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Stochastic partial differential equations in groundwater hydrology

Part 2: Application to Borden aquifer

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Abstract

Part I of this series of two papers (Unny, 1989) dealt with the theoretical derivation of the moment equations for the stochastic partial differential equation in the water table depth forced by stochastic rainfall input. Part I also developed a maximum likelihood estimation procedure for parameter determination. The primary aim of the present manuscript is the application of the parameter estimation procedure to the Borden aquifer, an aquifer designated as an experimental site, where extensive field measurements have been carried out. Estimates of hydraulic conductivity and transmissivity for the Borden aquifer, derived from the maximum likelihood algorithm, have been compared with estimates obtained by “traditional” procedures. The paper also presents the simulated solution of the governing differential equation in the one dimensional problem applied to the Borden aquifer.

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Abbreviations

A :

partial differential operator

B :

event space

e i :

normalized basis vectors inL 2(R)

I,\(\bar {\rm I}\) :

nonhomogenous input term

l :

log-likelihood function

P 0,P 1 :

probability measures

Q :

correlation function of the solution field

T, S, α x α y ,K :

parameters in the flow equation

V, Y :

variables in the ordinary SDE

x, y :

space coordinates

θ:

unknown vector of parameters

\(\bar \Theta \) :

parameter space

Ω:

sample space

b :

scalar Weiner process

β:

Hilbert space valued Weiner process

F i :

σ-algebras formed by subsets ofB

L :

likelihood function

L 2(R):

space of real square integrable functions

q :

measured correlation matrix of the input

t :

time coordinate

ν:

observations of the variable

w :

the white noise term

z :

aquifer saturated thickness

\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\hat \theta } \) :

maximum likelihood estimate of\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\theta } \)

ω i :

eigen values of the space correlation matrixq

λ i :

eigen values of the operatorA

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Authors and Affiliations

  1. Dept. of Systems Design Engineering, University of Waterloo, N2L 3G1, Waterloo, Ontario, Canada

    P. Kumar, T. E. Unny & K. Ponnambalam

Authors
  1. P. Kumar
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  2. T. E. Unny
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  3. K. Ponnambalam
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Kumar, P., Unny, T.E. & Ponnambalam, K. Stochastic partial differential equations in groundwater hydrology. Stochastic Hydrol Hydraul 5, 239–251 (1991). https://doi.org/10.1007/BF01544060

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  • DOI: https://doi.org/10.1007/BF01544060

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