|Publisher||Society of Petroleum Engineers||Language||English|
|Content Type||Journal Paper|
|Title||History Matching Under Training-Image-Based Geological Model Constraints|
|Authors||Jef Caers, Stanford U.|
|Volume||Volume 8, Number 3||Pages||218-226|
|Copyright||2003. Society of Petroleum Engineers|
History matching forms an integral part of the reservoir modeling workflow process. Despite the existence of many history-matching tools, the integration of production data with seismic and geological continuity data remains a challenge. Geostatistical tools now are routinely employed for integrating large-scale seismic and fine-scale well/core data. A general framework for integrating production data with diverse types of geological/structural data is largely lacking. In this paper, we develop a new method for history matching that can account for production data constraint by prior geological data, such as the presence of channels, fractures, or shale lenses. With multiple-point (MP) geostatistics, prior information about geological patterns is carried by training images from which geological structures are borrowed, then anchored to the subsurface data. A simple Markov chain is proposed to iteratively modify the MP geostatistical realizations until history match. The method is simple and general in the sense that the procedure can be applied to a wide variety of geological environments without requiring a modification of the algorithm. The method does not make assumptions on the flow model. Synthetic cases are used to assess the flexibility of the proposed method.
Production data bring an important, yet indirect, constraint to the spatial distribution of reservoir variables. Pressure data provide information on the average pore volume and permeability connectivity near wells, while fractional flow data inform the extent of permeability connectivity between wells. Production data rarely suffice, however, to characterize heterogeneous reservoirs; a large amount of uncertainty still remains after history matching of geostatistical models.1
History matching is an ill-posed inverse problem attempting to invert reservoir properties/parameters from measured flow and pressure data. Solutions to such inverse problems are rarely unique. The question of nonuniqueness has to be addressed. Ducking the problem often leads to history-matched reservoir models that are too smooth and inconsistent with the existing permeability heterogeneity. Often, other sources of data, such as seismic surveys and geological interpretations, need to be used.
The nonuniqueness of the history-matching problem is well known, and various techniques have been developed that allow integrating production data with geological continuity information in fine-scale geostatistical models.2-7 Most of these prior geological models reproduce only the covariance as a measure of geological continuity.
Covariance models are rarely sufficient to depict patterns of geological continuity consisting of strongly connected, curvilinear geological objects such as channels or fractures (see, for example, Caers and Journel8 and Strebelle9). Ideally, one would like to possess a single history-matching algorithm that can handle diverse types of geological structures or scenarios.
We propose a pixel-based history-matching method that can account for a wide variety of complex styles of geological continuity, not necessarily limited to the two-point statistics of a variogram model. Complex styles of geological heterogeneity are characterized by MP patterns and corresponding statistics. In MP geostatistics, such MP patterns are inferred from a training image. A fast sequential simulation algorithm, termed snesim (single normal equation simulation), borrows patterns from the training image and anchors them to local subsurface data. Various papers and case studies on integrating well and seismic data with geological conceptual information in MP geostatistics have been recently published10-12; a short review is given in this paper.
We propose to extend the data integration framework of MP geostatistics to include production data. We first review some important concepts in MP geostatistics that allow definition of a large variety of prior geological models, then develop the proposed history-matching methodology.
Borrowing Structures From Training Images.
In geostatistics, geological continuity is traditionally captured through a variogram. A variogram measures the degree of correlation/connectivity or, conversely, variability between any two locations in space. Because the variogram is only a two-point statistic, it cannot model curvilinear structures such as channels, nor can it model strong contiguous patterns of connectivities such as fractures. The representation of such complex geological features requires MP statistics, involving jointly more than two locations. The idea behind MP geostatistics is to infer spatial patterns using many spatial locations.8,9 The training image serves as a conceptual reservoir analog depicting the desired geological heterogeneity and allows the inference of such higher-order statistics. Training images are merely conceptual and need not be constrained to any subsurface data. In most applications of MP geostatistics thus far, such training images are generated using unconstrained Boolean techniques (e.g., channels and elliptical bodies; see Fig. 1A).
The corresponding algorithm to generate geostatistical models constrained to reservoir data, termed snesim, is proposed in Strebelle. 9 It is essentially not different from existing, more traditional conditional simulation techniques13,14 in that it sequentially generates the numerical model, one grid cell after another. The difference lies in the probability distributions from which these pixel values are drawn; in snesim these probabilities are actual proportions inferred from the training image and made conditional to an MP data event. In traditional sequential simulation, these probabilities are derived by kriging using a variogram model.
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