Conditional Statistical Moment Equations for Dynamic Data Integration in Heterogeneous Reservoirs

Abstract

Abstract We present an inversion method for the integration of dynamic data directly into the statistical moments of the dependent variables, such as pressure, for flow in heterogeneous porous media. We consider the problem of incompressible Darcy flow in a heterogeneous domain. We assume that a few measurements of permeability are available and that the mean, variance, and correlation structure of the permeability field are known. In addition, we assume that a few measurements of the dependent variable, e.g. pressure, are available. The first two statistical moments of the dependent variable (pressure) are conditioned on all available information directly. The approach is based on applying an iterative inversion scheme to integrate dynamic data into the conditional statistical moment equations (CSME). That is, in addition to statistical information about the permeability field, measurements of static (permeability) and dynamic (pressure) variables are used to condition, or improve the estimates of, the mean and second moments of pressure and velocity directly. This is different from Monte Carlo (MC) based geostatistical inversion techniques, where conditioning on dynamic data is performed for one realization of the permeability field. In MC based schemes, estimates of the prediction uncertainty are obtained from statistical postprocessing of a large number of inversions, one per realization. Our moment-based dynamic data integration method is demonstrated using several examples of flow in heterogeneous domains in a quarter-five-spot setting. We show that as the number of pressure measurements increases, the conditional mean pressure exhibits more spatial variability while the conditional pressure variance becomes smaller. Our experience indicates that iteration of the CSME inversion loop is only necessary when the pressure measurements deviate significantly from the prior (conditional) expectations. We also discuss how the CSME flow simulator can be used to assess the value of both static and dynamic information, measured in terms of reduction of prediction uncertainty.