Two‐dimensional steady unsaturated flow through embedded elliptical layers

Abstract

[1] New analytic element solutions are presented for unsaturated, two-dimensional steady flow in vertical planes that include nonoverlapping impermeable elliptical layers and elliptical inhomogeneities. The hydraulic conductivity, which is represented by an exponential function of the pressure head, differs between the inside and outside of an elliptical inhomogeneity; both the saturated hydraulic conductivity and water retention parameters are allowed to differ between the inside and outside. The Richards equation is transformed, through the Kirchhoff transformation and a second standard transformation, into the modified Helmholtz equation. Analytic element solutions are obtained through separation of variables in elliptical coordinates. The resulting equations for the Kirchhoff potential consist of infinite sums of products of exponentials and modified Mathieu functions. In practical applications the series are truncated but still fulfill the differential equation exactly; boundary conditions are met approximately but up to machine accuracy, provided that enough terms are used. The pressure head, saturation, and flow may be computed analytically at any point in the vadose zone. Examples are given of the shadowing effect of an impermeable elliptical layer in a uniform flow field and funneltype flow between two elliptical inhomogeneities. The presented solutions may be applied to study transport processes in vadose zones containing many impermeable elliptical layers or elliptical inhomogeneities.

Enhancement of seepage and lateral preferential flow by biopores on hillslopes

“…However, a closer examination showed that when a streamline approaches a macropore, it diverts around the macropore because at this pressure the hydraulic conductivity of the macropores is essentially zero, and the macropore thereby offers a very resistant pathway to flow. This effect of streamline deflection around unsaturated highly permeable objects has been clearly demonstrated and described by Bakker & Nieber (2005).…”

How do disconnected macropores in sloping soils facilitate preferential flow?

“…This case is for the flux (rainfall) rate equal to 0Ð025 m/day Table I. The rise in pore water pressure at the blocked end of macropores M1 and M2 are illustrated in Figure 7a and b for some of the cases result for unsaturated flow around embedded texturally contrasting material shown by Bakker and Nieber (2005). For this case, the saturated zone occupies only a small volume near the toe of the slope, and all the macropores lie within the overlying unsaturated zone (Figure 6a).…”

Laplace-transform analytic-element method for transient porous-media flow

“…3.2]. Bakker [28,51] and Bakker and Nieber [8] derived elliptical AEM elements for the modified Helmholtz equation. A significant difference between their elliptical AEM solutions and that given here is the presence of the complex Laplace parameter, p, which becomes large at small time; this is because p and t are multiplicative arguments to the exponential in the definition of the Laplace transform (e.g.…”

“…The parameter q is specified through the aquifer properties, element geometry, and p, while ω is determined to make the solution to (26a) periodic on π ≤ ψ < −π . The special functions that are solutions to (26a) and (26b) are Mathieu functions; see [8,54,55] for characteristic functional plots. Solutions to (25) for Re(q) < 0 arē …”

“…Elements that are not interface conditions have their boundary conditions specified in terms of total discharge potential (6) or normal flux, (8); if there are at least two elements, their strengths must be determined simultaneously.…”