Sensitivity of Travelling Wave Solution to Richards Equation to Soil Material Property Functions

Abstract

Richards equation describes water transport in soils, but requires as input, soil material property functions specifically relative hydraulic conductivity and relative diffusivity typically obtained from the soil–water retention curve (SWRC) function (involving capillary suction head). These properties are often expressed via particular functional forms, with different soil types from sandstones to loams represented within those functional forms by a free fitting parameter. Travelling wave solutions (profile of height ${\hat{\xi }}$ against moisture content $\varTheta$) of Richards equation using van Genuchten’s form of the soil material property functions diverge to arbitrarily large height close to full saturation. The value of relative diffusivity itself diverges at full saturation owing to a weak singularity in the SWRC. If, however, soil material property data are sparse near full saturation, evidence for the nature of that divergence may be limited. Here we rescale the relative diffusivity to approach unity at full saturation, removing a singularity from the original van Genuchten SWRC function by constructing a convex hull around it. A piecewise SWRC function results with capillary suction head approaching zero smoothly at full saturation. We use this SWRC with the Brooks–Corey relative hydraulic conductivity to develop a new relative diffusivity function and proceed to solve Richards equation. We obtain logarithmic relationships between height ${\hat{\xi }}$ and moisture content $\varTheta$ close to saturation. Predicted ${\hat{\xi }}$ values are smaller than profile heights obtained when solving using the original van Genuchten’s soil material property functions. Those heights instead exhibit power law behaviour.

Appendices

A: Theoretical Basis for the Predictive Conductivity Models

This appendix outlines the basis for predictive conductivity models (PCM). Although we do not ultimately apply a PCM in our analysis, we include this appendix to demonstrate why a soil–water retention curve (SWRC) that goes to zero smoothly at high saturation is incompatible with convergence of a PCM hence the reason a PCM was not employed in our analysis. Conversely, we show why the SWRC (if it goes to zero at all at full saturation) must do so abruptly in order for the PCM to converge, and moreover, why the PCM approaches unity abruptly at full saturation in that case. Full details of the theory presented in this section are given in Burdine (1953); Brooks and Corey (1964, 1966); Mualem (1976).

The reason why PCM have been developed in the first place is that the experimental determination of relative hydraulic conductivity (RHC) is often considered complicated and expensive (Or and Assouline 2011). Thus, for many practical applications, it can be advantageous to attempt to determine RHC mathematically from SWRC data which are based on soil pore-size distribution. The more commonly used PCM (Burdine 1953; Mualem 1976) that link SWRC to RHC functions assume a simple pore geometry and link capillary properties of pores to RHC. However, as we will see, they also impose a constraint on the SWRC if the RHC is to converge.

A typical PCM leads to a general expression given as

$$\begin{aligned} K_r(\varTheta ) = \varTheta ^{\kappa } \left[ \int ^{\varTheta }_{0}\frac{1}{H_+^{\delta }} \mathrm {d}\varTheta \bigg / \int ^{1}_{0}\frac{1}{H_+^{\delta }} \mathrm {d}\varTheta \right] ^{\eta } \end{aligned}$$

(A.1)

where $\kappa , \, \delta , \, \eta$ are model parameters. It is found that $\kappa = 2$, $\delta = 2$, and $\eta = 1$ for the Burdine model (Burdine 1953), and $\kappa = 1/2$, $\delta = 1$, and $\eta = 2$ for the Mualem model (Mualem 1976). We focus on the derivation of the Mualem model.

