Richards' Equation and Its Constitutive Relations as a System of Differential-Algebraic Equations

Publication Date


Type of Culminating Activity


Degree Title

Master of Science in Mathematics



Major Advisor

Jodi L. Mead


The unsaturated zone is the portion of the earth between the land surface and the water table. The flow of water in the unsaturated zone can be described mathematically by Richards' equation (RE) [16]. RE is a single differential equation which describes the relationship between pressure head ψ soil moisture content θ(ψ), and hydraulic conductivity K(ψ). To form a closed system of three equations and three unknowns, θ(ψ) and K(ψ) are found using constitutive relations. In this work we view RE and its constitutive relations as a system of differential-algebraic equations, or DAEs. When RE and its constitutive relations were formulated as a DAE in the literature, we found that the authors did not include the hydraulic conductivity, K(ψ), as an independent variable in the DAE structure [7, 11, 17]. If K(ψ) is not an independent variable in the model, uncertainty information regarding K(ψ) cannot explicitly be included in the problem. We propose two new DAE formulations which include K(ψ) as an independent variable, and prove that they are index 1 DAEs. The linear DAE uses Picard iterations, while Newton iterations are used to solve the nonlinear formulations [11]. Standard numerical discretizations were applied to the proposed DAEs and solutions to a common test problem were compared to solutions of DAE formulations found in the literature. We conclude that K(ψ) can be included as an independent variable in the DAE structure without adversely affecting the forward numerical solution of RE and its constitutive relations.

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