Due to the chemically active nature of carbonates rocks they are highly susceptible to the phenomenon of rock dissolution, a diagenetic process by which minerals are dissolved and removed by the fluid that fills the porous space creating and modifying the porous medium. This phenomenon might occur in small and medium proportions generating the so-called vugs, regions with pore volumes orders of magnitude greater than the conventional intergranular spaces. The dissolutions might also occur in much larger scales, generating extensive networks of caves. The validity of Darcy's flow equation to model flow in vuggy-karstified petroleum reservoirs can be questionable since viscous stresses of the fluid are not contemplated. The Brinkman equation can be a more general approach to model the flow in the presence of both: open spaces and porous media, even though not commonly used in hydrocarbon (HC) flow problems. The importance of understanding and modeling viscous fluid stresses in HC flow is highlighted by the fact that commercial reservoir simulators do not contemplate such effects.

In this work, we developed a numerical simulator able to represent the single and two-phase flow (water-oil) using both Darcy and Brinkman equations. Equations were discretized using a finite difference method (FDM) and solved using the IMPES formulation. The simulator was verified against representative cases of 1D and 2D reservoirs submitted to water injection. Results from both equations are compared in terms of the velocity field, water saturation front, recovery factor, and water breakthrough in the producing well. In our results, we observe significant differences between the Brinkman and Darcy formulations when higher orders of permeabilities are attributed to high porosity media. Regions with high porosity (attributed to the karstified region) presented worse areal sweep efficiency and better displacement efficiency when the flow is modeled by Brinkman's equation. No influence of Brinkman's viscous term on conventional porous media is observed. This work shows that the Brinkman-based simulator is able to couple the use of Darcy's formulation in regions with low Darcy numbers (Da) (conventional reservoirs) and the use of Stokes' formulation in regions with high Da numbers (regions of intense karstification).

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