Multiscale Methods for Modeling Fluid Flow Through Naturally Fractured Carbonate Karst Reservoirs

Abstract

Abstract Modeling and numerical simulations of Carbonate Karst reservoirs is a challenging problem due to the presence of vugs and caves which are connected via fracture networks at multiple scales. In this paper we propose a unified approach to this problem by using the Stokes-Brinkman equations which combine both Stokes and Darcy flows. These equations are capable of representing porous media (porous rock) as well as free flow regions (fractures, vugs, caves) in a single system of equations. The Stokes-Brinkman equations also generalize the traditional Darcy-Stokes coupling without sacrificing the modeling rigor. Thus, it allows us to use a single set of equations to represent multiphysics phenomena on multiple scales. The local Stokes-Brinkman equations are used to perform accurate scale-up. We present numerical results for permeable rock matrix populated with elliptical vugs. Both constant and variable background permeability matrices are considered and the effect the vugs have on the overall permeability is evaluated. Fracture networks connecting isolated vugs are also studied. It is shown that the Stokes-Brinkman equations provide a natural way of modeling realistic reservoir conditions, such as partially filled fractures. Introduction Naturally fractured karst reservoirs presents multiple challenges for numerical simulations of various fluid flow problems. Such reservoirs are characterized by the presence of fractures, vugs and caves at multiple scales. Each individual scale is an ensemble of porous media, with well defined properties (porosity and permeability) and a free flow region, where the fluid (oil, water, gas) meets no resistance from the surrounding rock [1]. The main difficulty in numerical simulations in such reservoirs is the co-existence of porous and free flow regions, typically at several scales. The presence of individual voids such as vugs and caves in a surrounding porous media can significantly alter the effective permeability of the media. Furthermore, fractures and long range caves can form various types of connected networks which change the effective permeability of the media by orders of magnitudes. An additional factor which complicates the numerical modeling of such systems is the lack of precise knowledge on the exact position of the interface between the porous media (rock) and the vugs/caves. Finally, the effects of cave/fracture fill in by loose material (sand, mud, gravel, etc.), the presence of damage at the interface between porous media and vugs/caves and the roughness of fractures can play very important role in the overall response of the reservoir. The modeling of fractured, vuggy media is traditionally done by using the coupled Stokes-Darcy equations [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. The porous regions is modeling by the Darcy equation [4,12], while the Stokes equation is used in the free flow region. At the interface between the two, various types of interface conditions are postulated [2, 3, 4, 5]. All of these interface conditions require continuity of mass and momentum across the interface. The difference comes when the tangential component of the velocities at the interface are treated. Each one of them proposes a different jump condition for the tangential velocities and/oir stresses, related in some way to the fluid stress. The selection of jump condition is subject to the fine structure of the interface and the flow type and regime (c.f. e.g. [13] and the references therein). Furthermore, these jump conditions introduce additional media parameters that need to be determined. These parameters can be obtained either experimentally, or computationally.