Fuzzification of the miscible displacement model in heterogeneous porous media

Typical realistic models for multiphase flow in heterogeneous formations are complex and random, and must incorporate the uncertainties inherent to the mixture process. These uncertainties can be modeled using differential equations coefficients, such as hydrodynamic dispersivity. In this work, the mathematical model is expressed in terms of a nonlinear coupled system of stochastic partial differential equations; a second order elliptic equation for the pressure, and a hyperbolic-dominated transport-diffusion equation for the solvent concentration in the mixture. Besides, the longitudinal dispersion coefficient is a fuzzy number. New perspective on the quantification of uncertainty for parameter estimation problems by means of numerical simulations and membership functions is the purpose of this research. In this way, a fuzzification of the semiclassical solution numerical approximation is built. In this regard, it is proved the continuity of the function that assigns the 3-tuple comprised by longitudinal dispersion, transverse dispersion, and molecular diffusion, to the corresponding value of the semiclassical solution, at a fixed point of the domain. The continuity result along with Zadeh's Extension Principle is applied to obtain the fuzzification. The relevance of this study resides in the novelty of the methodology that considers a model parameter as a fuzzy number, meanwhile, it is usually taken as a constant in literature. Other unprecedented result lays in the discovery of the link between theoretical concepts and numerical approximations to obtain a fuzzification.