Spatial Markov Model for the Prediction of Travel‐Time‐Based Solute Dispersion in Three‐Dimensional Heterogeneous Media

Abstract

In recent years, Spatial Markov Models have gained popularity in simulating solute transport in heterogeneous formations. They describe the transition times of particles between equidistant observation planes by statistical distributions, assuming correlation of the transit times of individual particles between subsequent steps. By this, the approach naturally captures preasymptotic solute dispersion. In this study, we analyze Spatial Markov Models assuming bivariate log‐normal distributions of the particle slowness (i.e., the inverse velocity) in subsequent transitions. The model is fully parameterized by the mean Eulerian velocity, the variance of the log‐slowness, and the correlation coefficient of log‐slowness in subsequent steps. We derive closed‐form expressions for distance‐dependent ensemble dispersion, which is defined in terms of the second‐central moments of the solute breakthrough curves. We relate the coefficients to the properties of the underlying log‐hydraulic conductivity field assuming second‐order stationarity. The results are consistent with linear stochastic theory in the limit of small log‐conductivity variances, while the approach naturally extends to high‐variance cases. We demonstrate the validity of the approach by comparison to three‐dimensional particle‐tracking simulations of advective transport in heterogeneous media with isotropic, exponential correlation structure for log‐conductivity variances up to five. This study contributes to relating solute dispersion to metrics of the porous‐medium structure in cases of strong heterogeneity.

Key Points:

We derive closed‐form expressions of ensemble dispersion in the spatial‐Markov framework of solute transport.

The expressions are consistent with linear theory in the limit of small log‐conductivity variances, but extend to high‐variance cases.

Comparison to particle‐tracking simulations of advective transport in 3‐D heterogeneous domains show excellent agreement.