Improving the Numerical Simulation of Soil Moisture–Based Richards Equation for Land Models with a Deep Or Shallow Water Table’’

1044 JOURNAL OF HYDROMETEOROLOGY VOLUME 11

Comments on ‘‘Improving the Numerical Simulation of Soil Moisture–Based Richards Equation for Land Models with a Deep or Shallow Water Table’’

GERRIT H. DE ROOIJ Department of Soil Physics, Helmholtz Centre for Environmental Research–UFZ, Halle (Saale), Germany

(Manuscript received 25 June 2009, in ﬁnal form 15 April 2010)

1. Introduction 2. Unsaturated flow: Theory and terminology Zeng and Decker (2009, hereafter ZD09) recently a. Water table depth discussed the effect of truncation errors in a numerical solution of Richards equation often used to model ver- The water table depth in ZD09 is defined as the depth tical soil water flow in land models for weather and cli- in the soil where the matric head c (units of length) is mate studies. They adopted a cross-disciplinary approach equal to the soil’s air-entry value csat. In soil physics, this by applying an established meteorological method to depth would usually be labeled the top of the capillary a soil physical problem. Their solution involved sub- fringe, as it indicates the depth below which all pores are tracting from the original equation the solution at hy- filled with water. However, if a monitoring well is in- drostatic equilibrium in order to eliminate the truncation stalled (usually a perforated tube of such radius that errors. capillary rise within it is negligible), the water level in There is some confusion in the terminology and in the this well would be located at the depth where the matric treatment of Richards equation in ZD09. This comment head is zero, since the water is at atmospheric pressure at is intended to clarify these issues since they can create that depth. This is the formal definition of the ground- misunderstandings in the communications between the water table that separates soil water from groundwater, various disciplines involved in the work of ZD09. More and which is used in both soil physics and groundwater seriously, the averaging equation for the matric head hydrology. It is obviously lower than the level where c 5 presented in ZD09 does not conserve energy and the csat. At hydrostatic equilibrium, the thickness of the expression for the average water content suffers from capillary fringe, in which the soil is saturated but the mathematically undefined terms that preclude its nu- water is at subatmospheric pressure, is equal to 2csat. merical evaluation for physically acceptable values. The The difference between the definitions for groundwater derivation is presented here of a complete set of energy- levels used by ZD09 and the groundwater community conserving averaging equations that can replace those of may become an issue if groundwater levels derived from ZD09. Finally, the point is made that ZD09 effectively water levels in monitoring wells are used as input for implemented a much-needed constant head lower bound- ZD09’s model. ary condition for soil water flow in their land model. The Note that, in the capillary fringe as well as in the improvement of the simulation results may well be at- groundwater, the volumetric water content u does not tributable to this boundary condition instead of the mod- depend on c,anddu/dc is necessarily zero if the po- ification of Richards’ equation. rous matrix is assumed inelastic. Of course the depth of c 5 csat 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 Corresponding author address: Gerrit H. de Rooij, Department of Soil Physics, Helmholtz Centre for Environmental Research– ZD09’s notation zw (units of length) for the level of the UFZ, Theodor-Lieser-Strasse 4, D-06120 Halle (Saale), Germany. top of the capillary fringe, and use zp for the phreatic E-mail: [email protected] level.

