Model Categories

Mathematical models in the discipline of hydrology (including surface and subsurface) generally fall into three main categories: empirical or statistical/metric, conceptual and physically-based (Wheater et al., 1993). The distinction between empirical, conceptual and physically-based models is sometimes not a clear one and is somewhat subjective. Models commonly contain a mix of modules from each category or may be a hybrid of two or more types. The three types of model categories commonly used are discussed below along with some of their strengths and limitations.

5.2.1.1 Empirical (or Metric) Models

Empirical models are generally the simplest of all the model types, and they are used to describe the behaviour between variables on the basis of observed data alone. Often the relationship between observed variables is described with a simple mathematical function without any assumptions made regarding the underlying physical processes (Wheater et al., 1993).

The equation describing baseflow recession, which is expressed as the exponential decay of groundwater discharge within a stream, is an example of a commonly used empirical model.

kt

e Q Q= ₀ −

where Q₀is the discharge at the start of the baseflow recession, Qis the discharge at any time over the recession period, k is the recession constant and time t is the time since the recession began. The value of the recession constant, k, is typically estimated empirically from continuous hydrograph records over an extended period (U.S. Army Corps of Engineers, 1999).

Another example of an empirical model is the unit hydrograph theory commonly applied in the simulation of catchment scale rainfall-runoff relationships (Chow, 1964). In this modelling approach the streamflow response to each unit of effective rainfall (the portion of the rainfall that becomes streamflow) is calculated as a linear, time invariant

function. A major strength of this modelling approach is that once the streamflow and effective rainfall components of an observed response have been separated, a unit hydrograph can be derived that allows for the event response to be characterised. The variability of the unit hydrograph can be established by analysing data from a range of events (Wheater et al., 1993).

Empirical models tend to be used equally by both surface and subsurface hydrologists because of their simplicity. Empirical models can have high predictive powers within the range of data available, but often lack explanatory depth. They are usually specific to the conditions under which data were collected and in these situations should not be extrapolated to other conditions or other catchments (Mulligan and Wainwright, 2004). These types of models are often criticised for not considering the physical attributes of the system and for ignoring the inherent non-linearity of a system.

5.2.1.2 Conceptual Models

The use of the term conceptual model can be somewhat confusing because of the various denotations. A conceptual model can be described as the basic idea or construct of how a system operates (Bredehoeft, 2004; Konikow and Bredehoeft, 1992). To a hydrogeologist, a conceptual model includes factors such as the geological framework, the location, types and characteristics of the aquifers in the study area, the identification of recharge and discharge sites and the direction of groundwater flow for the region to be modelled. An equivalent term commonly used in the field of hydrology is the

perceptual model (Beven, 2000a). In either case the terminology describes a mental model that is based on a researcher’s experience, prior knowledge, familiarity with datasets and knowledge about the study area; the formulation is qualitative.

In contrast, the use of the term conceptual model in the discipline of surface hydrological modelling refers to a type of model incorporating a relatively simple mathematical description for each of the processes being considered. A conceptual model typically represents a catchment by a series of internal storages, with individual storages representing key aspects of, or processes within, the system. Some examples of conceptual models that have been used to model groundwater-river interactions include Croke et al. (2000), Dietrich et al. (1989), Jakeman et al. (1989) and Moore and Bell (2002). This style of modelling is traditionally preferred by surface hydrologists because subsurface processes are mostly inferred from the stream hydrograph.

Nonetheless, conceptual models are increasingly being used by hydrogeologists to represent subsurface processes in data limited, highly heterogeneous and/or large catchments, and where only a final equilibrium response to a stressed system is of interest (Dawes et al., 2004; Dawes et al., 2001).

Traditionally, conceptual models for hydrological modelling apply a lumped modelling approach where the catchment (or storage components within the catchment) is represented as a single unit with state variables representing average values over the catchment area (Beven, 2000a). Conceptual models may also be applied in a semi- distributed manner by disaggregating a catchment into linked subcatchments over which the model is applied (Kokkonen et al., 2001; Merritt et al., 2003).

The observed relationship between variables (or storages) in conceptual models is described by functional forms that incorporate parameter values. Each of the model storage components are generally made up of empirical models, so many of the limitations of empirical models can apply to conceptual models. However, the configuration and relationships between storages can provide additional insights into the physical processes governing the system behaviour (Mulligan and Wainwright, 2004). Parameter values for conceptual models are typically obtained through calibration against observed data, such as stream discharge. Due to the requirement that parameter values are determined through calibration against observed data, conceptual models tend to suffer from problems associated with the identifiability and non-uniqueness of their parameter values (Jakeman and Hornberger, 1993; Merritt et al., 2003) because there may be several possible ‘best’ parameter fits that provide an equally good explanation of the data relationships or ‘equifinality’ (Beven, 2000b). Such problems with parameter identification can be minimised through limiting the number of parameters to be estimated and by using knowledge of the system to limit the range of possible values (Dunn, 1999; Koivusalo and Kokkonen, 2003; Seibert and McDonnell, 2002; Uhlenbrook and Sieber, 2005). The lack of uniqueness in parameter values for conceptual models means that the parameters in such models may have limited physical meaning (Wheater et al., 1993).

