Probability density function of steady state concentration in two-dimensional heterogeneous porous media
Article first published online: 23 NOV 2011
Copyright 2011 by the American Geophysical Union.
Water Resources Research
Volume 47, Issue 11, November 2011
How to Cite
2011), Probability density function of steady state concentration in two-dimensional heterogeneous porous media, Water Resour. Res., 47, W11523, doi:10.1029/2011WR010750., , , and (
- Issue published online: 23 NOV 2011
- Article first published online: 23 NOV 2011
- Manuscript Accepted: 8 OCT 2011
- Manuscript Revised: 27 SEP 2011
- Manuscript Received: 4 APR 2011
- concentration pdf;
- transverse mixing
 Spatial variability of hydraulic aquifer parameters causes meandering, squeezing, stretching, and enhanced mixing of steady state plumes in concentrated hot-spots of mixing. Because the exact spatial distribution of hydraulic parameters is uncertain, the spatial distribution of enhanced mixing rates is also uncertain. We discuss how relevant the resulting uncertainty of mixing rates is for predicting concentrations. We develop analytical solutions for the full statistical distribution of steady state concentration in two-dimensional, statistically uniform domains with log-hydraulic conductivity following an isotropic exponential model. In particular, we analyze concentration statistics at the fringes of wide plumes, conceptually represented by a solute introduced over half the width of the domain. Our framework explicitly accounts for uncertainty of streamline meandering and uncertainty of effective transverse mixing (defined at the Darcy scale). We make use of existing low-order closed-form expressions that lead to analytical expressions for the statistical distribution of local concentration values. Along the expected position of the plume fringe, the concentration distribution strongly clusters at its extreme values. This behavior extends over travel distances of up to tens of integral scales for the parameters tested in our study. In this regime, the uncertainty of effective transverse mixing is substantial enough to have noticeable effects on the concentration probability density function. At significantly larger travel distances, intermediate concentration values become most likely, and uncertainty of effective transverse mixing becomes negligible. A comparison to numerical Monte Carlo simulations of flow and solute transport show excellent agreement with the theoretically derived expressions.