Abstract
This paper considers the problem of estimating the shape of a petroleum reservoir on the basis of pressure data from wells within the boundaries of pressure data from wells within the boundaries of the reservoir. It is assumed that the reservoir properties, such as permeability and porosity, are properties, such as permeability and porosity, are known but that the location of the boundary is unknown. Thus, this paper addresses a new class of history-matching problems in which the boundary position is the reservoir property to be estimated. position is the reservoir property to be estimated. The problem is formulated as an optimal-control problem (the location of the boundary being the problem (the location of the boundary being the control variable). Two iterative methods are derived for the determination of the boundary location that minimizes a functional, depending on the deviation between observed and predicted pressures at the wells. The steepest-descent pressures at the wells. The steepest-descent algorithm is illustrated in two sample problems:the estimation of the radius of a bounded circular reservoir with a centrally located well, andthe estimation of the shape of a two-dimensional, single-phase reservoir with a constant-pressure outer boundary.
Introduction
A problem of substantial economic importance is the determination of the size and shape of a reservoir. Seismic data serve to define early the probable area occupied by the reservoir; however, probable area occupied by the reservoir; however, a means of using initial well-pressure data to determine further the volume and shape of the reservoir would be valuable. On the basis of representing the pressure behavior in a single-phase bounded reservoir in terms of an eigenfunction expansion, Gavalas and Seinfeld have shown how the total pore volume of an arbitrarily shaped reservoir can be estimated from late transient pressure data at the completed wells. We consider pressure data at the completed wells. We consider here the related problem of the estimation of the shape (or the location of the boundary) of a reservoir from pressure data at an arbitrary number of wells.
For reasons of economy, the time allowable for closing wells is limited. It is important, therefore, that any method developed for estimating the shape of a reservoir be applicable, in principle, from the time at which the wells are completed until the current time. Thus, the problem we consider here may be viewed as one in the general realm of history matching, but also one in which the boundary location is the property to be estimated rather than the reserved physical properties. The formulation in the present study assumes that everything is known about the reservoir except its boundary. In actual practice, the reverse is generally true. (By the time sufficient information is available regarding the spatial distribution of permeability and porosity, the boundaries may be fairly well known.) Nevertheless, relatively early in the life of a reservoir, when initial drillstem tests have served to identify an approximate distribution of properties, it may be of some importance to attempt to estimate the reservoir shape. Since knowledge of reservoir properties such as permeability and porosity is at properties such as permeability and porosity is at best a result of initial estimates from well testing, core data, etc., the assumption that these properties are known will, of course, lead only to an approximate reservoir boundary. As the physical properties are identified more accurately, the reservoir boundary can be more accurately estimated. It is the object of this paper to formulate in a general manner and develop and initially test computational algorithms for the class of history-matching problems in which the boundary is the unknown property.There are virtually no prior available results on the estimation of the location of the boundary of a region over which the dependent variable(s) is governed by partial differential equations. The method developed here, based on the variation of a functional on a variable region, is applicable to a system governed by a set of nonlinear partial differential equations with general boundary conditions. The derivation of necessary conditions for optimality and the development of two computational gradient algorithms for determination of the optimal boundary are presented in the Appendix. To illustrate the steepest-descent algorithm we present two computational examples using simulated reservoir data.
SPEJ
P. 19
A modification of steepest descent method for solving large-scaled unconstrained optimization problems