1.1 A.1: Mualem’s Predictive Conductivity Model

Mualem (1976) considers a homogeneous porous medium with a set of interconnected pores defined by their radius r, and a pore-water distribution function $f(r)\mathrm {d}r$. The contribution of filled pores of radii between r and $r + \mathrm {d}r$ to the volumetric moisture content $\theta$ (rescaled here as $\varTheta$) is

$$\begin{aligned} f(r) \mathrm {d} r = \mathrm {d} \varTheta (r) . \end{aligned}$$

(A.2)

Considering a porous slab of thickness $\Delta x$, the pore area distribution at the two slab sides is assumed to be identical. The probability $a(r,\rho )$ of a pore of size r to $r + \mathrm {d}r$ at location x encountering a pore of radius $\rho$ to $\rho + \mathrm {d}\rho$ at $x + \Delta x$ is

$$\begin{aligned} a(r,\rho ) =f(r)f(\rho )\;\mathrm {d} r \; \mathrm {d} \rho . \end{aligned}$$

(A.3)

More generally, the probability of connection for a pore of radius between r and $r + \mathrm {d}r$ to a pore of radius between $\rho$ and $\rho + \mathrm {d}r$ may be given as

$$\begin{aligned} a(r,\rho ) = G(r,\rho ) f(r)f(\rho )\;\mathrm {d} r \; \mathrm {d} \rho , \end{aligned}$$

(A.4)

where the function $G(r,\rho )$ accounts for partial correlation between the pores r and $\rho$ at a given moisture content $\varTheta$. In what follows, two pores in series with radius r and $\rho$, respectively, are replaced by a single equivalent pore radius R and it is assumed that $G(r,\rho )$ can be expressed as a function G(R) . To determine conductivity, this connection probability needs to be weighted by a local hydraulic conductivity for each equivalent pore.

Mualem (1976) considers a pair of capillary elements whose lengths are proportional to their radii to replace the pore configuration and estimates the local hydraulic conductivity as proportional to $r \rho$ or more specifically as $T(r,\rho ) r \rho$ where $T(r,\rho )$ is a correction due to tortuosity. Again, it is assumed that $T(r,\rho )$ can be expressed as T(R) where R is a single equivalent radius. The term in $r\rho$ comes about as follows. Two pores in series (length $l_r$ and $l_{\rho }$, respectively) are replaced by a single equivalent pore (length L). The pressure drop across the equivalent pore is assumed to be the same as the total pressure drop along the two original pores in series. Since Poiseuille pressure drops scale as $(8/\pi ) \mu Q L / R^4$ (and $\mu$ is viscosity and Q is flow rate, which is the same in all cases), it follows

$$\begin{aligned} L/R^4 = l_r/r^4 + l_{\rho }/\rho ^4. \end{aligned}$$

(A.5)

Moreover, since the volume of the equivalent pore is assumed to be the same as total volume of the two original pores, it follows that

$$\begin{aligned} L R^2 = l_r r^2 + l_{\rho } \rho ^2. \end{aligned}$$

(A.6)

Significantly, the length of the equivalent pore need not be the same as the total length of the two original ones. Finally the aspect ratio of the pores is assumed fixed, hence

$$\begin{aligned} {l_r}/{r} = {l_{\rho }}/{\rho }. \end{aligned}$$

(A.7)

The above constitute 3 homogeneous linear Eqs. (A.5)–(A.7) in 3 unknowns L, $l_r$ and $l_{\rho }$. Non-trivial solutions only result if the determinant of the system of equations is zero. This provides a condition linking R to r and $\rho$, specifically

$$\begin{aligned} R = \sqrt{r \rho }. \end{aligned}$$

(A.8)

Taking the local hydraulic conductivity proportional to $r\rho$ as was suggested above is equivalent to taking the conductivity of equivalent pore properties to $R^2$ which is what is expected for a single pore. Solutions can now be obtained for $l_r$ and $l_{\rho }$ assuming L is given. It is necessary to fix the value of L since when the determinant vanishes, the equations are not all linearly independent.

$$\begin{aligned}&l_r = L r^2 \rho / (r^3 + \rho ^3),&l_{\rho } = L r \rho ^2 / (r^3 + \rho ^3). \end{aligned}$$

(A.9)

Note that

$$\begin{aligned} l_r + l_{\rho } = L r \rho ( r + \rho ) / (r^3 + \rho ^3) \end{aligned}$$

(A.10)

which is always smaller than L.

