VARIABLY SATURATED WATER FLOW
Governing Water Flow Equations
The modified Richards equation describes one-dimensional uniform water movement in a partially saturated rigid porous medium, assuming that the air phase has a minimal impact on liquid flow and that thermal gradient effects on water flow can be disregarded.
The equation presented, ∂∂ = ∂∂ ∂∂ + (2.1), describes the relationship between water pressure head (h), volumetric water content (θ), time (t), and spatial coordinate (x), with positive values indicating upward movement Additionally, it incorporates a sink term (S) that represents the rate of water loss, as well as the angle (α) between the flow direction and the vertical axis, where α ranges from 0° for vertical flow to 90° for horizontal flow The unsaturated hydraulic conductivity function (K) is also a crucial element of this equation, influencing the movement of water through unsaturated soil.
K h x = K x K h x (2.2) where K r is the relative hydraulic conductivity [-] and K s the saturated hydraulic conductivity [LT -1 ]
2.1.2 Uniform Water Flow and Vapor Transport
The Richards equation (2.1) primarily addresses liquid phase water flow, overlooking the influence of vapor phase on water mass balance Although this simplification is suitable for many scenarios, there are instances where vapor flow significantly impacts overall water movement, particularly in drier soils Research by Scanlon et al (2003) indicates that in the deep vadose zone of arid and semiarid regions in the western U.S., thermal vapor fluxes often dominate water fluxes The integration of nonisothermal liquid and vapor flow in HYDRUS is elaborated upon by Saito et al.
The total volumetric water content (θ T), expressed as the sum of volumetric liquid water content (θ) and volumetric water vapor content (θ v), is crucial for understanding water flow dynamics in soil This relationship is represented in equation (2.3), which incorporates various types of hydraulic conductivities, including isothermal and thermal for both liquid and vapor phases The overall water flow includes contributions from isothermal liquid flow, isothermal vapor flow, gravitational liquid flow, thermal liquid flow, and thermal vapor flow Due to the temperature dependence of several terms in the equation, it is essential to solve it concurrently with the heat transport equation to accurately capture the temporal and spatial variations in soil temperature.
2.1.3 Flow in a Dual-Porosity System
Dual-porosity models propose that water movement is primarily confined to fractures and larger pores, while the water within the matrix remains static This means that the matrix, composed of immobile water pockets, is capable of exchanging, retaining, and storing water, but does not allow for convective flow As a result, these models facilitate the understanding of two-region, dual-porosity flow and transport dynamics.
[Philip, 1968; van Genuchten and Wierenga, 1976] that partition the liquid phase into mobile (flowing, inter-aggregate), θ mo , and immobile (stagnant, intra-aggregate), θ im , regions: mo im
In this study, we examine the exchange of water and solutes between two regions, represented by the subscripts m for fractures, inter-aggregate pores, or macropores, and im for the soil matrix, intra-aggregate pores, or rock matrix The exchange is typically calculated using a first-order rate equation, allowing for a clear understanding of the dynamics involved in these interactions.
The dual-porosity formulation for water flow in HYDRUS-1D employs a mixed approach, utilizing Richards equation to model water movement in fractures (macropores) while applying a straightforward mass balance equation to capture moisture dynamics in the matrix, as outlined by Šimůnek et al (2003).
( ) cos mo mo w im im w
(2.5) where S m and S im are sink terms for both regions, and Γ w is the transfer rate for water from the inter- to the intra-aggregate pores
Germann (1985) and Germann and Beven (1985) proposed an alternative dual-porosity approach that utilizes a kinematic wave equation to model the gravitational movement of water in macropores, which has not yet been implemented in HYDRUS-1D While dual-porosity models have gained popularity in solute transport studies, such as those by van Genuchten (1981), their application in water flow problems remains limited.
2.1.4 Flow in a Dual-Permeability System
Dual-porosity models typically assume that water within the matrix remains stagnant, whereas dual-permeability models permit water flow within the matrix This distinction is highlighted in the approach developed by Gerke and van.
Genuchten (1993a, 1996) utilized Richards' equations to analyze two distinct pore regions, which is integrated into the HYDRUS-1D model This method defines the flow equations for both macropore or fracture systems (denoted by subscript f) and matrix systems (denoted by subscript m).
Solute im mo im im mo mo
Water im mo im im mo mo
= + a Uniform Flow b Mobile-Immobile Water c Dual-Porosity d Dual-Permeability e Dual-Permeability with MIM
Equilibrium Model - Non-Equilibrium Models -
Solute im mo im im mo mo
Solute im mo im im mo mo
Water im mo im im mo mo
Water im mo im im mo mo
= + a Uniform Flow b Mobile-Immobile Water c Dual-Porosity d Dual-Permeability e Dual-Permeability with MIM
Equilibrium Model - Non-Equilibrium Models -
Figure 2.1 illustrates conceptual models of physical nonequilibrium for water flow and solute transport The water content, denoted as θ, is divided into mobile (θ mo) and immobile (θ im) flow regions, as shown in plots (b) and (c) In plot (d), θ M and θ F represent the water contents in the matrix and macropore (fracture) regions, respectively Plot (e) further distinguishes the mobile and immobile flow regions within the matrix and macropore domains, indicated by θ M,mo, θ M,im, and θ F The concentrations of solutes in these regions are represented by c, with subscripts corresponding to their respective water contents The total solute content in the liquid phase is denoted as S, while w indicates the volume ratio of the macropore or fracture domain to the total soil system It is important to note that the water contents θ f and θ m in equation (2.6) differ from those in equation (2.5), as they refer to the specific pore domains rather than the total pore space, leading to the relationship θ = wθ f + (1-w)θ m, where θ M and θ F signify the absolute water contents in the matrix and macropore regions, respectively.
Root Water Uptake
2.2.1 Root Water Uptake Without Compensation
The sink term, S, is defined as the volume of water removed from a unit volume of soil per unit time due to plant water uptake Feddes et al [1978] defined S as
The root-water uptake response function, denoted as α(h), is a dimensionless function that varies with the soil water pressure head, ranging from 0 to 1 This function plays a crucial role in determining the potential water uptake rate (S p) in plants Figure 2.1 illustrates the stress response function as outlined by Feddes et al., highlighting its significance in understanding how water stress impacts plant water uptake.
In the context of water uptake, it is crucial to note that water absorption is assumed to be zero when soil moisture is near saturation, particularly above a specific threshold known as the "anaerobiosis point" (h1) Additionally, water uptake is also considered negligible below the wilting point pressure head (h4) Optimal water absorption occurs between the pressure heads h2 and h3, while uptake fluctuates linearly between h3 and h4, as well as between h1 and h2 The variable Sp represents the water uptake rate during periods without water stress, where the function α(h) equals 1.
Van Genuchten [1987] expanded formulation of Feddes et al [1978] by including osmotic stress as follows
The osmotic head (h φ) is defined as a linear combination of solute concentrations (c i), represented by the equation h = a c φ i, where a i are experimental coefficients that convert concentrations into osmotic heads van Genuchten (1987) introduced an S-shaped function to characterize the water uptake stress response, indicating that the effect of reduced osmotic head can be either additive or multiplicative.
