Stochastic Modeling Of Space Dependent Reservoir-Rock Properties
Authors: Dimitrakopoulos R. 
Published: March 2010

Abstract
Numerical modeling of space dependent and variant reservoir-rock properties such as porosity, permeability, etc., are routinely used in the oil industry. The techniques commonly applied are simplistic resulting in poor estimates. Alternatively, a stochastic framework in terms of the theory of Intrinsic Random Functions of order k may be used. Then, "best" estimators in the well established stochastic sense and their associated estimation variance can be derived. In addition, reservoir-rock properties may be simulated in the form of stochastic random fields and subsequently conditioned to reproduce the data measurements and their statistical properties. The advantages of stochastic approach are bared in the use of patterns of variation within the existing data in terms of generalized covariance or variograms.
Stochastic models can be used differently depending on their properties. Estimated models may be used in describing and assessing the performance of the reservoir or the estimation of reserves. Simulated ones can provide alternatives to test the sensitivity of flow simulation or to tackle estimation of effective block permeabilities.

Introduction
The actual performance of a reservoir during the exploitation time is reproduced-predicted from reservoir simulation studies. A controlling factor in the quality of predictions is the numerical description of reservoir-rock properties such as porosity, permeability and fluid saturations(1,2,3). Furthermore, while the sophistication of fluid transport formulations, numerical methods and numerical models continues to increase, quantitative geological modeling remains significantly less advanced. As a result, prediction problems arise from unreliable reservoir description(4). For these reasons, it has already been suggested(3,4) that more emphasis be given to increasing the sophistication and improving the certainty of quantitative reservoir characterization.
During the last few years, attempts at reservoir description(5) have had two major aspects. The first is a qualitative-geological and the second quantitative-numerical. Although attention was given and progress has been made in the former(6,7), there is much room for improvement in the latter, so that qualitative information provided by the first aspect can be expressed quantitatively.
The present study is a natural step in the continuing effort to improve geological reservoir description. It attempts to provide basic elements of a generalized stochastic framework(8,9) which allows the effective transformation of geological descriptions to computer processable numerical equivalents. The reasons why stochastic models, as opposed to deterministic, chosen are: (1) geological variables exhibit a locally random behavior (Fig. I), but on average there is a structural aspect in spatial variability (correlation structure) which is expressed with the similarity of neighbouring values (Fig. 2); the stochastic approach (2) enables the estimation of large volumes from core samples using best (optimal) estimators; (3) provides a measurement of uncertainty in the above estimation; and (4) allows the generation (simulation) of fields of values which "look like" reality in terms of their variation pattern. The stochastic framework in modeling geological variables was originally introduced in the mining industry(10,11) and found extensive use in ground water hydrology(12,13). In reservoir engineering, few authors have attempted to use stochastic methods for parameter estimation(14,15) or simulation(16,17,18) using restricting