Applied Mathematics and Plasma Physics

D. M. Tartakovsky, S. P. Neuman and Z. Lu, "Conditional stochastic averaging of steady state
unsaturated flow by means of Kirchhoff transformation", *Water Resour. Res.*, vol. 35, no. 3, pp. 731-745, 1999

We consider the effect of measuring randomly varying soil hydraulic properties on one's ability to predict steady state unsaturated flow subject to random sources and/or initial and boundary conditions. Our aim is to allow optimum unbiased prediction of system states (pressure head, water content) and fluxes deterministically, without upscaling and without linearizing the constitutive characteristics of the soil. It has been shown by Neuman et al. [1999] that such prediction is possible by means of first ensemble moments of system states and fluxes, conditioned on measured values of soil properties, when the latter scale in a linearly separable fashion as proposed by Vogel et al. [1991]; the uncertainty associated with such predictions can be quantified by means of the corresponding conditional second moments. The derivation of moment equations for soils whose properties do not scale in the above manner requires linearizing the corresponding constitutive relations, which may lead to major inaccuracies when these relations are highly nonlinear, as is often the case in nature. When the scaling parameter of pressure head is a random variable independent of location, the steady state unsaturated flow equation can be linearized by means of the Kirchhoff transformation for gravity-free flow. Linearization is also possible in the presence of gravity when hydraulic conductivity varies exponentially with pressure head. For the latter case we develop exact conditional firstand second-moment equations which are nonlocal and therefore non-Darcian. We solve these equations analytically by perturbation for unconditional vertical infiltration and compare our solution with the results of numerical Monte Carlo simulations. Our analytical solution demonstrates in a rigorous manner that the concept of effective hydraulic conductivity does not apply to ensemble-averaged unsaturated flow except when gravity is the sole driving force.

@article{tartakovsky-1999-conditional,

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

title = {Conditional stochastic averaging of steady state
unsaturated flow by means of Kirchhoff transformation},

year = {1999},

urlpdf = {http://math.lanl.gov/~dmt/papers/unsatWRR.pdf},

journal = {Water Resour. Res.},

volume = {35},

number = {3},

pages = {731-745}

}