MEDIA
Chapter 9 Chapter 9 Applications of MsFEM to upscaling of displacements in heterogeneous porous media
9.1 Introduction
Enhanced oil recovery methods generally involve the injection of fluids that alter the flow properties of the natural rock-fluid system in reservoir. To study the motion of this combined mixture given the distribution of pores is a goal of a reservoir engineer. In this chapter we consider the coarse models for two-phase immiscible flow. A typical example considered here are one injection well and one production well.
Through the use of sophisticated geological and geostatistical modeling tools, engineers and geologists can now generate highly detailed, three dimensional representations of reser- voir properties. Such models can be particularly important for reservoir management, as fine scale details in formation properties, such as thin, high permeability layers or thin shale barriers, can dominate reservoir behavior. The direct use of these highly resolved models for reservoir simulation is not generally feasible because their fine level of detail (several millions) places prohibitive demands on computational resources. Therefore, the ability to coarsen these highly resolved geologic models to levels of detail appropriate for reservoir simulation (tens of thousands grid blocks), while maintaining the integrity of the model for purposes of flow simulation (i.e., avoiding the loss of important details), is clearly needed.
In this chapter we discuss the applications of MsFEM to the scale-up of displacement processes in heterogeneous cross-sectional models (2-D). Moreover, the disadvantages of these scale-up models and their improvements will be considered. We note that the coarse models described in this chapter are not the exact homogenized limits of detailed equations.
The homogenization of transport phenomena is a complicated problem. Our goal in this chapter is to approximate the average characteristics of transport flow (e.g., production rate) on a coarser grid. For example, given the fine description of the reservoir in 2000 x 2000 grid we would like to describe the reservoir displacements in a coarser grid, for example 100 x 100 grid.
The coarse models discussed in this chapter designed to generate a coarsened model that
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is capable of providing simulation predictions in close agreement with results using the orig- inal, detailed reservoir description. More specifically, it requires an agreement in the global pressure-flow rate behavior of the reservoir, the breakthrough (the time when oil reaches the production well) characteristics of the displacing fluid and, the post-breakthrough fractional flows (the production rate of oil later times) of all reservoir fluids.
The coarse model introduced in [19] achieves an efficient scale-up result by identifying high fluid velocities (via single phase flow calculation). These high flow areas lead to an early breakthrough of displacing fluids. The non-uniform coarsening idea developed in [19]
coarsens non-uniformly the fine grid geological descriptions in such a way that the high flow regions are finely gridded and the low flow regions are more coarsely gridded. The resulting coarsened reservoir description is able to model both average reservoir behavior and some important effects due to extremes in reservoir properties (such as early breakthrough of injected fluids), without prior knowledge of the global flow field. But there is a rather definite limit to the scale up that can be achieved through. non-uniform coarsening alone.
Indeed, given many high flow areas, this coarse model needs to resolve all these areas.
This decreases the efficiency of the scale-up. Let's note that MsFEM can be used for the calculation of average velocity field in this coarse model.
The coarse models we discuss in this chapter for two-phase flow speed up the scale up process and c ~ n he efficiently combined with MsFEM to describe the flow properties of the reservoir. The main idea of this method is to incorporate higher order moments (corre- lations) into coarse models. These ideas have been used before in material sciences and turbulent flow problems. The calculation of higher moments require fine detailed informa- tion of the velocity field or some robust approximation of it. Using the base functions of MsEM which do contain a robust approximation of details, we can calculate these higher moments. Numerical experiments show that our coarse models improve the results of ex- isting non-uniform coarsening approach. In this chapter we also simplify rigorous upscaled models derived in [50, 201 for periodic (or layered) non-ergodic flow and apply the results for flow in a typical reservoir cross-section.
This chapter is organized as follows. In the next section we present the governing equations for two-phase flow. In section 3 we discuss initial and boundary conditions for governing equations and a special case of two-phase flow, unit mobility case, used in this chapter. The flow features and their simulation have been discussed in section 4. In section 5
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we present the existing non-uniform coarsening approach along with some numerical exam- ples. We discuss the coarse model in section 6. In section 7 we derive rigorous homogenized equations for layered and non-ergodic periodic flow using the results of [50, 201. Sections 8, 9, 10, 11 are devoted to numerical implementations and results for the coarse model in the unit mobility case and their comparisons with the existing non-uniform coarsening approach. In sections 12 and 13 we extend the results of the unit mobility case to more general two-phase flow when the velocity weakly depend on time and present numerical results. We conclude the chapter with section 14 where my current research in upscaling of two-phase flow is discussed.
9.2 Governing equations
We consider a heterogeneous system which represent two-phase immiscible flow. Our inter- est is in the effect of permeability heterogeneity on two-phase flow. Therefore, we neglect the effect of gravity, compressibility, and capillary pressure, and consider porosity to be constant
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This system can be described by writing Darcy's law for each phase (all quantities are dimensionless) :
where vj are the Darcy's velocity for the phase j ( j = o, w oil, water), p is pressure, S is water saturation, k is the permeability tensor, kPj is the relative permeabilities of each phase and P j is the viscosity of the phase j. The Darcy's law for each phase coupled with mass conservation, can be manipulated to give the pressure and saturation equations:
which can be solved subject to the boundary and initial conditions (see next sections). The
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k r j ( S ) are referred as the relative permeabilities of the fine scale. A single set of relative permeability curves are assumed to describe the entire domain. Permeability is usually highly variable with the different value in each fine grid block.