Use of Upscaling Techniques in Digital Rock Physics: Methodologies and Problems

Abstract

Transport equations have been mostly introduced in reservoir engineering through heuristic proposals. The use of various upscaling techniques has been widespread to attack problems arising when considering Darcy-scale heterogeneities: effective permeability for heterogeneous media, pseudo-functions, etc... Such multi-scale approaches were in the past less applied for the first upscaling problem, i.e., from the pore-scale to the Darcy-scale, mainly due to the lack of quantitative data about the pore-scale structure. The current rapid development and availability of pore-scale investigation tools (X-ray and NMR tomography, confocal microscopy, etc…) changed the trend. Direct numerical simulations, or more sophisticated upscaling techniques, are now widely used. Their implementation, however, is not necessarily straightforward. DNS requires to solve explicitly a given initial boundary value problem, which in turn constrains the result interpretation. Which macro-scale equation should be associated? What is the impact of the peculiar conditions used for the calculations, in particular boundary conditions if the medium is anisotropic? How to calculate the effective properties? Upscaling techniques such as volume averaging (Whitaker, 1999;...), homogenization theory (Bensoussan et al., 1978;...), stochastic analysis (Dagan & Neuman, 1997;…), to name a few, offer a better insight into the kind of macro-scale equations which may be associated to a particular pore-scale transport problem. However, this is not necessarily straightforward if one consider non-linear and/or highly coupled problems. Questions also arise about the kind of boundary conditions one may use in the calculations of the effective properties. These questions are examined in this paper for simple diffusive problems and the flow of a single phase in non-linear cases, such as the flow of a non-Newtonian fluid, or flow with inertia effects. The various upscaling techniques which lead to the determination of effective properties through the resolution of pore-scale closure problems are briefly introduced and compared. The question of closure is examined, in particular for the non-linear cases: which boundary conditions, how to handle anisotropic effects, etc.... Applications are provided for simple periodic systems as well as for more complex images of real rock materials obtained from tomography. This is illustrated in Fig. 1 taken from Zami-Pierre et al. (2016). Figs 1a, b, c, d, f, g, h, i shows different types of pore-scale structures used in the upscaling process. Figs 1e and j, the problematic of mesh refinement and numerical convergence, which is a crucial point in the quantitative analysis. Fig. 1K represents the pore-scale shear rate as obtained from numerical simulations in the case of a non-Newtonian fluid. Finally, the results are used to provide some insight into the controversial discussion about the use of tomographic images for the determination of effective properties to be used in reservoir numerical models.