The well and the trench are modelled within a wadi reservoir, including the saturated and unsaturated zones. The models are in all cases conceptualised as 2D, corresponding to a cross-section or axisymmetric domain. A vertical cross-section is considered within the reservoir located far away from the borders of a channel or any structural barrier (e.g. a dam). Topography within the reservoir or in the underlying bedrock is not considered, and the cross-sections are oriented perpendicular to the hydraulic gradient of the wadi channel.
The scenarios are conceptualised using hydrogeological parameters representing arid and semiarid regions where a thick VZ or a CL pose a hindrance to infiltration. The selected values are representative for typical wadis in the KSA. As a significant variation, two different sets of models are evaluated:
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Shallow water Table (WT) models. The thickness of the vadose zone is 10 m: this is a representative value for wadi aquifers in Saudi Arabia (Missimer et al. 2012).
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Deep WT models. The thickness of the vadose zone is 50 m: this set of scenarios allows assessment of free drainage in comparison to the horizontal discharge of infiltrated water for shallow WT scenarios.
For both WT options, well and trench infiltration with simultaneous recharge through the surface of the reservoir (symbolised as “+ SF”) are investigated. These correspond to model setups 1 (shallow WT), 2 (shallow WT), 4 (deep WT), and 5 (deep WT) in Table 1. Additionally, these setups are compared to sole surface water (SF) infiltration in setups 3 (shallow WT) and 6 (deep WT). The scenarios in Table 1 will be referred to as the base scenarios, which are slightly modified in some sections.
Table 1 Base scenarios used in this study. Letter D is used as a prefix to denote a deep WT setting Two indicators are used to assess the effectiveness of the trench and the well to bypass the CL and the VZ: (1) the time at which recharge starts and, (2) the cumulative groundwater recharge over a year. The recharge rates are accounted for by a line located at the outer boundary of the domain which extends either vertically along the saturated thickness (shallow setups) or covers the lower horizontal domain boundary (deep setups). The quasi-steady-state recharge rates (QSSRR) of the scenarios are also estimated. They are used to carry out two additional analyses, namely the validation of the conceptual models and a sensitivity analysis.
The general structure of the different analyses carried out is the following—the scenarios in Table 1 are run, and the start of groundwater recharge is analysed (simulations of continuous infiltration). Subsequently, the complexity of the models is increased to transient conditions to allow for an assessment of the cumulative groundwater recharge (analysis of cumulative recharge). Next, a sensitivity analysis is carried out, exploring how the geometry of the well and the trench impact the QSSRR. This section is supplemented with a short analysis of the cost of infiltrating water in terms of the well and trench construction costs. Successively, simplified versions of the models presented in Table 1 are built, and their QSSRRs are compared against the infiltration rates obtained from analytical solutions. Note that, if quasi-steady-state conditions prevail, the recharge rate is equivalent to the infiltration rate. Finally, the analysed solutions are compared in terms of the cumulative recharge produced by the maximum number of wells and trenches fitted in a delimited area. Results and discussion are provided in the same section.
Conceptual background
The movement of water through the soil under unsaturated conditions is described by Richards’ equation (Richards 1931, Eq 1).
$$ \frac{\partial \theta (h)}{\partial t}=\frac{\partial }{\partial z}\left[K(h)\frac{\partial h}{\partial z}\right]+\frac{\partial K(h)}{\partial z}-S(h) $$
(1)
where h is the hydraulic head, K(h) the unsaturated hydraulic conductivity and S(h) a sink function that accounts for sinks as sources of water. This equation is a partial differential nonlinear equation which in this case represents one-dimensional (1D) vertical flow (z-direction) in a mixed form, i.e., there are two dependent variables, namely Ɵ and h. These two variables are linked through the water retention curve.
The water retention curve can be represented by mathematical expressions. One of the most frequently used ones was formulated by van Genuchten (1980; Eq. 2).
$$ \theta (h)=\frac{\theta_{\mathrm{s}}-{\theta}_{\mathrm{r}}}{{\left[1+{\left(-\alpha h\right)}^n\right]}^m}+{\theta}_{\mathrm{r}} $$
(2)
where α, n and m are fitting parameters (often referred to as the van Genuchten parameters), Ɵ(h) is the volumetric water content, Ɵs is the saturated volumetric water content, and Ɵr is the residual volumetric water content. It is commonly assumed that m = 1–1/n.
The parameter α is related to the inverse of the air entry value, and n is a measure of the pore size distribution. The residual volumetric water content is the water content measured at very negative pressure heads when the water retention curve approximates a flat region. Van Genuchten (1980) suggests that this value can be equitable to the wilting point.
