Fractal Distributions of Reservoir Properties and Their Use in Reservoir Simulation

Abstract

Geostatistics has become a popular way to distribute reservoir properties between wells. One of the geostatistical methods being used is fractal geostatistics. Most porosity well logs have been found to have a fractal character. An analysis of vertical and horizontal logs, core photos, and outcrop photos has led to a rather simple model to describe porosity and permeability distributions. This representation has been tested in reservoir simulation of a mature waterflood and found to match production history with very little history matching. Fractal distributions were found to require much less history matching than classical layer cake models. Conoco is now actively applying this new technology to a number of its reservoirs. Introduction We would like to have exact answers to where oil is and how it can be produced. Based on our limited core and log information, however, this is not possible. Usually, we have measurements on less than one billionth of a reservoir. With this small a sample set, all distributions are suspect. Still, the economics of oil recovery force us to get the best estimates we can. That means using as much of the data as wisely as possible. The approach here is to use statistical analyses of our data and a rather simple statistical model to leverage these data. This paper illustrates how statistical tests can be used to identify fractal distributions. The results of these tests are fed into the statistical model to generate distributions of reservoir properties between wells. These distributions are then used in reservoir simulations. The statistical model is developed from examining core photographs and is also consistent with many porosity well logs. This similarity is used to generate porosity distributions for reservoir simulations. Porosity-permeability relationships are used to assign permeability. With the resulting distributions, we compare three different reservoir simulation methods on a mature waterflood in southwestern United States. The first reservoir simulation method is a hybrid finite-difference/streamtube method that starts with two-dimensional (2-D) vertical cross sections with relatively fine grids. The second method uses pseudorelative permeabilities based again on 2-D cross sections. The third method uses three-dimensional (3-D) distributions. A mature waterflood in a carbonate reservoir serves as a field test case. The average cumulative oil recovery from 20 realizations is nearly the same for all three methods. P. 619^

Determination of Fractal Parameters of Fracture Networks Using Pressure-Transient Data

“…It is also well known that most porosity well logs have been found to have a fractal character; in fact, the fractal dimension can be obtained by using techniques like rescaled range, variogram, and spectral methods. [5][6][7][8][9] Thus, to determine the fractal model parameters, we either use information from porosity well logs, or we try to attain this information from another type of test.…”

Numerical Estimation of the Degree of Reservoir Permeability Heterogeneity Using Pressure Drawdown Tests

“…Then, the fractally distributed permeability values for three different fractal dimensions, D (2.1, 2.5, and 2.9) were entered as data to the simulator. It has been suggested previously that the permeability distribution in x-y direction (horizontal) represents a fBm (Hewett, 1986;Hewett and Behrens, 1990;Perez and Chopra, 1991;Berta et al, 1994;Tubman and Crane, 1995). This might be fGn (Hewett, 1986;Hewett and Behrens, 1990;Perez and Chopra, 1991;Berta et al, 1994;Kelkar and Shibli, 1994;Tubman and Crane, 1995) or fBm (Perez and Chopra, 1991) in x-z direction (vertical).…”

“…In practice, however, this range of fractal dimension value of permeability distribution is quite unrealistic and the fractal dimension of horizontal permeability distribution is expected to be less than 2.9 (Hewett, 1986;Hewett and Behrens, 1990;Perez and Chopra, 1991;Berta et al, 1994;Kelkar and Shibli, 1994;Tubman and Crane, 1995). The following representation of Equations (7) and (8), rounded-off to the nearest decimal point, was tested using the numerical model and found to be applicable with the acceptable margin of error…”

“…Permeability distributions can be generated using stochastic methods (Sagar et al, 1993;Yadavelli et al, 1994Yadavelli et al, , 1995, simulated annealing (Deutsch and Journel, 1994) and fractal geometry (Chang and Yortsos, 1990;Babadagli, 1999). In a quantitative description of these distributions, the fractal dimension of the distribution (Hewett, 1986;Hewett and Behrens, 1990;Chang and Yortsos, 1990;Berta et al, 1994;Babadagli, 1999) and Dykstra-Parson's coefficient (Sagar et al, 1993;Yadavelli et al, 1994) have been used previously. Once the permeability distribution is quantified one way or another, it can then be correlated to the pressure response.…”

“…Fractal techniques have been useful in generating representative permeability fields of carbonate systems (Emanuel et al, 1989;Perez and Chopra, 1991;Berta et al, 1994;Kelkar and Shibli, 1994;Babadagli, 1999). If the fractal characteristics of the permeability distribution in a reservoir are available, the representative models can be generated using these fractal properties and these models are used to simulate any transport processes.…”