Vugs, caves, and fracture can significantly alter the effective permeability of carbonate reservoirs and should be accurately accounted for in a geomodel. Accurate modeling of the interaction between free-flow and porous regions is essential for flow simulations and detailed production engineering calculations. However, flow simulation of such reservoirs is very challenging because of the co-existence of porous and free-flow regions on multiple scales that need to be coupled.

Multiscale methods are conceptually well-suited for this type of modeling as they allow varying resolution and provide a systematic procedure for coarsening and refinement. However, to date there are hardly no multiscale methods developed for problems with both free-flow and porous regions. Our work is a first step to make a uniform multiscale framework, where we develop a multiscale mixed finite-element (MsMFE) method for detailed modeling of vuggy and naturally-fractured reservoirs. The MsMFE method uses a standard Darcy model to approximate pressure and fluxes on a coarse grid, but capatures fine-scale effects through basis functions determined from numerical solutions of local Stokes-Brinkman flow problems on the underlying fine-scale geocellular grid. The Stokes-Brinkman equations give a unified approach to simulating free-flow and porous regions using a single system of equations, avoid explicit interface modeling, and reduce to Darcy or Stokes flow by appropriate choices of parameters.

In the paper, the MsMFE solutions are compared with fine-scale Stokes-Brinkman solutions for test cases including both short- and long-range fractures. The results demonstrate how fine-scale flow in fracture networks can be represented within a coarse-scale Darcy flow model by using multiscale elements computed solving the Stokes-Brinkman equations. The results indicate that the MsMFE method is a promising path toward direct simulation of highly detailed geocellular models of vuggy and naturally-fractured reservoirs.


Naturally fractured and carbonate reservoirs are composed of porous material, but will typically also contain relatively large void spaces in the form of fractures, small cavities, and caves, which are called vugs in the geological literature. Flow simulation of such formations is very challenging because of the co-existence of porous and free-flow domains on multiple scales that require coupling (Wu et al. 2006).

The Darcy-Stokes equations have been used to model industrial infiltration processes and coupled surface and subsurface flow, for which the porous and the free-flow domains are well separated. The Darcy-Stokes model consists of Darcy's law combined with mass conservation in the porous subdomain and the Stokes equations in the free-flow subdomain. To close the model, one must specify conditions on the interface between the Darcy and Stokes subdomains. All these conditions require continuity of mass and momentum over the interface, but differ in the way they allow the tangential component to jump across the interface.

In a carbonate reservoir, the porous and free-flow domains are not well separated: vugs and rock matrix are intertwined throughout the reservoir, often on multiple scales. This means that the coupled Darcy-Stokes approach is not feasible for several reasons. First of all, precise information about the location and geometry of the interface between vugs and the porous matrix is required and also experimentally determined values related to the interface conditions. This information may be possible to obtain for an engineered medium or a small rock sample, but is not possible to obtain for a sector or a full reservoir model. Second, explicit representation of the medium on a centimeter scale, as required to resolve vugs and fractures, would make the flow problem computationally intractable. Finally, the free-flow domains may contain loose fill-in material or particle suspensions in the fluids filling the void space.

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