There are several formulations to represent the unsaturated hydraulic conductivity, (K(h), as a function of the saturated hydraulic conductivity (Ks; e.g. Brooks and Corey 1964). In the numerical simulations conducted in this study, the equation by van Genuchten-Mualem is employed (Eq. 3).
$$ K(h)={K}_{\mathrm{s}}{S}_{\mathrm{e}}^I{\left[1-{\left(1-{S}_{\mathrm{e}}^{1/m}\right)}^m\right]}^2 $$
(3)
where K(h) is the unsaturated hydraulic conductivity, Ks the saturated hydraulic conductivity, Se the effective water saturation (Eq. 4), and I the pore-connectivity parameter. The latter parameter is on average 0.5 for diverse soil types (Mualem 1976).
$$ {S}_{\mathrm{e}}=\frac{\theta -{\theta}_{\mathrm{r}}}{\theta_{\mathrm{s}}-{\theta}_{\mathrm{r}}} $$
(4)
Conceptual models
The conceptual models employed in this study are identical to the ones used by Henao Casas (2019), who provides further details on the justification of the model parameters and characteristics.
Model dimensions
The trench is represented in a two-dimensional (2D) cross-section and the well in a 2D axisymmetric domain (pseudo-3D domain). Figure 1 shows the conceptual representation of the shallow and deep models. In the shallow scenarios, a WT is established from the bedrock and up to 10 m above the base of the models, while above, an unsaturated zone extends for 9 m. Finally, capping the VZ, there is a CL of 1 m of thickness. The dimensions of the parts of the system are based on the average values reported by Missimer et al. (2012) and the ranges provided by other authors for wadi deposit thickness (Al-Turbak et al. 1993; Masoud et al. 2019), depth to WT (Al-Shaibani 2008; Al-Turbak et al. 1993) and saturated thickness (Al-Turbak et al. 1993; Sorman et al. 1997).
The trench has a depth of 3 m and a width of 1 m based on average values mentioned in the literature (Bouwer 2002; Schueler 1987). The vadose zone well is 5 m in depth and 1 m in diameter using the same criteria (Bouwer 2002; Liang et al. 2018; Sasidharan et al. 2019). In the shallow WT scenarios, the domain dimensions are set as 20 m in the vertical direction and 25 m in the horizontal one. In the deep WT scenarios, the domain corresponds to 49 m of VZ capped by 1 m of CL. In this case, the saturated thickness is excluded.
Boundary conditions
Boundary conditions (BCs) are shown in Fig. 1. In the shallow WT scenarios, a constant pressure head of 2.5 m is assigned to the horizontal upper infiltration BC, representing a constant water level within the reservoir. The infiltration from the trench and the well are represented using constant pressure heads in equilibrium with the upper BC. The vertical upper right boundary is set as the seepage face to allow the passage of excess flow, in accordance with Heilweil et al. (2015). The left BC is conceptually a symmetry boundary (and no flux technically), assuming in all cases axial symmetry. The horizontal lower BC is a no-flow boundary to represent the bedrock. The scenarios involving a deep WT are simulated using the same characteristics of the shallow WT models, except that the bottom of the former scenarios is limited by a water table (i.e. a constant pressure head of 0 m).
Hydraulic parameters
Material properties are summarised in Table 2. The material of the wadi aquifer is selected as ‘loamy sand’ from the soil catalogue of HYDRUS 2D/3D (Carsel and Parrish 1988; Šimunek and Sejna 2018a), whose hydraulic parameters are in the range of values found in the literature (Al-Shaibani 2008; Hussein et al. 1993; Kalwa 2013; Masoud et al. 2019; Missimer et al. 2015, Missimer et al. 2012; Rosas 2013, 2013; Sorman et al. 1997). The hydraulic conductivity is set as anisotropic and based on the ratio between the vertical and horizontal components reported by Missimer et al. (2012) and Sorman et al. (1997). The parameters of the ‘silty clay’ from the soil catalogue of HYDRUS 2D/3D are assigned to the CL in order to resemble the CL at the Al-Alb Dam (Kalwa 2013).
Table 2 Hydraulic parameters defined for the baseline scenario Numerical realisation
The software used to model the scenarios is HYDRUS 2D/3D version 3.01 (Šimunek and Sejna 2018a, b). The modelling area is discretised with an irregular mesh. The average element edge length is 0.7 m. Mesh refinements are applied along the well/trench contour (0.24 m), the WT (0.09 m), and the lower (0.13 m) and upper boundaries of the CL (0.37 m). The total number of nodes ranges between 5,960 and 6,950 for the set of models with a deep WT, and between 21,124 and 22,332 for those with a shallow WT (note: number of nodes is lower in deep WT scenarios as the saturated part of the simulation domain was left out).
