Integrated modelling of potential recharge in small recharge ponds

Abstract

Integrated models that simultaneously simulate potential recharge and time delay for wetting front to reach water table are necessary for comprehensive and coordinated planning of managed aquifer recharge. Most of the existing models do not simultaneously simulate the potential recharge, length of advancement of wetting front, and time delay for wetting front to reach water table except the few numerical models, which are data intensive, high initial software and data setup cost, spatial and temporal discretisation depended accuracy, and time-consuming. This paper presents an integrated potential recharge simulation (IPRS) and time delay for wetting front to reach water table (TDWF) models for simultaneously simulating the potential recharge and time delay for wetting front to reach water table under time-variant inflows and outflows, respectively. The models have been derived by integrating the modified Green-Ampt model for variable water depth into the water balance equation of recharge pond. The numerical calculations of the IPRS and TDWF models are performed by the computer program written in Fortran-95. The performance comparison of the IPRS and TDWF models with HYDRUS-1D model for a hypothetical recharge pond over variety of soils showed its ability and comparability with the HYDRUS model. The models have also been successfully demonstrated for a recharge pond in a watershed in a semi-arid region of India. The IPRS model was found to be sensitive to the changes in inflow and saturated hydraulic conductivity of the bed soil of the recharge pond. The derived models will be helpful in simultaneous simulating the potential groundwater recharge and time delay to reach water table and optimum design and evaluating the effectiveness of recharge ponds and/or other surface artificial groundwater recharging facilities of any shape and size in the field.

Appendices

Appendix 1. Estimation of area related model’s variables

The area related model’s variables, ${\overline{A}}_{es}(t)$, ${\overline{A}}_{rs}(t)$, A_vs(t), and A_vs(t-Δt) are estimated by:

$$ {\overline{A}}_{es}(t)=0.5\left[{A}_{es}\left(t-\varDelta t\right)+{A}_{es}(t)\right] $$

(1)

$$ {\overline{A}}_{rs}(t)={\overline{A}}_{es}(t)=0.5\left[{A}_{es}\left(t-\varDelta t\right)+A{e}_{es}(t)\right] $$

(2)

$$ {A}_{vs}(t)=\frac{1}{3}\left[{A}_b+{A}_{es}(t)+\sqrt{A_b\;{A}_{es}(t)\;}\right] $$

(3)

$$ {A}_{vs}\left(t-\varDelta t\right)=\frac{1}{3}\left[{A}_b+{A}_{es}\left(t-\varDelta t\right)+\sqrt{A_b\;{A}_{es}\left(t-\varDelta t\right)}\right] $$

(4)

where ${\overline{A}}_{es}(t)$ is the average water surface area for evaporation between time t-Δt and t [L²]; A_es(t) is the water surface area for evaporation at time t [L²]; A_es(t-Δt) is the water surface area of for evaporation at time t-Δt [L²]; ${\overline{A}}_{rs}(t)$ is the average wetted planner area for the potential recharge between time t-Δt and t [L²]; A_vs(t) is the surface area for water storage volume at time t [L²]; A_vs(t-Δt) is the surface area for water storage volume at time t-Δt [L²]; and A_b is the bottom surface area of the recharge pond [L²].

Appendix 2.

2.A HYDRUS model

In Hydrus-1D modelling code, Richards’ model (Simunek et al. 1998) with a sink term for water flow in a homogeneous or uniform soil is defined as:

$$ C(h)\frac{\partial h}{\partial t}=\kern1em \frac{\partial }{\partial z}\left[K(h)\;\left(\frac{\partial h}{\partial z}+1\right)\right]-S\left(z,t\right) $$

(B.1)

$$ h={h}_0<0,\kern1em 0\le z\le \infty, \kern1em t=0\kern1em $$

(B.2a)

$$ h={h}_t>0,\kern1em z\ge 0,\kern2em t>0\kern1em $$

(B.2b)

in which, C(h) = dθ/dh = specific water capacity function [L^-1]; h = hydraulic head [L]; K(h) = hydraulic conductivity of soil [LT^-1], z = soil depth [L]; S(z, t) = sink ( evaporation rate) in space and time, and h₀ and h_t are the initial and boundary condition potentials, respectively.

