Multiphase Flow in Porous Media

Abstract

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re·use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks.The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. 216113020·543210 PREFACEThe past decade has seen remarkable growth in research related to petroleum reseIVoir simulation. This growth reflects several developments, not the least of which is the increased interest in oil recovery technologies requiring sophisticated engineering. Augmenting this interest has been the broader availability of supercomputers capable of handling the tremendous computational demands of a typical reseIVoir simulator.The field of reseIVoir simulation incorporates several major facets of applied mathematics. First, in view of the varieyt and complexity of the processes encountered, it is imperative that the modeler adopt a systematic approach to establishing the equations governing reseIVoir flows. Second, the mathematical structure of these flow equations needs to be carefully analyzed in order to develop appropriate and efficient numerical methods for their solution. Third, since some aspects of the discretized flow equations are typically stiff, one must develop efficient schemes for solving large sparse systems of linear equations.This monograph has three parts, each devoted to one of these three aspects of reseIVoir modeling. The text grew out of a set of lectures presented by the authors in the autumn of 1986 at the IBM Scientific Center in Bergen, Norway. We feel that it is only appropriate to caution the reader that many of the ideas that we present in this monograph do not reflect standard approaches in petroleum reseIVoir simulation. In fact, our aim is to outline promising new ways of attacking reseIVoir simulation problems, rather than to compile another textbook for the mainstream.We owe a debt of gratitude to the IBM Scientific Center for allowing us the opportunity to present these lectures. In particular, we wish to thank Bengt Aspvall and Trond Steihaug for helping to organize the lecture series. Dr. Aspvall, as manager of the Institute, deseIVes special credit for his diligence and sacrifice on our behalf. Such hospitality makes scientific exchange a joy. We also wish to thank the many attendees at our lectures; their questions and discussions contributed greatly to this monograph.Finally, we wish to thank those who provided support for our work in writing this monograph.

Biodecolorization and Bioremediation of Denim Industrial Wastewater by Adapted Bacterial Consortium Immobilized on Inert Polyurethane Foam (PUF) Matrix: A First Approach with Biobarrier Model

“…The four bacterial species present in the porous medium develops strong biofilm forming microbes with the nutrients. The fundamental equation for saturated transient ground water flow of constant density, in horizontal direction, can be written in the form (Allen et al, 1988), Here h (x, t) denotes the hydrolic head, S s is the specific storage, K is the saturates hydraulic conductivity and f (x, t) represents source. The specific discharge vector υ (x, t) called Darcy velocity represents the speed of the water.…”

A Numerical Model for Trickle Bed Reactors

“…It is well known that capillary pressure acts as a diffusive force [1]; in these simulations, we note that capillary pressure causes the saturation front to "smear." In addition, in the region downstream from the front, the capillary pressure causes the liquid phase to diffuse from the high-porosity region to the low-porosity region.…”

“…Figure 3 shows an enlarged view of these solutions around the shock. Here we can see that the simulation using van Leer slopes predicts an incorrect location of the rarefaction and the shock (this problem is alluded to in [1,6]). On the other hand, the simulation using modified van Leer slopes (i.e., the van Leer slopes with the χ correction) more accurately predicts the location of the rarefaction and the shock.…”

“…Several authors [1,6] have suggested that in the presence of non-convexity, the secondorder Godunov method can produce shocks that violate the entropy condition (see Section 4.1 for an example). To prevent these entropy-violating shocks from occurring in this work, we use a variation of the approach in [6] to modify the algorithm in [10] slightly and apply it to our problem to handle regions around saturation fronts.…”

“…Solving the Riemann problem is central to Godunov's method; as a result, it has been widely studied in the literature. Allen et al [1] and LeVeque [15] provide a general description of Riemann problems for scalar conservation laws.…”