In the context of salinity stress, experimental constants p, p1, and p2 are utilized, with the exponent p approximately equal to 3, as noted by van Genuchten (1987) The parameter h50 indicates the pressure head at which water extraction rates decrease by 50% under minimal osmotic stress, while hφ50 denotes the osmotic head for a similar reduction during negligible water stress This highlights the differing sensitivities of root water uptake to water and salinity stresses, which can be effectively modeled using the additive formulation presented in the equations.
(2.12) where a 1 is a coefficient accounting for different response to different stresses (=h 50/h φ50 )
Unlike the formulation by Feddes et al [1978], the S-shape function for the stress response, α(h,h φ), does not account for the reduction in transpiration during periods of saturation This simplification is deemed reasonable as saturated conditions typically last for only brief durations.
When the potential water uptake rate is equally distributed over the root zone, S p becomes
The potential transpiration rate (T p) is expressed in units of [LT -1], while L R represents the depth of the root zone in [L] This concept can be further generalized by incorporating a non-uniform distribution of the potential water uptake rate across a root zone with any arbitrary shape.
The schematic representation of the plant water stress response function, α(h), utilized by Feddes et al (1978) and van Genuchten (1987), illustrates the spatial variation of the potential extraction term, S p, across the root zone This function incorporates the normalized water uptake distribution, b(x), measured in [L -1], and is derived by normalizing any arbitrarily measured or prescribed root distribution function, bΝ(x).
∫ ′ (2.15) where L R is the region occupied by the root zone Normalizing the uptake distribution ensures that b(x) integrates to unity over the flow domain, i.e.,
Fig 2 3 Schematic of the potential water uptake distribution function, b(x), in the soil root zone
There are many ways to express the function b(x): constant with depth, linear [Feddes et al.,
1978], or the following function [Hoffman and van Genuchten, 1983]:
In HYDRUS, users can define any shape for the water uptake distribution function, as long as it remains constant throughout the simulation However, when the rooting depth changes over time, only the method by Hoffman and van Genuchten (1983) is applicable It is important to note that in this context, the soil profile is defined with the bottom at x = 0 and the soil surface at x = L, where L represents the x-coordinate of the soil surface and L R denotes the root depth.
From (2.14) and (2.16) it follows that S p is related to T p by the expression
The actual water uptake distribution is obtained by substituting (2.14) into (2.7):
S h h x φ =α h h x b x T φ (2.19) whereas the actual transpiration rate, T a , is obtained by integrating (2.19) as follows
The root depth (L R) in simulations can be constant or variable, and for annual vegetation, a growth model is necessary to represent the changes in rooting depth over time HYDRUS operates under the assumption that the actual root depth is determined by the maximum rooting depth (L m).
[L], and a root growth coefficient, f r (t) [-] [Šimůnek and Suarez, 1993a]:
For the root growth coefficient, f r (t), we use the classical Verhulst-Pearl logistic growth function
The rooting depth at the start of the growing season, denoted as L0, is influenced by a growth rate, r, which is expressed in time units This growth rate can be determined by the assumption that 50% of the rooting depth will be achieved halfway through the growing season or by utilizing specific data.
2.2.2 Root Water Uptake With Compensation
The ratio of actual to potential transpiration of the root uptake without compensation is defined as follows:
The equation T T ∫ ∫α φ =ω (2.23) defines a dimensionless water stress index (ω) as introduced by Jarvis in 1989 Jarvis also proposed a critical value for this index, known as the root adaptability factor (ω c), which indicates a threshold beyond which reduced water uptake in stressed areas of the root zone is fully compensated by increased uptake in other regions Although some reduction in potential transpiration is observed below this threshold, it is less significant compared to scenarios where there is no compensation for water uptake.
Fig 2 4 Ratio of actual to potential transpiration as a function of the stress index ω
Thus, for the interval when ω is larger than the threshold value ω c (Fig 2.3), one obtains
While for the interval when ω is smaller than the threshold value ω c , one has
When the parameter ω c is equal to one we hence have noncompensated root water uptake, and when ω c is equal to zero we obtain fully compensated uptake.
The Unsaturated Soil Hydraulic Properties
Unsaturated soil hydraulic properties, such as moisture content (θ(h)) and hydraulic conductivity (K(h)), exhibit significant nonlinearity concerning pressure head HYDRUS offers the flexibility to utilize five distinct analytical models to characterize these hydraulic properties, including those developed by Brooks and Corey (1964) and van Genuchten (1980).
The soil water retention, θ(h), and hydraulic conductivity, K(h), functions according to
Brooks and Corey [1964] are given by
K =K S + + (2.27) respectively, where S e is effective saturation:
In the equation =θ θ (2.28), θ r represents the residual water content while θ s indicates the saturated water content Key parameters include K s, the saturated hydraulic conductivity, and α, which is the inverse of the air-entry value or bubbling pressure Additionally, n serves as a pore-size distribution index, and l is a pore-connectivity parameter set to 2.0 based on the original research by Brooks and Corey in 1964 The parameters α, n, and l in the HYDRUS model are considered empirical coefficients that significantly influence the shape of the hydraulic functions.
HYDRUS utilizes the soil-hydraulic functions developed by van Genuchten in 1980, which are based on Mualem's statistical pore-size distribution model from 1976 This approach allows for a predictive equation that describes the unsaturated hydraulic conductivity function in relation to soil water retention parameters The specific expressions formulated by van Genuchten in 1980 are integral to this process.
The above equations contain five independent parameters: θ r , θ s , α, n, and K s The pore- connectivity parameter l in the hydraulic conductivity function was estimated [Mualem, 1976] to be about 0.5 as an average for many soils
A third set of hydraulic equations implemented in HYDRUS are those by Vogel and
Císlerová (1988) enhanced van Genuchten's (1980) equations to provide greater flexibility in describing hydraulic properties close to saturation The soil water retention, θ(h), and hydraulic conductivity, K(h), functions developed by Vogel and Císlerová (1988) are illustrated in Figure 2.3.
The equations discussed enable a non-zero minimum capillary height, h_s, by substituting the parameter θ_s in van Genuchten's retention function with a slightly larger fictitious parameter, θ_m Although this adjustment has minimal impact on the retention curve, it significantly influences the hydraulic conductivity function's shape and value, particularly in fine-textured soils with a small n value (1.0 < n < 1.3) To enhance the analytical flexibility, the parameter θ_r in the retention function is replaced by a fictitious parameter θ_a, which remains less than or equal to θ_r, preserving the physical meaning of both θ_r and θ_s as measurable quantities The approach assumes that the predicted hydraulic conductivity function aligns with a measured hydraulic conductivity value, K_k = K(θ_k), at a water content θ_k that is less than or equal to the saturated water content (θ_k ≤ θ_s and K_k ≤ K_s) The hydraulic characteristics consist of nine unknown parameters: θ_r, θ_s, θ_a, θ_m, α, n, K_s, K_k, and θ_k When θ_a equals θ_r, θ_m equals θ_k and θ_s, and K_k equals K_s, the soil hydraulic functions of Vogel and Císlerová revert to the original expressions of van Genuchten.