DOI: 10.1175/2010JHM1189.1

Ó 2010 American Meteorological Society Unauthenticated | Downloaded 10/02/21 08:30 PM UTC AUGUST 2010 N O T E S A N D C O R R E S P O N D E N C E 1045 b. Richards equation: Water-content-based form the mass balance of a elementary volume with the u-based and mixed form version of Darcy’s Law [Eq. (2)]. Because the gradients in Eq. (2) vanish in saturated soils while flow remains Equation (1) of ZD09 contains both the volumetric possible, the purely u-based form becomes invalid there; water content and the matric head. It is therefore termed this u-based version of the equation is not considered the mixed form of Richards equation, not the water- by ZD09. 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 equa- c. Relationship between Darcy’s Law and Richards tion is that the flux density is proportional to the gradient equation in the hydraulic head through Darcy’s Law: Contrary to the suggestion of the statement preceding Eq. (6) in ZD09, Richards equation [Eq. (1) in ZD09] ›(c 1 z) q 5ÀK , (1) can be derived by combining the mass balance with ›z Darcy’s Law [Eq. (2) in ZD09] (see e.g., Jury et al. 1991, in which the notation is taken from ZD09. In the u-based p. 105). One therefore does not solve ZD09’s Eqs. (1) form, the gradient of u is used: and (2) simultaneously, but only Eq. (1) subject to initial and boundary conditions, since Eq. (1) of ZD09 already ›u du incorporates their Eq. (2). q 5ÀD 1 , (2) ›z dc Note, incidentally, that ZD09’s equation for gravitational drainage [Eq. (4) in ZD09] and their alternative with D [units of (length)2 (time)21] the soil water dif- formulation of Darcy’s Law [Eq. (14) in ZD09] lack a fusivity, which can be expressed as K(dc/du) (Hillel minus sign. For gravitational drainage, the vertical gra- 1998, p. 216) The term du/dc is the slope of the soil water dient in the matric head (›c/›z) equals zero, and the flow characteristic defined by Eq. (5) of ZD09 (or any other is strictly gravity driven, with the gradient in the gravitational head equal to 1 (hence the frequently used term expression one chooses). Below zw, both gradients in Eq. (2) are zero, and the u-based form of Darcy’s Law is ‘‘unit-gradient flow’’) since the vertical coordinate z is of no use. defined positive upward. With the flux opposite to the In contrast, the hydraulic head gradient is equally well gradient, it follows from Darcy’s Law that qb 52K(zb) defined in the saturated and unsaturated zones of a soil. (the subscript b indicates the lower boundary of the Since ZD09 ignore hysteresis, the conversion of their model domain). The minus sign correctly indicates that Eq. (1) to the matric head–based form is straightfor- the flow is downward. ward: it requires the expansion of the storage change term: d. The lower boundary condition ›u du ›c The unit-gradient lower boundary condition [Eq. (4) 5 , (3) ›t dc ›t in ZD09] is typically invoked when groundwater is too deep to affect the hydraulic head in the model domain. where t denotes time. Given the remarks above about The condition forces a perpetual but nonconstant down- the behavior of du/dc below the depth where c 5 csat, ward flux at zb, thereby draining the profile. The moti- Eq. (3) implies that in the capillary fringe and in the vation for this boundary condition lies in the damping groundwater, the temporal gradient (›u/›t) vanishes and deeper in the profile of the infiltration/evapotranspiration the solution to the flow equation for time-varying bound- forcing that takes place at the soil surface. The ampli- ary conditions becomes a sequence of steady-state solu- tude of the fluctuations between upward and downward tions: the saturated system responds instantaneously to fluxes reduces with depth, until it eventually becomes changes in the boundary conditions. By taking this into zero and the downward flux density is equal to the long- account, model codes based on Richards equation can term net infiltration. The profile dries to the point where adequately handle saturated-unsaturated flow, as is well the hydraulic conductivity deep in the profile is equal to documented in the literature. The appearance of a time that net-infiltration rate. Figures 3–6 in ZD09 show the derivative of u does not affect its applicability to satu- tendency toward that steady-state flow, but the runs were rated regions. too short to achieve it. ZD09’s objections against the use of the u-based form From the above it is clear that relying on the unit- of Richards equation for partially saturated soils hold gradient boundary condition when the groundwater ta- only for the strictly u-based form derived by combining ble is shallow is not warranted, and ZD09 provide ample

