Reservoir Performance Prediction Methods Based on Fractal Geostatistics

Abstract

Summary This paper describes a method for blending fractal statistics, detailed geologic data, finite-difference simulation, and streamtube models into a systematic approach for prediction of reservoir performance. The objective is to make accurate predictions for large-scale projects by detailed accounting of reservoir heterogeneity with reduced history-matching effort at a low overall cost. The method has been tested for waterflood and miscible gas injection projects with balanced injection/production volumes. Example applications are shown for four field cases. Introduction Recent papers1,2 have described methods for applying the concepts of geostatistics to reservoir modeling. Ref. 2, in particular, has shown how fractal distributions can be used to describe reservoir heterogeneity in simulation models. These papers demonstrate that detailed descriptions of reservoir heterogeneity can improve the accuracy of fluid-flow models. Because of computer limitations, however, increased detail limits the applicability of these methods for performance predictions of large-scale projects. Ref. 3 describes a hybrid finite-difference/streamtube model for calculating the performance of large-scale chemical flood projects. The concept of the method is to use a finite-difference model to represent displacement efficiency and vertical sweep. A stream-tube model completes the calculation for areal conformance. The work described in this paper generalizes the hybrid model approach by incorporating fractal geostatistics to improve the estimation of vertical sweep efficiency and an improved streamtube model formulated for more accurate calculation of areal conformance. This paper also describes procedures for coupling these methodologies into a system for large-scale project performance predictions and example applications of the procedures. The theoretical bases are presented in a brief overview, with detailed explanations of the underlying principles included in the references. Example applications are presented for three CO2 flood projects and a mature waterflood. Procedure for Calculating Reservoir Performance The procedure for performance calculation is based on the following steps.Establish the porosity/permeability character of the reservoir from well logs and cores and determine the statistical structure with the concept of random fractals.Use a random fractal-interpolation scheme based on the fractal characteristics determined from the well logs to project well data to the interwell region.Establish fluid-flow and displacement parameters from PVT, relative permeability, and, if available, coreflood data.Assemble geologic and fluid data into a highly detailed finite-difference cross-sectional model representing reservoir flow between a typical injector/producer pair. The cross-sectional model is highly detailed in the vertical direction. This model represents the geology on a scale much more detailed than conventional methods. The intent is to model heterogeneity near the same level of detail for which the data are available. A typical well-log resolution or sampling frequency of laboratory core measurement is 1 to 2 ft [0.3 to 0.6 m]. Therefore, this scale should be approached for the grid-block size in the vertical direction.Run the finite-difference model for projected flood conditions and develop a dimensionless characteristic solution that relates phase fractional flow at the producer to PV of fluid injected.Develop a streamtube model of the reservoir to represent areal conformance. The formulation of the streamtubes should incorporate variable mobility ratios, permeability trends, no-flow boundaries, etc.Couple the streamtube model with the characteristic solution to estimate field-wide project performance. Adjust gross fluid voidage to history value to check the model against known reservoir performance. Impose planned injection rate to forecast future performance. Each of these steps is described in detail below. Fractal Distributions. Petroleum engineers have long recognized the need to represent heterogeneity in reservoir fluid-flow calculations. Dykstra and Parsons4 described the reservoir as distinct layers of varying permeability. More recent papers have treated permeability distributions1 and discontinuous shales.5 This work uses fractal statistics to represent reservoir heterogeneity between wells as a random fractal variation superimposed on a smooth interpolation of correlated well-log values. The characteristics of the random fractal variation are determined from an analysis of the well logs or core properties used as the staring points for the interpolation. This amounts to a smooth interpolation witha superimposed texture. Fractals are characterized by the fact that they exhibit variations at all scales of observation and have partial correlations over all scales. Every attempt to divide such a geometry into smaller, more uniform regions results in the resolution of even more structure or roughness; the closer you look, the more detail you see. The variation of properties of many natural systems has been shown to be fractal in character. For instance, varve thickness in lake sediments and the flooding cycles of the Nile River have been shown to have fractal variation. The assumption of the method in this work is that the natural processes that created oil reservoirs yielded porosity/permeability distributions with a fractal character. The geometries of fractal distributions are characterized by their intermittent or "spotty" nature. This characteristic is quantified by a parameter called the intermittency exponent, H. Ref. 2 describes the theory and application of fractal statistics with particular reference to reservoir description. Step 1 - Analyzing Data for Statistical structure. To construct a heterogeneous reservoir cross-sectional model, the well-log and/or core data are analyzed for their intermittency exponent. This can be accomplished by testing the well-log or core data for their degree of correlation using the rescaled range (R/S) procedure.6 Log analysis using the R/S procedure typically indicates an average exponent, H, of 0.6 to 0.9. A value of H=0.5 indicates a totally random structure, while a value close to 1.0 implies a highly correlated structure. A layered sandstone might have an H between 0.85 and 0.9. Fractal Distributions. Petroleum engineers have long recognized the need to represent heterogeneity in reservoir fluid-flow calculations. Dykstra and Parsons4 described the reservoir as distinct layers of varying permeability. More recent papers have treated permeability distributions1 and discontinuous shales.5 This work uses fractal statistics to represent reservoir heterogeneity between wells as a random fractal variation superimposed on a smooth interpolation of correlated well-log values. The characteristics of the random fractal variation are determined from an analysis of the well logs or core properties used as the staring points for the interpolation. This amounts to a smooth interpolation witha superimposed texture. Fractals are characterized by the fact that they exhibit variations at all scales of observation and have partial correlations over all scales. Every attempt to divide such a geometry into smaller, more uniform regions results in the resolution of even more structure or roughness; the closer you look, the more detail you see. The variation of properties of many natural systems has been shown to be fractal in character. For instance, varve thickness in lake sediments and the flooding cycles of the Nile River have been shown to have fractal variation. The assumption of the method in this work is that the natural processes that created oil reservoirs yielded porosity/permeability distributions with a fractal character. The geometries of fractal distributions are characterized by their intermittent or "spotty" nature. This characteristic is quantified by a parameter called the intermittency exponent, H. Ref. 2 describes the theory and application of fractal statistics with particular reference to reservoir description. Step 1 - Analyzing Data for Statistical structure. To construct a heterogeneous reservoir cross-sectional model, the well-log and/or core data are analyzed for their intermittency exponent. This can be accomplished by testing the well-log or core data for their degree of correlation using the rescaled range (R/S) procedure.6 Log analysis using the R/S procedure typically indicates an average exponent, H, of 0.6 to 0.9. A value of H=0.5 indicates a totally random structure, while a value close to 1.0 implies a highly correlated structure. A layered sandstone might have an H between 0.85 and 0.9.

