Abstract
Since the number of parameters to be estimated in a reservoir history match is potentially quite large, it is important to determine which parameters can be estimated with reasonable accuracy from the available data. This aspect can be called determining the identifiability of the parameters. The identifiability of porosity and absolute parameters. The identifiability of porosity and absolute and relative permeabilities on the basis of flow and pressure data in a two-phase (oil/water) reservoir is pressure data in a two-phase (oil/water) reservoir is considered. The question posed is: How accurately can one expect to estimate spatially variable porosity and absolute permeability and relative permeabilities given typical permeability and relative permeabilities given typical production and pressure data" To gain insight into this production and pressure data" To gain insight into this question, analytical solutions for pressure and saturation in a one-dimensional (1D) waterflood are used. The following, conclusions are obtained.Only the average value of the porosity can be determined on the basis of water/oil flow measurements.The permeability distribution can be determined from pressure drop data with an accuracy depending on the pressure drop data with an accuracy depending on the mobility ratio.Exponents in a power function representation of the relative permeabilities can he determined from WOR data alone but not nearly so accurately as when pressure drop and flow data are used simultaneously.
Introduction
The utility of reservoir simulation in predicting reservoir behavior is limited by the accuracy with which reservoir properties can be estimated. Because of the high costs properties can be estimated. Because of the high costs associated with coring analysis, reservoir engineers must rely, on history matching as a means of estimating reservoir properties. In this process a history match is carried out by choosing the reservoir properties as those that result in simulated well pressure and flow data that match as closely as possible those measured during production. In general, reservoir properties at each gridblock in the simulator represent the unknown values to be determined. Although there are efficient methods for estimating such a large number of unknowns, it has long been recognized from the results of single phase history matching exercises that many different sets of parameter values may yield a nearly identical match of observed and predicted pressures. The conventional single phase predicted pressures. The conventional single phase history matching problem is in fact a mathematically illposed problem, which explains its nonunique behavior. Such a situation is, in short, the result of the large number of unknowns to be estimated on the basis of the available data and the lack of sensitivity of the simulator solutions to the parameters. Because of this lack of sensitivity, the need to reduce the number of unknown Parameters or to introduce some additional constraints, such as "smoothness" of the estimated parameters, has been recognized. A problem as important as that of choosing which minimization method to employ in history matching is that of choosing, on the basis of the available well data. which properties actually should be estimated. This selection properties actually should be estimated. This selection depends on the relationship of the unknown parameters to the simulated well data. Ideally one would want to knowwhich parameters can be determined uniquely if the measurements were exact, andgiven the expected level of error in the measurements, how accurately we can expect to be able to estimate the parameters.
The first question, that of establishing uniqueness of the estimated parameters, is notoriously difficult to answer, and for a parameters, is notoriously difficult to answer, and for a problem as complicated as reservoir history matching, problem as complicated as reservoir history matching, there are virtually no general results available that allow one to establish uniqueness for permeability or porosity. Thus, it is not possible in general to base our choice of which parameters to estimate on rigorous mathematical uniqueness results. In lieu of an answer to Question 1, the selection of parameters to be estimated can be based on Question 2, parameters to be estimated can be based on Question 2, which is amenable to theoretical analysis. If the expected errors in estimation of any of the parameters, or any linear combination of the parameters, are extremely large, then that parameter or set of parameters can be judged as not identifiable. In such a case, steps may be taken to reduce the number of unknown parameters. In summary, the reservoir history matching problem is a difficult parameter estimation problem, and understanding the relationship between the unknown parameters and the measured data is essential to obtaining meaningful estimates of the reservoir properties. Quantitative studies regarding the accuracy of estimates for single-phase history matching problems have been reported by Shah et al. and Dogru et al. Shah et al,. investigated the optimal level of zonation for use with 1D single-phase (oil) situations.
SPEJ
P. 697
Formulating the history matching problem with consistent error statistics