A stochastic approach to nonlinear unconfined flow subject to multiple random fields

Abstract

In this study, the KLME approach, a moment-equation approach based on the Karhunen–Loeve decomposition developed by Zhang and Lu (Comput Phys 194(2):773–794, 2004), is applied to unconfined flow with multiple random inputs. The log-transformed hydraulic conductivity F, the recharge R, the Dirichlet boundary condition H, and the Neumann boundary condition Q are assumed to be Gaussian random fields with known means and covariance functions. The F, R, H and Q are first decomposed into finite series in terms of Gaussian standard random variables by the Karhunen–Loeve expansion. The hydraulic head h is then represented by a perturbation expansion, and each term in the perturbation expansion is written as the products of unknown coefficients and Gaussian standard random variables obtained from the Karhunen–Loeve expansions. A series of deterministic partial differential equations are derived from the stochastic partial differential equations. The resulting equations for uncorrelated and perfectly correlated cases are developed. The equations can be solved sequentially from low to high order by the finite element method. We examine the accuracy of the KLME approach for the groundwater flow subject to uncorrelated or perfectly correlated random inputs and study the capability of the KLME method for predicting the head variance in the presence of various spatially variable parameters. It is shown that the proposed numerical model gives accurate results at a much smaller computational cost than the Monte Carlo simulation.

Appendix: The derivation of second-order equations

For notation brevity, we set $\left\{ {\psi_{1} ,\psi_{2} ,\psi_{3} ,\psi_{4} } \right\} = \left\{ {\xi ,\eta ,\varphi ,\varsigma } \right\}$ in the following derivation. By collecting the terms with p + q + r + s=2 $\left( {p,q,r,s \ne 1} \right),$ we have

$$ S\frac{\partial }{\partial t}\left( {\sum\limits_{k,l = 1}^{4} {\sum\limits_{i,j = 1}^{\infty } {h_{ij}^{{\psi_{k} \psi_{l} }} \left( {\psi_{k} } \right)_{i} \left( {\psi_{l} } \right)_{j} } } } \right) = \nabla \left[ {K_{G} h^{\left( 0 \right)} \nabla \left( {\sum\limits_{k,l = 1}^{4} {\sum\limits_{i,j = 1}^{\infty } {h_{ij}^{{\psi_{k} \psi_{l} }} \left( {\psi_{k} } \right)_{i} \left( {\psi_{l} } \right)_{j} } } } \right)} \right] + g_{ij} $$

$$ \begin{aligned} g_{ij} = & \nabla \left[ {K_{G} \left( {\sum\limits_{k,l = 1;k \le l}^{4} {\sum\limits_{i,j = 1}^{\infty } {h_{ij}^{{\psi_{k} \psi_{l} }} \left( {\psi_{k} } \right)_{i} \left( {\psi_{l} } \right)_{j} } } } \right)\nabla h^{\left( 0 \right)} } \right] + \nabla \left[ {K_{G} h^{\left( 0 \right)} \left( {\sum\limits_{k = 1}^{4} {\sum\limits_{i,j = 1}^{\infty } {k_{i} \xi_{i} \left( {\psi_{k} } \right)_{j} \nabla h_{j}^{{\psi_{k} }} } } } \right)} \right] \\ & + \nabla \left[ {K_{G} \left( {\sum\limits_{k = 1}^{4} {\sum\limits_{i,j = 1}^{\infty } {k_{i} \xi_{i} h_{j}^{{\psi_{k} }} \left( {\psi_{k} } \right)_{j} } } } \right)\nabla h^{\left( 0 \right)} } \right] + \nabla \left[ {K_{G} \left( {\sum\limits_{k,l = 1}^{4} {\sum\limits_{i,j = 1}^{\infty } {\left( {h_{i}^{{\psi_{k} }} \nabla h_{j}^{{\psi_{l} }} } \right)\left( {\psi_{k} } \right)_{i} \left( {\psi_{k} } \right)_{j} } } } \right)} \right] \\ & + \nabla \left[ {K_{G} \frac{1}{2}\left( {\sum\limits_{i,j = 1}^{\infty } {k_{i} k_{j} \xi_{i} \xi_{j} } } \right)h^{\left( 0 \right)} \nabla h^{\left( 0 \right)} } \right] \\ \end{aligned} $$

