Spatial Markov-Process Modeling of Solute Dispersion with Log-Normal Velocity Distributions

Spatial Markov processes have become efficient methods to simulate solute transport in heterogeneous formations. The approach follows solute particles from one observation plane to the next, assuming that the particle velocity of an individual travel-distance increment depends on the velocity of the preceding increment. The approach can be seen as a correlated continuous-time random walk with deterministic spatial jumps, or as correlated time-domain random-walk method. The first-order Markov property allows simulating the pre-asymptotic regime with a limited set of rules. The transition of velocities from one step to the next can be formulated by a discrete transition matrix, or approximated with a parametric joint distribution. For the latter, we use the bivariate log-normal distribution. For this distribution, we show that the pdf of the normalized flux-weighted slowness (= inverse velocity) is identical with pdf of the volume-weighted normalized Eulerian velocity. For a flux-weighted injection, we derive analytical expressions of the travel-time variance and associated dispersion coefficient both for discrete travel-time increments and in the continuous limit of infinitesimally small increments over the same distance. The analytical solution reproduces the first-order solution of perturbative methods in the limit of small velocity variances at the limits of small and very large travel distances, but it provides natural extensions for large variances of the log-velocity. In the case of a volume-weighted injection, the mean log-slowness relaxes exponentially to the asymptotic mean, while the variance of log-slowness remains constant. The associated analytical expressions for injection into the volume involve integrals requiring numerical quadrature. We compare the derived expressions with particle-tracking simulations in 3-D heterogeneous media with isotropic exponential covariance function testing variances of log-conductivity up to 5. We observe that the variance of the log-velocity scales linearly with that of log-conductivity and that the integral scale of the log-velocity remains fairly constant. The parameters of the spatial-Markov-process model can be related to parameters of the log-conductivity field with minimal adjustments to first-order results while being applicable to cases of large velocity variability.