### Cite Details

D. M. Tartakovsky and S. P. Neuman, "Transient flow in bounded randomly heterogeneous domains
1. Exact conditional moment equations and
recursive approximations", *Water Resour. Res.*, vol. 34, no. 1, pp. 1-12, 1998

### Abstract

We consider the effect of measuring randomly varying local
hydraulic conductivities *K(***x**) on
one's ability to predict transient flow within bounded
domains, driven by random sources, initial head, and boundary
conditions. Our aim is to allow optimum unbiased prediction of
local hydraulic heads *h(***x**,t) and
Darcy fluxes **q**(**x**,t) by
means of their ensemble moments,
*〈h(***x**,t)〉_{c} and
*〈***q**(**x**,t)〉_{c},
conditioned on measurements of
*K(***x**). We show that these predictors
satisfy a compact deterministic flow equation which contains a
space-time integrodifferential "residual flux"
term. This term renders
*〈***q**(**x**,t)〉_{c}
nonlocal and non-Darcian so that the concept of effective
hydraulic conductivity looses meaning in all but a few special
cases. Instead, the residual flux contains kernels that
constitute nonlocal parameters in space-time that are
additionally conditional on hydraulic conductivity data and
thus nonunique. The kernels include symmetric and nonsymmetric
second-rank tensors as well as vectors. We also develop
nonlocal equations for second conditional moments of head and
flux which constitute measures of predictive uncertainty. The
nonlocal expressions cannot be evaluated directly without
either a closure approximation or high-resolution conditional
Monte Carlo simulation. To render our theory workable, we
develop recursive closure approximations for the moment
equations through expansion in powers of a small parameter
which represents the standard estimation error of natural
*ln K(***x**). These approximations are
valid to arbitrary order for either mildly heterogeneous or
well-conditioned strongly heterogeneous media. They allow, in
principle, evaluating the conditional moments numerically on
relatively coarse grids, without upscaling, by standard
methods such as finite elements.

### BibTeX Entry

@article{tartakovsky-1998-transient,

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

title = {Transient flow in bounded randomly heterogeneous domains
1. {E}xact conditional moment equations and
recursive approximations},

year = {1998},

urlpdf = {http://maeresearch.ucsd.edu/Tartakovsky/Papers/tartakovsky-1998-transient.pdf},

journal = {Water Resour. Res.},

volume = {34},

number = {1},

pages = {1-12}

}