Initial conditions
The bottom of the saturated thickness has a pressure head of 10 m. This pressure decreases linearly up to the WT, where the pressure head is 0 m. For the vadose zone, a pressure head of −2.3 m is assigned, and the scenarios are run until quasi-steady-state conditions are reached.
Simulations of continuous infiltration
The six scenarios presented in Table 1 are run and the time in which groundwater recharge starts is assessed (i.e. first arrival time of infiltration at the water table) and compared with the different means of infiltration. All scenarios are run until a quasi-steady state is attained. The respective recharge rates obtained (i.e. QSSRRs) are used as a reference in the sensitivity analysis.
Analysis of cumulative recharge
Events of periodic water infiltration into the wadi aquifer are simulated by considering the natural conditions in which wadi dams operate, and then translating them into the scenarios. The operation of the wadi dams implies input of water into the reservoir from rainfall and runoff, with simultaneous evaporation and infiltration into the wadi aquifer. The periodic simulations are configured with the same climatological conditions every year, and therefore they provide cumulative recharge representative of the long-term tendency of a year.
Yearly cumulative groundwater recharge from the well, the trench, and the SF are derived from the periodic simulations. Note that the recharge from these means of infiltration is in different domains, i.e., the well in volumes and the rest in areas. Therefore, the trench and the SF recharge are converted into volumes, and subsequently compared with the well recharge.
Two scenarios are considered to convert the trench recharge into volume: (1) the trench has the same storage volume as the well; (2) the trench has the same infiltration area (i.e. the area available for infiltrating water) as the well. Thus, the 2D cumulative recharge of the trench is multiplied by the lengths that assure the equivalence of volume (1.6 m) and area (2.7 m) to the well. In the case of the SF, the recharge is computed from a circular area which has the same radius of influence as the well (25 m).
Analytical solutions
Simplified versions of the models are set up to compare with analytical equations. These simplified models are equal to the ones described in the previous subsection, with a few exceptions. They do not have a clogging layer, nor anisotropy (the whole domain has a single hydraulic conductivity of 4 m/day). Besides, the entire length of the well and the trench up to the surface is set as a constant pressure head, and the hypothetical water level of the reservoir is established at the soil surface (pressure head of 0 m).
The simplified scenarios are run until quasi-steady-state conditions are attained. The QSSRRs of the well and the trench are compared with the infiltration rates calculated through the analytical equations.
Three analytical expressions are used to represent the well in a shallow WT setting. They correspond to the equation by Elrick et al. (1989; Eq. 5) and the method 1 of the USBR (1977; Eq. 6). The former equation is developed for the one-ponded height technique, which includes a term for the unsaturated water flow, while the latter is devised for a packer test in unconsolidated material at settings in which A ≥ 10r and A = H. The meaning and selected values of the variables in all the equations used in this section are presented in Table 3, and the schematic representation of the variables is depicted in Fig. 2.
$$ Q{\displaystyle \begin{array}{c}={K}_{\mathrm{s}}\left(\frac{2\ \uppi\ {H}^2}{C}+\uppi\ {r}^2+\frac{2\ \uppi\ H}{\upalpha\ C}\right)\\ {}C={H}^2\left(\frac{\frac{A}{H}{\sinh}^{-1}\left(\frac{A}{r}\right)}{A^2}-\frac{\sqrt{{\left(\frac{r}{H}\right)}^2+{\left(\frac{A}{H}\right)}^2}+\frac{r}{H}}{A^2}\right)\end{array}} $$
(5)
$$ {\displaystyle \begin{array}{c}Q=\frac{K_{\mathrm{s}}\ \left({C}_{\mathrm{s}}+4\right)\ r\ {T}_{\mathrm{u}}}{2}\\ {}{C}_{\mathrm{s}}=\frac{2\pi\ \left(\raisebox{1ex}{$A$}\!\left/ \!\raisebox{-1ex}{$r$}\right.\right)}{\ln\ \left(\raisebox{1ex}{$A$}\!\left/ \!\raisebox{-1ex}{$r$}\right.\right)}\end{array}} $$
(6)
Table 3 Variables of the analytical equations and the selected values The well in a deep WT setting is typified with the analytical equations of Glover (1953; Eq. 7) for a packer test and Bouwer’s equation (Bouwer 2002) for a vadose zone well (Eq. 8). Both equations are formulated for a deep WT setting in which A ≥ 10r. Glover’s equation is further restricted to configurations where H = A.