Note, however, that the volume of the equivalent pore is the same as the sum of the volumes of the two original ones by construction. Once we have established the size of a single equivalent pore, we have also established the liquid saturation, since we consider that the medium is filled up to pores of a given size but no further. Supposing we can re-write the tortuosity and connectivity factors as functions of $\varTheta$ (instead of in terms of r and $\rho$, or in terms of R), it follows that the hydraulic conductivity K is (assuming $r_{min}$ and $\rho _{min}$ are minimum sizes)

$$\begin{aligned} K \propto T(\varTheta ) G(\varTheta ) \int _{r_{min}}^{r} r f(r) \mathrm {d}r \int _{\rho _{min}}^{\rho } \rho f(\rho ) \mathrm {d}\rho , \end{aligned}$$

(A.11)

and relative hydraulic conductivity is (assuming $R_{min}$ and $R_{max}$ are minimum and maximum sizes)

$$\begin{aligned} K_r(\varTheta ) = T(\varTheta ) G (\varTheta ) \left[ \int _{R_{min}}^{R} r f(r) \;\mathrm {d}r \Bigg / \int _{R_{min}}^{R_{max}} r f(r) \; \mathrm {d}r \right] ^{2}. \end{aligned}$$

(A.12)

Here we have exploited the fact that the integrals over r and $\rho$ are the same, and the denominator of Eq. (A.12) is a normalisation condition. The important point here is that unless f(r) decays quite rapidly as r becomes large (decaying faster than $1/r^2$), K does not converge, or equivalently the denominator of $K_r$ diverges, making $K_r$ itself tend to zero.

If too much volume is associated with the large pores, which only fill near saturation, effectively all the flow at saturation is dominated by transport through those large pores, and in relative terms, below saturation flow is negligible. In situations like that, we can expect to see large variations in conductivity between rock samples, since the presence of slightly different numbers of large pores in different samples may influence hydraulic conduction.

The tortuosity correction factor $T(\varTheta )$ that is applied, and the connectivity correlation factor $G(\varTheta )$ are assumed to be power-law functions of $\varTheta$, and are replaced by a single factor $\varTheta ^{\kappa }$. Substituting $\kappa =1/2$ (fit by Mualem (Mualem 1976; Assouline 2001) to 45 soil samples) within that term, applying the capillary law relating pore radius to capillary head $r = C/H_+$ where C is a constant (independent of geometry), and using Eq. (A.2), we obtain for Eq. (A.12)

$$\begin{aligned} K_r(\varTheta ) = \varTheta ^{1/2} \left[ \int _{0}^{\varTheta } \frac{\mathrm {d} \varTheta }{H_+} \Bigg / \int _{0}^{1} \frac{\mathrm {d} \varTheta }{H_+} \right] ^{2}. \end{aligned}$$

(A.13)

As noted in the main text, there are convergence issues here if $H_+$ is too small for too large a fraction of the volume in the limit as $H_+ \rightarrow 0$. If, for example, $H_+ \approx |\mathrm {d}H_+/\mathrm {d}\varTheta |_{\varTheta = 1}{(1-\varTheta )}$ for some finite $|\mathrm {d}H_+/\mathrm {d}\varTheta |_{\varTheta = 1}$, then $\int _{0}^{1} \mathrm {d}\varTheta /H_+$ diverges. As mentioned earlier, the issue is that there is now so much volume in large radius (small $H_+$) pores that almost all conduction occurs through large pores. A larger (i.e. singular) $|\mathrm {d}H_+/\mathrm {d}\varTheta |$ resolves the issue by having less volume in such small $H_+$ pores. The SWRC selected by van Genuchten (van Genuchten 1980) typically has for $\varTheta$ close to 1

$$\begin{aligned} H_+ \approx \left( {(1-\varTheta )}/{m}\right) ^{1-m}; \qquad \varTheta \approx (1-mH_+^{1/{(1-m)}}), \end{aligned}$$

(A.14)

for some value $m < 1$ (typically with m close to 1 for sandstones, and m significantly less than 1 for loams). Here we recognise via Eq. (A.2), that

$$\begin{aligned} f(r)&= \frac{\mathrm {d}\varTheta }{\mathrm {d}r} = \frac{\mathrm {d}\varTheta }{\mathrm {d}H_+} \cdot \frac{\mathrm {d}H_+}{\mathrm {d}r}, \end{aligned}$$

