Estimation of two-phase petroleum reservoir properties

Abstract

The estimation of petroleum reservoir properties on the basis of production rate and pressure observations at the wells is an essential component in the prediction of reservoir behavior. The reservoir properties to be estimated appear as parameters in the partial differential equations describing the flow of fluids in the reservoir. The estimation of these properties is referred to variously as the inverse or identification problem or as history matching. In this dissertation, new results have been obtained pertaining to the estimation of petroleum reservoir properties. Most of the prior analysis of the reservoir parameter estimation problem has been confined to reservoirs containing a single fluid phase, e.g., oil. We consider here reservoirs that contain two fluid phases, e.g., oil and water. The parameters to be estimated in such a case are the porosity and permeability, which depend on spatial location, and the saturation-dependent relative permeabilities. In this work we treat two basic problems in reservoir parameter estimation: (1) establishing the ability to estimate the desired parameters (so-called identifiability), and (2) developing and testing a new algorithm, based on optimal control theory, to carry out the estimation. In regard to problem (1), we have extended the classic analytical (Buckley-Leverett) solution for incompressible flow to heterogeneous reservoirs. Analysis for an incompressible water flooding situation shows that the spatially varying properties at locations behind the saturation front have an effect on the pressure solution. The spatially varying properties can be uniquely determined based on data taken up to the time of water breakthrough. Only an integral value of the porosity can be determined from the water-oil ratio data alone; however, the spatially varying porosity may be determined when the initial saturation varies with location. The values of the relative permeabilities which are identifiable, and the information about the relative permeabilities obtained for other intervals of saturation, is established. Analytical expressions are derived for the sensitivity of the pressure and water-oil ratio observations to parameters appearing in functional forms of the relative permeabilities. When the relative permeabilities are represented as exponential functions, the coefficients and exponents can be uniquely determined. For problem (2), an algorithm is developed for the estimation of porosity, permeability and the relative permeabilities for two-phase, compressible reservoirs. This work represents the first study for which relative permeabilities have been estimated based on a model generally used to represent fluid flow in petroleum reservoirs. An objective function, composed of the weighted sum of squares of the deviations between the observed and calculated values of pressure and water-oil ratio, is minimized by a first-order gradient method based on optimal control theory. The algorithm is tested for one and two-dimensional hypothetical water floods. The algorithm performed well for problems in which the porosity, permeability and relative permeability exponents were simultaneously estimated. The increase from one to two spatial variables does not appear to change the properties of the estimation problem. Small observation errors are shown not to significantly affect the convergence of the estimates.