sections of predicted parameters within any reservoir. In addition, sets of point
values generated by the models can be loaded into visualization software to
provide three-dimensional representations of the predicted parameters.
that paleofaulting and paleofracturing were responsi-ble for the calcite distribution, thus providing no vari-ables that could be used directly to predict calcite distribution at the desired scale prior to drilling. As a result, statistical models were sought that used values of variables accessible before drilling as proxies for the unattainable values of the cause-and-effect vari-ables. For the purpose of predicting calcite, linear regression models were used.
Although prediction of calcite improved reservoir-quality prediction in the case of the Zechstein 2 Carbon-ate, other factors influence porosity and permeability distribution. Thus, models were developed to predict porosity and permeability distribution directly. Because of the functional complexity of the porosity and perme-ability distributions, artificial neural network models (a form of artificial intelligence) and linear regression models were used. The objective of the models was to predict porosity and permeability in as much detail as possible ahead of the drill. Although the models are not capable of replicating the high-frequency variations of porosity and permeability that occur within a facies, trends within facies can be predicted.
DATA
A statistical study of factors useful for predicting reservoir-quality distribution requires a database con-taining variables likely to be either directly or indirectly related to reservoir quality. For the Zechstein 2 Carbon-ate study, an existing database at BEB Erdgas und Erdöl GmbH was expanded to include data for hypothesized reservoir-quality controls. Core data included mineral-ogy, facies, subfacies, porosity, and permeability from 287 wells. The cores provided good coverage of the facies present in a given area, and data from core plugs generally were available every 15 cm throughout a core.
Although each core did not necessarily cover the entire Ca2 interval, enough data from surrounding wells were available to adequately represent all facies present in a given area. Well log, structural, geochemical, thickness, and location data also were available. Because of the large amount of core available, all porosity and perme-ability values used for model development came from measurements on core plugs, rather than from well logs. For reasons discussed later, the data were divided into ten subsets, ranging in number of wells from 7 to 81, and in number of samples from 616 to 6990.
Figure 1. Location of study area (outlined in black) within the Upper Permian Zechstein 2 Carbonate (Ca2) of the Southern Zechstein Basin in northern Germany.
LSW = lowstand wedge.
Data Smoothing
An important problem in predicting porosity and permeability using core-plug data is the large varia-tion of the values over small distances within a core (high-frequency, high-excursion data). For example, porosity values commonly differ by an order of mag-nitude (e.g., from 2% to 20%) among several core plugs separated by <1 m. These differences may be due to geologic factors (subfacies changes or diagenesis), sampling bias (choosing the unusual specimen for analysis), or erroneous measurements.
No existing method has the ability to predict such abrupt, centimeter-scale excursions, but prediction at this level is generally not necessary. However, because models cannot predict the abrupt change, the residual values (the observed porosity/permeability value minus the model-predicted value) from these models will be large. To make the variance of the calibration data set more commensurate with the predicted data set, data can be “smoothed.” Kacewicz (1994) found that smoothing data improved the performance of a neural network. A simple way to accomplish data smoothing involves using an average value of porosity or perme-ability for each well or for each major subdivision within a well such as facies; however, models typically can pre-dict a higher frequency of variation than this, so infor-mation would be lost through such “oversmoothing.”
To avoid oversmoothing, a smaller window of observations for averaging can be used. In addition, a moving average can be calculated. Although this diminishes the abruptness of the changes in porosity-permeability values, a single high-excursion value still has a large influence. Thus, we used a tapered moving average. This weights the values closer to the middle of the window more heavily than those at the ends. For the Zechstein 2 Carbonate, a weighted moving average of five measurements (a window typically <1 m) was calculated so that the middle value was weighted most heavily (0.4), then the two adjacent values less (0.2 each), and the top and bottom values least (0.1 each).
The same procedure was used with the five depths associated with the five porosity and permeability val-ues to obtain an average depth value. The averaging procedure did not cut across facies boundaries.
METHODS
Prediction Techniques
Although statistical procedures can use different vari-ables to predict the distribution of a parameter, the ability to predict does not imply cause and effect. Establishing cause and effect is not necessary to accomplish the goal of prediction. All variables that significantly predict parameters of interest may aid in understanding cause and effect, but their utility in prediction models depends on the ability to estimate their values away from well control. If, for example, the percentage of anhydrite cement in the Zechstein 2 Carbonate significantly pre-dicts carbonate mineralogy, this might give clues about the calcitization process; however, cement content would be difficult to estimate in undrilled localities, so its input value in an equation to predict calcite vs.
dolomite is uncertain, and the variable is not useful in practical estimation. In fact, a poor estimate of the input value may adversely affect predictive capabilities.
