Errors in History Matching
- Z. Tavassoli (Imperial College) | Jonathan N. Carter (Imperial College) | Peter R. King (Imperial College)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- September 2004
- Document Type
- Journal Paper
- 352 - 361
- 2004. Society of Petroleum Engineers
- 5.5 Reservoir Simulation, 5.6.5 Tracers, 5.1.5 Geologic Modeling, 2.4.3 Sand/Solids Control, 3.3.6 Integrated Modeling, 7.6.2 Data Integration, 5.1.2 Faults and Fracture Characterisation, 5.6.9 Production Forecasting, 5.5.8 History Matching, 4.3.4 Scale, 5.8.6 Naturally Fractured Reservoir, 5.1 Reservoir Characterisation, 5.3.2 Multiphase Flow
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The usual procedure in history matching is to adopt a Bayesian approach with an objective function that is assumed to have a single simple minimum at the "correct" model. In this paper, we use a simple cross-sectional model of a reservoir to show that this may not be the case. The model has three unknown parameters: high and low permeabilities and the throw of a fault. We generate a large number of realizations of the reservoir and choose one of them as a base case. Using the weighted sum of squares for the objective function, we find both the best production- and best parameter-matched models. The results show that a good fit for the production data does not necessarily have a good estimation for the parameters of the reservoir, and therefore it may lead to a bad forecast for the performance of the reservoir. We discuss the idea that the "true" model (represented here by the base case) is not necessarily the most likely to be obtained using conventional history-matching methods.
Reservoir history matching is a difficult inverse problem arising in the petroleum industry. The aim of history matching is to find a model such that the difference between the performance of the model and the history of a reservoir is minimized. Traditionally, this is done by hand. But the task of varying the parameters of a reservoir description by hand until a satisfactory match is obtained is extremely onerous and time-consuming. Therefore, gradient-based optimization techniques are increasingly adopted by the oil industry for computer-aided history matching. This is because of the great time-saving benefits they can offer over conventional trial-and-error approaches. Starting with an initial reservoir description, these optimization techniques automatically vary reservoir parameters until stopping criteria are achieved and a history match of field performance is obtained. This mechanistic activity is referred to as "automatic history matching." In Refs. 1 through 18, there are samples of studies that use the method of gradients for automatic history matching in order to find the best-matched model of a reservoir. Many other methods have been studied and applied to the history-matching problem. Examples include optimal control theory,19-22 stochastic modeling techniques,23-26 sensitivity analysis techniques,27-29 and gradual deformation method.31 In Refs. 31 through 45, some other studies on history matching are listed. The use of 3D streamline paths to assist in history matching has become increasingly common, because of their speed compared to that of the conventional methods. Samples of such studies are given in Refs. 46 through 52. These techniques are designated as "assisted history matching" to distinguish them from automated history-matching techniques.
In most of the studies above, independent of the method used for the history matching, there is usually an assumption that there exists a simple unique solution at the "correct" model. They therefore neglect the inherent nonuniqueness of the solution of the underlying inverse problem. This, consequently, leads to the assumption that a good history-matched model is a good representation of the reservoir and therefore gives a good forecast.
In this paper, we want to challenge these assumptions and show that the problem may have multiple solutions, and subsequently, a good history-matched model might have geological properties quite far from those of the "truth" and therefore could lead to a bad forecast. There have been some studies that attempt to tackle the problem of nonuniqueness of the solution, for example Refs. 1, 6, and 53 through 55. Ref. 53 presents a field story in which the risk arising from such nonuniqueness is highlighted. In Ref. 6, an approach is proposed that couples a chaotic sampling of parameter space with a local minimization technique. Through the evolution of a nonlinear dynamical system, several points are successively used as initial guesses for a local gradient-based optimizer. This provides a series of alternative matched models with different production forecasts that improve the understanding of the possible reservoir behaviors. In Ref. 1, the application of a global optimization algorithm called the tunneling method is presented for two test cases in which a series of minima was found. In Ref. 54, there is a study on the parameter estimation in the oil industry carried out by applying a modified genetic algorithm on a simple reservoir model. It demonstrated that it is difficult to obtain a history match using simple optimization methods. The simple model used in Ref. 54 was a 2D cross section of a reservoir with a sequence of alternating good- and poor-quality sands and a simple fault. The three unknown parameters were the high and low permeabilities of the sands, and the throw of the fault. In a later work by J.N. Carter,55 there is an attempt to capture the effects of modeling errors in inverse problems using Bayesian statistics. He then used a similar cross-sectional model of a reservoir to test the results. It appeared that the proposed error model yields a multimodal objective function that leads to multiple acceptable solutions. The approach used in Carter55 was to generate a large number of models, which could be searched to find the best match according to the criteria chosen.
In this paper, we use the same cross-sectional model of a reservoir as in Refs. 54 and 55, and we also use an approach similar to the one in Carter55 to generate a large number of realizations of the reservoir. We then choose one of these models as a base case, and use the sum of squares as the measure of fit between the performance of the base case and each realization. We also find the "distance" between the two models on the parameter space. By obtaining both the best production- and the best parameter-matched models, we then study the history matching and also the prediction of the both models. The results show that a model with a good history matching could have parameters quite different from those of the "true" case, and it may give a bad forecast. To show that the results are generic, a general study of the forecast then is carried out that confirms the robustness of the results. For the rest of the paper, firstly in the next section, the reservoir model is described and objective functions are defined. Then, a base case is chosen in the best-models section, and the best production- an.parameter-matched models are obtained. Then, in the following section, the prediction of the best models is assessed, and then a generalized discussion of the forecast is given, followed by conclusions.
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