Geostatistical Modeling of Transmissibility for 2D Reservoir Studies

Abstract

Summary A geostatistical approach is used to characterize reservoir transmissibility with the aim of assigning simulator parameters in 2D models. Transmissibility is represented as a spatial random function where heterogeneity is described by the probability distribution and the variogram of sample values. The key element of the geostatistical model is the definition of block transmissibilities as spatial geometric averages. Published analytical results have shown that the effective transmissibility of infinite, statistically isotropic flow fields is equal to the ensemble geometric mean. Numerical results presented here show that a spatial geometric average is an excellent approximation of effective transmissibility in such finite fields as simulator gridblocks. The geostatistical model for transmissibility is used to show that the mean and variance of block-averaged values depend on the averaging area. As the averaging area is increased, mean block transmissibility decreases toward the ensemble geometric mean while the block variance decreases to zero. The geostatistical model is also used to investigate the kriging of block transmissibilities from well data. The current method of correcting bias in kriged values is found to cause artifacts of gridblock size in flow simulation results. The simpler, uncorrected kriging estimator is shown to preserve overall flow-field transmissibility, regardless of gridblock size. Introduction Ever since the landmark paper of Warren and Price,1 there has been a growing awareness that computer models of fluid flow in reservoirs have become much more sophisticated than actual reservoir descriptions. Reservoir characterization and the assignment of simulator parameters are increasingly regarded as essential elements of a successful reservoir study.2 This paper addresses the very specific, yet important, problem of determining gridblock transmissibilities for 2D areal models. Reservoir transmissibility characterization is approached here within the framework of geostatistics.3 The geostatistical approach for studying heterogeneous porous media is well established in the hydrology literature4 and is gaining increasing acceptance in petroleum applications.5 In this approach, transmissibility is viewed as a regionalized variable, representing an outcome from a spatial random function. The heterogeneity of transmissibility is characterized by the probability distribution function and the spatial autocovariance (or the variogram) of the random function, as estimated from sample data. The main goal of this paper is to develop basic geostatistical concepts and procedures to estimate gridblock transmissibilities used in 2D simulators. Block transmissibilities are defined as spatial geometric averages that, under certain conditions, are shown to give excellent approximations to true effective block values. This paper differs from previous work6–8 in regard to the geostatistical estimation of permeability or transmissibility because it examines the important change of support-scale effects caused by nonlinear spatial averaging. This paper also shows that the use of accepted bias-correction procedures for kriged block values introduces artifacts of gridblock size in flow simulation results. Theoretical concepts are illustrated as they are developed with examples from an actual reservoir amenable to a 2D study. The data used here come from the "H" pool of the Lower Cretaceous Crystal Viking field in South Central Alberta.9 The Crystal Viking reservoir is interpreted as an estuarine tidal-channel/bay complex lying uncomformably on the regional inner-shelf/lower-shoreface facies of the Viking formation. The H pool reservoir consists primarily of carbonaceous, fine- to medium-grain sandstones believed to be shallow channel-bar deposits. The reservoir is hydrodynamically separated from the tidal-Channel sandstones of the main "A" pool by impermeable estuary/bay-fill mudstones. The H sandstone is elongated north/south and extends over 1500 ha, reaching a maximum thickness of 13 m. Core-derived transmissibility values are available from the 23 wells shown in Fig. 1. Geostatistical Model for Transmissibility This section presents the basic elements of a geostatistical model for transmissibility. The important concept of block-averaged transmissibility is defined, and its relationship to effective transmissibility is discussed. The transmissibility T(x) at a point x=(u, v) in the horizontal plane is defined as the integral of permeability, k(u, v, w) over the reservoir thickness w2-w1 divided by viscosity:Equation 1 Point permeabilities and transmissibilities are assumed to be scalar. Viscosity, µ, is assumed to be constant and equal to unity. Consider Y(x)=ln T(x), the natural logarithm of T(x). Following the usual geostatistical approach,3Y(x) is modeled as a second-order stationary and ergodic spatial random function. The expected value (mean), variance, and autocovariance function of Y(x) are given byEquations 2–5 where s(h) and ?(h) are the autocovariance and variogram functions of the distance lag h, respectively. The ensemble geometric mean of T(x), TG, is, defined byEquation 6 The cumulative distribution plot of transmissibilities (Fig. 2) shows that the data are represented adequately by a log-normal, frequency-distribution model represented by the straight line. This observation is fairly typical of permeability or transmissibility measurements10 and serves to justify the study of the variable Y(x). Table 1 summarizes statistics of the transmissibility data. The minor discrepancies between data and model values result mainly from a single outlier value (743 md·m/Pa·s) that does not fit the model. This value is associated with a high-permeability conglomeratic streak that may act as a preferential flow channel in the reservoir. Unfortunately, insufficient data exist to characterize the important spatial continuity of such high-transmissibility zones.