A Comparison of Mapping Schemes for Reservoir Characterization
Authors: Zwahlen Eric D. , Patzek Tad W. 
Published: December 2007

Abstract
Often one is interested in the relationships between various types of geological data, such as depth, formation thickness, position, permeability, porosity, etc. However, geological data are difficult to analyze due to their spatial complexity, and because obvious analytical maps of one type of data onto another seldom exist. Typically, multivariate regression techniques are used to correlate geological data. This approach requires often stringent assumptions about the functional form of the data. Here, we compare a method that does not require such assumptions: Alternating Conditional Expectation (ACE), with two other approaches that do: Geostatistics and MATLAB. The ACE method, developed by Breiman and Friedman (1985), is an iterative optimal transformation that maximizes the correlation between the transformed dependent variable and the sum of the transformed independent variables. Even with multiple independent variables, the specific functional dependence of each variable can be found. A thorough geostatistical analysis of the full data set is treated as the ground truth, against which the two other methods are compared. The field example from the South Belridge diatomite, Kern County, CA, involves correlating the true vertical depths of diatomite layer boundaries with their × (EW) and y (NS) coordinates. The geological data from approximately 70 wells in an area about 1000 feet by 1000 feet have been used. We compare how well the three methods can smooth and interpolate the data to produce layer mappings. In the end, the most laborious of the three methods, geostatistics with Surfer as a plotting tool, wins hands down. The other two methods are far more limited.

Introduction
Geologic data may be a function of many variables. The data also may contain scatter due to inherent heterogeneity and data collection errors. One must perform some type of regression to correlate the data with each of the variables. The functional form of a specific variable may give physical insight into the data, such as faulting or the location of a different type of rock.
In most regression schemes, a functional form is assumed, and then the equation is fit to the data to find the optimal coefficients. This introduces a bias into the regression. In this paper, we examine a nonparametric regression scheme for fitting data and generating surface plots of the data. The purpose of this paper is to compare various methods for visualizing reservoir layer elevation ASL, z, as a function of × (easting) and y (northing). We use layer depth data from 70 wells in the Shell Phase III Steam Drive Pilot in the South Belridge Diatomite near Bakersfield, California (Fig. 1). We investigate geostatistics, the Alternating Conditional Expectation (ACE) method, and MATLAB.
Geostatistics. The usual geostatistical procedure has been applied to the field data. The top surfaces of three cycles (G, K and M) are used here as an illustration.
An experimental variogram of the G-Cycle top is shown in Fig. 2. The white color at the corners denotes lack of data. The maximum data variation occurs near the corners. From this variogram, it is obvious that a SE-NW trend exists in the data, approximately at 45–55 degrees counterclockwise from the EW axis. The experimental variograms along three different directions (55, 145 and 100 degrees), Fig. 3, are then fit with a nested model. A linear and a spherical semivariogram have been invoked to match the data in the direction of 55.
(1)
and
(2)
and (h) = 210 otherwise, Anisotropy ratio = 1. The fit is quite good and matches the field data in several other directions (e.g., 100 and 145).
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