Unauthenticated | Downloaded 10/02/21 08:30 PM UTC 1046 JOURNAL OF HYDROMETEOROLOGY VOLUME 11 proof of simulations producing unrealistic results as a 3. Averaging of the water content and matric head consequence of the ill-advised use of this boundary over depth condition. Zero flux at the lower boundary is not always To determine the average water content and matric realistic either, as ZD09 correctly state. ZD09 present potential within grid cells, ZD09 first average u and then their Eq. (14) as a novel boundary condition. If we add directly apply the local u(c) relationship to the average the missing minus sign and replace c by C 2 z ac- E water content to estimate the average c of grid cell i. cording to ZD09’s Eq. (7), this boundary condition re- i This approach conserves mass during the averaging oper- duces to Darcy’s Law, evaluated at z . b ation but fails to conserve energy. Obviously, any aver- However, with Richards equation [ZD09’s Eq. (1) and aging operation for any type of application should neither its equivalent Eq. (11a)] being a second-order partial create nor destroy energy or mass (e.g., Gray 2002), but differential equation, its solution requires boundary con- schemes intended for operation on large areas involving ditions that prescribe the dependent variable, its gradient, vast quantities of water should be particularly meticu- or a linear combination of both. Equation (14) of ZD09 lous in obeying the fundamental conservation laws. is none of these and therefore does not constitute a math- The matric head c represents potential energy of the ematically valid boundary condition. But the text pre- soil solution expressed by weight (e.g., Hillel 1998, ceding ZD09’s Eq. (14) is enlightening: it explains that p. 153). To conserve energy, averaging over a soil volume Eq. (14) of ZD09 arises from prescribing a hydraulic (or, in one dimension, over a depth interval) therefore head at a given depth below z . Such a prescribed head b requires the local values of c to be weighted by the weight does represent a valid boundary condition, and is aptly rgu of the local water (with r as the density of the soil named fixed-head boundary condition in the ground- solution and g the gravitational acceleration). If the water literature. As ZD09 correctly state, it allows an soil solution is assumed to have uniform density (in- exchange of water in two directions across the boundary compressible fluid, isothermal conditions), this sim- for which it is defined. plifies to weighting by u (de Rooij 2009). The general This boundary condition can, and probably should, be expression for the average of c over a depth interval implemented in land models. But that can be done di- then becomes rectly by specifying a suitable hydraulic head at zb (which can be derived from a specified groundwater level if ! ð ð À1 z z hydrostatic equilibrium is assumed between z and z ). i11/2 i11/2 p b c 5 uc dz u dz , (4) The implementation by ZD09, through an additional i ziÀ1/2 ziÀ1/2 model layer below the model domain, is cumbersome and cloaks the true nature of the boundary condition, but where the overbar denotes an average, subscript i in- it seems to be the best way to introduce it in their land dicates the depth interval, and the notational convention model. of the limits of depth interval i was adopted from ZD09. For areas where the phreatic aquifer can easily dis- For z $ z (the entire interval above the capillary charge into a sufficiently dense surface water network i21/2 w fringe) and at hydrostatic equilibrium we have: and the climate ensures a precipitation surplus, the phreatic level may vary little, making the fixed-head boundary ! ð ð À1 z z condition valid, even if the prescribed head is constant i11/2 i11/2 c 5 u (z)(z À z) dz u (z) dz , (5) over time. The annual net infiltration is then entirely i E p E ziÀ1/2 ziÀ1/2 converted to river discharge (and perhaps recharge to deeper aquifers) because the storage change in the aqui- where u (z) indicates the equilibrium water content at z, fer is zero. Under less favorable conditions, the fixed-head E in accordance with ZD09’s notation. For z # z (the lower boundary condition can produce unacceptable i11/2 w entire interval in or below the capillary fringe) we have amounts of groundwater recharge or capillary flow into for uniform saturated water content u : the soil if applied to the scale of climate models. Nev- sat ! ertheless, this type of boundary condition is potentially ð ð À1 z z very valuable for land models. If the need arises, the i11/2 i11/2 ci 5 usat(zp À z) dz usat dz magnitude of the fluxes passing zb can be manipulated ziÀ1/2 ziÀ1/2 by varying K(zb) in the calibration mode. Also, by pre- 1 scribing the hydraulic head directly at z (possibly re- 5 z À (z 1 z ). (6) b p 2 iÀ1/2 i11/2 quiring modifications in the model code), it can be made time dependent (to reflect seasonal and other variations) For zi21/2 , zw , zi11/2 (top of the capillary fringe in a transparent way. within the depth interval) we note that, at hydrostatic

Unauthenticated | Downloaded 10/02/21 08:30 PM UTC AUGUST 2010 N O T E S A N D C O R R E S P O N D E N C E 1047 equilibrium, zw 5 zp 2 csat. The correct value of ci is using the appropriate integration boundaries, and sub- found by computing averages for the unsaturated and sequently averaging those averages by weighting by the saturated regions from Eqs. (5) and (6), respectively, by amounts of water in both regions:

ð z 1 i11/2 zp À (ziÀ1/2 1 zp À csat) usat(zp À csat À ziÀ1/2) 1 uE(z)(zp À z) dz 2 z Àc c 5 ð p sat . (7) i zi11/2 usat(zp À csat À ziÀ1/2) 1 uE(z) dz zpÀcsat