(27)

Rearranging (27) as

$$ \sum\limits_{k,l = 1;k \le l}^{4} {\sum\limits_{i,j = 1}^{\infty } {L_{ij}^{{\psi_{k} \psi_{l} }} \left( {h_{ij}^{{\psi_{k} \psi_{l} }} } \right)\left( {\psi_{k} } \right)_{i} \left( {\psi_{l} } \right)_{j} } } = 0 $$

(28)

where

$$ \begin{gathered} L_{ij}^{{\psi_{k} \psi_{k} }} \left( {h^{{\psi_{k} \psi_{k} }} } \right) = S\frac{{\partial h_{ij}^{{\psi_{k} \psi_{k} }} }}{\partial t} - \nabla \left[ {K_{G} h^{\left( 0 \right)} \nabla h_{ij}^{{\psi_{k} \psi_{k} }} } \right] - \nabla \left[ {K_{G} h_{ij}^{{\psi_{k} \psi_{k} }} \nabla h^{\left( 0 \right)} } \right] - \nabla \left[ {K_{G} h_{i}^{{\psi_{k} }} \nabla h_{j}^{{\psi_{k} }} } \right] - \hfill \\ \delta \left( {k,1} \right)\nabla \left[ {\frac{1}{2}K_{G} k_{i} k_{j} h^{\left( 0 \right)} \nabla h^{\left( 0 \right)} } \right] - \delta \left( {k,1} \right)\nabla \left[ {K_{G} k_{i} h_{j}^{\xi } \nabla h^{\left( 0 \right)} } \right] - \delta \left( {k,1} \right)\nabla \left[ {K_{G} h^{\left( 0 \right)} k_{i} \nabla h_{j}^{\xi } } \right]\quad k \ge 1 \hfill \\ L_{ij}^{{\psi_{k} \psi_{l} }} \left( {h^{{\psi_{k} \psi_{k} }} } \right) = - \nabla \left[ {K_{G} \left( {h_{i}^{{\psi_{k} }} \nabla h_{j}^{{\psi_{l} }} + h_{i}^{{\psi_{l} }} \nabla h_{j}^{{\psi_{k} }} } \right)} \right] - \hfill \\ \delta \left( {k,1} \right)\nabla \left[ {K_{G} k_{i} h_{j}^{{\psi_{l} }} \nabla h^{\left( 0 \right)} } \right] - \delta \left( {k,1} \right)\nabla \left[ {K_{G} h^{\left( 0 \right)} k_{i} \nabla h_{i}^{{\psi_{l} }} } \right]\quad k,l \ge 1;k < 1 \hfill \\ \end{gathered} $$

(29)

By considering the following properties of ξ _i and η _i , we have

$$ \begin{gathered} \left\langle {\xi_{i} \xi_{j} \xi_{m} \xi_{n} } \right\rangle = \delta_{ij} \delta_{mn} + \delta_{im} \delta_{jn} + \delta_{in} \delta_{jm} \hfill \\ \left\langle {\eta_{i} \eta_{j} \xi_{m} \xi_{n} } \right\rangle = \delta_{ij} \delta_{mn} \hfill \\ \left\langle {\varepsilon_{i} \eta_{j} \xi_{m} \xi_{n} } \right\rangle = 0 \hfill \\ \end{gathered} $$

(30)

