A joint stochastic simulation method using the Bernstein copula as a flexible tool for modeling nonlinear dependence structures between petrophysical properties

The statistical dependence between petrophysical properties (porosity, permeability, water saturation, etc.) in heterogeneous formations is usually nonlinear and complex; therefore, traditional statistical techniques based on assumptions of linearity are not appropriate for modeling these dependence relationships. Also, these methods may not reproduce the extreme values and data variability, which may represent impermeable barriers or high permeability zones. A modern way to model the petrophysical dependence structure between random variables is using copulas. Copula functions have been previously applied to this kind of problems, but it seems to be very restrictive that a single copula family be flexible enough to model the nonlinear dependence structure between petrophysical properties in highly heterogeneous porous media. For this reason, in this work we have resorted to a nonparametric approach, where the Bernstein copula is used to model the empirical petrophysical relationship without imposing any distributional constraint. The copula based stochastic method proposed here, basically consists on applying the simulated annealing method with a joint probability distribution model estimated by a nonparametric Bernstein copula. This approach has several advantages, among others we can mention that does not require the assumption of normality or other probability distribution, and is not restricted to the case of linear dependence between the variables. The proposed method provides a very flexible tool to model the complex dependence relationships between pairs of petrophysical properties. It is shown a case study where this tool is applied to model the permeability–porosity nonlinear relationship in carbonate double-porosity formations with complex microstructure of pore. It is discussed a comparative study between methods already established and the proposed one. ► The proposed method is a flexible tool to model nonlinear dependence relationships. ► Allows us to model petrophysical properties without assume any distribution function. ► The proposed method gives us an easy way to simulate bivariate data. ► There is no need to make logarithmic transformations of permeability. ► The proposed method has low computational cost in contrast with the other methods.