Model Description

History

The Soil Moisture Accounting and Routing for Transport [SMART] model (Mockler et al., 2016) is a bucket-type rainfall-runoff model.

SMART is an enhancement of the Soil Moisture Accounting and Routing with Groundwater [SMARG] model, a lumped rainfall–runoff model developed at National University of Ireland, Galway (Kachroo, 1992), and based on the soil layers concept (O’Connell et al., 1970; Nash and Sutcliffe, 1970). Separate soil layers were introduced to capture the decline with soil depth in the ability of plant roots to extract water for evapotranspiration. SMARG was originally developed for flow modelling and forecasting and was incorporated into the Galway Real-Time River Flow Forecasting System [GFFS] (Goswami et al., 2005).

The SMART model reorganised and extended SMARG to provide a basis for water quality modelling by separating explicitly the important flow pathways in a catchment.

Conceptual model

Fig. 1: Conceptual representation of the bucket-type rainfall-runoff model SMART.

Model inputs and outputs

Tab. 1: SMART model inputs
Input	Description	Unit
\(P\)	Precipitation as rainfall	mm per time step
\(E_P\)	Potential evapotranspiration	mm per time step

Tab. 2: SMART model outputs
Output	Description	Unit
\(Q\)	River discharge	m^-3 s^-1
\(E_A\)	Actual evapotranspiration	mm per time step

Model parameters

Tab. 3: SMART model parameters
Parameter	Description	Unit	Typical range*
\(\theta_T\)	Rainfall aerial correction factor	–	0.9–1.1
\(\theta_C\)	Evaporation decay coefficient	–	0–1
\(\theta_H\)	Quick runoff ratio	–	0–0.3
\(\theta_D\)	Drain flow ratio	–	0–1
\(\theta_S\)	Soil outflow coefficient	–	0–0.013
\(\theta_Z\)	Effective soil depth	mm	15–150
\(\theta_{SK}\)	Surface reservoir residence time	timestep	1–240h
\(\theta_{FK}\)	Interflow reservoir residence time	timestep	48–1440h
\(\theta_{GK}\)	Groundwater reservoir residence time	timestep	1200–4800h
\(\theta_{RK}\)	Channel reservoir residence time	timestep	1–96h

*: After Mockler et al. (2016).

Model equations

The SMART model forcings are precipitation \(P\) [mm/time step] and potential evapotranspiration \(E_P\) [mm/time step]. The precipitation input is first transformed into the corrected precipitation \(p_C\) [mm/time step] using the aerial correction parameter \(\theta_{T}\) [–] (1).

(1)\[p_C = \theta_{T} P\]

The difference between corrected precipitation and potential evapotranspiration determines whether the modelling time step is under energy-limited conditions (condition \(\gamma\) is true) or water-limited conditions (\(\gamma\) is false) (2). Then, the effective precipitation \(p_E\) [mm/time step] (3) and the precipitation contribution to the actual evapotranspiration \(e_A\) [mm/time step] (4) are determined accordingly.

(2)\[\gamma: p_C \geq E_P\]

(3)\[\begin{split}p_E = \begin{cases} \theta_{T}P - E_P, & \text{if } \gamma \\ 0, & \text{otherwise} \end{cases}\end{split}\]

(4)\[\begin{split}e_A = \begin{cases} E_P, & \text{if } \gamma \\ \theta_{T}P, & \text{otherwise} \end{cases}\end{split}\]

The two parameters for quick runoff ratio \(\theta_H\) [–] and soil outflow coefficient \(\theta_S\) [–] are adjusted according to the antecedent soil moisture conditions to become \(\theta_{H'}\) [–] (5) and \(\theta_{S'}\) [–] (6), respectively. The six soil moisture layers are of equal depths and sum up to a total field capacity defined by the parameter \(\theta_Z\) [mm].

