a. Water table depth

The water table depth in ZD09 is defined as the depth in the soil where the matric head ψ (units of length) is equal to the soil’s air-entry value ψ_sat. In soil physics, this depth would usually be labeled the top of the capillary fringe, as it indicates the depth below which all pores are filled with water. However, if a monitoring well is installed (usually a perforated tube of such radius that capillary rise within it is negligible), the water level in this well would be located at the depth where the matric head is zero, since the water is at atmospheric pressure at that depth. This is the formal definition of the groundwater table that separates soil water from groundwater, and which is used in both soil physics and groundwater hydrology. It is obviously lower than the level where ψ = ψ_sat. At hydrostatic equilibrium, the thickness of the capillary fringe, in which the soil is saturated but the water is at subatmospheric pressure, is equal to −ψ_sat. The difference between the definitions for groundwater levels used by ZD09 and the groundwater community may become an issue if groundwater levels derived from water levels in monitoring wells are used as input for ZD09’s model.

Note that, in the capillary fringe as well as in the groundwater, the volumetric water content θ does not depend on ψ, and dθ/dψ is necessarily zero if the porous matrix is assumed inelastic. Of course the depth of ψ = ψ_sat can vary with time, and as soon as air enters pores that were previously saturated, that part of the soil no longer is part of the capillary fringe or the groundwater domain. In the following we retain ZD09’s notation z_w (units of length) for the level of the top of the capillary fringe, and use z_p for the phreatic level.

b. Richards equation: Water-content-based form and mixed form

Equation (1) of ZD09 contains both the volumetric water content and the matric head. It is therefore termed the mixed form of Richards equation, not the water-content-based form, as ZD09 suggest (see Jury et al. 1991, 105–107 for the relation between the various forms of Richards equation). The crucial property of this equation is that the flux density is proportional to the gradient in the hydraulic head through Darcy’s Law:

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in which the notation is taken from ZD09. In the θ-based form, the gradient of θ is used:

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with D [units of (length)² (time)⁻¹] the soil water diffusivity, which can be expressed as K(dψ/dθ) (Hillel 1998, p. 216) The term dθ/dψ is the slope of the soil water characteristic defined by Eq. (5) of ZD09 (or any other expression one chooses). Below z_w, both gradients in Eq. (2) are zero, and the θ-based form of Darcy’s Law is of no use.

In contrast, the hydraulic head gradient is equally well defined in the saturated and unsaturated zones of a soil. Since ZD09 ignore hysteresis, the conversion of their Eq. (1) to the matric head–based form is straightforward: it requires the expansion of the storage change term:

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where t denotes time. Given the remarks above about the behavior of dθ/dψ below the depth where ψ = ψ_sat, Eq. (3) implies that in the capillary fringe and in the groundwater, the temporal gradient (∂θ/∂t) vanishes and the solution to the flow equation for time-varying boundary conditions becomes a sequence of steady-state solutions: the saturated system responds instantaneously to changes in the boundary conditions. By taking this into account, model codes based on Richards equation can adequately handle saturated-unsaturated flow, as is well documented in the literature. The appearance of a time derivative of θ does not affect its applicability to saturated regions.

ZD09’s objections against the use of the θ-based form of Richards equation for partially saturated soils hold only for the strictly θ-based form derived by combining the mass balance of a elementary volume with the θ-based version of Darcy’s Law [Eq. (2)]. Because the gradients in Eq. (2) vanish in saturated soils while flow remains possible, the purely θ-based form becomes invalid there; this θ-based version of the equation is not considered by ZD09.

c. Relationship between Darcy’s Law and Richards equation

Contrary to the suggestion of the statement preceding Eq. (6) in ZD09, Richards equation [Eq. (1) in ZD09] can be derived by combining the mass balance with Darcy’s Law [Eq. (2) in ZD09] (see e.g., Jury et al. 1991, p. 105). One therefore does not solve ZD09’s Eqs. (1) and (2) simultaneously, but only Eq. (1) subject to initial and boundary conditions, since Eq. (1) of ZD09 already incorporates their Eq. (2).