Equations (5)–(7) rely on a uniform saturated water Eqs. (5), (9), and (10) gives the average matric head in content but are ﬂexible with respect to the choice of a depth interval above the capillary fringe: the soil water characteristic. ZD09 employ Clapp and Hornberger’s (1978) expression [see Eq. (5) of ZD09]: 1 "# csat À 1 2À1/B 2À1/B B zp À zi11/2 zp À ziÀ1/2 c 5 À ÀB i 1 c c u À 2 sat sat c 5 c . (8) B sat u "# sat À1 z À z 1À1/B z À z 1À1/B 3 p i11/2 À p iÀ1/2 . Inverting this relationship and making use of the hy- csat csat drostatic equilibrium condition, cE 5 zp 2 z, yields (11a) ! 1/B If ZD09’s Eqs. (8) and (9) are combined, a different csat uE(z) 5 usat , z $ zp À csat. (9) expression for c above the capillary fringe can be found, zp À z i but this equation too suffers from terms that cannot be evaluated. Combining ZD09’s Eq. (8) instead with the With this we can evaluate the integral of uE in Eq. (5) to correct expression for ui, Eq. (10) above, yields an esti- ﬁnd the average water content ui for regions above the mate of c based directly on u without conserving energy: capillary fringe: i i 8 ð > 2 ! z <> 1/BÀ1 i11/2 c c sat 4 sat uE(z) dz c 5 c z u c i sat> 1 iÀ1/2 sat sat :> zp À zi11/2 ui 5 5 (zi11/2 À ziÀ1/2) À 1 zi11/2 À ziÀ1/2 1 B (zi11/2 À ziÀ1/2) À 1 9 B !3>ÀB 2 ! !3 1/BÀ1 => 1/BÀ1 1/BÀ1 c c c À sat 5 . (11b) 34 sat À sat 5. (10) > zp À ziÀ1/2 ;> zp À zi11/2 zp À ziÀ1/2

Note that this equation is a correction of Eq. (9) of The average matric head for a depth interval that in- ZD09. The equation derived here avoids negative num- cludes the top of the capillary fringe can be found by bers raised to noninteger powers [every power term in combining Eqs. (7), (10) (with adjusted integration ZD09’s Eq. (9)], which cannot be evaluated. Combining boundaries), and (11a):

8 9 "# <> => 1 u c2 z À z 2À1/B c 5 (z 1 c À z )(z À c À z )u 1 sat sat p i11/2 À 1 i :>2 p sat iÀ1/2 p sat iÀ1/2 sat 1 c ;> À 2 sat 8 B 9 "#À1 <> => u c z À z 1À1/B 3 (z À c À z )u 1 sat sat p i11/2 À 1 . (12) :> p sat iÀ1/2 sat 1 c ;> À 1 sat B

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For completeness, the expression for the average water content of a depth interval enclosing the top of the capillary fringe is also given: 8 <> u c u 5 (z À c À z )u 1 sat sat i :> p sat iÀ1/2 sat 1 À 1 9B "# => z À z 1À1/B 3 p i11/2 1 (z z )À1. À > i11/2 À iÀ1/2 csat ;

(13)

In summary, the average matric head is given by Eq. (6) for saturated regions of the soil, by Eq. (11a) for unsaturated regions, and by Eq. (12) for partially saturated regions. The average water content is simply usat for saturated regions, is given by Eq. (10) for unsaturated regions, and is given by Eq. (13) for partially saturated regions. Although these equations are somewhat more elaborate than those of ZD09 they possess the same qualities: they are explicit and rely on the same param- eters as ZD09’s equations. Their implementation should therefore be equally straightforward. The distinct ad- vantage of Eqs. (6), (11a), and (12) and Eqs. (10) and (13) is their applicability anywhere in the profile and FIG. 1. Values of the average matric head in two 4-m-thick their accuracy irrespective of the grid size. In contrast, profiles of uniform soils divided into equally sized grid cells. The number of grid cells used is indicated in the legend. The exact so- ZD09’s Eq. (8) for ci breaks down if a grid cell contains even the smallest saturated region, and is expected to lutions are based on weighting the matric head at any depth by the volumetric water content at that depth. The label ZD09 refers to an perform poorer with increasing gridcell size. approximation [Eq. (10)] based on ZD09’s scheme. The vertical The performance of the equations for ci is demon- coordinate z is positive upward and zero at the soil surface. Cal- strated in Fig. 1 for the soil used by ZD09 (a clay loam), culated values for individual cells are plotted at the depth of the and for a Dutch dune sand (de Rooij 1995) to which the cell’s center. parameter vector (usat, csat, B)5(0.410, 20.265 m, 0.889) of Clapp and Hornberger (1978) was fitted. For both 4. The source of the mass-balance errors cases, a hydrostatic moisture profile was calculated with a. Hydrostatic equilibrium simulations the groundwater at 4-m depth. The approximation by ZD09 is quite accurate, even in the sandy soil. Its main Figure 2 in ZD09 shows initial moisture profiles that problem is the limited range of applicability. Below zw are consistent with the depths of the capillary fringe (23.77 m for the clay loam and 23.73 m for the sand), specified on input (termed water table depths by ZD09). the equation cannot be evaluated, which leaves out 25% The authors ran the Community Land Model version 3 of the profile for a 1-m cell size, and the exact single-cell (CLM3) with these initial conditions while implement- value (which represents the u-weighted matric head ing zero flux as the top and bottom boundary conditions. above the groundwater table) cannot be calculated by This implies that the profile should converge to hydro-