After multiplying Eq. 28 with ξ _m ξ _n and taking expectation of the resulting equations, we have

$$ \begin{gathered} \sum\limits_{i,j = 1}^{\infty } {L_{ij}^{\xi \xi } \left( {h_{ij}^{\xi \xi } } \right)\left( {\delta_{ij} \delta_{mn} + \delta_{im} \delta_{jn} + \delta_{in} \delta_{jn} } \right)} + \sum\limits_{i,j = 1}^{\infty } {L_{ij}^{\eta \eta } \left( {h_{ij}^{\eta \eta } } \right)\delta_{ij} \delta_{mn} } + \sum\limits_{i,j = 1}^{\infty } {L_{ij}^{\varphi \varphi } \left( {h_{ij}^{\varphi \varphi } } \right)\delta_{ij} \delta_{mn} } + \hfill \\ \sum\limits_{i,j = 1}^{\infty } {L_{ij}^{\varsigma \varsigma } \left( {h_{ij}^{\varsigma \varsigma } } \right)\delta_{ij} \delta_{mn} } = 0\quad m,n = 1,2, \ldots \hfill \\ \end{gathered} $$

(31)

Similarly, by multiplying Eq. 31 with η _m η _n , φ _m φ _n and ς _m ς _n , respectively, and taking expectation of the resulting equations, we can obtain the three sets of equations similar to Eq. 31. These equations can be expressed as a uniform representation

$$ \begin{gathered} \sum\limits_{i,j = 1}^{\infty } {L_{ij}^{{\psi_{k} \psi_{k} }} \left( {h_{ij}^{{\psi_{k} \psi_{k} }} } \right)\left( {\delta_{ij} \delta_{mn} + \delta_{im} \delta_{jn} + \delta_{in} \delta_{jn} } \right)} + \hfill \\ \left( {1 - \delta \left( {k,l} \right)} \right)\sum\limits_{l = 1}^{4} {\sum\limits_{i,j = 1}^{\infty } {L_{ij}^{{\psi_{l} \psi_{l} }} \left( {h_{ij}^{{\psi_{l} \psi_{l} }} } \right)\delta_{ij} \delta_{mn} } } = 0\quad m,n = 1,2, \ldots \hfill \\ \end{gathered} $$

(32)

After accumulating Eq. 32 for k = 1, 2, 3 and 4 and because of the symmetry of $L_{mn}^{\varepsilon \varepsilon } ,L_{mn}^{\eta \eta } ,L_{mn}^{\varphi \varphi }$ and $L_{mn}^{\varsigma \varsigma } ,$ we have

$$ L_{mn}^{\varepsilon \varepsilon } \left( {h_{mn}^{\varepsilon \varepsilon } } \right) + L_{mn}^{\eta \eta } \left( {h_{mn}^{\eta \eta } } \right) + L_{mn}^{\varphi \varphi } \left( {h_{mn}^{\varphi \varphi } } \right) + L_{mn}^{\varsigma \varsigma } \left( {h_{mn}^{\varsigma \varsigma } } \right) = 0,\quad m \ge n \ge 1 $$

(33)

By substituting Eq. 29 into Eq. 33, we obtain the second-order equations

$$ \begin{gathered} S\frac{{\partial h_{ij} }}{\partial t} = \nabla \left[ {K_{G} h^{\left( 0 \right)} \nabla h_{ij} } \right] + g_{ij} \hfill \\ g_{ij} = \nabla \left[ {K_{G} h_{ij} \nabla h^{\left( 0 \right)} } \right] + \sum\limits_{k = 1}^{4} {\left( {\nabla \left[ {K_{G} h_{i}^{{\psi_{k} }} \nabla h_{j}^{{\psi_{k} }} } \right]} \right)} + \nabla \left[ {K_{G} h^{\left( 0 \right)} k_{i} \nabla h_{j}^{\xi } } \right] + \hfill \\ \nabla \left[ {\frac{1}{2}K_{G} k_{i} k_{j} h^{\left( 0 \right)} \nabla h^{\left( 0 \right)} } \right] + \nabla \left[ {K_{G} k_{i} h_{i}^{\xi } \nabla h^{\left( 0 \right)} } \right]\quad i \ge j \ge 1 \hfill \\ \end{gathered} $$

(34)

where $h_{ij} = h_{ij}^{\xi \xi } + h_{ij}^{\eta \eta } + h_{ij}^{\varphi \varphi } + h_{ij}^{\varsigma \varsigma } .$