On transport through heterogeneous media: application of conjugated reciprocal transformations

Abstract

Conjugation of reciprocal transformations is used to solve a class of boundary value problems involving a source term relevant to water transport through a heterogeneous medium with a volumetric extraction mechanism. The main emphasis is the solution method that involves conjugation of reciprocal transformations, as well as other changes of variable, applied to a newly identified integrable model. The transport equation is a version of the nonlinear Richards equation, based on Buckingham’s extension of Darcy’s law to unsaturated media, now allowing for heterogeneous transport coefficients. In addition, there is a plant-root water extraction term that depends on both water content and position. The nonlinear boundary conditions have prescribed flux at one boundary and zero flux at the other barrier boundary of a one-dimensional heterogeneous medium.

Appendix A

1.1 Relationship with soil–water dynamics

Here, it is summarised how boundary value problems such as (14) involving a nonlinear evolution process with a source term can arise in the transport of water through heterogeneous soil with an extraction process.

In continuum models of an unsaturated soil, hydrological properties are commonly specified by the independent hydraulic conductivity $K(\varTheta )$ and capillary potential $h(\varTheta )$, both of which are dependent on the saturation level [2]. In practice, soils are not homogeneous but $K(\varTheta )$ and $h(\varTheta )$ may have explicit dependence on location, in 1 + 1-dimensional models here designated by the depth z. A tractable way to represent soil heterogeneity is via the notion of scale heterogeneity leading to the representations [39]

$$\begin{aligned} h(\varTheta ,z)=h^*(\varTheta )/\rho ^*(z),~~~K(\varTheta )=K^*(\varTheta )\rho ^*(z)^2, \end{aligned}$$

(A1)

where $\rho ^*(z)$ is a scale factor. Then, in the classical Darcy–Buckingham formulation the volumetric water flux density becomes

$$\begin{aligned} V=-K(\varTheta )\frac{\partial }{\partial z} [h(\varTheta ,z)-z] \end{aligned}$$

(A2)

whence the scale-heterogeneity relations in (A1) show that

$$\begin{aligned} V=K^*(\varTheta )\rho ^*(z)^2\left[ \ 1+h^*(\varTheta )\rho ^{*'}(z)/\rho ^*(z)^2-h^{*'}(\varTheta )\varTheta _z/\rho ^*(z)\ \right] . \end{aligned}$$

(A3)

Hence, the soil–water diffusivity scales as

$$\begin{aligned} D(\varTheta )=D^*(\varTheta )\rho ^*(z) \end{aligned}$$

(A4)

where $D^*(\varTheta )=K^*(\varTheta )h^{*'}(\varTheta ).$ The 1 $+$ 1-dimensional water mass balance equation is given by

$$\begin{aligned} \frac{\partial \varTheta }{\partial t}+\frac{\partial V}{\partial z}+Q=0, \end{aligned}$$

(A5)

wherein Q denotes an extraction rate term. In the context of [40], this is due to the presence of a web of plant roots in which case Q generally increases with $\varTheta$ and decreases with z. Insertion of V as given by (A3) into (A5) produces a scale-heterogeneous extension of Richards’ equation. Integrable cases with $Q=0$ were given in [41]. The present work involves an integrable nonlinear model with an extraction process so that $Q\ne 0$. Nonlinear integrable models in soil mechanics typically have [4, 41]

$$\begin{aligned} D^*(\varTheta )=\frac{C(C-1)}{(C-\varTheta )^2} \end{aligned}$$

(A6)

where C is a parameter that typically ranges from 1.05 to 1.4 over all soil types [42].

A standard boundary value problem that arises in the analysis of water transport through a porous medium such as soil in the absence of an extraction process invokes a constant flux at one boundary $z=z_1$ and zero flux at a second boundary $z=z_2$ (see, e.g. [4]) so that

$$\begin{aligned} \left. V=K^*(\varTheta )\rho ^*(z)^2+K^*(\varTheta )h^*(\varTheta )\rho ^{*'}(z)+ \rho ^*(z)\dfrac{C(C-1)}{(C-\varTheta )^2}\varTheta _z= \begin{array}{l} R \ \ \text {at} \ \ z=z_1, \\ 0 \ \ \text {at} \ \ z=z_2. \end{array} \right\} \end{aligned}$$

(A7)

With V corresponding to a specialisation of $K^*(\varTheta )$, $h^*(\varTheta )$ associated with $D^*(\varTheta )$ given by (A6) and $\rho ^*(z)=b/(z+b)$, on insertion into the mass balance equation (A5) with constant extraction rate Q and an appropriate change of variables, a boundary value problem of the type (14) results. It is remarked that, in the absence of an extraction process so that $Q=0$, the relevance of boundary conditions such as (A7) to water transport due to infiltration of rain at a boundary was investigated in [43, 44]. Two-phase transport with boundary infiltration under gravity has been analysed via conjugation of a reciprocal and Bäcklund transformation in [33].