We demonstrate a pore-to-reservoir simulation methodology and apply it to singe-phase flow. Traditional numerical methods are based on the discretization of partial differential equations with known spatially-dependent coefficients, such as porosity and permeability. However, in porous media flow we do not know a priori what the governing transport equations are − for instance, single-phase transport cannot be accurately described by an advection-dispersion equation − nor do we know the reservoir properties everywhere.
We propose a different approach that does not pre-suppose the functional form of the upscaled transport equations and which automatically accounts for uncertainty in the reservoir description. Single-phase transport is modeled as a continuous time random walk. Particles make a series of transitions between nodes with a probability ψ(t)dt that a particle will first arrive at a node from a nearest neighbor in a time t to t+dt. A top-down multiscale approach is used to find the flow field. At the micron scale, ψ(t) for particle transitions from pore to pore are found from modeling advection and molecular diffusion in a geologically representative network model. This ψ(t) is used to compute transport on the mm to cm scale. At larger scales, we represent the reservoir as a network of nodes connected by links. For each node-to-node transition, we compute an upscaled ψ(t) from a simulation of transport at the smaller scale. We account for small-scale uncertainty by interpreting ψ(t) probabilistically and running simulations for different possible realizations of the reservoir model. To make the number of computations manageable, ψ(t) is parameterized in terms of sub-scale heterogeneity and Peclet number, meaning that only a few representative simulations are required.
We demonstrate the methodology by finding ψ(t) for pore-scale flow and using it in a million-cell reservoir model. We show that the macroscopic behavior can be very different from that predicted by assuming that the advection-dispersion equation operates at the small scale. Small-scale structure does impact macroscopic transport; increasing the pore-level heterogeneity delays breakthrough and leads to longer late-time tails of the production since the solute spends more time in slow-flowing regions of the domain. We discuss extensions to multiphase flow and the development of a novel network-based probabilistic reservoir simulation approach.