5.2.1.3 Physically-Based Models

Physically-based models operate through solving mathematical equations describing fundamental physical principals. These models often apply a distributed modelling

approach whereby a catchment is descretised into a large number of elements or grid squares, and model predictions are distributed in space (Beven, 2000a). The spatially distributed, physically-based type of groundwater model is the only type of model available that allows for the groundwater system to be modelled in two or three dimensions, and hence is a powerful tool in groundwater management. State variables representing local averages, such as hydraulic conductivity for example, are applied to each descretised unit of space, and mathematical equations representing physical processes are solved for each region. In theory, the parameters used within physically- based models are measurable, for example, through field or laboratory measurement and sampling, and hence have known values. In practice, the large number of parameters required to represent key physical processes and characteristics arising from spatial heterogeneities render many parameters unknown (Beven, 2000b).

Where parameters cannot be measured in a catchment they must be determined through calibration with observed data. Parameter estimates that are calibrated often have non- unique values, a problem also found when using empirical and conceptual models, and hence the physical interpretability of the values may be questionable. Where measurements can be made, point data from localised areas are often used to represent large areas, and the data sets are aggregated to the scale required for descretisation. Differences between that scale at which measurements were made and the scale at which the model algorithms apply sometimes creates additional uncertainty in model outcomes (Wheater et al., 1993).

The industry-standard model code used for modelling hydrogeological systems is a spatially distributed, physically-based model code entitled MODFLOW (McDonald and Harbaugh, 1988), which solves the governing differential equation for groundwater flow in three dimensions:

t h S W z h K z y h K y x h K x x y z s ∂ ∂ = − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ + ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂

where Kx,Ky and Kzare values of hydraulic conductivity along the x,y and z coordinate

axes; h is the hydraulic head; W is a flux term that accounts for pumping, recharge , or other sources and sinks; Ss is the specific storage; and t is time. The MODFLOW

capability (and that of other spatially distributed physically-based models) is continually being developed, with the current focus on increasing the functionality of integrated

surface and groundwater modelling (integrated types of models are also discussed in sections 5.2.3 and 5.3). MODFLOW continues to be a commonly used tool for modelling the management problems associated with surface-groundwater flow, and as such the applications are too vast to review. Some examples include research by Sophocleous et al. (1998), Rodríguez et al. (2006) and several of the reports listed in Table 3-1.

Physically-based models make a good attempt to represent the physical processes that occur within a catchment to the limit possible given data limitations and the validity of the assumptions built into the model, and they are extremely useful in developing an understanding of how a system works. The risk of physically-based modelling approaches is that they become over-parameterised (Beven, 1993; Beven, 2001). Overly complicated models with large numbers of processes considered together with the associated parameters run the risk of having a high degree of uncertainty associated with model inputs, which can be translated through to the model outputs resulting in lower predictive capability, particularly at larger catchment scales.

Konikow and Bredehoeft (1992) argue that groundwater models when applied to a field study area cannot truly be validated because of inadequate parameter estimation, perceptual model deficiencies, and numerical errors, and hence the models can only be tested and invalidated. However, it is through this model testing and evaluation of errors that models are improved and a better understanding of the problems and associated model conceptualisation are gained.

Another type of physically-based modelling approach is one based on solving analytical equations. There are numerous equations that have been derived, for example, to calculate the impact of groundwater extraction on streamflow depletion rates and volumes (Anderson, 2003; Boulton, 1942; Fox et al., 2002; Glover and Balmer, 1954; Hantush, 1965; Jenkins, 1968; Kirk and Herbert, 2002; Knight et al., 2005; Pulido- Velazquez et al., 2005). The assumptions required to solve the most commonly used analytical equations such as those of Jenkins (1968), which assume the presence of an unconfined aquifer and a river which fully penetrates the aquifer (in other words it is not possible to drawdown groundwater levels below the base of the river), make these types of streamflow depletion calculations limited in practice.

The trade-offs between the modelling approaches tend to be that of parsimony versus complexity, the associated predictive versus explanatory powers, and the data/computational requirements versus the costs.