to deal with the large computational expense and the numerical stability due to the significant scale difference between fractures and formation blocks. It is shown that the scale-up results based on the Stokes-Brinkman model agree with fine-scale solutions. Furthermore, global scale-up gives much better results than local scale-up, especially when the caves are interconnected by fracture networks that make the correlation length bigger than the scale of coarse scale blocks.;The proposed model flexibly adapts to the different flow physics in naturally fractured carbonate karst reservoirs in a simple and effective way. It extends modeling and predicting capabilities in the efficient development of this important type of reservoir.