Fig 2.5 Schematics of the soil water retention (a) and hydraulic conductivity (b) functions as given by equations (2.32) and (2.33), respectively
The Vogel and Císlerová model (1988) calculates the air entry value (h_s) at -2 cm, and is implemented as "van Genuchten-Mualem with an air-entry value of -2 cm." This model is particularly recommended for use with heavy textured soils, such as clays.
Version 4.0 of HYDRUS allows the soil hydraulic properties to be defined also according to Kosugi [1996], who suggested the following lognormal distribution model for S e (h):
Application of Mualem's pore-size distribution model [Mualem, 1976] now leads to the following hydraulic conductivity function:
Note that in this manual we use the symbol α instead of h 0 and n instead of σ as used in Kosugi
Durner [1994] divided the porous medium into two (or more) overlapping regions and suggested to use for each of these regions a van Genuchten-Mualem type function [van
Genuchten, 1980] of the soil hydraulic properties Linear superposition of the functions for each particular region gives then the functions for the composite multimodal pore system [Durner et al., 1999]:
Combining this retention model with Mualem’s [1976] pore-size distribution model leads now to:
+ (2.40) where w i are the weighting factors for the two overlapping regions, and α i , n i , m i (=1-1/n i ), and l are empirical parameters of the separate hydraulic functions (i=1,2)
Figure 2.5 illustrates the composite retention and hydraulic conductivity functions for two overlapping porous media, with pressure head axes displayed on a logarithmic scale, highlighting the significant enlargement of near-saturated values Despite the fracture domain constituting only 2.5% of the total pore space, it contributes nearly 90% of the hydraulic conductivity at near-saturation Similar curves have been utilized in studies of fractured rock by researchers such as Peters and Klavetter (1988), Pruess and Wang (1987), and Flint et al.
Log( C onduc tiv it y [ c m /da y s ])
Fig 2.6 Example of composite retention (left) and hydraulic conductivity (right) functions (θ r =0.00, θ s =0.50, α1 =0.01 cm -1 , n 1 =1.50, l=0.5, K s =1 cm d -1 , w 1 =0.975, w 2 =0.025, α2 =1.00 cm -1 , n 2 =5.00)
2.3.2 Uniform Water Flow and Vapor Transport System
The thermal hydraulic conductivity function, K LT , in (2.3) may be defined as (e.g.,
Noborio et al [1996ab], Saito et al [2006]):
The gain factor (G wT), valued at 7 for sand, represents the temperature influence on the soil water retention curve, as established by Nimmo and Miller in 1986 Additionally, γ denotes the surface tension of soil water, measured in MT^-2.
Jm -2 ], and γ0 is the surface tension at 25 o C (= 71.89 g s -2 ) The temperature dependence of γ is as follows (γ is in [g s -2 ] and T in [ o C]):
The isothermal, K vh , and thermal, K vT , vapor hydraulic conductivities are described as
(e.g., Nassar and Horton [1989], Noborio et al [1996b], Fayer [2000]): v vh vs r w u
= ρ (2.44) where D is the vapor diffusivity in soil [L 2 T -1 ], ρ vs is the saturated vapor density [ML -3 ], M is the molecular weight of water [M mol -1 ] (=0.018015 kg mol -1 ), g is the gravitational acceleration
[LT -2 ] (=9.81 m s -2 ), R u is the universal gas constant [J mol -1 K -1 , ML 2 T -2 mol -1 K -1 ] (=8.314 J mol -
1K -1 ), η e is the enhancement factor [-] [Cass et al., 1984], and H r is the relative humidity [-] The vapor diffusivity, D v , in soil is defined as: v g v a
D =τ a D (2.45) where a v is the air-filled porosity [-], τ g is the tortuosity factor as defined by Millington and Quirk
(1961) and D a is the diffusivity of water vapor in air [L 2 T -1 ] at temperature T [K]:
The saturated vapor density, ρ vs [ML -3 ] (in kg m -3 ), as a function of temperature may be expressed as:
3716 6014 31 exp ρ 10 (2.47) and the relative humidity, H r [Philip and de Vries, 1957]: r exp
In soil, when the liquid and vapor phases of water achieve equilibrium, the vapor density can be calculated by multiplying the saturated vapor density by the relative humidity This relationship is expressed mathematically as ρ = ρvs Hr.
The volumetric water vapor content, θ v , is given here in terms of an equivalent water content [L 3 L -3 ] as follows: s s v v vs r w w θ θ H θ θ θ ρ ρ ρ ρ
The HYDRUS-1D code incorporates the enhancement factor, η e, to quantify the increases in thermal vapor flux resulting from liquid-island effects and heightened temperature gradients in the air phase Initially defined by Cass et al in 1984, this enhancement factor can be articulated as outlined by Campbell in 1985.
(2.51) where f c is the mass fraction of clay in the soil [-].
Scaling in the Soil Hydraulic Functions
HYDRUS employs a scaling procedure that streamlines the characterization of spatial variability in unsaturated soil hydraulic properties within the flow domain It approximates the variability in hydraulic properties of a soil profile through linear scaling transformations that connect the soil hydraulic characteristics θ(h) and K(h) of individual layers to reference characteristics θ*(h*) and K*(h*) This technique is grounded in the similar media concept proposed by Miller and Miller in 1956, which applies to porous media with differing internal geometries The approach was further refined by Simmons et al in 1979 to include materials with varying morphological properties but scale-similar soil hydraulic functions HYDRUS incorporates three independent scaling factors, enabling the definition of a linear model for the actual spatial variability in soil hydraulic properties, as outlined by Vogel et al in 1991.
In the general case, the scaling factors for water content (α θ), pressure head (α h), and hydraulic conductivity (α K) are independent of each other However, less general scaling methods can be derived by establishing specific relationships among these factors For instance, the original Miller-Miller scaling method is achieved by setting α θ to 1 (where θ r * equals θ r) and defining α K as α h - 2 A comprehensive analysis of these scaling relationships and their relevance to the hydraulic characterization of heterogeneous soil profiles can be found in the work of Vogel et al [1991].
Temperature Dependence of the Soil Hydraulic Functions
In HYDRUS, a scaling technique is employed to represent the temperature dependence of soil hydraulic functions, based on capillary theory This theory posits that temperature's effect on soil water pressure head can be quantitatively predicted through its impact on surface tension Philip and de Vries (1957) formulated an equation to express this relationship, which includes temperature (T) in Kelvin and surface tension (σ) at the air-water interface.
The equation \( h_{T} \) and \( h_{ref} \) (or \( \sigma_{T} \) and \( \sigma_{ref} \)) represents the relationship between pressure heads at a given temperature \( T \) and a reference temperature \( T_{ref} \), as defined in equation (2.42) Additionally, \( \alpha_{h}^{*} \) serves as the temperature scaling factor for the pressure head.
