Assigning Effective Values to Simulator Gridblock Parameters for Heterogeneous Reservoirs

Abstract

Summary Characterization of heterogeneous reservoirs for simulation studies by use of a statistical component to the reservoir description combined with calculation of effective permeabilities is discussed through a case study of a fluvial reservoir. Inspection of whole core and logs from the Sherwood reservoir revealed that it is extremely heterogeneous, containing short shales embedded in a mixture of sands and silts. A statistical approach was used to quantify the distribution of core-plug porosity and permeability measurements to discriminate between significantly different rock types. A successive rescaling procedure was then adopted in which we first calculated effective values for each rock type from the core-plug data. These results were combined with statistics on the spatial distribution of the rock types to calculate effective values for the nonshale part of each layer. Finally, the effect of the shales on the vertical permeability was incorporated. The resulting anisotropy ratios, on the order of 10 - 3, were used successfully in black-oil model studies, a good match between the simulator predictions and observed production data being achieved with minimal fine tuning of the model parameters. Introduction The aim of this paper is to describe how the variability and uncertainty associated with heterogeneous reservoirs can be characterized for simulation by a statistical component to the reservoir description combined with some recently developed techniques for calculating effective permeabilities. The techniques are illustrated and commented upon by application to the Sherwood reservoir (at Wytch Farm, Dorset, the U.K.'s largest onshore oil field), which accumulated in rocks deposited in a fluvial environment. The particular intention of the paper, however, is to discuss the merits of these techniques and, more generally, to point out some of the factors that influence effective permeabilities that are not always apparent to the unwary. At the time that the Sherwood study began, only five wells had been cored or logged. Detailed correlation was not possible owing to a combination of three factors: lack of data, large well spacing (up to 1500 m [4,920 ft]) and the limited lateral extent of lithofacies in a fluvial environment. Thus, only a broad subdivision of the reservoir into Lower, Middle, and Upper members was made. The sequence is interpreted as a transition from mainly channel facies in the Lower member, through sheetflood facies in the Middle, to mainly playa margin facies in the Upper. The Upper Sherwood, which contains most of the oil, was divided into three layers of equal thickness to allow better modeling of fluid displacement. Inspection of whole cores and logs showed that the Sherwood was composed of a heterogeneous mixture of shales (which were uncorrelatable between wells) embedded in a variety of different rock types. Early simulation studies indicated that the choice of vertical/horizontal permeability anisotropy ratio was a key parameter in understanding the oil-displacement mechanism and thus influenced future development plans for the reservoir. A study was therefore initiated to determine the effect of reservoir heterogeneity on this ratio. This was done by calculating effective horizontal and vertical permeabilities, kve and kHe, for each layer of the simulation model. The lack of a precise reservoir description, combined with a high degree of variability, suggested that the spatial distribution of heterogeneities should be quantified statistically. The wide range of scales on which heterogeneities were observed (core-plug to shales) necessitated the use of a scaling-up procedure in which we first calculated effective permeabilities for the finest-scale heterogeneities and then used these as the data required for the next scale-up. The methodology described here is very flexible and can accommodate the acquisition of new data at a later date. It can also incorporate more complex models of the spatial distribution of the heterogeneities if such information becomes available at a later date. Although the example relates to a fluvial reservoir, we suggest that the techniques and the way in which they are used may find wider application. An essential condition for the success of the approach is close cooperation between geologists and engineers from the beginning of the study. Effective Values The effective permeability, k, of a heterogeneous medium is the permeability of an equivalent homogeneous medium that, for the same boundary conditions, would give the same flux. Thus, the k, of a heterogeneous medium depends on both the boundary conditions and the distribution of heterogeneities, which in turn depends on the volume being considered. Effective permeability is not therefore an intrinsic property of the medium in the same way that, for example, porosity is. Further, this argument also applies to direct measurements of permeability. Thus, the value obtained from a heterogeneous core plug is really an effective permeability for the imposed boundary conditions of no flow through the sides of the core and uniform pressures over the inlet and outlet faces. Unless the heterogeneities are insignificant or their scale length is small compared with the core, such a value is not a unique absolute permeability for that piece of rock because, under a different pressure regime, a different value would be obtained. We must therefore try to make any calculation (or measurement) of k, independent of, or at least insensitive to, the boundary conditions. This can be done by calculating effective permeabilities for volumes of rock that are large compared with the scale of the heterogeneities that they contain, This is particularly important when a scaling-up procedure is used to get the k, of a larger volume of rock (such as of a simulator gridblock) from measurements at a smaller scale. If significant heterogeneities occur on the same scale as the "averaging" volume, however, it will not be possible to calculate the k, of this volume under the reservoir pressure regime because the latter is unknown at the scale that affects ke-i.e., the subgridblock scale. In this study, effective values were calculated for each layer of the model to ensure that the scale of heterogeneities was small compared with the scale of the averaging volume and because there was no observable areal variation in the nature of the heterogeneity across the layers. Methods of Calculating Effective Permeabilities In this section, we briefly review two recently developed methods of calculating effective permeabilities for heterogeneous reservoirs. Both are for the flow of a single-phase, incompressible fluid and are derived by equating expressions for the flux through a volume of the real heterogeneous medium with the flux through an equal volume of an equivalent homogeneous medium. The boundary conditions in both cases are constant pressures over the inlet and outlet faces, pI and po, and no flow through the sides of the model, which are normal to the imposed pressure gradient. Rotation of the boundary conditions gives effective permeabilities in the three mutually perpendicular directions, which are aligned with the axes of the model. SPERE P. 455^