A finite difference, fully implicit scheme is used for the solution to Eq. (B.1). K(h) is determined by the potential gradient. For each time step, the changes in the head are calculated iteratively. The mass balance is calculated at each node following each time step, and its error never exceeded 10^-3 in simulations.

2.B Water depth simulation model

Water depth simulation model for the estimation of time variant water depth in the pond by Ali et al. (2015) which is:

$$ {H}_w\;\left(n\;\varDelta\;t\right)\kern1.12em =\kern1em \frac{1}{A_{vs}\;\left(n\;\varDelta\;t\right)}\left\{\begin{array}{l}{H}_w\;\left[\;\left(n-1\right)\;\varDelta\;t\;\right]\;{A}_{vs}\left[\;\left(n-1\right)\;\varDelta\;t\;\right]\kern1em \\ {}\\ {}+\left[\;{Q}_i\;\left(n\;\varDelta\;t\right)\;{A}_{ws}+{P}_i\kern0.24em \left(n\;\varDelta\;t\right)\kern0.24em {A}_t-{E}_p\;\left(n\;\varDelta\;t\right)\kern0.36em {\overline{A}}_{es}\left(n\;\varDelta\;t\right)-{Q}_0\;\left(n\;\varDelta\;t\right)\;\right]\;\varDelta t\\ {}\\ {}-{K}_s\;\left[1+\frac{H_w\;\left(n\;\varDelta\;t\right)+{\psi}_f}{\;{Z}_f\;\left(n\;\varDelta\;t\right)}\right]\;{\overline{A}}_{rs}\left(n\;\varDelta\;t\right)\kern0.36em \varDelta t\end{array}\right\}\kern1.12em $$

(B.3)

The variables of Eq. (B.3) are defined previously.

Recharge pond with rectangular shape cross section without inflows is expressed as (Ali et al. 2015):

$$ {\displaystyle \begin{array}{l}{H}_w\;\left(n\;\varDelta\;t\right)=\frac{Z_f\left(n\kern0.24em \varDelta\;t\right)}{Z_f\left(n\kern0.24em \varDelta\;t\right)+{K}_s\;\varDelta t}\kern1em \left\{{H}_w\;\left[\kern0.24em \left(n-1\right)\;\varDelta\;t\;\right]-{E}_p\;\left(n\;\varDelta\;t\right)\kern0.24em \varDelta\;t-{K}_s\;\left[\;1+\frac{\;{\psi}_f}{\;{Z}_f\left(n\kern0.24em \varDelta\;t\right)\;}\right]\;\varDelta\;t\right\}\\ {}\;\end{array}} $$

(B.4)

Appendix 3. Performance evaluation indicators

Nash-Sutcliffe efficiency (E) (Nash and Sutcliffe 1970) is given by:

$$ E=\kern1em 1\kern1em -\kern1em \frac{\sum \limits_{i=1}^n{\left({R}_{oi}-{R}_{pi}\right)}^2\;}{\sum \limits_{i=1}^n{\left({R}_{oi}-\overline {R_o}\right)}^2\;} $$

(C.1)

where E is the Nash-Sutcliffe efficiency; R_oi is the ith observed potential recharge rate estimates; R_pi is the ith predicted potential recharge rate estimates; $\overline {R_o}$= mean of the observed potential recharge rate estimates; i is an integer varying from 1 to n, i = 1, 2, 3,..., n; and n is the total number of potential recharge rate estimates.

The index of agreement (d) (Legates and McCabe 1999) is given by:

$$ d\kern1.12em =\kern1.12em 1-\frac{\sum \limits_{i=0}^n{\left({R}_{oi}-{R}_{pi}\right)}^2}{\sum \limits_{i=1}^n{\left(|{R}_{pi}-\overline {R_o}|+|{R}_{oi}-\overline {R_o}|\right)}^2} $$

(C.2)

The root mean square error (RMSE) (Schneider and MacCuen 2005) is given by:

$$ RMSE=\sqrt{\frac{1}{n}\sum \limits_{i=1}^n{\left({R}_{pi}-{R}_{oi}\right)}^2} $$

(C.3)

Relative bias (PB) is defined (Van Liew et al. 2007) as:

$$ RB\kern1em =\kern1em \frac{\sum \limits_{i=1}^n\left({R}_{pi}-{R}_{oi}\right)}{n}\kern1em $$

(C.4)