In this paper, a novel technique has been developed to implicitly estimate the relative permeability by history matching three-phase production data with the ensemble Kalman filter (EnKF). Power law repetitive of the relative permeability curves is utilized, while both endpoints and shape of the relative permeability curve are included in the state vectors which are updated by assimilating the observed data sequentially. The newly developed technique has been validated with accurately evaluating the relative permeability in a synthetic reservoir with two-dimension and three-phase flow. It is shown from the synthetic case that good estimation of the relative permeability curves can be obtained by assimilating the observed oil rates, gas-oil ratios and well bottomhole pressures of the production wells. Both the shape factors and the endpoints of the relative permeability curves are accurately evaluated; however, a larger ensemble size is needed to avoid the filter divergence. Compared to the existing implicit methods, the newly developed technique does not require the gradient of the objective function and thus makes it easy to implement.
Relative permeability is not only one of the most important parameters used in reservoir characterization, but also crucial for predicting reservoir performance throughout the life of a reservoir. In general, relative permeability is obtained from the displacement experiments with core samples. However, due to the huge scaling difference between the core samples and the reservoir as well as the difference between the experimental conditions and the formation conditions, direct application of estimated results generated from core samples to the whole reservoir may induce significant errors in evaluating the reservoir performance. Furthermore, interpretation of the laboratory experiment data may also add further uncertainty to the process of reservoir simulation. Therefore, it is of practical and fundamental importance to accurately evaluate the relative permeability in hydrocarbon reservoirs.
In principle, estimation of the relative permeability curve can be obtained inversely by history matching the production data obtained from the displacement experiments or field operations[1–6]. A brief description of the process of implicit relative permeability estimation method is provided as follows. Prior to the history matching, a relative permeability representation model is selected and initialized. Then, reservoir simulation is conducted by using the initial relative permeability curve to generate the simulated production data. Subsequently, the relative permeability curves are adjusted using a robust algorithm to minimize the discrepancy between the simulated production data and the field observation data. Once the discrepancy is minimized, the corresponding relative permeability curve is regarded as the approximation of the real relative permeability curves.
The relative permeability representation models are classified into two categories: the functional model and the nonfunctional model. Among the functional models, the power law model, which determines the relative permeability curve by the endpoints and the exponential factors, is the most widely used model[4, 6]. Compared to the functional models, the nonfunctional models are more flexible and show more degrees of freedom, among which the cubic or B-spline curve are commonly applied to represent the relative permeability.
