Identifying and Estimating Significant Geologic Parameters With Experimental Design
- Christopher D. White (Louisiana State U.) | Brian J. Willis (BP Canada Energy Co.) | Keshav Narayanan (Object Reservoir Inc.) | Shirley P. Dutton (Bureau of Economic Geology)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- September 2001
- Document Type
- Journal Paper
- 311 - 324
- 2001. Society of Petroleum Engineers
- 1.14 Casing and Cementing, 5.5.8 History Matching, 5.1.5 Geologic Modeling, 5.6.3 Deterministic Methods, 5.4.6 Thermal Methods, 7.1.9 Project Economic Analysis, 5.7.2 Recovery Factors, 7.1.10 Field Economic Analysis, 5.1.3 Sedimentology, 4.1.5 Processing Equipment, 5.5 Reservoir Simulation, 1.2.3 Rock properties, 5.1 Reservoir Characterisation, 5.6.9 Production Forecasting, 4.3.4 Scale, 4.1.2 Separation and Treating, 5.6.4 Drillstem/Well Testing, 5.6.5 Tracers
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Variability in production predictions caused by geologic heterogeneity and uncertainty was examined with a new method. The experimental design maximized the information derived from the flow simulation of various geologic models. Empirical response surface models were fit to simulation results to assess the consequences of varying several geologic factors. Posterior distributions of geologic parameters were estimated from flow responses. Improvements in the accuracy of the reservoir model were quantified by computing changes in factor uncertainties resulting from response measurements. Monte Carlo simulations with prior and posterior factor distributions illustrate uncertainty reduction. This approach can be used for model comparison, sensitivity analysis, and estimation of probability distributions of geologic model parameters.
Reservoir simulation is used widely to investigate the effects of geologic heterogeneity and engineering parameter variability on reservoir production performance. Because many geologic and engineering factors interact to affect recovery predictions, an exhaustive examination of recovery behavior for all possible parameter combinations is prohibitively time-consuming and expensive. The factors that most strongly influence production behavior should be identified to focus analyses and measurements. If reservoir-simulation studies are conducted with an experimental design, response surface models can estimate how the variation of input factors affects reservoir behavior with a relatively small number of reservoir-simulation models. Response surface models can test the relative importance of the factors in experimental designs statistically. Because response surfaces are accurate and simple to evaluate, they are efficient proxies for reservoir simulators.
A method based on response surfaces, Monte Carlo simulation, and Bayes' theorem (RSMCB) was used to examine the effects of geologic variability and different models for permeability fields. The method was demonstrated in an analysis of outcrop data from a heterolithic tide-influenced deltaic sandstone. The data set includes probe permeameter measurements, detailed bedding maps, shale diagrams, cement maps, facies maps, facies-specific rock properties, and variograms. Because of the importance of geostatistical methods for reservoir model construction, designed simulations examined the effects of varying geostatistical parameters and compared stochastic and deterministic geologic models.
Experimental design1,2 has been used in reservoir engineering applications, including performance prediction,3 uncertainty modeling,4-6 sensitivity studies,7-9 upscaling,10,11 history matching,12 and development optimization.13
The first step in a designed simulation study is to identify the factors that may influence flow responses. These factors may be geologic (e.g., cement nodule permeability) or engineering (e.g., skin factor) parameters; they may be deterministic (e.g., well spacing), stochastic (e.g., permeability fields), or controllable (e.g., injection rate).
Factor ranges should include all feasible factor values. Factors are usually scaled to span the range of -1,1. Factor-response relationships should be as linear as possible. However, linear factor-response relations are difficult to guarantee a priori and are impossible to obtain when many responses are modeled. This scaling difficulty is a motivation for using quadratic rather than linear models.7,11 Factor scaling should reflect the range of factor variability. For example, although a permeability range from 1 md to 1 darcy could be mapped linearly to the range of -1,+1, a logarithmic scaling to the same range gives a more nearly linear response in most cases.
A design is a set of factor-value combinations for which responses are measured.1,2 In a two-level factorial design, each factor is assigned to its maximum or minimum value (±1) in all possible combinations with other factors (Fig. 1). For three factors, this requires eight experiments; for K factors, 2K experiments are needed. Similarly, three-level factorial designs assign each factor its minimum, centerpoint, or maximum value (-1,0,+1) in all possible combinations with other factors (Fig. 1b); this design requires 27 experiments for three factors, or 3K experiments for K factors. Box-Behnken14 designs are modified three-level factorials (Fig. 1c). This design requires 15 experiments for three factors, including three at the factor centerpoint (all factors assigned to their centerpoint values). Centerpoint replicates make the design more nearly orthogonal, which improves the precision of estimates of response surface coefficients. There is no simple formula relating the number of required experiments to the number of factors for Box-Behnken designs; however, the number of experiments needed will always be between 2K and 3K.
Box-Behnken designs were used in this study. This design has several advantages relative to alternatives. Compared with a three-level full-factorial design, a Box-Behnken design reduces the number of required experiments by confounding higher-order interactions. This reduction becomes more significant as the number of factors increases. For five factors, a Box-Behnken design requires 41 experiments, compared to 243 experiments required for a full three-level factorial and 32 for a full two-level factorial. Box-Behnken designs have the desirable qualities of being nearly orthogonal and rotatable for many cases.14 Unlike D-Optimal designs,1,2,4 Box-Behnken designs neither require nor depend on prior specification of the model. Further, unlike first-order (e.g., two-level factorial) designs,8,9 Box-Behnken designs allow estimation of quadratic terms and do not imply constant sensitivities of responses to factors. Most two-level designs do not include experiments at the design centerpoint. By including the centerpoint, Box-Behnken designs reduce estimation error for the most likely responses. Box-Behnken designs require only a few more experiments than two-level designs and allow construction of more versatile and accurate models.
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