Pore-to-field simulation of single-phase transport using continuous time random walks

We demonstrate a pore-to-reservoir simulation methodology 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 hops between nodes with a probability ψ(t)dtψ(t)dt that a particle will first arrive at a node from a nearest neighbor in a time t to t+dtt+dt. We describe transport at four scales: pore, core, grid-block, and field. At each scale the system is represented as a lattice of nodes with an appropriate transition time probability ψψ derived from a simulation at the next smaller scale. At the micron scale, we fit a truncated power law ψ(t)ψ(t) for the distribution of transition times between pores. We use the transport algorithm of Rhodes and Blunt [Rhodes ME, Blunt MJ. An exact particle tracking algorithm for advective-dispersive transport in networks with complete mixing at nodes. Water Resour Res 2006;42:W04501. doi:10.1029/2005WR004504] on a network model representation of Berea sandstone whose results are in good agreement with experiment. This ψpψp is then used to calculate transport at the core scale and a ψcψc is found that accounts for cm-scale transitions. Similarly we use ψcψc to derive a meter-scale ψgbψgb from simulations of grid-block level transport. At the field scale, we use the upscaled ψgbψgb from simulations at the meter-scale. We demonstrate the methodology by considering transport in a channeled sandstone reservoir, from a pore-scale representation of the microscopic structure to a million-cell field-scale geological model. Effectively we simulate transport in a model containing of order 10121012 cells while accounting for uncertainty in the reservoir description. Heterogeneity at all scales impacts transport and tends to retard the advance of the plume with particles becoming trapped in slow-moving regions, increasing breakthrough times by up to an order of magnitude compared to those predicted using a traditional advection dispersion model.