Unsteady flow in unsaturated soils from a cylindrical source of finite radius

Abstract

A solution to the two‐dimensional diffusion equation for unsteady and unsaturated flow in soils has been obtained. The physical problem is that of flow from a cylindrical source of finite radius and infinite length. Neglecting gravity, the diffusion equation is reduced to a simple form which can be transformed to a nonlinear ordinary differential equation. This transformation is achieved after substituting an empirical relation for the diffusivity of the soil. This ordinary differential equation is then solved by numerical techniques. Two soils, Yolo light clay and Pachappa loam, have been used for examples. Infiltration rates have been computed in each case; these are found to be almost constant for this problem. A computer model in extended Algol language has been developed.

Reply [to “Comments on the paper by R. Singh and J. B. Franzini, ‘Unsteady flow in unsaturated soil-water movement from a cylindrical source’”]

“…As a consequence of this error, our results [Singh and Franzini, 1967] are incorrect. It appears that Philip's conclusion that the flow velocity from a cylindrical source decreases with time is correct.…”

“…Philip [1959], Yeh [1967], Singh [1967], and others have used this transformation for the onedimensional case. Yeh [1967] used the numerical method outlined in Singh and Franzini [1967] to solve the case of horizontal absorption. This approach requires that the diffusivity of the soil be expressible in exponential form.…”

“…This approach requires that the diffusivity of the soil be expressible in exponential form. It should be noted that our solution [Singh and Franzini, 1967] would apply to the situation where • --constant, i.e. the case where ro is proportional to t TM.…”

“…

Philip is indeed correct when he states that there is an error in our analysis [Singh and Franzini, 1967]. As he pointed out, a similarity substitution usefully reduces the number of independent variables in a partial differential equation only when the variables removed from the equation are also removed from the governing conditions.

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Dynamics of Water in the Soil-Plant-Atmosphere System: A Model Named Troika

“…The single-root models have all a) assumed cylindrical geometry in a soil homogeneous with depth, so that one-dimensional flow results, b) neglected gravity, and c) assumed constant diffusivity and/or steady-state flow. Analytical solutions for unsteady flow of water from a fiaite cylinder assumed constant diffusivity (SHIH, 1969) or that diffusivity is an exponential function of moisture content (SINGH and FRANZINI, 1967). PHILIP (1957) and GARDNER (1960) assumed a constant flux into the root as a boundary condition to obtain water potentials in the root-soil system.…”