Following Constantz [1982], the temperature dependence of the hydraulic conductivity can be expressed as
(2.55) where K ref and K T denote hydraulic conductivities at the reference temperature T ref and soil temperature T [ o C], respectively; à ref and à T (ρ ref and ρ T ) represent the dynamic viscosity [ML -1 T -
1] (density of soil water [ML -3 ]) at temperatures T ref and T, respectively; and α K * is the temperature scaling factor for the hydraulic conductivity.
Hysteresis in the Soil Hydraulic Properties
Unsaturated flow models typically rely on unique, single-valued (non-hysteretic) functions for θ(h) and K(h) to define soil hydraulic properties at specific points in the profile While this simplification is often sufficient for various flow simulations, some scenarios necessitate a more accurate representation that includes hysteresis in these properties The HYDRUS code addresses this need by integrating the empirical model proposed by Scott et al [1983], which was later adapted by Kool and Parker [1987] to include air entrapment effects Building on the work of Vogel et al [1996], the current version of HYDRUS further enhances the Kool and Parker model by incorporating hysteresis into the hydraulic conductivity function.
To effectively model hysteresis in the retention function, it is essential to have both the main drying and main wetting curves available These curves are defined using a specific equation, incorporating parameter vectors that are crucial for accurate representation.
, α w , n w ), respectively, where the subscripts d and w indicate wetting and drying, respectively The following restrictions are expected to hold in most practical applications: d w , d w r = r θ θ α ≤α (2.56)
We also invoke the often assumed restriction d w n =n (2.57)
If data are lacking, one may use α w = 2α d as a reasonable first approximation [Kool and Parker, 1987; Nielsen and Luckner, 1992] We further assume
Fig 2.7 Example of a water retention curve showing hysteresis Shown are the boundary wetting curve, θ w (h), and the boundary drying curve, θ d (h)
The parameters θs and α are the sole independent factors defining hysteresis in the retention function, as described by equation (2.58) In the hysteresis model, drying scanning curves are derived from the main drying curve, while wetting scanning curves originate from the main wetting curve To determine the scaling factors for the drying scanning curves, the main drying curve serves as a reference in the scaling equation (2.52), maintaining αh at 1 to focus exclusively on scaling in the water content direction.
( )h r ' θ [ d ( ) -h r d ] θ =θ +α θ θ (2.59) and forcing each scanning curve, θ(h), to pass through the point (θ ∆ , h ∆ ) characterizing the latest reversal from wetting to drying Substituting this reversal point into (2.59), and assuming that θ r =θ r d , leads to
The scaling procedure yields a fictitious value for the parameter θ s ’ in the drying scanning curve, which may fall outside the primary hysteresis loop This scaling relationship also applies to the wetting scanning curves.
The equation ( )h r ' θ [ w ( ) -h r ] θ =θ +α θ θ (2.61) introduces the fictitious parameter θ r ’, which may be scaled outside the main loop To determine the scaling factor α θ for a specific scanning curve, one can substitute the reversal point (θ ∆ , h ∆ ) and the full saturation point (θ s , 0) into equation (2.61) By subtracting the resulting equations, θ r ’ can be eliminated, allowing for the calculation of α θ.
The parameter θ r ’ is calculated using the formula θ r ’ = θ s - α θ (θ s w - θ r ) In cases where the main hysteresis loop does not close at saturation, the water content at saturation for a specific wetting scanning curve can be determined through the empirical relationship established by Aziz and Settari (1979).
An analogous hysteretic approach can be utilized for the unsaturated hydraulic conductivity function K(h), where the primary branches K d (h) and K w (h) of the hysteresis loop are defined by the same parameters as the corresponding retention curves θ d (h) and θ w (h) Additionally, these branches are characterized by the saturated conductivities K s d and K s w, as outlined in Eq (2.29) For drying scanning curves, we derive results from Eq (2.52).
From knowledge of the reversal point (h ∆ , K ∆ ) we obtain
For a wetting scanning curve we have now
K h =K +α K h (2.66) where K r ’ is a fictitious parameter Substituting the reversal point (h ∆ , K ∆ ) and the saturation point (0, K s ) into (2.66) and solving for α K yields
The fictitious conductivity parameter K r ’ may be obtained from (2.66) as K r ’ = K s - α K K s w
If the main hysteresis loop is not closed at saturation, the hydraulic conductivity at saturation for a wetting scanning curve is evaluated using equations similar to (2.63), i.e.,
The previously implemented model, although straightforward, exhibits a pumping effect that causes hysteresis loops to shift to unrealistic areas of the retention function To address this issue, we integrated the hysteresis model proposed by Lenhard et al (1991) and Lenhard and Parker (1992) into HYDRUS, which effectively eliminates the pumping effect by monitoring historical reversal points We extend our sincere gratitude to Robert Lenhard for his valuable assistance in this endeavor.
Initial and Boundary Conditions
The solution of Eq (2.1) requires knowledge of the initial distribution of the pressure head or water content within the flow domain:
The initial condition for the simulation can be established at the time t₀, with hᵢ[L] defined as a function of x This condition can be set to match the water content at field capacity, calculated according to the methodology outlined by Twarakavi et al (2009).
The van Genuchten model (1980) defines key soil hydraulic parameters, including water content at field capacity (θ fc) and saturation (S fc), alongside residual water content (θ r), saturated water content (θ s), porosity (n), and saturated hydraulic conductivity (K s) At field capacity, the hydraulic conductivity is approximately 0.01 cm/d, as noted by Twarakavi et al This model is essential for understanding soil moisture dynamics and water retention characteristics in agricultural and environmental studies.
2009] The initial pressure head at field capacity is calculated from the water content at field capacity using the the van Genuchten [1980] retention curve model
One of the following boundary conditions must be specified at the soil surface (x=L) or at the bottom of the soil profile (x=0):
(2.71) where h 0 [L] and q 0 [LT -1 ] are the prescribed values of the pressure head and the soil water flux at the boundary, respectively
In addition to the system-independent boundary conditions outlined in (2.71), we examine two system-dependent boundary conditions that cannot be predetermined One such condition pertains to the soil-air interface, which is influenced by atmospheric factors The potential fluid flux across this interface is solely governed by external conditions, yet the actual flux is also affected by the transient soil moisture conditions near the surface Consequently, the soil surface boundary condition can fluctuate between a prescribed flux and a prescribed head type condition The numerical solution of (2.1) is achieved by constraining the absolute value of the surface flux through two specific conditions as described by Neuman et al (1974).
In the context of soil water dynamics, the maximum potential rate of infiltration or evaporation (E) is influenced by atmospheric conditions, represented by the equation A S at h h h ≤ ≤ x=L (2.73) The minimum (h A) and maximum (h S) pressure heads at the soil surface are determined by the equilibrium between soil water and atmospheric vapor, with h S typically set to zero unless there's a small ponded layer during heavy rainfall HYDRUS offers an option to remove any excess water above zero on the soil surface immediately When one endpoint of the equation is reached, a prescribed head boundary condition is applied to calculate the actual surface flux Additionally, methods for calculating E and h A based on atmospheric data have been detailed by Feddes et al [1974], with h A being computable from air humidity (H r [-]) using specific logarithmic relations.
