Although principal-component analysis (PCA) has been widely applied to effectively reduce the number of parameters characterizing a reservoir, its disadvantages are well-recognized by researchers. First, PCA may distort the probability-distribution function (PDF) of the original model, especially for non-Gaussian properties such as facies indicator or permeability field of a fluvial reservoir. Second, it smears the boundaries between different facies. Therefore, the models reconstructed by traditional PCA are generally unacceptable.
In this paper, a work flow is proposed to integrate cumulative-distribution-function (CDF) mapping with PCA (CDF/PCA) for assisted history matching on a two-facies channelized reservoir. The CDF/PCA is developed to reconstruct reservoir models by use of only a few hundred principal components. It inherits the advantage of PCA to capture the main features or trends of spatial correlations among properties, and more importantly, it can properly correct the smoothing effect of PCA. Integer variables such as facies indicators are regenerated by truncating their corresponding PCA results with thresholds that honor the fraction of each facies at first, and then real variables such as permeability and porosity are regenerated by mapping their corresponding PCA results to new values according to the CDF curves of different properties in different facies. Therefore, the models reconstructed by CDF/PCA preserve both geological (facies fraction) and geostatistical (non-Gaussian distribution with multipeaks) characteristics of their original or prior models.
The CDF/PCA method is first applied to a real-field case with three facies to quantify the quality of the models reconstructed. Compared with the traditional PCA results, the integration of CDF-based mapping with PCA can significantly improve the quality of the reconstructed reservoir models. Results for the real-field case also reveal some limitations of the proposed CDF/PCA, especially when it is applied to reservoirs with three or more facies. Then, the CDF/PCA together with an effectively parallelized derivative-free optimization method is applied to history matching of a synthetic case with two facies. The geological facies, reservoir properties, and uncertainty characteristics of production forecasts of models reconstructed with CDF/PCA are well-consistent with those of the original models. Our results also demonstrate that the CDF/PCA is applicable for conditioning to both hard data and production data with minimal compromise of geological realism.