Regression
Regression analysis was used extensively to identify which variables significantly predict mineralogy, poros-ity, and permeability in the Zechstein 2 Carbonate. In addition, predictive models were developed using for-ward stepwise regression with backfor-ward elimination, which is a method to select a few predictor variables from a large number of potential predictor variables.
Using this procedure, individual variables enter into and exit from an evolving model (Draper and Smith, 1981; Bowerman and O’Connell, 1990). For the Zechstein modeling, a significance level of 0.15 was chosen for a variable to enter into and to remain in the model.
Figure 2. Schematic cross section through the Zechstein 2 Carbonate showing distribution of depositional facies [plat-form, platform-LSW (low-stand wedge), upper slope, middle slope, lower slope, and basin] and mineralogy (dolomite vs. calcite). The distribution of calcite with-in the slope is not facies dependent.
Although stepwise regression helps to reduce the num-ber of predictor variables, it does not necessarily provide the best regression model, nor can the remaining predic-tor variables be considered the most important.
The regression analyses made extensive use of indicator variables [that is, variables that assume dis-crete values (e.g., 0 and 1)] to identify different cate-gories of a variable (Kleinbaum and Kupper, 1978). In this study, for example, an indicator variable was cre-ated for mineralogy, where mineralogy = 0 if the sam-ple (a core plug) is dolomite, but mineralogy = 1 if the sample is calcite. If regression models are constructed such that a 0,1 variable is the dependent (predicted) variable, and hypotheses about the regression will be tested, it may be advisable to use a logistic function.
However, if the purpose of the regressions is only to pre-dict, as is the case in this study, such transforms are not mandatory.
Indicator variables allow the inclusion of such qual-itative data as mineral type and facies in quantqual-itative models. The number of (0,1) indicator variables required to represent one type of information (e.g., facies) is n – 1, where n is the number of different cate-gories for the information (e.g., upper slope, platform, etc.) (Kleinbaum and Kupper, 1978). If five different facies occur, which is the case for the Zechstein 2 Car-bonate, then four indicator variables are required to represent all the facies, as the “missing” variable is represented when the other four variables equal 0. For the Zechstein data, the lower slope facies was not used as a facies variable; thus, when the platform, platform-lowstand-wedge, upper slope, and middle slope facies variables were all 0 for a particular sample, that sam-ple represented the lower slope facies.
If the relationship between the predicted and the predictor variables is complex, continuous variables, such as depth or thickness, can be transformed to new variables to possibly improve the regression model.
Common transformations include logs, squares, square roots, and reciprocals of the original variables.
Because of the complex spatial distribution of porosity and permeability in the Zechstein 2 carbonate, several different transformations of the location variables were made in attempts to accommodate nonplanar variations. For example, logarithms, squares, and reci-procals of the spatial data were offered to the predic-tion models, and commonly provided improved predictive ability (e.g., ln(depth), X2, 1/Y).
Artificial Neural Networks
Although prediction of calcite in the Zechstein 2 Carbonate was relatively straightforward using linear models, prediction of porosity and permeability using linear regression was commonly improved by the addition of terms higher than second order. This sug-gested a high level of complexity (nonlinearity?); for this reason, an adaptive nonparametric prediction method was sought that might better predict the poros-ity/permeability distribution. One such method is an artificial neural network. A network consists of inter-connected computing cells; weights are assigned to the connections between cells. These weights are used in
conjunction with input data to predict some outcome.
The network learns to predict a desired outcome by iteratively modifying the weights and comparing the predicted result with the actual result (Haykin, 1994).
This study used BPNET, a back-propagation neural network developed by author A. Woronow. The pro-gram randomly splits an input data file into a training data set (from which the network learns how to predict porosity or permeability) and a test data set that is not used during training (to assess the prediction effective-ness of the network as learned from the training data set). The test data consisted of ~10% of a data set. As with regression, the withheld data constitute a critical part of the evaluation of the predictive capabilities;
they provide the only means to evaluate how well the predictive tool will work when presented with new data. The program was run on a 486-33 PC, and the neural network used one hidden layer, eight nodes, and a sigmoidal logistic function (Haykin, 1994).
For each data set, the network was allowed to learn until no further effective improvement occurred. The time required to reach this state ranged from ~30 min to 3 hr, depending on the size and complexity of the data set, although the program was allowed to run beyond the cessation of improvement to ensure that a later
“breakthrough” in learning did not occur. In two cases, the program was allowed to run for ~14 hr to further check for this possibility; no additional learning occurred. The normal training times corresponded to between 500 and 3000 learning cycles, where a cycle is one pass through each case in the training data set
Unlike for the regression models, formed variables were not important for the neural network models, because a network can effectively develop its own linear and nonlinear transformations of variables to provide better prediction. If, however, a known relationship occurs (e.g., it is known that one variable is related to another by a particular function), the introduction of that transformed variable to the neural network could help the network learn faster. However, such functional relationships are not known in this case.