$$ {\displaystyle \begin{array}{c}Q={K}_{\mathrm{s}}\ r\ H\ {C}_{\mathrm{u}}\\ {}{C}_{\mathrm{u}}=\frac{2\uppi\ \left(\raisebox{1ex}{$H$}\!\left/ \!\raisebox{-1ex}{$r$}\right.\right)}{\sinh^{-1}\left(\raisebox{1ex}{$H$}\!\left/ \!\raisebox{-1ex}{$r$}\right.\right)-1}\end{array}} $$
(7)
$$ Q=\frac{2\ \uppi\ {K}_{\mathrm{s}}{H}^2}{\ln \left(\raisebox{1ex}{$2H$}\!\left/ \!\raisebox{-1ex}{$r$}\right.\right)-1} $$
(8)
For the trench, the equations used in all cases are the ones by Bouwer (2002; Eq. 9) and the empirical equation by Heilweil et al. (2015; Eq. 10). Note that trench infiltration according to Bouwer’s equation (Eq. 9) is 20% of the well infiltration calculated through Eq. 8.
$$ Q=\frac{0.4\ \uppi\ {K}_{\mathrm{s}}\ {H}^2}{\ln \left(4\ \frac{H}{W}\right)-1} $$
(9)
$$ q={K}_{\mathrm{s}}\ \left[2.79\ \ln (T)-5.47\right] $$
(10)
When choosing the values of the variables, the main factor to consider is the range of values in which the equations are valid and the resemblance to the base scenarios (Table 1).
Sensitivity analysis of well and trench dimensions and cost analysis
Sensitivity analysis
The dimensions of the trench and the well are modified with respect to the base scenarios. The modifications are performed in multiples of the original sizes; thus, the length and radius of the well are doubled and halved, and the trench depth and the width are modified in the same way. Subsequently, the modified scenarios are run until quasi-steady-state conditions are reached. The obtained QSSRRs are compared with those of the base scenarios.
Analysis of the construction costs
Costs of construction are allocated to the well and the trench base scenarios and all the variations considered in the sensitivity analysis (e.g. a well with twice the original diameter). The cost of construction is, in broad terms, a function of the well/trench volume. Based on the range of prices provided by Brown and Schueler (1997) for a vadose zone well and an infiltration trench (4–9$ per ft3 and a selected price of 7.5$ per ft3), volumes and construction costs are computed for each of the scenarios of the sensitivity analysis.
An indirect approach is used to estimate the cumulative recharge of the scenarios involving modifications in the original dimensions (note that the periodic simulations are strictly constructed for the base scenarios). The coefficient between the QSSRRs of the scenarios in the sensitivity analysis (numerator) and the base scenarios (denominator) is calculated. Subsequently, the obtained coefficient is multiplied by the cumulative recharge of the base scenarios and then divided by the respective construction cost to obtain the yearly cumulative recharge per dollar. Only the scenarios in a shallow WT setting are considered.
In addition to the scenarios of the sensitivity analysis, two more scenarios are evaluated. They are termed the “optimum” scenarios, and there is one for a well and one for a trench. They consist of the combination of vertical and horizontal dimensions, which ensure the highest QSSRRs per unit of volume. In the case of the well, this happens when the diameter is half of the original one, which is 0.5 m, and the length twice as long (10 m). The trench in the optimum scenario has both dimensions halved, so its depth is 1.5 m and the width 0.5 m.
Well and trench performance under areal constraints
The infiltration potential of wells and trenches on a fixed square area is assessed to mimic conditions with limited infiltration area. The maximum number of wells and trenches that can fit in a 1-km2 dam is calculated in accordance with the optimum distributions shown in Fig. 3. In these hypothetical distributions, the wells and trenches are positioned in such a way that their areas of influence do not overlap, enabling the arithmetic addition of the individual cumulative recharge. For the trench, the lateral influence extent (Li, Fig. 3) is 50 m for the shallow WT setting and 23 m for the deep WT configuration. In the case of the well, the influence radii (ri, Fig. 3) chosen were 25 and 17.5 m for shallow and deep WT settings, respectively.
The computed number of wells/trenches is multiplied by the yearly cumulative recharge of one well/trench. The resulting figures correspond to the annual cumulative recharge yielded by a battery of wells/trenches for both deep and shallow WT.