(2.74) where M is the molecular weight of water [M mol -1 ] (=0.018015 kg mol -1 ), g is the gravitational acceleration [LT -2 ], (=9.81 m s -2 ), and R is the gas constant [J mol -1 K -1 ] (=8.314 J mol -1 K -1 ) [ML 2 T -2 mol -1 K -1 ]
HYDRUS-1D can model variations in potential evaporation and transpiration throughout the day by assuming that hourly values from 0-6 a.m and 6-12 p.m account for 1% of the total daily value, while a sinusoidal pattern is followed during the remaining hours This method, as noted by Fayer (2000), allows for a more accurate representation of daily water dynamics.
(2.75) where T p is the daily value of potential transpiration (or evaporation) Similarly, variation of precipitation can be approximated using a cosine function as follows:
= + ∆ − (2.76) where P is the average precipitation rate of duration ∆t
Potential evaporation and transpiration fluxes can be derived from potential evapotranspiration by applying Beer’s law, which divides the solar radiation component of the energy budget based on canopy interception.
Potential evapotranspiration (ET p), transpiration (T p), and evaporation fluxes (E p) are crucial components in understanding water movement in ecosystems, measured in units of [LT -1 ] The leaf area index (LAI) and soil cover fraction (SCF) play significant roles in this process, while the constant k influences radiation extinction by the canopy This constant varies based on factors such as sun angle, plant distribution, and leaf arrangement, typically ranging from 0.5 to 0.75.
In version 4.16, interception can be considered when the Leaf Area Index (LAI) is entered Interception (I) is defined as follows (e.g., Von Hoyningen-Hüne [1983], Braden [1985], van Dam et al [1997]):
(2.78) where P is precipitation (m/d), I is interception (m/d), and a and b are empirical constants (for ordinary agricultural crops a≈0.025 cm d -1 , b≈SCF)
Another option in HYDRUS is to permit water to build up on the surface If surface ponding is expected, a "surface reservoir" boundary condition of the type [Mls, 1982]
The net infiltration rate, represented as flux q0, is defined as the difference between precipitation and evaporation According to Equation (2.79), the height h(L,t) of the surface water layer rises with precipitation while decreasing due to infiltration and evaporation.
In HYDRUS, a third system-dependent boundary condition is the seepage face located at the bottom of the soil profile, allowing water to exit the saturated flow domain This boundary condition operates under a zero-flux assumption when the local pressure head at the bottom (x = 0) is negative or below a specified threshold (hSeep) Once the bottom of the profile reaches saturation, a zero pressure head or the specified value hSeep is applied This condition is commonly relevant to finite lysimeters that drain under the influence of gravity.
A system-dependent boundary condition for modeling soil profiles is the flow to horizontal subsurface tile drains, which can be approximated using HYDRUS-1D This software offers two analytical solutions for tile drainage, with the first being the Hooghoudt equation, originally proposed by Hooghoudt in 1940 and further referenced by van Hoorn in 1998 and van Dam et al in 1997.
8 hBot eq dr 4 hTop dr dr drain dr entr
The equation \( q_{dr} = K_{hTop} \cdot \frac{h_{dr}}{L_{dr}} - K_{hBot} \cdot \frac{h_{dr}}{L_{dr}} - \gamma_{entr} \cdot D_{eq} \) describes the drain discharge rate per unit surface area, where \( K_{hTop} \) and \( K_{hBot} \) represent the horizontal saturated hydraulic conductivities above and below the drain system, respectively The watertable height \( h_{dr} \) at the midpoint between drains is crucial for calculating subsurface flow, while \( L_{dr} \) denotes the spacing between drains The equivalent depth \( D_{eq} \), defined by Hooghoudt, depends on the drain spacing, the depth to an impervious layer, and the drain radius In cases where drains are positioned in a homogeneous soil profile just above an impervious layer, the equation can be simplified HYDRUS-1D utilizes a numerical scheme similar to that of the SWAT model, facilitating effective modeling of subsurface drainage dynamics.
4 h dr dr drain dr entr
K h h q = L +γ (2.81) where K h is the horizontal saturated hydraulic conductivity [LT -1 ]
The second analytical solution in HYDRUS was derived by Ernst [Ernst, 1962; van
Hoorn, 1997] for a layered soil profile:
8 ( ) ln dr v dr h r dr dr drain dr r v dr entr dr
The equation = + ∑ + + (2.82) defines the relationship between saturated hydraulic conductivities (K v' for vertical flow and K r' for radial flow) and the thicknesses of the respective layers (D v for vertical and D r for radial flow) It also incorporates the transmissivity of soil layers (∑(KD) h) relevant to horizontal flow, represented in units of [L²T⁻¹] Additionally, the wet perimeter of the drain (u) and a geometry factor for radial flow (a dr) are included, with the latter's value varying based on flow conditions, as detailed in Table 12.2.
In certain scenarios, a system-dependent lower boundary condition can be applied when a functional relationship between the water table's position and soil drainage is identified This relationship is further explored in Section 10.3.
HYDRUS enables the simulation of snow accumulation at the atmospheric boundary by incorporating air temperature data It assumes that when temperatures are below -2°C, all precipitation falls as snow, while above +2°C, it is entirely liquid, with a linear transition between these temperatures Additionally, when air temperatures exceed zero, any existing snow melts in proportion to the temperature, governed by the Snow Melting Constant, which indicates the amount of snow that melts per day for each degree Celsius Furthermore, potential evaporation from the snow is adjusted using the Sublimation Constant.
Potential evapotranspiration may be calculated in HYDRUS-1D using either the FAO recommended Penman-Monteith combination equation for evapotranspiration (ET 0 ) [Monteith, 1981; Monteith and Unsworth, 1990; FAO, 1990] or the Hargreaves equation [Hargreaves, 1994;
Jensen et al., 1997] With the Penman-Monteith approach, ET 0 is determined using a combination equation that combines the radiation and aerodynamic terms as follows [FAO, 1990]:
The evapotranspiration rate (ET₀) is influenced by various factors, including the radiation term (ET_rad), the aerodynamic term (ET_aero), and specific constants such as the latent heat of vaporization (λ) and net radiation at the surface (R_n) Additionally, soil heat flux (G), atmospheric density (ρ), and the specific heat of moist air (cₚ) play crucial roles in this process The vapor pressure deficit, represented by (eₐ - e_d), is determined by the saturation vapor pressure (eₐ) and actual vapor pressure (e_d) Crop canopy resistance (r_c) and aerodynamic resistance (r_a) are also key components The slope of the vapor pressure curve (∆) and the psychrometric constant (γ) are essential for understanding these dynamics.
Water Mass Transfer
The mass transfer rate, Γ_w, for water between fracture and matrix regions in dual-porosity studies is modeled as proportional to the difference in effective saturations of these regions, as described by the first-order rate equation (2.99) In this equation, θ_im represents the matrix water content, ω is a first-order rate coefficient, and S_e^m and S_e^im denote the effective fluid saturations of the mobile (fracture) and immobile (matrix) regions, respectively This approach, which focuses on the difference in effective water contents rather than pressure heads, offers a more realistic representation of the exchange rate between the fracture and matrix regions.