A Generalized Streamline Method to Predict Reservoir Flow

“…A one-dimensional solution, consistent witti the computed fractional flow, was then propagated along each tube using the Higgins and Leighton method . Excellent matches for nine field cases, including a C02 flood, veere reported using this method [Emanuel et al, 1989] .…”

“…For tracer flows the veldcity field does not change with time, and hence a streamline follows the particle pathway if there is no molecular diffusion or sub-gridblock-scale dispersion . The use of streamlines to describe single phase flow i s standard in the theo ry of flow in porous media [Bear, 1972] One of the most successful applications of streamtubes used a hybrid method to predict reservoir performance [Emanuel et al, 1989;Hewett and Behrens, 1991 ;Lake et al, 19811 . Effective fractional flows veere computed from conventional fine grid simulation of a representative vertical cross-section of the reservoir .…”

A Streamline Simulator to Model Field Scale Three-Dimensional Flow

“…For a given gridblock we must ultimately determine a single property, whether saturation or composition, at a given time t. For each streamline passing through a gridblock we first calculate (7) as the average between the inlet and exit TOF's for the streamline. We then use Eq.…”

“…Renard [20] proposed a similar idea to also account for changing well rates. Emanuel et al [7] extended the 2D streamtube model of Higgins & Leighton to a hybrid 3D model. Instead of mapping an analytical model to the areal streamtubes, they used a general finite difference 1D solution generated from a representative 2D cross-section, as proposed by Lake et al [14].…”

Wavelet upscaling based on piecewise bilinear approximation of the permeability field

“…The uncertainty in the description of a reservoir is normally dealt with by statistical methods (Gelhar, 1986;Dagan, 1989), or fractals (Emanuel et al, 1989;Hewett, 1986). The appropriate method is determined by considering the range of the correlations in the medium.…”