ZD09’s equation at all. Note that ci for a single cell of static equilibrium. Since the initial soil profiles reflected 4-m height is much higher for sand than it is for clay the hydrostatic equilibrium profiles, the nodal water con- loam. This reflects the fact that a large fraction of the tent changes should be minimal, only reflecting minor water in the coarse textured sand is stored in large pores round-off errors. The moisture profiles in Fig. 2 instead at high matric head. The finer pores of the clay loam can show a dramatic drying, as they evolve toward another retain much more water at lower matric heads: for the hydrostatic equilibrium profile. ZD09 state that the mass- sand, the water content at 1 m above the groundwater conservative numerical scheme somehow manufactures table is only 0.09; for ZD09’s soil, the water content still water contents in excess of usat. Such extreme values is 0.32 at 4 m above the groundwater table. usually indicate numerical oscillations, and the possibility

Unauthenticated | Downloaded 10/02/21 08:30 PM UTC AUGUST 2010 N O T E S A N D C O R R E S P O N D E N C E 1049 therefore exists that the calculated water contents else- of the soil. Thus, subtracting an equilibrium value only where in the profile are too low (although not neces- modifies the gravitational head, as Eq. (11) of ZD09 sarily smaller than zero). This is to be expected if the confirms. This implies that the subtraction of the hy- numerical scheme is indeed mass conservative, as ZD09 draulic head corresponding to hydrostatic equilibrium is state. By removing the excess water as runoff (by topping- entirely equivalent to changing the reference height for off down to usat or lower), the positive oscillations are the gravitational head, and of no consequence for the corrected while the negative oscillations are allowed to numerical solution. exert their full effect on the total storage in the profile. The question then remains why the simulations based This could explain the profound drying visible in Fig. 2 on the ‘‘corrected’’ Richards equation suddenly produce of ZD09, but it is not clear why the drying stops when zw the expected output. Since the cause of the errors ap- reaches the top of the lowest grid cell. parent in Fig. 2 of ZD09 remains unclear, this question The cause of these oscillations is not immediately cannot not be addressed with certainty without access clear since the simulations started with and should con- to the original numerical code and the input files. But vert to hydrostatic equilibrium without any flow occur- together with the modified Richards equation, the new ring. The fact that the excessive drying only occurred fixed-head boundary condition is introduced. This bound- when the top of the capillary fringe was within the model ary condition not only permits fluxes through the lower domain may point to convergence problems related to boundary but also places zw, and thus the singularity at the singularity at csat, where the soil water characteris- csat, below the model domain. Therefore, the improve- tics of Clapp and Hornberger (1978) and Brooks and ment in the results may be due to the new boundary