Equation (2.99) assumes that the water retention properties of both the matrix and fracture domains are the same, which necessitates cautious application, particularly in dual-porosity models Despite this limitation, the equation has been effectively utilized in various studies, including those by Kửhne et al (2004, 2005).
One significant advantage of the dual-porosity model based on the mass transfer equation (2.99) is that it requires fewer parameters, as it does not necessitate explicit knowledge of the retention function for the matrix region; instead, only the residual and saturated water contents are needed By integrating (2.99) with a dual-porosity nonequilibrium flow model, it provides the essential soil hydraulic parameters for the equilibrium model, along with two additional parameters that characterize the matrix region: the residual water content (θ r im) and the saturated water content (θ s im).
The first-order mass transfer coefficient, ω, along with the water contents, plays a crucial role in modeling By assuming that the residual water content in the fracture region is zero, and that any residual water exists solely in the immobile region, the number of model parameters can be effectively reduced.
The exchange rate of water between fracture and matrix regions is considered to be proportional to the pressure head differences between these two pore areas.
Genuchten, 1993a], the coupling term, Γ w , becomes:
The first-order mass transfer coefficient, α_w, is essential for understanding pressure heads in different pore regions of porous media, as defined by Gerke and van Genuchten (1993b) This coefficient can be expressed through the equation α_w = (2K_a / d) * β * γ_w, where d represents the effective diffusion pathlength, β is a geometry-dependent shape factor, and γ_w is a scaling factor set at 0.4 This formulation aligns the first-order approach with the numerical solutions of horizontal infiltration equations at the cumulative infiltration curve's half-time level Furthermore, Gerke and van Genuchten (1996) assessed the effective hydraulic conductivity, K_a, at the fracture-matrix interface using a straightforward arithmetic average of the hydraulic heads in both regions, h_f and h_m.
The use of (2.101) implies that the medium contains geometrically well-defined rectangular or other types of macropores or fractures (e.g Edwards et al [1979], van Genuchten and Dalton
Geometrically based models, although conceptually appealing, can be challenging to apply in field settings due to the complexity of structured soils and rocks, which often consist of diverse aggregates and matrix blocks of varying sizes and shapes Additionally, the parameters involved may not be easily identifiable As an alternative to directly using the complex equation (2.101), it is possible to consolidate the parameters β, d, and γw into a single effective hydraulic conductivity K a * for the fracture-matrix interface.
*( ) w K h a α = (2.103) in which case K a * can be used as a calibration parameter (this variable is an input parameter to HYDRUS).
NONEQUILIBRIUM TRANSPORT OF SOLUTES INVOLVED IN SEQUENTIAL FIRST-ORDER DECAY REACTIONS
Governing Solute Transport Equations
Solutes can exist in liquid, solid, and gaseous phases, with distinct decay and production processes in each Interactions between solid and liquid phases are described by nonlinear nonequilibrium equations, while those between liquid and gas phases are considered linear and instantaneous In the liquid phase, solutes are transported through convection and dispersion, whereas in the gas phase, diffusion plays a key role The system of first-order decay reactions for three solutes (A, B, and C) is outlined in the research by Šimůnek and van Genuchten (1995).
In the study of phase interactions, the concentrations in the liquid, solid, and gaseous phases are denoted as c, s, and g, respectively The subscripts s, w, and g indicate the solid, liquid, and gaseous phases Zero-order (γ) and first-order (à, à') rate reactions are represented by straight arrows, while circular arrows (k g, k s) illustrate the equilibrium distribution coefficients among the phases.
Typical examples of sequential first-order decay chains are:
(NH 2 ) 2 CO → NH 4 + → NO 2 - → NO 3 - c 1 s 1 c 2 s 2 c 3 c 4
3 Pesticides [Wagenet and Hutson, 1987]: a) Uninterrupted chain - one reaction path:
Parent Daughter Daughter pesticide → product 1 → product 2 → Products c 1 s 1 c 2 s 2 c 3 s 3
Product Product Product b) Interrupted chain - two independent reaction paths:
Parent Daughter Parent pesticide 1 → product 1 → Product Pesticide 2 → Product c 1 s 1 c 2 s 2 c 3 s 3 c 4 s 4
Other examples of chemicals involved in sequential biodegradation chains are hormones
HYDRUS currently accommodates up to ten solutes, including chlorinated aliphatic hydrocarbons and explosives, allowing for both unidirectional coupling in a chain and independent movement.
The governing partial differential equations for one-dimensional nonequilibrium chemical transport of solutes in a sequential first-order decay chain are derived during transient water flow in a variably saturated rigid porous medium, as outlined by Šimůnek and van Genuchten (1995).
(3.2) where c, s, and g are solute concentrations in the liquid [ML -3 ], solid [MM -1 ], and gaseous [ML -
The article discusses the various phases involved in solute transport, highlighting key parameters such as volumetric flux density (q), first-order rate constants (à w, à s, à g) for liquid, solid, and gas phases, and zero-order rate constants (γ w, γ s, γ g) for different phases It also defines soil bulk density (ρ), air content (a v), and the sink term (S) in the water flow equation, alongside the root nutrient uptake term (r a), which relates to passive uptake Additionally, it mentions the dispersion coefficient (D w) for the liquid phase and the diffusion coefficient (D g) for the gas phase The subscripts denote the respective phases, while the parameters can represent various reactions, including biodegradation, volatilization, and precipitation, through nine zero- and first-order rate constants.
HYDRUS models the nonequilibrium interaction between solution (c) and adsorbed (s) concentrations while assuming equilibrium between solution (c) and gas (g) concentrations of the solute within the soil system The relationship between adsorbed concentration (s) and solution concentration (c) is characterized by a generalized nonlinear adsorption isotherm equation.
Equation (3.3) incorporates empirical coefficients k s,k [L 3 M -1 ], β k [-], and η k [L 3 M -1 ], with the Freundlich, Langmuir, and linear adsorption equations being specific instances of this equation When β k equals 1, it simplifies to the Langmuir equation; when η k equals 0, it becomes the Freundlich equation; and when both conditions are met, it results in a linear adsorption isotherm Additionally, solute transport without adsorption is represented by k s,k = 0 Although the coefficients k s,k, β k, and η k are considered independent of concentration, they can vary over time based on temperature changes, a topic that will be explored further.
The concentrations g k and c k are related by a linear expression of the form
, (1, ) k g k k s g = k c k ε n (3.4) where k g,k is an empirical constant [-] equal to (K H R u T A ) -1 [Stumm and Morgan, 1981], in which
K H is Henry's Law constant [MT 2 M -1 L -2 ], R u is the universal gas constant [ML 2 T -2 K -1 M -1 ] and T A is absolute temperature [K]
3.1.1 Two-Site Sorption Model (Chemical Nonequilibrium)
The two-site sorption concept, introduced by Selim et al (1977) and van Genuchten and Wagenet (1989), is integrated into HYDRUS to account for nonequilibrium adsorption-desorption reactions This model posits that sorption sites can be categorized into two distinct fractions, enhancing the understanding of soil-water interactions.
