Modeling preferential water flow and solute transport in unsaturated soil using the active region model

Abstract

Preferential flow and solute transport are common processes in the unsaturated soil, in which distributions of soil water content and solute concentrations are often characterized as fractal patterns. An active region model (ARM) was recently proposed to describe the preferential flow and transport patterns. In this study, ARM governing equations were derived to model the preferential soil water flow and solute transport processes. To evaluate the ARM equations, dye infiltration experiments were conducted, in which distributions of soil water content and Cl⁻ concentration were measured. Predicted results using the ARM and the mobile–immobile region model (MIM) were compared with the measured distributions of soil water content and Cl⁻ concentration. Although both the ARM and the MIM are two-region models, they are fundamentally different in terms of treatments of the flow region. The models were evaluated based on the modeling efficiency (ME). The MIM provided relatively poor prediction results of the preferential flow and transport with negative ME values or positive ME values less than 0.4. On the contrary, predicted distributions of soil water content and Cl⁻ concentration using the ARM agreed reasonably well with the experimental data, with ME values higher than 0.8. The results indicated that the ARM successfully captured the macroscopic behavior of preferential flow and solute transport in the unsaturated soil.

Appendices

Appendix A: Derivation of the governing equation for water flow

Figure 1 schematically shows the vertical water flow occurring in a rectangular soil unit volume. Because the horizontally averaged soil water content is used in the ARM (Liu et al. 2005; Sheng et al. 2009), it is assumed that the soil water flows in the z-direction only. According to the mass conservation principle, we obtain the water conservation equation for the system during an arbitrary small time period Δt between t and t + Δt as

$$ V_{\text{in}} = V_{\text{out}} + V_{\text{inc}} + V_{\text{sk}} $$

(11)

Here V _in is the water volume flowing into the soil volume during Δt; V _out is the water volume flowing out from the soil volume during Δt; V _inc is the increase of the water volume stored in the soil volume during Δt; V _sk is the water volume extracted (e.g., plant root, positive) or supplied (e.g., irrigation, negative) from other processes during Δt.

For the one-dimensional vertical flow, the water volume flowing into the soil volume during Δt can be expressed as

$$ V_{\text{in}} = q_{\text{a}} \left( {x,z,t + \frac{1}{2}\Updelta t} \right)f\left( {x,z,t + \frac{1}{2}\Updelta t} \right)\Updelta x\Updelta t $$

(12)

where q _a (flow per unit area per unit time) is the water flux within the active region in the z-direction (upward positive).

Similarly, the water volume flowing out from the soil volume during Δt is expressed as

$$ V_{\text{out}} = q_{\text{a}} \left( {x,z + \Updelta z,t + \frac{1}{2}\Updelta t} \right)f\left( {x,z + \Updelta z,t + \frac{1}{2}\Updelta t} \right)\Updelta x\Updelta t $$

(13)

The increase of the water volume stored in the soil volume during Δt is expressed by

$$ \begin{gathered} V_{\text{inc}} = \theta_{\text{a}} \left( {x,z + \frac{1}{2}\Updelta z,t + \Updelta t} \right)f\left( {x,z + \frac{1}{2}\Updelta z,t + \Updelta t} \right)\Updelta x\Updelta z \hfill \\ \begin{array}{*{20}c} {} & {} & {} & { - \theta_{\text{a}} } \\ \end{array} \left( {x,z + \frac{1}{2}\Updelta z,t} \right)f\left( {x,z + \frac{1}{2}\Updelta z,t} \right)\Updelta x\Updelta z \hfill \\ \begin{array}{*{20}c} {} & {} & {} & - \\ \end{array} \theta_{\text{i}} \left[ {f\left( {x,z + \frac{1}{2}\Updelta z,t + \Updelta t} \right) - f\left( {x,z + \frac{1}{2}\Updelta z,t} \right)} \right]\Updelta x\Updelta z \hfill \\ \end{gathered} $$

(14)

where θ _i is the soil water content within the inactive region.