(5)\[\theta_{H'} = \theta_H \frac{\sum_{\lambda=1}^{6} S_{\lambda}}{\theta_Z}\]

(6)\[\theta_{S'} = \theta_S \frac{\sum_{\lambda=1}^{6} S_{\lambda}}{\theta_Z}\]

Under energy-limited conditions

The infiltration flux \(q_0\) [mm/time step] and the percolation fluxes through the soil layers \(q_{\lambda}\) [mm/time step] are then calculated as described in Equations (7) and (8), respectively.

(7)\[q_0 = (1 - \theta_{H'}) p_E\]

(8)\[\begin{split}q_{\lambda} = \begin{cases} q_{\lambda - 1} - \Big(\frac{\theta_{Z}}{6} - S_{\lambda}\Big), & \text{if } q_{\lambda - 1} + S_{\lambda} > \frac{\theta_{Z}}{6} \\ 0, & \text{otherwise} \end{cases}\end{split}\]

If all soil layers reach saturation, the saturation excess flux \(q_6\) [mm/time step] is divided into quick runoff as drainflow \(r_{DF}\) [mm/time step] (9) and slow runoff as interflow. The outflow from the six soil layers contributes to the three runoff pathways: interflow \(r_{IF}\) [mm/time step], shallow groundwater flow \(r_{SGW}\) [mm/time step], and deep groundwater flow \(r_{DGW}\) [mm/time step]. First, the soil outflow contributes to the interflow runoff following a power law from the top layer to the bottom layer (11). Then, the soil outflow contributes to the shallow groundwater runoff following an inverse law from the top layer to the bottom layer (13). Finally, the soil outflow contributes to the deep groundwater runoff following a power law from the bottom layer to the top layer (15). The parameter \(\theta_{S'}\) is used in each of the three law distributions to determine the fraction of each soil layer that contributes to runoff during the modelling time step.

(9)\[r_{DF} = \theta_D q_6\]

(10)\[\begin{split}{s_{IF}}_{\lambda} = \begin{cases} S_{\lambda}(\theta_{S'})^{\lambda}, & \text{if } \gamma \\ 0, & \text{otherwise} \end{cases}\end{split}\]

(11)\[\begin{split}r_{IF} = \begin{cases} \sum_{\lambda=1}^{6} {s_{IF}}_{\lambda}, & \text{if } \gamma \\ 0, & \text{otherwise} \end{cases}\end{split}\]

(12)\[\begin{split}{s_{SGW}}_{\lambda} = \begin{cases} S_{\lambda}\Big(\frac{\theta_{S'}}{\lambda}\Big), & \text{if } \gamma \\ 0, & \text{otherwise} \end{cases}\end{split}\]

(13)\[\begin{split}r_{SGW} = \begin{cases} \sum_{\lambda=1}^{6} {s_{SGW}}_{\lambda}, & \text{if } \gamma \\ 0, & \text{otherwise} \end{cases}\end{split}\]

(14)\[\begin{split}{s_{DGW}}_{\lambda} = \begin{cases} S_{\lambda}(\theta_{S'})^{7 - \lambda}, & \text{if } \gamma \\ 0, & \text{otherwise} \end{cases}\end{split}\]

(15)\[\begin{split}r_{DGW} = \begin{cases} \sum_{\lambda=1}^{6} {s_{DGW}}_{\lambda}, & \text{if } \gamma \\ 0, & \text{otherwise} \end{cases}\end{split}\]

Under water-limited conditions

The water deficit \(d_0\) [mm/time step] (16) to meet the potential evapotranspiration demand is totally or partially provided by the available soil moisture evapotranspiration fluxes \(e_{\lambda}\) [mm/time step] (18). The variable \(b_{\lambda}\) [–] (20) acts as a boolean to stop the soil moisture contribution to evapotranspiration as soon as the remaining water deficit \(d_{\lambda}\) [mm/time step] (19) has been fully met by a given soil layer; it is initiated with a value of 1 if a deficit exists (17). The first soil layer can fully meet the water deficit if it contains enough water, otherwise the second layer can contribute to meeting the remaining water deficit with a depleted rate using the evaporation decay parameter \(\theta_C\) [–] (19), and so forth for the next downward layer down to the bottom layer. Effectively, a lesser fraction of the available soil moisture in a layer can meet the remaining water deficit moving downwards following a power law (18)–(19).