[MM -1 ], on one fraction of the sites (the type-1 sites) is assumed to be instantaneous, while sorption, s k k
[MM -1 ], on the remaining (type-2) sites is considered to be time-dependent At equilibrium we have for the type-1 (equilibrium) and type-2 (kinetic) sites, respectively
The equation e k k s s = fs k nε (3.6) represents the relationship between the fraction of exchange sites in equilibrium with the solution phase, denoted as f Since type-1 sorption sites are consistently at equilibrium, differentiating this equation provides the sorption rate specifically for these type-1 equilibrium sites.
Sorption on type-2 nonequilibrium sites is considered a first-order kinetic rate process According to Toride et al (1993), the mass balance equation for these type-2 sites accounts for both production and degradation factors.
∂∂ = + + + (3.9) where ω k is the first-order rate constant for the kth solute [T -1 ]
Substituting (3.3) through (3.4) into (3.1) and (3.2) leads to the following equation
∂ ∂ + ∂∂ =∂∂ ∂∂ ∂∂ + + = (3.10) in which E k [L 2 T -1 ] and B k [LT -1 ] are an effective dispersion coefficient and effective velocity given by
∂ (3.12) respectively The coefficients F k and G k in (3.10) are defined as
(3.14) where the variable g k accounts for possible changes in the adsorption parameters caused by temperature changes in the system as follows (see also section 3.4):
To address numerical and programming considerations, we separated the total retardation factor R k [-] used in (3.10) into two components: R k1, which pertains to the liquid and gaseous phases, and R k2, which is related to the solid phase.
3.1.2 Attachment-Detachment Model (Two Kinetic Sites Model)
In this study, we define the mass balance equation for virus, colloid, and bacteria transport and fate models, which typically utilize a modified version of the convection-dispersion equation.
The equation presented illustrates the relationship between colloid, virus, and bacteria concentrations in both aqueous and solid phases In this context, 'c' denotes the concentration of these entities in the liquid phase, while 's' represents their concentration in the solid phase The subscripts e, 1, and 2 indicate different states of equilibrium and two distinct kinetic sorption sites Additionally, the terms à w and à s signify the inactivation and degradation processes occurring in the liquid and solid phases, respectively.
Sorption to equilibrium sites can be analyzed using the previously established equation, while the mass transfer between the aqueous phase and solid kinetic phases is expressed as a differential equation, indicating the relationship between concentration, density, and time.
The first-order deposition coefficient (k_a) and the first-order entrainment coefficient (k_d) are crucial parameters in understanding colloid retention, represented by the dimensionless function ψ Research indicates that these coefficients are highly influenced by water content, with a notable increase in attachment occurring as water content decreases.
Initial and Boundary Conditions
The solution of (3.10) requires knowledge of the initial concentration within the flow region, Ω, i.e.,
In the context of the theoretical development and numerical solutions presented, the equation (3.42) involves prescribed functions of x, specifically c i [ML -3 ], c im,i [ML -3 ], and s i k [-] It is important to note that the initial condition for s i k is only necessary when considering nonequilibrium adsorption For simplicity, the subscript k is omitted throughout this report, indicating that the transport-related equations are applicable to all solutes within the decay chain.
Initial conditions for solute transport can be defined using total concentration S [ML -3; mass of solute per volume of soil] rather than liquid concentration c [ML -3; mass of solute per volume of water] For linear sorption, the liquid phase concentration is subsequently calculated based on this total concentration.
(3.43) and for nonlinear sorption by finding a root of the following nonlinear equation:
For the two kinetic sorption sites model (only a linear case is implemented, i.e., without blocking) the distribution coefficient k s for (3.42) is assumed to be defined as: a s d k k k θ
In nonequilibrium phase models, instead of directly defining concentrations, the initial nonequilibrium phase concentration can be aligned with the equilibrium phase concentration Consequently, for dual porosity models, the concentration of immobile water is established to be equal to that of mobile water Additionally, the sorbed concentration at the kinetic sorption sites is defined accordingly.
(1 ) s s= − f k c (3.46) for the linear sorption and:
+ (3.47) for the nonlinear sorption For the two kinetic sorption sites model, the sorbed concentrations are set equal to: a d s k c k θ
= ρ (3.48) where k a and k d are the attachment and detachment coefficients, respectively
Boundary conditions, specifically Dirichlet and Cauchy types, can be applied to the upper or lower boundaries in a system Dirichlet boundary conditions, also known as first-type conditions, specify the concentration at a given boundary.
( , ) 0( , ) at 0 or c x t =c x t x= x=L (3.49) whereas third-type (Cauchy type) boundary conditions may be used to prescribe the concentration flux at the upper or lower boundary as follows:
The equation D c + q_c = q_c x x L θ ∂ x ∂ (3.50) describes the relationship between fluid flux and concentration, where q_0 indicates the upward fluid flux and c_0 signifies the concentration of the incoming fluid [ML -3] In scenarios such as an impermeable boundary (where q_0 = 0) or when water flow is directed out of the region, this equation simplifies to a Neumann-type boundary condition.
When volatile solutes exist in both liquid and gas phases, a distinct soil surface boundary condition is necessary This scenario demands a third-type boundary condition that includes an extra term to consider gaseous diffusion through a stagnant boundary layer of thickness d [L] on the soil surface The additional solute flux is directly proportional to the concentration difference of gases above and below the boundary layer, as outlined by Jury et al (1983).
The equation D qc q c k c g x L x d θ ∂ + = + (3.52) describes the relationship between the molecular diffusion coefficient in the gas phase (D g) and the gas concentration above the stagnant boundary layer (g atm), measured in [L² T⁻¹] and [ML⁻³] respectively According to Jury et al (1983), g atm can be assumed to be zero Additionally, when water flow is absent or directed out of the region, equation (3.52) simplifies to a second-type (or Neumann type) boundary condition.
Equations (3.52) and (3.53) can only be used when the additional gas diffusion flux is positive
Jury et al [1983] discussed how to estimate the thickness of the boundary layer, d; they recommended a value of 0.5 cm for d as a good average for a bare surface.
Effective Dispersion Coefficient
The dispersion coefficient in the liquid phase, D w , is given by [Bear, 1972] w
The effective dispersion coefficient in the soil matrix for one-dimensional transport is expressed as D = D | q | + D θ θ τ, where D_w represents the molecular diffusion coefficient in free water, τ_w is the tortuosity factor in the liquid phase, |q| denotes the absolute value of the Darcian fluid flux density, and D_L indicates the longitudinal dispersivity Additionally, the diffusion contribution from the gas phase is incorporated into this equation.
D = D | q |+ D + a D k θ θ τ τ (3.55) where D g is the molecular diffusion coefficient in the gas phase [L 2 T -1 ] and τ g is a tortuosity factor in the gas phase [-]
The tortuosity factors for both phases are evaluated in HYDRUS as a function of the water and air contents using the relationship of Millington and Quirk [1961]:
In versions 4.05 and higher of HYDRUS-1D, alternative relationships are introduced to define tortuosity coefficients in both gaseous and liquid phases According to Moldrup et al (2000), a specific formulation is proposed for calculating the tortuosity factor in the gaseous phase, applicable to sieved and repacked soils.