The ARM assumes that water flow occurs in the active region exclusively. Therefore, it is assumed that the sink and source terms also appear in the active region only and are expressed in the form of

$$ V_{\text{sk}} = r_{\text{aw}} \left( {x,z + \frac{1}{2}\Updelta z,t + \frac{1}{2}\Updelta t} \right)f\left( {x,z + \frac{1}{2}\Updelta z,t + \frac{1}{2}\Updelta t} \right)\Updelta x\Updelta z\Updelta t $$

(15)

where r _aw is the rate of water consumed and supplied in the active region of the soil volume.

Substituting Eqs. (12) to (15) into Eq. (11), dividing both sides by ΔxΔzΔt, taking the limit and rearranging the terms, we obtain (upward positive)

$$ {\frac{{\partial \left( {fq_{\text{a}} } \right)}}{\partial z}} + {\frac{{\partial \left( {f\theta_{\text{a}} } \right)}}{\partial t}} - \theta_{\text{i}} {\frac{\partial f}{\partial t}} + fr_{\text{aw}} = 0 $$

(16)

Equation (16) is the governing equation for water flow. Including Darcy’s law in Eq. (16) yields

$$ - {\frac{\partial }{\partial z}}\left[ {fK_{\text{a}} \left( {{\frac{{\partial h_{\text{a}} }}{\partial z}} + 1} \right)} \right] + {\frac{{\partial \left( {f\theta_{\text{a}} } \right)}}{\partial t}} - \theta_{\text{i}} {\frac{\partial f}{\partial t}} + fr_{\text{aw}} = 0 $$

(17)

where K _a and h _a are the unsaturated hydraulic conductivity and the pressure head of the active region, respectively. The h _a and K _a can be expressed as the functions of θ _a (Liu et al. 2005; van Genuchten 1980):

$$ h\left( {S_{\text{a}} } \right) = {\frac{1}{\alpha }}\left[ {\left( {S_{\text{a}} } \right)^{1/m} - 1} \right]^{1/n} = {\frac{1}{\alpha }}\left[ {\left( {S_{\text{e}}^{*} } \right)^{(\gamma-1)/m} - 1} \right]^{1/n} $$

(18)

and

$$ K_{\text{a}} = K_{\text{s}} S_{\text{a}}^{1/2} \left[ {1 - \left\{ {1 - S_{\text{a}}^{1/m} } \right\}^{m} } \right]^{2} = K_{\text{s}} \left( {S_{\text{e}}^{*} } \right)^{(1-\gamma)/2} \left[ {1 - \left\{ {1 - \left( {S_{\text{e}}^{*} } \right)^{(1-\gamma)/m}} \right\}^{m} } \right]^{2} $$

(19)

where α(1/cm), n and m = 1 − 1/1n are parameters; K _s is the saturated hydraulic conductivity of the active region.

Appendix B: Derivation of the governing equation for solute transport

Governing equation for solute transport is also derived using the procedure similar to the soil water flow. According to the mass conservation principle of solute transport, we have

$$ M_{\text{in}} = M_{\text{out}} + M_{\text{inc}} + M_{\text{sk}} $$

(20)

where M _in is the solute mass transporting into the soil volume during Δt; M _out is the solute mass transporting out from the soil volume during Δt; M _inc is the increase of the solute mass stored in the soil volume during Δt; M _sk is mass of solute extracted (e.g., plant root, positive) or supplied (e.g., irrigation, negative) from other processes during Δt.

For the one-dimensional transport (upward positive), the solute mass transporting into the soil volume during Δt is expressed as

$$ M_{\text{in}} = J_{\text{a}} \left( {x,z,t + \frac{1}{2}\Updelta t} \right)f\left( {x,z,t + \frac{1}{2}\Updelta t} \right)\Updelta x\Updelta t $$

(21)

where J _a (transport per unit area per unit time) is the solute flux within the active region in the z-direction, expressed as the sum of convection and dispersion/diffusion as follows

$$ J_{\text{a}} = q_{\text{a}} c_{\text{a}} - \theta_{\text{a}} D_{\text{a}} {\frac{{\partial c_{\text{a}} }}{\partial z}} $$

(22)

in which c _a (mass of solute per volume of soil solution) is the dissolved solute concentration in the active region; D _a is the dispersion–diffusion coefficient of the active region.