(A.15)

and also recognise that (as alluded to earlier),

$$\begin{aligned} H_+ = \frac{C}{r}. \end{aligned}$$

(A.16)

It follows then that in the limit of large r (i.e. small $H_{+}$)

$$\begin{aligned} f(r)&\approx \frac{m}{(1-m)}H_+^{m/(1-m)} \cdot \frac{C}{r^2} \nonumber \\&\approx \frac{m}{(1-m)}\frac{C^{m/(1-m)}C}{r^{m/(1-m)}r^2} \nonumber \\&\approx \frac{m}{(1-m)}\frac{C^{1/(1-m)}}{r^{(2-m)/(1-m)}} . \end{aligned}$$

(A.17)

Notice how this function decays in the large r limit. If m is close to 1 (e.g. sandstone), f(r) decays very rapidly at large r, so there are comparatively few pores that are much larger than the sample average. If m is rather smaller than unity (e.g. loam), the population of pores that are significantly larger than the sample average increases. Certainly, the average pore size in loam tends to be much smaller than in sandstone but that effect is already accounted for in our dimensionless system. What we are considering here is not the average pore size, but rather the pore size distribution relative to that average.

Note that in the limit as $m\rightarrow 0$, hence with a non-singular $H_{+}\propto (1-\varTheta )$, it follows via Eq. (A.17) that f(r) becomes proportional to $r^{-2}$, and (as noted earlier) the denominator of Eq. (A.12) then fails to converge as $R_{max}\rightarrow \infty$. Yet again this demonstrates that there are now, in relative terms, so many large pores that almost all the flow is passing through them, so the (total) conductivity we compute is sensitive to what exactly the largest pore size $R_{max}$ is. As $R_{max}\rightarrow \infty$, the hydraulic conductivity up to any finite pore size $R<R_{max}$ is then negligibly small compared to the conductivity through the very largest pores, so that relative conductivity $K_{r}$ (considered only up to that finite pore size) tends to zero.

Assuming on the other hand that $0<m<1$ so that Eq. (A.13) does indeed converge, notice, however,

$$\begin{aligned} \frac{\mathrm {d}K_r}{\mathrm {d}\varTheta } \approx ~ \frac{1}{2 \, \varTheta } K_r + \frac{2 \varTheta ^{1/2} }{H_+} \int _0^{\varTheta } \frac{\mathrm {d}\varTheta }{H_+} \bigg / \left( \int _0^1 \frac{\mathrm {d}\varTheta }{H_+}\right) ^2. \end{aligned}$$

(A.18)

The consequence of having $H_+ \rightarrow 0$ abruptly as $\varTheta \rightarrow 1$, e.g. $H_+$ scaling as proportional to $(1-\varTheta )^{\alpha }$ for some $0< \alpha < 1$ with $\alpha = 1 - m$ here (see Eq. (A.14)), is that $\mathrm {d}K_r/\mathrm {d}\varTheta$ also diverges as $(1-\varTheta )^{-\alpha }$ in that same limit. A singular $H_+$ vs $\varTheta$ relation leads to a convergent $K_r$ vs $\varTheta$ but not a convergent $\mathrm {d}K_r / \mathrm {d}\varTheta$ vs $\varTheta$, at least in the PCM adopted here. The only way to keep $\mathrm {d}K_r/\mathrm {d}\varTheta$ from diverging is to take $\alpha \rightarrow 0$, i.e. $m\rightarrow 1$. However, that means according to (A.14) that $H_{+}$ must stay nonzero as full saturation is approached. Returning to the case $0<m<1$, however, the value of $D_r(\varTheta )$ predicted by the PCM meanwhile is divergent, since $D_r = K_r |\mathrm {d}H_+/\mathrm {d}\varTheta |$ (van Genuchten 1980), and since we require $\mathrm {d}H_+/\mathrm {d}\varTheta$ to diverge in order to have a convergent $K_r$, it necessarily follows that $D_r$ diverges.