Werner et al (2004) demonstrated that this formulation offers enhanced predictions for various porosity-based relationships In a similar vein, Moldrup et al (1997) proposed an alternative method for determining the tortuosity coefficient in the liquid phase.
The Millington-Quirk tortuosity models, developed in 1961, are anticipated to be effective for sands due to their foundation on the assumption of randomly distributed particles of uniform size In contrast, Modrup’s tortuosity models are expected to demonstrate superior performance across various soil types.
Temperature and Water Content Dependence of Transport and Reaction
The coefficients related to diffusion (D w, D g), zero-order production (γ w, γ s, γ g), first-order degradation (à w ’, à s ’, à g ’, à w, à s, à g), and adsorption (k s, k g, β, η, ω) are significantly influenced by temperature HYDRUS models this temperature dependency using the Arrhenius equation, as noted by Stumm and Morgan (1981) With some modifications, this equation can be represented in a more general form, as described by Šimůnek and Suarez (1993a).
(3.59) where a r and a T are the values of the coefficient being considered at a reference absolute temperature T r A and absolute temperature T A , respectively; R u is the universal gas constant, and
E a [ML 2 T -2 M -1 ] is the activation energy of the particular reaction or process being modeled
The water content dependence of degradation coefficients is implemented using a modified equation of Walker [1974]:
In soil science, the coefficient value at a reference water content (θ ref) is denoted as 'a', while 'r' represents the coefficient at the actual water content (θ) The solute-dependent parameter 'B' typically has a value of 0.7 It's important to note that θ ref can vary across different soil layers and is determined based on a constant reference pressure head (h ref) specific to each compound.
Root Solute Uptake
The root solute uptake models implemented in HYDRUS-1D was developed by Simunek and Hopmans [2009] and only a brief description is given below
To distinguish between point and root domain nutrient uptake rates, we use lower case variables for point root nutrient uptake rates [ML -3 T -1 ] and upper case variables for nutrient uptake rates [ML -2 T -1 ] across the entire root zone, L R Both point and root domain nutrient uptakes are considered to be the total of their passive and active components.
The equation R_t = P_t + A_t (3.62) defines the total actual root nutrient uptake rates, where r_a, p_a, and a_a represent the rates of total, passive, and active nutrient uptake [ML -3 T -1] at any given point Additionally, R_a, P_a, and A_a indicate the actual total, passive, and active root nutrient uptake rates [ML -2 T -1] specifically for the root zone domain.
Passive nutrient uptake is simulated by multiplying root water uptake (compensated or uncompensated) with the dissolved nutrient concentration, for concentration values below a priori defined maximum concentration (c max), or
The maximum solution concentration for passive root uptake, denoted as c max, plays a crucial role in determining how nutrients are absorbed by plant roots When c max exceeds the dissolved nutrient concentration (c), all nutrients are passively absorbed; however, if c max is zero, only active uptake occurs This parameter influences the balance between passive and total nutrient uptake, allowing for flexibility in how different nutrients are absorbed For instance, sodium (Na) uptake can be entirely excluded by setting c max to zero, while calcium (Ca) uptake can be restricted with a finite c max value Conversely, a high c max value enables the passive uptake of all available phosphorus (P) or nitrogen (N) in the soil solution It is important to note that c max serves as a control model parameter and may not have a direct physiological interpretation.
Passive actual root nutrient uptake for the whole root domain, P a [ML -2 T -1 ], is calculated by integrating the local passive root nutrient uptake rate, p a , over the entire root zone:
Defining R p as the potential (subscript p) nutrient demand [ML -2 T -1 ], the potential active nutrient uptake rate, A p [ML -2 T -1 ], is computed from:
Active nutrient uptake is triggered only when passive root nutrient uptake fails to meet the plant's potential nutrient demand Notably, passive uptake can be minimized or entirely halted, resulting in the potential active nutrient uptake being equal to the plant's nutrient demand Once the active nutrient uptake is determined, the specific rates of potential active nutrient uptake can be calculated.
The potential root zone active nutrient uptake rate, A p [ML -2 T -1 ], is distributed across the root zone by utilizing a predefined spatial root distribution, b(x, t), similar to the method employed for root water uptake.
Using Michaelis-Menten kinetics (e.g., Jungk [1991]) provides for actual distributed values of active nutrient uptake rates, a a [ML -3 T -1 ], allowing for nutrient concentration dependency, or:
The Michaelis-Menten constant (K m), measured in [ML -3], is a crucial parameter in understanding nutrient uptake in plants Research has documented the Michaelis-Menten constants for various essential nutrients such as nitrogen (N), phosphorus (P), and potassium (K) across multiple plant species, including corn, soybean, wheat, tomato, pepper, lettuce, and barley For detailed values and further insights, refer to the literature, including the work of Bar-Yosef (1999).
The total active uncompensated root nutrient uptake rate, denoted as A_a [ML -2 T -1], is determined by integrating the actual active root nutrient uptake rate, a_a, throughout the root zone L_R This method parallels the approach used for calculating the non-compensated root water uptake term.
The above nutrient uptake model includes compensation of the passive nutrient uptake, by way of the root water uptake compensation term, s c , and root adaptability factor, ω c , in Eq (3.64)
A similar compensation concept as used for root water uptake above, was implemented for active nutrient uptake rate, by invoking a so-called nutrient stress index π:
After substitution of the active total root nutrient uptake rate value from Eq (3.68) above, this newly defined nutrient stress index (π) is equal to:
The critical value of the nutrient stress index, π c, indicates the threshold above which active nutrient uptake is fully compensated by uptake from less-stressed soil regions Consequently, the local compensated active root nutrient uptake rate, denoted as a ac [ML -3 T -1 ], is derived by incorporating the nutrient stress index function into the denominator of the relevant equation.
The total compensated active root nutrient uptake rate, A ac [ML -2 T -1], within the two-dimensional root domain, L R, is calculated similarly to the compensated root water uptake term.
Equation (3.65) suggests that a decrease in root water uptake leads to an increase in active nutrient uptake while maintaining total nutrient uptake despite soil water stress However, this perspective is unrealistic, as it is expected that water-stressed plants would have reduced nutrient requirements To address this, the uptake model incorporates additional flexibility by adjusting the potential nutrient demand (R p [ML -2 T -1]) in proportion to the reduction in root water uptake, as determined by the actual to potential transpiration ratio.
The root nutrient uptake model with compensation requires several key inputs, including the potential nutrient uptake rate (R p), the spatial root distribution function (b(x,z,t)), the Michaelis-Menten constant (K m), the maximum passive nutrient concentration (c max), the minimum concentration for active uptake (c min), and the critical nutrient stress index (π c) The model allows for the passive nutrient uptake term to be disabled by setting c max to zero, while active uptake can be turned off by assigning a zero value to R p or a high value to c min Parameter values are likely specific to the nutrient and plant type, and π c is expected to be higher for agricultural crops compared to natural plants, which may better compensate for soil stresses Calibration of parameters like c max is necessary for accurate predictive modeling under specific conditions.