An inversion method for the integration of dynamic (pressure) data directly into statistical moment equations (SMEs) is presented. The method is demonstrated for incompressible flow in heterogeneous reservoirs. In addition to information about the mean, variance, and correlation structure of the permeability, few permeability measurements are assumed available. Moreover, few measurements of the dependent variable are available. The first two statistical moments of the dependent variable (pressure) are conditioned on all available information directly. An iterative inversion scheme is used to integrate the pressure data into the conditional statistical moment equations (CSMEs). That is, the available information is used to condition, or improve the estimates of, the first two moments of permeability, 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 at a time. In the MC approach, estimates of the prediction uncertainty are obtained from statistical post-processing of a large number of inversions, one per realization.
Several examples of flow in heterogeneous domains in a quarter-five-spot setting are used to demonstrate the CSME-based method. We found that as the number of pressure measurements increases, the conditional mean pressure becomes more spatially variable, while the conditional pressure variance gets smaller. Iteration of the CSME inversion loop is necessary only when the number of pressure measurements is large. Use of the CSME simulator to assess the value of information in terms of its impact on prediction uncertainty is also presented.
The properties of natural geologic formations (e.g., permeability) rarely display uniformity or smoothness. Instead, they usually show significant variability and complex patterns of correlation. The detailed spatial distributions of reservoir properties, such as permeability, are needed to make performance predictions using numerical reservoir simulation. Unfortunately, only limited data are available for the construction of these detailed reservoir-description models. Consequently, our incomplete knowledge (uncertainty) about the property distributions in these highly complex natural geologic systems means that significant uncertainty accompanies predictions of reservoir flow performance.
To deal with the problem of characterizing reservoir properties that exhibit such variability and complexity of spatial correlation patterns when only limited data are available, a probabilistic framework is commonly used. In this framework, the reservoir properties (e.g., permeability) are assumed to be a random space function. As a result, flow-related properties such as pressure, velocity, and saturations are random functions. We assume that the available information about the permeability field includes a few measurements in addition to the spatial correlation structure, which we take here as the two-point covariance. This incomplete knowledge (uncertainty) about the detailed spatial distribution of permeability is the only source of uncertainty in our problem. Uncertainty about the detailed distribution of the permeability field in the reservoir leads to uncertainty in the computed predictions of the flow field (e.g., pressure).