In conclusion, using the predictive conductivity model PCM we have a choice between a system in which $H_{+}$ remains nonzero at full saturation (in which case $K_{r}$ converges to unity at $\varTheta \rightarrow 1$ with finite $\mathrm {d}K_{r}/\mathrm {d}\varTheta$ there), or a system in which $H_{+}$ approaches zero smoothly at full saturation (in which case $K_{r}$ is not well defined in the PCM, since conduction is dominated by the flow through the largest pores in the sample, hence is extremely sensitive to the particular sample), or else a system in which $H_{+}\rightarrow 0$ and $K_{r}\rightarrow 1$ at full saturation, but necessarily with mild singularities in that limit (both $|\mathrm {d}H_{+}/\mathrm {d}\varTheta |$ and $|\mathrm {d}K_{r}/\mathrm {d}\varTheta |$ diverge at full saturation). In the main text, we have avoided this issue by abandoning the PCM.

B: Appendix B

This appendix summarises some of the findings from Boakye-Ansah and Grassia (2021) regarding how travelling wave solutions of Richards equation behave when using the original van Genuchten (1980) soil material property functions. The relevant solutions are presented here for ease of comparison with those presented in the main text that use modified soil material property functions.

The profile of the travelling wave to Richards equation previously obtained within the limit of the special case we consider is

$$\begin{aligned} \frac{\mathrm {d} {\xi }}{\mathrm {d}\varTheta } = \frac{D_r(\varTheta )}{\varTheta - K_r(\varTheta )}, \end{aligned}$$

(B.1)

which is analogous to Eq. (14) (identical to Eq. (15)). In the case of Eq. (B.1) above, the original $D_r$ and $K_r$ functions given by van Genuchten are used in its solution.

Near $\varTheta \approx 1$,

$$\begin{aligned} D_r(\varTheta ) \approx \left| \frac{\mathrm {d}H_+}{\mathrm {d}\varTheta } \right| \approx \frac{(1-m)}{m^{(1-m)}} (1-\varTheta )^{-m}, \end{aligned}$$

(B.2)

is the approximation used which is the derivative of the original van Genuchten SWRC in the same limit (Boakye-Ansah and Grassia 2021). The denominator of Eq. (B.1) is also scaled as

$$\begin{aligned} \varTheta - K_r \approx (1 - K_r) - (1 - \varTheta ). \end{aligned}$$

(B.3)

If $K_r \rightarrow 1$ abruptly as $\varTheta \rightarrow 1$, then typically, $1-\varTheta \ll 1-K_r$. As explained in Boakye-Ansah and Grassia (2021) moreover

$$\begin{aligned} 1 - K_r \approx \frac{2}{m^m}(1-\varTheta )^m. \end{aligned}$$

(B.4)

Hence,

$$\begin{aligned} \frac{\mathrm {d} \xi }{\mathrm {d}\varTheta } \approx \frac{|\mathrm {d}H_+/\mathrm {d}\varTheta |}{1-K_r(\varTheta )}; \qquad \frac{\mathrm {d} \xi }{\mathrm {d}\varTheta } \approx \frac{(1-m)}{2m^{1-2m}} (1-\varTheta )^{-2m}, \end{aligned}$$

(B.5)

for which we obtain

$$\begin{aligned} \xi = c_0 + \frac{(1-m)m^{2m-1}}{2(2m-1)} (1-\varTheta )^{1-2m}, \end{aligned}$$

(B.6)

where $c_0$ is an integration constant. Here we are primarily interested in m values $0.5< m < 1$ so that $-1< 1-2m < 0$. Note that this is a power law for $\xi$, hence it grows more rapidly than the logarithmic law discussed in the main text. We have expressed this in terms of a variable $\xi$ rather than a rescaled variable ${\hat{\xi }}$ which was not used in Boakye-Ansah and Grassia (2021) owing to Eq. (B.2) diverging in the $\varTheta \rightarrow 1$ limit. However, for comparison we can convert from $\xi$ to ${\hat{\xi }}$ by dividing through by the capped maximum value of $|\mathrm {d}H_+/\mathrm {d}\varTheta |$ given in Table 1.