Improving the Numerical Simulation of Soil Moisture–Based Richards Equation for Land Models with a Deep Or Shallow Water Table’’
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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 gravita- tional 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 gravi- tational 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.