Title:

Transport in heterogeneous porous media

We present a new algorithm for modelling single phase transport of a tracer in porous media which demonstrates that structure on all scales affects macroscopic transport behaviour. We marry the robustness of the continuous time random walk (CTRW) framework with the simplicity of a Monte Carlo approach to reservoir simulation. We simulate transport as a series of particles transitioning between nodes with probability (t).dt that a particle will first arrive at a nearest neighbor in a time t to t + dt. To this end we first determine the mixing rules and transition probability ADE(t) for transport governed by the advectiondispersion equation (ADE) (Rhodes and Blunt, 2006). We validate our algorithm by simulating advective transport in bond percolation clusters at the critical point. We compute the histogram of flow speeds using the velocities from the bonds on the backbone and find the multifractal spectrum for twodimensional lattices with linear dimension L _ 2000 and in three dimensions for L _ 250. We demonstrate that in the limit of large systems all the negative moments of the velocity distribution become illdefined. However, to model transport, the velocity histogram should be weighted by the flux to obtain a welldefined mean travel time. Finally, we use CTRWtheory to demonstrate that anomalous transport is observed whose characteristics can be related to the multifractal properties of the system. We next demonstrate a poretoreservoir simulation methodology which is consistent across all scales of interest. At the micron scale, we fit a truncated power law (t) for the distribution of particle transition times from pore to pore simulations. To do this we use our transport algorithm on a geologically representative network model of Berea sandstone and compare the results to the explicit modelling of advection and molecular diffusion by Bijeljic and Blunt (2006). We find that the results are similar. We then demonstrate the effect of increasing pore scale heterogeneity on the power law exponent (_) by stretching the distribution of throat radii in our network model. We show that by increasing the spread of velocities within the network we decrease _ making the transport more anomalous  in keeping with the consensus currently in the literature. This (t) is then used to calculate transport on the mm to cm scale. We can then move up to the metre/grid block scale by using the transit time distribution from the mmcm simulation to model transport in an explicit, geologically representative model of heterogeneity found within a grid block of the reservoir. From these numerical experiments we determine the (t) appropriate for transport on grid block scale systems characterized by Peclet (Pe) number and the type of heterogeneity within the system. This allows us to account for small scale uncertainty by interpreting (t) probabilistically and running simulations for different possible realizations of the reservoir heterogeneity. At the field scale, we represent the reservoir as an unstructured network of nodes connected by links. For each nodetonode transition, we use our upscaled (t) from a simulation of transport at the smaller scale. We account for smallscale uncertainty by parameterising (t) in terms of subscale heterogeneity and Peclet number. We demonstrate the methodology by finding a (t) for each scale of interest taking into consideration the relevant physics at that scale and using the appropriate function in a millioncell reservoir model. We show that the macroscopic behaviour can be very different from that predicted by assuming that the ADE operates at the small scale. Smallscale structure dramatically retards the advance of the plume with the particles becoming trapped in the slow moving pores/regions increasing breakthrough times by an order of magnitude compared to those predicted by the ADE.
