Recently, we have shown that inverse problem theory can be applied to generate realizations of a two-dimensional permeability field conditioned to the variogram and pressure data. In this work, we extend the approach to the generation of well skin factors and two-dimensional porosity and permeability fields that are conditioned to porosity and permeability variograms and multiwell pressure data. Multiwell production data as well as multiwell interference test data are considered. We show that in certain instances the incorporation of "hard" porosity and permeability data reduces the uncertainty in our estimates of rock properties and well skin factors.
It is now common knowledge that reservoir heterogeneities can have a dominate effect on oil production achieved by waterflooding and/or gas injection. It is also known that classical history matching does not yield a unique reservoir description, because the number of gridblock permeabilities and porosities typically far exceeds the number of independent data and production data may be very insensitive to some gridblock values of porosity and permeability. Moreover, the distribution of rock property fields obtained by classical reservoir description often violates some of the prior information inferred from cores, logs or geologic description. Inverse problem theory provides methodology to determine realizations of rock property fields that honor all information in a statistical sense. The incorporation of all data is important, not only because no data should be ignored without valid reasons, but also because the incorporation of additional independent data reduces both the ill-posedness of the inverse problem and the uncertainty in the reservoir descriptions obtained.
The basic inverse theory techniques used in this work rest on the assumption that the prior reservoir model satisfies a Gaussian distribution and that errors in pressure measurements may be considered as Gaussian random variables with zero mean. Thus, the a posteriori probability density function for the model is Gaussian and the most probable model can be obtained by maximizing the a posteriori probability density function. This maximum, which is referred to as the maximum a posteriori estimate, can be obtained by a gradient method. In this work the Gauss-Newton algorithm is used.
Gradient methods have been used in history matching for a long time. Moreover, as early as 1976, Gavalas et al. used intuitive empirical expressions for the auto correlations of porosity and permeability, the cross correlation between them and the mean values of porosity and permeability to incorporate prior information when history matching pressure data obtained at several wells in a one-dimensional synthetic reservoir under single-phase flow conditions.