HR: 1340h
AN: H13B-1243    [Abstracts]
TI: Using Analytical Solution Methods to Analyze and Combat Alarming Growth of Errors in Traditional Unsaturated Flow Numerical Computations
AU: * Tracy, F T
EM: Fred.T.Tracy@erdc.usace.army.mil
AF: Engineer Research and Development Center, 3909 Halls Ferry Road, Vicksburg, MS 39180, United States
AB: Recently (WRR, 2006), analytical solutions for Richards' equation, \begin{equation} \nabla · \left (k \nabla h \right ) + \frac{∂ k}{∂ z} = \frac{∂ θ}{∂ t} \end{equation} where h is pressure head, k is the hydraulic conductivity, θ is moisture content, z is the z coordinate, and t is time, were derived for unsaturated flow in a homogeneous three-dimensional (3-D) box region. Also, \begin{equation} k = k_r k_s \end{equation} where k_r is the relative hydraulic conductivity, and k_s is the saturated hydraulic conductivity. Here, k_r and θ are both functions of h, thus creating a strong nonlinearity in the partial differential equation (PDE). Tools such as the quasi-linear approximation, \begin{equation} \ln k = \ln k_s + α h \end{equation} separation of variables, Fourier series, and the change of variable \begin{equation} \bar h = e^{α h} - e^{α h_d} \end{equation} were all used. α is a constant, and h_d is the pressure head when the soil is very dry. The bigger α is, the more nonlinear the problem is. This 3-D problem and a two-dimensional (2-D) version of the problem in the x-z plane were then solved by traditional finite element and finite volume methods, respectively. It was startling to observe how much the error increased with increasing nonlinearity. This presentation will first summarize the box-shaped problem and its solution. It will then highlight the rapid growth of errors as α is increased (a phenomenon not readily understood by many practicing engineers), even when high performance computing and parallel processing are used to solve for 501 × 501 grids in 2-D and 161 × 161 × 161 grids in 3-D. The presentation will then show great improvement in these errors by using the analytical/numerical combination from first doing the following change of variables: \begin{equation} \bar h = e^{α h} - 1, ~~~~~ \bar k_r = k_r e^{-α h}, ~~~~~ h ≤ 0 \end{equation} \begin{equation} \bar h = h, ~~~~~ \bar k_r = k_r = 1,~~~~~ h ≥ 0 \end{equation} before doing the numerical solution. To illustrate, Eq. 1 for steady state becomes \begin{equation} \nabla · \left (\bar k_r \nabla \bar h \right ) + α \frac {∂}{∂ z} \left [\left (\bar h +1 \right ) \bar k_r \right ] = 0, ~~~~~ h ≤ 0 \end{equation} \begin{equation} \nabla² \bar h = 0, ~~~~~ h > 0 \end{equation} Using the quasi-linear approximation, Eq. 7 becomes the linear PDE \begin{equation} \nabla² \bar h + α \frac {∂ \bar h}{∂ z} = 0 \end{equation} Eq. 9 is similar to the constant diffusivity moisture-content-based equation often used in the analytic finite element. Finally, this presentation will show great improvement in convergence of the nonlinear system of equations generated from Eq. 7 as compared to the original nonlinear system. ERDC's Adaptive Hydrology (ADH) 3-D finite element program will be improved based on this work.
DE: 1805 Computational hydrology
DE: 1829 Groundwater hydrology
DE: 1847 Modeling
DE: 1849 Numerical approximations and analysis
DE: 1875 Vadose zone
SC: Hydrology [H]
MN: 2007 Fall Meeting