Estimation of Effective Permeabilities in the Lower Stevens Formation of the Paloma Field, San Joaquin Valley, California

Abstract

Summary. Simulation programs used in reservoir studies require effective gridblock permeability values representing spatial averages over large volumes of formation. Within such volumes, most reservoir rocks exhibit a considerable degree of heterogeneity, with permeability measurements often varying over several orders of magnitude. A satisfactory method for averaging permeabilities must consider not only the magnitude of permeability variations, but also the variations of permeability in permeability variations, but also the variations of permeability in space. Using experimental histograms and variograms to describe the heterogeneous permeability field within gridblocks, a Monte Carlo technique is used to estimate effective permeability. The method is illustrated by an example with actual permeability data from the lower Stevens formation of the Paloma field in Kern County, CA. Experimental downhole variograms of permeability showed a large nugget effect at short lags and a longer correlation structure with a range of 100 ft [30 m]. For the particular distribution and spatial correlation structure of these data, effective horizontal and vertical permeabilities were found to be respectively higher and lower than the geometric mean. Calculated effective permeabilities were found to be relatively insensitive to assumptions regarding the horizontal range of spatial correlation. This lack of sensitivity is caused by the large nugget effect or random short-range correlation structure. Effective permeabilities obtained with an approximate analytical relation were compared with effective permeabilities calculated numerically. Introduction Black-oil simulation programs used in reservoir studies require that average horizontal and vertical permeabilities be assigned to gridblocks representing volumes of formation often hundreds of acres in extent by tens of feet thick. Over such areas, most reservoir rocks exhibit a considerable degree of heterogeneity with core-sample permeability measurements sometimes varying by several orders of magnitude. To provide values representative of the portion of reservoir being modeled, average permeabilities must portion of reservoir being modeled, average permeabilities must account for this heterogeneity by considering not only variations in permeability magnitude but also variations in spatial arrangement. For the simple case of a heterogeneous formation consisting of parallel beds of uniform permeability (Fig. 1), the effective parallel beds of uniform permeability (Fig. 1), the effective permeability of the medium is the arithmetic mean for flow parallel permeability of the medium is the arithmetic mean for flow parallel to bedding and the harmonic mean for flow perpendicular to bedding. It can be shown that in the general case of a heterogeneous formation with an arbitrary spatial arrangement of permeabilities, the effective permeability of the medium must lie between permeabilities, the effective permeability of the medium must lie between the harmonic and arithmetic means. A more precise determination of effective permeability requires a description of the spatial arrangement of permeabilities within the formation. Because variations of permeability in a real reservoir cannot be described in permeability in a real reservoir cannot be described in a deterministic manner, statistical methods must be used. One possible method of describing reservoir heterogeneity in a statistical sense is to consider the histogram and variogram (Appendix A) of permeability measurements. The histogram characterizes the relative abundance and magnitude of sample permeabilities within the formation, and the variogram provides permeabilities within the formation, and the variogram provides a quantitative measure of statistical similarity between two permeability values separated by a given distance in a given direction. permeability values separated by a given distance in a given direction. Once a statistical description of permeability spatial variations is obtained, the effective permeability of a flow field may be estimated by the following numerical procedure.Using a Monte Carlo technique, generate permeability values on a three-dimensional (3D) nodal grid discretizing the rock volume so that the histogram and spatial correlation structure of the data are reproduced.By finite-difference methods, solve the equation for steady-state, single-phase flow at each node of the discretized flow field.From the solution of the flow equation, calculate the effective permeability of the system as a whole by dividing the volumetric flux through the system by the pressure gradient imposed on the boundaries of the system.Repeat the preceding steps to determine an average effective permeability for the specified permeability characteristics and flow-field dimensions. The technique is similar to the one used by Warren and Price; however, it has been generalized to take into account permeability spatial correlation. Other workers have also used this numerical approach to study the effect of aquifer heterogeneity on head and specific discharge. Analytical methods for estimating effective permeability have also been proposed, but they are applicable only under restricted conditions of permeability distribution and spatial-correlation structure. In the remainder of this paper, this numerical method for estimating effective permeability is illustrated by an example with data from a well in the Paloma field, Kern County, CA. Effective permeabilities calculated numerically are compared with those obtained from the analytical results of Gelhar and Axness. Geology Data for this study come from the lower Stevens formation in Well PA 26 × 3, located on the edge of the Paloma field. The data PA 26 × 3, located on the edge of the Paloma field. The data consist of 580 measurements of permeability from core samples taken every 1 ft [0.3 m] of the formation. The Stevens formation is a thick, mainly sandstone sequence of upper Miocene Age underlying the southern end of the San Joaquin Valley. The sandstones are usually coarse, poorly sorted, and of quartzose to arkosic composition. Graded bedding is common and paleowater depths of 2,000 to 6,000 ft [610 to 1830 m] are paleowater depths of 2,000 to 6,000 ft [610 to 1830 m] are estimated from foraminiferal faunas. The Stevens is interpreted as a sequence of sediment gravity-flow deposits that accumulated in a deep-sea fan system. The lower Stevens intersected by Well PA 26 × 3 is a very thick and massive sequence consisting of medium- to coarse-grained sandstones believed to belong to the middle fan facies of a deep-sea fan depositional environment. SPERE P. 1301