Upscaling of Unstable Miscible Displacements and Multiphase Flows Using Multiresolution Wavelet Transformation
Authors: Sahimi M. , Rasaei M.R. , Ebrahimi F. , Haghighi M. 
Published: December 2007

Abstract
We describe a novel approach for up-scaling of geological models of heterogeneous reservoirs. The method, which is based on the use of wavelet transformations, incorporates all the relevant data on the permeability, porosity, and other important properties of a reservoir at all the important scales. It generates a nonuniform computational grid which preserves its resolved structure in the regions around the wells and in the high-permeability zones, but is scaled up in the rest of the geological model. To demonstrate the accuracy and efficiency of the method, we apply it to the study of three important problems, namely, the pressure transient analysis, unstable miscible displacements, and two-phase flows. In the study of multiphase flows, we apply a new and robust front detection method. The numerical dispersion is eliminated by a high-order total variation diminishing (TVD) method which yields results that are as accurate as those obtained by the dual-mesh methods, but with a greatly-reduced computational cost. The speed-up in the computations is up to three orders of magnitude.

1. Introduction
Computer simulation of unstable miscible displacements and multiphase flows in petroleum reservoirs involves intensive computations1. Advances in measurement and estimation techniques, together with considerable progress in the development of methods for characterization of petroleum reservoirs, can provide significant amounts of data on various properties of the reservoirs.
Using such data, modern geostatistical techniques generate highly resolved geological models of heterogeneous petroleum reservoirs. To carry out reservoir simulations, the governing flow and transport equations are discretized, and the flow domain is divided into grid blocks which are characterized by the associated rock properties. Due to computational limitations (mainly computation time) it is not feasible to carry out multiphase flow simulations with a computational grid that corresponds to the highly resolved geological model. It is, therefore, necessary to up-scale the properties of the grid blocks of the geological model in order to develop a coarsened grid that can be used in reservoir simulation with a reasonable amount of computation time.
Up-scaling also plays a critical role in reservoir planning and management. In the past, it was common to forecast production based on reservoir simulations that used deterministic models of reservoirs - those that did not take into account the effect of the uncertainties in the distributions of the reservoirs' properties. However, deterministic models yield predictions that may be more harmful to reservoir planning and management than not having a model at all, since the appraisal of the potential production of a petroleum field must take into account the effect of the uncertainties in the properties of the field. Under such conditions, production forecast obtained by a stochastic model of a reservoir allows the quantification of the effect of the uncertainties on the reservoir performance by simulation of fluid flow in many possible realizations of the same reservoir, which are necessarily very time consuming. To reduce the simulation time, up scaling of the geological model is a natural and, perhaps, the only feasible way.
For simulation of single-phase flows, the most important parameter is the absolute permeability, for which up-scaling methods are well-established. However, simulation of multiphase flows involves, in addition to computing the equivalent absolute permeabilities of the coarsened blocks, adjustments to the phase flow through the connected blocks of the coarsened grid. To accomplish this many methods have been proposed, including up-scaling techniques that employ pseudo-functions, the so-called dual-mesh approaches, and simpler non-uniform coarsening methods.