Note, incidentally, that ZD09’s equation for gravitational drainage [Eq. (4) in ZD09] and their alternative formulation of Darcy’s Law [Eq. (14) in ZD09] lack a minus sign. For gravitational drainage, the vertical gradient in the matric head (∂ψ/∂z) equals zero, and the flow is strictly gravity driven, with the gradient in the gravitational head equal to 1 (hence the frequently used term “unit-gradient flow”) since the vertical coordinate z is defined positive upward. With the flux opposite to the gradient, it follows from Darcy’s Law that q_b = −K(z_b) (the subscript b indicates the lower boundary of the model domain). The minus sign correctly indicates that the flow is downward.

d. The lower boundary condition

The unit-gradient lower boundary condition [Eq. (4) in ZD09] is typically invoked when groundwater is too deep to affect the hydraulic head in the model domain. The condition forces a perpetual but nonconstant downward flux at z_b, thereby draining the profile. The motivation for this boundary condition lies in the damping deeper in the profile of the infiltration/evapotranspiration forcing that takes place at the soil surface. The amplitude of the fluctuations between upward and downward fluxes reduces with depth, until it eventually becomes zero and the downward flux density is equal to the long-term net infiltration. The profile dries to the point where the hydraulic conductivity deep in the profile is equal to that net-infiltration rate. Figures 3–6 in ZD09 show the tendency toward that steady-state flow, but the runs were too short to achieve it.

From the above it is clear that relying on the unit-gradient boundary condition when the groundwater table is shallow is not warranted, and ZD09 provide ample proof of simulations producing unrealistic results as a consequence of the ill-advised use of this boundary condition. Zero flux at the lower boundary is not always realistic either, as ZD09 correctly state. ZD09 present their Eq. (14) as a novel boundary condition. If we add the missing minus sign and replace ψ_E by C − z according to ZD09’s Eq. (7), this boundary condition reduces to Darcy’s Law, evaluated at z_b.

However, with Richards equation [ZD09’s Eq. (1) and its equivalent Eq. (11a)] being a second-order partial differential equation, its solution requires boundary conditions that prescribe the dependent variable, its gradient, or a linear combination of both. Equation (14) of ZD09 is none of these and therefore does not constitute a mathematically valid boundary condition. But the text preceding ZD09’s Eq. (14) is enlightening: it explains that Eq. (14) of ZD09 arises from prescribing a hydraulic head at a given depth below z_b. Such a prescribed head does represent a valid boundary condition, and is aptly named fixed-head boundary condition in the groundwater literature. As ZD09 correctly state, it allows an exchange of water in two directions across the boundary for which it is defined.

This boundary condition can, and probably should, be implemented in land models. But that can be done directly by specifying a suitable hydraulic head at z_b (which can be derived from a specified groundwater level if hydrostatic equilibrium is assumed between z_p and z_b). The implementation by ZD09, through an additional model layer below the model domain, is cumbersome and cloaks the true nature of the boundary condition, but it seems to be the best way to introduce it in their land model.

For areas where the phreatic aquifer can easily discharge into a sufficiently dense surface water network and the climate ensures a precipitation surplus, the phreatic level may vary little, making the fixed-head boundary condition valid, even if the prescribed head is constant over time. The annual net infiltration is then entirely converted to river discharge (and perhaps recharge to deeper aquifers) because the storage change in the aquifer is zero. Under less favorable conditions, the fixed-head lower boundary condition can produce unacceptable amounts of groundwater recharge or capillary flow into the soil if applied to the scale of climate models. Nevertheless, this type of boundary condition is potentially very valuable for land models. If the need arises, the magnitude of the fluxes passing z_b can be manipulated by varying K(z_b) in the calibration mode. Also, by prescribing the hydraulic head directly at z_b (possibly requiring modifications in the model code), it can be made time dependent (to reflect seasonal and other variations) in a transparent way.

a. Hydrostatic equilibrium simulations

Figure 2 in ZD09 shows initial moisture profiles that are consistent with the depths of the capillary fringe specified on input (termed water table depths by ZD09). The authors ran the Community Land Model version 3 (CLM3) with these initial conditions while implementing zero flux as the top and bottom boundary conditions. This implies that the profile should converge to hydrostatic equilibrium. Since the initial soil profiles reflected the hydrostatic equilibrium profiles, the nodal water content changes should be minimal, only reflecting minor round-off errors. The moisture profiles in Fig. 2 instead show a dramatic drying, as they evolve toward another hydrostatic equilibrium profile. ZD09 state that the mass-conservative numerical scheme somehow manufactures water contents in excess of θ_sat. Such extreme values usually indicate numerical oscillations, and the possibility therefore exists that the calculated water contents elsewhere in the profile are too low (although not necessarily smaller than zero). This is to be expected if the numerical scheme is indeed mass conservative, as ZD09 state. By removing the excess water as runoff (by topping-off down to θ_sat or lower), the positive oscillations are corrected while the negative oscillations are allowed to exert their full effect on the total storage in the profile. This could explain the profound drying visible in Fig. 2 of ZD09, but it is not clear why the drying stops when z_w reaches the top of the lowest grid cell.

