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 advection-dispersion 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 two-dimensional
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 ill-defined. However, to model transport, the velocity histogram
should be weighted by the flux to obtain a well-defined 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 pore-to-reservoir 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 mm-cm 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 node-to-node transition, we use our upscaled (t) from a
simulation of transport at the smaller scale. We account for small-scale uncertainty by
parameterising (t) in terms of sub-scale 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 million-cell 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.
Small-scale 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.