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 (\(t\)).
The response function in this example models for all soil types.
The parameter ‘average soil particle diameter’ (diameter) is used to characterize a soil type. Values used are those derived from the particle diameter ranges in the USDA soil classification system.
The parameters in the response function are functions of the soil diameter parameter. This allows for direct incorporation of soil types into the response function model, and provides a model across all soil types rather than a separate model for each soil type (as in the previous example).
Four different soil types are considered, with names derived from the USDA soil classification system.
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.
A predicted \(S_e\) vs. \(t\) curve is generated for the very fine sand, with a \(z\) value of 0.4.
Resulting \(S_e\) (effective saturation) curves are generated for the differential equation and response function models, for the following sets of data
\(z\) values of 0.8, 0.16, 0.24, and 0.32, and predicted \(z\) value of 0.4 (very fine sand)
\(z\) values of 0.6, 1.2, 1.8, 2.4, and predicted \(z\) value of 3.0 (fine sand)
\(z\) values of 6.0, 12.0, 18.0, and 24.0, and predicted \(z\) value of 30.0 (coarse sand)
\(z\) values of 12.0, 24.0, 36.0, and 48.0, and predicted \(z\) value of 60.0 (very coarse sand)
Results for all the soil types are shown here.
Conclusions
The response function is able to give somewhat accurate time curve fits and predictions for all the four soil types across the entire spatial domain. Accuracy in this example is somewhat hindered by the fact that only one property (average particle diameter) was used to describe a given soil type. This is an underparameterization of what would be required to more uniquely characterize a specific soil type.
The response function not only provides an accurate model for spatially-variant nonlinear dynamics, but is also able to incorporate complex, highly nonlinear constitutive relationships that are used in the nonlinear partial differential equation model.
The ability to automatically incorporate soil properties into the model eliminates the need for constitutive relationships that are experimentally-derived for a specific soil type. Often, the soil types tested in the lab do not match the more heterogeneous conditions found in situ.
The response function approach, with its ability to incorporate properties that can characterize heterogenous soil ttypes, allows for applicability in a wider range of real-world settings, where observed data can be used to build an accurate predictive model.
The relatively accurate results from this example validate the ability of the response function approach to capture nonlinear dynamics across spatial domains and across varying properties that define the system being modeled. This flexibility will allow for its use in a vast array of applications, where nonlinear dynamics can be learned from observed data.