Abstract
The pressure and flow statistics of Darcy flow through a three-dimensional random permeable medium are expressed as a path integral in a form suitable for evaluation by simulated annealing. There are several advantages to using simulated annealing for this problem: (i) any probability distribution can be used for the permeability, (ii) there is no need to invert the transmissibility matrix which, while not a factor for single-phase flow, offers distinct advantages for multiphase flow, and (iii) the action used for simulated annealing, whose extremum yields Darcy’s law, is eminently suitable for coarse graining by integrating over the short-wavelength degrees of freedom. We show that the pressure and flow statistics obtained by simulated annealing are in excellent agreement with those obtained from standard finite-volume calculations.
Highlights
The study of flow in porous media has a wide range of applications, including hydrology (e.g. [1, 2]), geothermal engineering (e.g.[3]), materials science (e.g. [1]), and the medical sciences (e.g. [4])
The rock heterogeneities exert a significant influence on the flow, from the pore scale up to the kilometer scale
Calculation of the Darcy pressure statistics depends on an explicit description of the permeability K at the mesoscopic scale
Summary
The study of flow in porous media has a wide range of applications, including hydrology (e.g. [1, 2]), geothermal engineering (e.g.[3]), materials science (e.g. [1]), and the medical sciences (e.g. [4]). The study of flow in porous media has a wide range of applications, including hydrology [1, 2]), geothermal engineering (e.g.[3]), materials science Another application, and the focus of the work reported here, is the flow of oil in a reservoir. There are two basic ways to conceptualize the flow through a porous material. The flow of oil through a rock, by contrast, calls for flow descriptions on the kilometer scale. Solving the Navier–Stokes equations on the later scale is not feasible because of limited information about the rock permeability and the matrix form in which to cast the problem. Even if all of this information was available, the computational requirements would be prohibitive
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