Richards’ Equation

• Problem Description

  • Richards’ equation (RE) is a nonlinear PDE that describes the movement of water in variably saturated porous media. Almost any practical application of RE requires a numerical solution, yet RE remains challenging to solve reliably and accurately for a given set of boundary conditions and soil hydraulic properties.

  • This example considers the one-dimensional RE, where movement of water is modeled in the vertical (\(z\)) direction.

  • Four different soil types are considered in this example.

• Model Descriptions

  • RE is derived from the mass continuity equation applied to a control volume of soil, and Darcy’s law to descrive vertical flux.

  • There are various forms of RE. The \(\theta\)-based form in one spatial dimension (\(z\)) describes the volumetric water content in the soil (\(\theta\)) as a function of space (\(z\)) and time (\(t\)).

  • The \(\theta\)-based form of RE is given by:

\[\begin{array}{rl} {d \theta \over d t} \: = \: & {\partial \over \partial z} \left [ {K \left ( \theta \right ) \over C \left ( \theta \right )} {\partial \theta \over \partial z} \: - \: K \left ( \theta \right ) \right ] \\ \\ & \mbox{where:} \\ & \ \ K \mbox{ is the hydrodynamic conductivity} \\ & \ \ C \mbox{ represents the rate of change of } \theta \mbox{ with respect} \\ & \ \ \ \ \ \ \mbox{to the hydraulic head (measure of liquid pressure)} \\ & \ \ \mbox{The constitutive relationships } C(\theta) \mbox{ and } K(\theta) \mbox{ are} \\ & \ \ \ \ \ \ \mbox{determined experimentally for various soil types} \end{array}\]

• • The Van Genuchten and Mualem models are used to define the constitutive relationships used in this example. These are nonlinear functions, which can be highly nonlinear for certain soil types.

  • The parameters in the constitutive relationships that define a particular soil type are \(\theta_r\) (residual soil water content), \(\theta_s\) (saturated soil water content), \(\alpha\) (parameter of the van Genuchten equation corresponding to approximately the inverse of the air-entry value), \(n\) (emprical shape-defining paramter in the van Genuchten and Mualem models), and \(K_s\) (effective saturated hydraulic condictivity).

  • A response function is built to model the the volumetric soil water content in the soil (\(\theta\)) as a function of space (\(z\)) and time (\(t\)).

  • Four different soil types are considered, with names derived from the USDA soil classification system.

  1. Very Fine Sand
  • The very fine sand example is defined by constutive parameters \(\theta_r = 0.100\), \(\theta_s = 0.380\), \(\alpha = 0.027\), \(n = 1.23\), and \(K_s = 2.88\). A spatial domain of \(0<z<0.4\) and time domain of \(0<t<1\) were used.
  2. Fine Sand
  • The fine sand example is defined by constutive parameters \(\theta_r = 0.100\), \(\theta_s = 0.390\), \(\alpha = 0.059\), \(n = 1.48\), and \(K_s = 31.44\). A spatial domain of \(0<z<3.0\) and time domain of \(0<t<1\) were used.
  3. Coarse Sand
  • The very fine sand is defined by constutive parameters \(\theta_r = 0.057\), \(\theta_s = 0.410\), \(\alpha = 0.124\), \(n = 2.28\), and \(K_s = 350.2\). A spatial domain of \(0<z<30.0\) and time domain of \(0<t<1\) were used.
  4. Very Coarse Sand
  • The very coarse sand is defined by constutive parameters \(\theta_r = 0.102\), \(\theta_s = 0.381\), \(\alpha = 0.0335\), \(n = 2.00\), and \(K_s = 796.6\). A spatial domain of \(0<z<60.0\) and time domain of \(0<t<1\) were used.

• Results

  • \(S_e\) (effective saturation) vs. \(t\) curves are generated for the differential equation and response function models, where \(S_e\) is defined by the equation \(S_e = {\theta - \theta_r \over \theta_s - \theta_r}\).

  • Differential equation model values are shown as data points, and response function model values are shown as lines.

  1. Very Fine Sand

  • Resulting \(S_e\) vs. \(t\) curves are generated for the differential equation and response function models, for \(z\) values of 0.8, 0.16, 0.24, and 0.32.

  • A predicted \(S_e\) vs. \(t\) curve is generated for a \(z\) value of 0.4.

  • Results for the very fine sand are shown here.

  2. Fine Sand

  • Resulting \(S_e\) (effective saturation) curves are generated for the differential equation and response function models, for \(z\) values of 0.6, 1.2, 1.8, and 2.4.

  • A predicted \(S_e\) vs. \(t\) curve is generated for a \(z\) value of 3.0.

  • Results for the fine sand are shown here.

  3. Coarse Sand

  • Resulting \(S_e\) (effective saturation) curves are generated for the differential equation and response function models, for \(z\) values of 6.0, 12.0, 18.0, and 24.0.

  • A predicted \(S_e\) vs. \(t\) curve is generated for a \(z\) value of 30.0.

  • Results for the coarse sand are shown here.

  4. Very Coarse Sand

  • Resulting \(S_e\) (effective saturation) curves are generated for the differential equation and response function models, for \(z\) values of 12.0, 24.0, 36.0, and 48.0.

  • A predicted \(S_e\) vs. \(t\) curve is generated for a \(z\) value of 60.0.

  • Results for the very coarse sand are shown here.

• Conclusions

  • The response function shows accurate fits to the data derived from solution of the partial differential equation.

  • The response function is able to accurately predict the time curves for extrapolated \(z\) values in each of the four soil types.

  • The response function provides an accurate model for time curves over the entire spatial domain, thereby extending its applicability in modeling nonlinear dynamics to spatially-variant nonlinear dynamics.