American Institute of Mining, Metallurgical, and Petroleum Engineers, Inc.



As an introduction to a session on variational methods in reservoir modeling, the purpose of this paper is to emphasize the rich variety inherent in the structuring of such methods. We illustrate this with a few examples, chosen partly to suggest that there may remain more such methods yet to be discovered and tried than methods already known and tested. This paper is not intended to be a survey, nor is it intended to present a full account even of the ideas illustrated. Indeed, a complete compendium of known variational methods potentially applicable to reservoir simulation and all the details of the methods discussed would be needlessly lengthy and distract from our main purpose: inspiring the imagination of the reader to the exciting potential of that which may yet come.

The variety of variational methods can best be illustrated by showing examples of a number of ways to treat the same problem. For this it is helpful to choose a simple system which retains the character of reservoir simulation problems. For this we consider the equations on a bounded region Rn for 0 t T



where a and are strictly positive smooth functions, Q(x,t) is flow data, = u if Q 0, and (x,t) is data for Q greater than 0.

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