### Cite Details

Z. Lu, S. P. Neuman, A. Guadagnini and D. M. Tartakovsky, "Conditional moment analysis of steady state unsaturated
flow in bounded, randomly heterogeneous soils",

*Water Resour. Res.*, vol. 38, no. 4, doi:

10.1029/2001WR000278, 2002

### Abstract

We consider steady state unsaturated flow in bounded,
randomly heterogeneous soils under the influence of random
boundary and source terms. Our aim is to predict pressure heads
and fluxes without resorting to Monte Carlo simulation,
upscaling, or linearization of the constitutive relationship
between unsaturated hydraulic conductivity and pressure head. We
represent this relationship through Gardner's exponential model
while treating its exponent α as a random
constant and saturated hydraulic conductivity
*K*_{s}
as a spatially correlated random field. We linearize the steady
state unsaturated flow equations by means of the Kirchhoff
transformation and integrate them in probability space to obtain
exact integro-differential equations for the conditional mean
and variance-covariance of transformed pressure head and
flux. After approximating these equations recursively to second
order in the standard deviation
σσ_{Y} of *Y
= ln K*_{s}, we solve them by finite elements for
superimposed mean uniform and divergent flows in the vertical
plane, with and without conditioning on measured *Y*
values. Comparison with Monte Carlo solutions demonstrates that
whereas our nonlocal solution is nominally restricted to mildly
nonuniform media with
σ_{Y}^{2}
« 1, it yields
remarkably accurate results for strongly nonuniform media with
σ_{Y}^{2}
at least as large as 2. This accords well with a previous theoretical analysis, which shows that the
solution may remain asymptotic for values of
σ_{Y}^{2}
as large as 2.

### BibTeX Entry

@article{lu-2002-conditional,

author = {Z. Lu and S. P. Neuman and A. Guadagnini and D. M. Tartakovsky},

title = {Conditional moment analysis of steady state unsaturated
flow in bounded, randomly heterogeneous soils},

year = {2002},

urlpdf = {http://maeresearch.ucsd.edu/Tartakovsky/Papers/lu-2002-conditional.pdf},

journal = {Water Resour. Res.},

volume = {38},

number = {4},

doi = {10.1029/2001WR000278}

}