(16)\[\begin{split}d_0 = \begin{cases} 0, & \text{if } \gamma \\ E_P - e_A, & \text{otherwise} \end{cases}\end{split}\]

(17)\[\begin{split}b_0 = \begin{cases} 0, & \text{if } \gamma \\ 1, & \text{otherwise} \end{cases}\end{split}\]

(18)\[\begin{split}e_{\lambda} = \begin{cases} b_{\lambda - 1} d_{\lambda - 1}, & \text{if } S_{\lambda} \geq d_{\lambda - 1} \\ b_{\lambda - 1} S_{\lambda}, & \text{otherwise} \end{cases}\end{split}\]

(19)\[\begin{split}d_{\lambda} = \begin{cases} 0, & \text{if } S_{\lambda} \geq d_{\lambda - 1} \\ \theta_{C} ( d_{\lambda - 1} - S_{\lambda} ), & \text{otherwise} \end{cases}\end{split}\]

(20)\[\begin{split}b_{\lambda} = \begin{cases} 0, & \text{if } S_{\lambda} \geq d_{\lambda - 1} \\ b_{\lambda - 1}, & \text{otherwise} \end{cases}\end{split}\]

Under water-limited and energy-limited conditions

At each time step, whether under water-limited or energy-limited conditions, the soil layer states \(S_{\lambda}\) [mm] are updated given their inward and outward fluxes during the time step as described in (21).

(21)\[\frac{dS_{\lambda}}{dt} = q_{\lambda - 1} - {s_{IF}}_{\lambda} - {s_{SGW}}_{\lambda} - {s_{DGW}}_{\lambda} - e_{\lambda}\]

The routing for the five runoff pathways is conceptualised as five linear reservoirs which are characterised by three residence time parameters \(\theta_{SK}\) [time step], \(\theta_{FK}\) [time step], and \(\theta_{GK}\) [time step] (22)–(26). The five runoff pathways contribute to a final linear reservoir for channel routing which is characterised by the residence time parameter \(\theta_{RK}\) [time step] to compute the catchment total flow \(Q\) [mm/time step] (27).

(22)\[q_{OF} = \frac{S_{OF}}{\theta_{SK}}\]

(23)\[q_{DF} = \frac{S_{DF}}{\theta_{SK}}\]

(24)\[q_{IF} = \frac{S_{IF}}{\theta_{FK}}\]

(25)\[q_{SGW} = \frac{S_{SGW}}{\theta_{GK}}\]

(26)\[q_{DGW} = \frac{S_{DGW}}{\theta_{GK}}\]

(27)\[Q = \frac{S_R}{\theta_{RK}}\]

Finally, the reservoir states [mm] are updated given their inward and outward fluxes during the time step as described in Equations (28)–(33).

(28)\[\frac{dS_{OF}}{dt} = \big( p_E - q_0 \big) - q_{OF}\]

(29)\[\frac{dS_{DF}}{dt} = r_{DF} - q_{DF}\]

(30)\[\frac{dS_{IF}}{dt} = \big( q_6 - r_{DF} \big) + r_{IF} - q_{IF}\]

(31)\[\frac{dS_{SGW}}{dt} = r_{SGW} - q_{SGW}\]

(32)\[\frac{dS_{DGW}}{dt} = r_{DGW} - q_{DGW}\]

(33)\[\frac{dS_{R}}{dt} = q_{OF} + q_{DF} + q_{IF} + q_{SGW} + q_{DGW} - Q\]

Procedural model

The python package smartpy is a numerical implementation of the equations described in the conceptual model presented above. This procedural model numerically solves these equations using an explicit forward Euler method, and an operator splitting technique (i.e. the procedural model solves the equations sequentially following the order presented above).