The cause of these oscillations is not immediately clear since the simulations started with and should convert to hydrostatic equilibrium without any flow occurring. The fact that the excessive drying only occurred when the top of the capillary fringe was within the model domain may point to convergence problems related to the singularity at ψ_sat, where the soil water characteristics of Clapp and Hornberger (1978) and Brooks and Corey (1964) are nondifferentiable. The grid size and/or the time step may have been excessively large around z_w to adequately deal with this. Unfortunately, Oleson et al. (2004) do not provide details of the iteration scheme of their Richards’ solver and the criteria used in CLM3 to modify the iteration process (e.g., by changing the time step or stopping the program if some set of convergence criteria is not satisfied).

b. The modified Richards equation

The benefits of subtracting the solution for hydrostatic equilibrium from Richards’ equation are not apparent from the structure of the equation [ZD09’s Eq. (1)]. From the equation preceding Eq. (11) in ZD09 it is clear that the same constant is subtracted from the hydraulic head at any depth. Since Richards equation only requires the gradient of the hydraulic head, changing it by a constant has no effect. This can also be seen from the fact that one can define the z coordinate with respect to an arbitrary but constant elevation (ZD09 implicitly define z=0 at the soil surface). The choice of this reference elevation obviously affects the magnitude of the gravitational head, but since only gradients are required, this is immaterial.

When simulations are nonsteady and the groundwater level changes, the hydrostatic equilibrium hydraulic head that is to be subtracted from the actual head should be updated for every time step. Even then, the numerical equivalence of the original and modified versions of Richards’ equation is unlikely to be compromised because the spatial gradients, which are not affected, are evaluated within and not between the time steps. It is crucial that the matric head is not affected by the operation because that would change the water content and the hydraulic conductivity in the unsaturated part of the soil. Thus, subtracting an equilibrium value only modifies the gravitational head, as Eq. (11) of ZD09 confirms. This implies that the subtraction of the hydraulic head corresponding to hydrostatic equilibrium is entirely equivalent to changing the reference height for the gravitational head, and of no consequence for the numerical solution.

The question then remains why the simulations based on the “corrected” Richards equation suddenly produce the expected output. Since the cause of the errors apparent in Fig. 2 of ZD09 remains unclear, this question cannot not be addressed with certainty without access to the original numerical code and the input files. But together with the modified Richards equation, the new fixed-head boundary condition is introduced. This boundary condition not only permits fluxes through the lower boundary but also places z_w, and thus the singularity at ψ_sat, below the model domain. Therefore, the improvement in the results may be due to the new boundary condition, since earlier simulations with z_w below z_b gave correct results (ZD09’s Fig. 1). This view is supported by ZD09’s statement in their section 4 that “if the same free drainage bottom boundary condition is applied, the use of the original or modified form of the Richards equation does not make much difference.” In the next paragraph the authors state that the fixed-head boundary condition is important in resolving the mass balance problems, but needs to be used in conjunction with the modified Richards equation to achieve its full effect. The paper does not offer direct evidence for these additional benefits in a test that separates the effect of the new boundary condition from that of the modified Richards equation.

c. Constant flux density simulations

ZD09 report that the soil model cannot handle high infiltration and drainage flux densities. These high flow rates were imposed on a soil initially at hydrostatic equilibrium, with a fairly dry top soil. Modeling such an infiltration process is numerically challenging even for the most sophisticated unsaturated flow models, so the reported numerical difficulties are not surprising, given the coarse discretization. Interestingly, the modified version of Richards equation performs poorly below 1.75-m depth (ZD09’s Fig. 7). For the largest infiltration rate, q is still smaller than the saturated hydraulic conductivity, and the steady-state solution should therefore maintain an unsaturated soil profile. ZD09 nevertheless report water contents in excess of saturation for both version of Richards’ equation, indicating numerical instabilities that may have arisen from the nature of the flow problem in combination with the numerical discretization, rather than the singularity of the soil water characteristic.