Corey (1964) are nondifferentiable. The grid size and/or condition, since earlier simulations with zw below zb gave the time step may have been excessively large around correct results (ZD09’s Fig. 1). This view is supported zw to adequately deal with this. Unfortunately, Oleson by ZD09’s statement in their section 4 that ‘‘if the same et al. (2004) do not provide details of the iteration scheme free drainage bottom boundary condition is applied, of their Richards’ solver and the criteria used in CLM3 the use of the original or modified form of the Richards to modify the iteration process (e.g., by changing the time equation does not make much difference.’’ In the next step or stopping the program if some set of convergence paragraph the authors state that the fixed-head bound- criteria is not satisfied). ary condition is important in resolving the mass balance problems, but needs to be used in conjunction with the b. The modified Richards equation modified Richards equation to achieve its full effect. The The benefits of subtracting the solution for hydrostatic paper does not offer direct evidence for these additional equilibrium from Richards’ equation are not apparent benefits in a test that separates the effect of the new from the structure of the equation [ZD09’s Eq. (1)]. From boundary condition from that of the modified Richards the equation preceding Eq. (11) in ZD09 it is clear that equation. the same constant is subtracted from the hydraulic head c. Constant flux density simulations at any depth. Since Richards equation only requires the gradient of the hydraulic head, changing it by a constant ZD09 report that the soil model cannot handle high has no effect. This can also be seen from the fact that one infiltration and drainage flux densities. These high flow can define the z coordinate with respect to an arbitrary rates were imposed on a soil initially at hydrostatic but constant elevation (ZD09 implicitly define z50at equilibrium, with a fairly dry top soil. Modeling such an the soil surface). The choice of this reference elevation infiltration process is numerically challenging even for obviously affects the magnitude of the gravitational head, the most sophisticated unsaturated flow models, so the but since only gradients are required, this is immaterial. reported numerical difficulties are not surprising, given When simulations are nonsteady and the groundwater the coarse discretization. Interestingly, the modified ver- level changes, the hydrostatic equilibrium hydraulic head sion of Richards equation performs poorly below 1.75-m that is to be subtracted from the actual head should be depth (ZD09’s Fig. 7). For the largest infiltration rate, q updated for every time step. Even then, the numerical is still smaller than the saturated hydraulic conductivity, equivalence of the original and modified versions of and the steady-state solution should therefore maintain Richards’ equation is unlikely to be compromised be- an unsaturated soil profile. ZD09 nevertheless report wa- cause the spatial gradients, which are not affected, are ter contents in excess of saturation for both version of evaluated within and not between the time steps. It is Richards’ equation, indicating numerical instabilities that crucial that the matric head is not affected by the op- may have arisen from the nature of the flow problem in eration because that would change the water content combination with the numerical discretization, rather and the hydraulic conductivity in the unsaturated part than the singularity of the soil water characteristic.

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5. Conclusions the modified Richards equation with the fixed-head lower boundary condition [ZD09’s Eq. (11) and (14)]. Prescrib- The analysis by ZD09 appears to point to problems in ing the hydraulic head directly at z would facilitate an the numerical solution of Richards equation, possibly b easier and more valid comparison between the original triggered by the nondifferentiable singularity in the soil and modified version of Richards equation by allowing it water characteristic at c . Together with a modified sat to implement all combinations of boundary conditions Richards equation to solve this problem, ZD09 intro- and versions of Richards equation. duce a new lower boundary condition, which is shown here to be a fixed-head condition. This type of boundary REFERENCES condition can be of considerable value in land models. The superior performance of the modified Richards Brooks, R. H., and A. T. C. Corey, 1964: Hydraulic properties of equation may be linked to the fact that the current for- porous media. Colorado State University Hydrology Papers 3, 27 pp. mulation of the fixed-head boundary condition removes Clapp, R. B., and G. M. Hornberger, 1978: Empirical equations for the singularity in the soil water characteristic from the some hydraulic properties. Water Resour. Res., 14, 601–604. simulated section of the soil profile. de Rooij, G. H., 1995: A three-region analytical model of solute Averaging water contents and matric heads over ver- leaching in a soil with a water-repellent top layer. Water Re- tical intervals in the soil and the groundwater can and sour. Res., 31, 2701–2707. ——, 2009: Averaging hydraulic head, pressure head, and gravi- should be done in such a way that both mass and potential tational head in subsurface hydrology, and implications for energy of the water are conserved in the averaging op- averaged fluxes, and hydraulic conductivity. Hydrol. Earth erations. A complete set of equations is presented here Syst. Sci., 13, 1123–1132. that ensures this conservation of mass and energy, and Gray, W. G., 2002: On the definition and derivatives of macroscale hopefully will be used in place of Eqs. (8) and (9) of ZD09. energy for the description of multiphase systems. Adv. Water Resour., 25, 1091–1104. In addition, one equation in ZD09 contained terms that Hillel, D., 1998: Environmental Soil Physics. Academic Press, 771 pp. cannot be evaluated, and a corrected version is provided. Jury, W. A., W. R. Gardner, and W. H. Gardner, 1991: Soil Physics. A more direct implementation of the fixed-head lower 5th ed. John Wiley & Sons, 328 pp. boundary condition is proposed as an alternative for Oleson, K. W., and Coauthors, 2004: Technical description of the the zero flux and unit-gradient boundary conditions. Community Land Model (CLM). NCAR Tech. Note NCAR/ TN-4611STR, 173 pp. ZD09 mostly compared the tandem of the conventional Zeng, X., and M. Decker, 2009: Improving the numerical solution Richards’ equation with the unit-gradient lower bound- of soil moisture–based Richards equation for land models with ary condition [ZD09’s Eq. (1) and (4)] to the tandem of a deep or a shallow water table. J. Hydrometeor., 10, 308–319.

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