A Multiscale Method Coupling Network and Continuum Models in Porous Media I: Steady-State Single Phase Flow

We propose a numerical multiscale method for coupling a conservation law for mass at the continuum scale with a discrete network model that describes the microscale flow in a porous medium. In this work we focus on coupling pressure equations. Evaluating pressure from a detailed network model over a large physical domain is typically computationally very expensive. We assume that over the same physical domain there exists an effective mass conservation equation at the continuum scale which could have been solved efficiently if the equation was explicitly given. Our coupling method uses local simulations on sampled microscale domains to evaluate the continuum equation and thus solve for the pressure in the full domain. We allow nonlinearity in the network model as well as the mass conservation equation. Convergence of the coupling method is analyzed. In the case where classical homogenization applies, we prove convergence of the proposed multiscale solutions to the homogenized equations. Numerical simulations are presented.