Similarly, the solute mass leaving from the soil volume during Δt is expressed as

$$ M_{\text{out}} = J_{\text{a}} \left( {x,z + \Updelta z,t + \frac{1}{2}\Updelta t} \right)f\left( {x,z + \Updelta z,t + \frac{1}{2}\Updelta t} \right)\Updelta x\Updelta t $$

(23)

The increase of the mass of solute stored in the soil volume during Δt can be expressed as

$$ \begin{gathered} M_{\text{inc}} = C_{\text{a}} \left( {x,z + \frac{1}{2}\Updelta z,t + \Updelta t} \right)f\left( {x,z + \frac{1}{2}\Updelta z,t + \Updelta t} \right)\Updelta x\Updelta z \hfill \\ \begin{array}{*{20}c} {} & {} & {} & { - C_{\text{a}} } \\ \end{array} \left( {x,z + \frac{1}{2}\Updelta z,t} \right)f\left( {x,z + \frac{1}{2}\Updelta z,t} \right)\Updelta x\Updelta z \hfill \\ \begin{array}{*{20}c} {} & {} & {} & - \\ \end{array} C_{\text{i}} \left[ {f\left( {x,z + \frac{1}{2}\Updelta z,t + \Updelta t} \right) - f\left( {x,z + \frac{1}{2}\Updelta z,t} \right)} \right]\Updelta x\Updelta z \hfill \\ \end{gathered} $$

(24)

where C _a and C _i are the total concentrations of solute in all forms (including the solute dissolved in soil solution and soil gas, and adsorbed by soil particles) in the active and the inactive regions, respectively. By assuming that the solute only exists in the solid (absorbed by soil particles) and liquid (dissolved in the soil solution) phases, the terms C _a and C _i can be expressed as

$$ C_{\text{a}} = \theta_{\text{a}} c_{\text{a}} + \rho_{\text{b}} s_{\text{a}} $$

(25)

and

$$ C_{\text{i}} = \theta_{\text{i}} c_{\text{i}} + \rho_{\text{b}} s_{\text{i}} $$

(26)

where c _i (mass of solute per volume of soil solution) is the dissolved solute concentration in the inactive region; s _a and s _i (mass of sorbent per mass of dry soil) are the absorbed solute concentrations in the active and inactive regions, respectively; ρ _b is the soil bulk density.

By assuming that the sink and source terms of solute occur in the active region exclusively, the solute mass extracted (sink, positive) and/or supplied (source, negative) from other processes is expressed as

$$ M_{\text{sk}} = c_{\text{as}} r_{\text{aw}} \left( {x,z + \frac{1}{2}\Updelta z,t + \frac{1}{2}\Updelta t} \right)f\left( {x,z + \frac{1}{2}\Updelta z,t + \frac{1}{2}\Updelta t} \right)\Updelta x\Updelta z\Updelta t $$

(27)

where c _as is the concentration of the sink term of soil water of the active region.

Substituting Eqs. (21) to (27) into Eq. (20), dividing both sides by ΔxΔzΔt, taking the limit and rearranging the terms, we have (upward positive) the following governing equation for solute transport

$$ - {\frac{\partial }{\partial z}}\left( {f\theta_{\text{a}} D_{\text{a}} {\frac{{\partial c_{\text{a}} }}{\partial z}}} \right) + {\frac{{\partial \left( {fq_{\text{a}} c_{\text{a}} } \right)}}{\partial z}} + {\frac{{\partial \left( {f\theta_{\text{a}} c_{\text{a}} } \right)}}{\partial t}} + {\frac{{\partial \left( {f\rho_{\text{b}} s_{\text{a}} } \right)}}{\partial t}} - \left( {\theta_{\text{i}} c_{\text{i}} + \rho_{\text{b}} s_{\text{i}} } \right){\frac{\partial f}{\partial t}} + fr_{\text{aw}} c_{\text{as}} = 0 $$

(28)