Borehole representation in distributed models

Existing hydro-economic models that implement distributed groundwater components typically have used relatively coarse discretisation of the groundwater model domain to minimise compu- tational demands. For example, the MODFLOW groundwater model used by Mulligan et al. (2014), and which also is actively used to inform groundwater management in the Republican River Basin (Republican River Compact Administration, 2003), has a spatial grid resolution of 1.6 km and is temporally discretised into monthly stress periods. The use of distributed groundwater models at this resolution, while substantially more detailed than single-cell and semi-distributed aquifer models, will lead to errors in the estimation of drawdown within ab- straction wells as pumping is assumed to be distributed evenly over a single grid cell that is orders of magnitude larger than the diameter of the actual borehole. The water level in the bore- hole is one of the primary determinants of both well yield and economic costs of groundwater extraction, and, therefore, of farmers’ irrigation decision making. Consequently, errors in the prediction of borehole head may induce inaccuracies in subsequent hydro-economic simulations and estimations of policy effectiveness that have not been considered in previous analyses. A number of potential methods to refine predictions of groundwater head in the vicinity of abstrac- tion wells exist, which are described below along with a discussion of their suitability for use in integrated hydro-economic modelling frameworks.

Distributed groundwater models that use finite difference solutions may provide the ability to refine the model grid locally in the areas of rapidly varying hydrological conditions, such as around abstraction boreholes or surface water features. Both MODFLOW (Mehl and Hill, 2005) and ZOOMQ3D (Jackson and Spink, 2004) models contain options for local grid refinement that enable local-scale hydrological processes to be represented using a finer-scale model sub-grid that

is nested within the coarser grid of the regional scale groundwater model. Local grid refinement reduces overall computational demands as it is not necessary to refine the discretisation of the full regional model domain. Nevertheless, a significant limitation of this approach is that, even with local refinement, model cell sizes are still likely to be substantially larger than the diameter of an abstraction borehole. Furthermore, flow convergence to a well is radial in nature and, therefore, is best simulated using a radial coordinate system (Anderson and Woessner, 1992). However, local grid refinement techniques typically require both the regional and local grids to be based on a square or rectangular mesh to be enable coupling of the two grids and, as a result, are unable to represent adequately radial flow process in the vicinity of an abstraction borehole. An alternative approach to the local grid refinement techniques described above is the recently developed MODFLOW Unstructured Grid Package (MODFLOW-USG) (Panday et al., 2013). MODFLOW-USG enables both structured and unstructured grids to be coupled using a control volume finite difference solution scheme. As a result, this technique can improve significantly the representation of boreholes in regional spatially-distributed groundwater models compared to models that use only structured grids. However, the design of highly complex unstructured model grids requires substantial expertise and may impose significant computational demands for both pre- and post-processing. Moreover, MODFLOW-USG, as with structured local grid refinement, does not use a radial coordinate system that is the most accurate method for sim- ulating convergence of flow to an abstraction borehole. Given this, it seems that the predictive improvements that could be gained from either MODFLOW-USG or local-grid refinement are unlikely to be sufficient to balance the increased computational costs that implementation of these methods in integrated, multidisciplinary analyses would entail.

As discussed above, flow to abstraction boreholes is most appropriately described using a radial flow model. A range of analytical solutions exist to estimate radial flow to wells, some of which have been used previously in hydro-economic analysis. For example, Brozović et al. (2010) and Madani and Dinar (2012a) both use the Theis equation (Theis, 1935) to evaluate the effects of spatially distributed well pumping within economic analyses of optimal groundwater management. However, analytical models commonly require a large number of assumptions to be made about the hydrogeological system that do not match the complex heterogeneity found in real-world aquifers. Radial flow to boreholes is simulated more adequately using numerical solution schemes that employ a cylindrical coordinate systems and are able to incorporate well storage effects, non-linear flow, and seepage face development that may influence the effect of pumping on borehole heads. For example, the Darcy-Forchheimer equation, which introduces an additional quadratic term to Darcy’s law to account for non-linear flow effects (Forchheimer,

1901), has been shown to provide an improved representation of radial flow in the vicinity of abstraction boreholes (Mathias et al., 2008; Mathias and Todman, 2010; Wen et al., 2011).

Radial flow models offer advantages for simulating flow around abstraction wells, but do not represent regional groundwater flow processes in a computationally efficient manner. The ability to analyse spatially distributed groundwater pumping over large areas is often important in hydro-economic analysis due to the presence of multiple water users, who each may be affected by externalities generated by the others pumping (Brozović et al., 2010; Pfeiffer and Lin, 2012). Therefore, to be useful in integrated hydro-economic analysis, radial flow models must have the capacity to be embedded within regional groundwater modelling frameworks. A number of multi- scale methods exist to represent radial flow to a well within regional groundwater models. Finite element and finite volume techniques have been used in petroleum and atmospheric modelling (Hiebert et al., 1993), but have yet to be employed in groundwater modelling due to the difficulties associated with defining the complex grid structures and significantly increased computational

demands. Alternatively, hybrid radial-cartesian modelling frameworks can enable numerical

radial flow models to be embedded within existing regional groundwater flow models (Upton et al., 2013). In a hybrid radial-cartesian model, the radial model grid replaces a section of the cartesian grid of the regional-scale groundwater model, enabling the incorporation of both local- and regional-scale groundwater flow processes in a consistent framework. As a result, hybrid radial-cartesian models may offer a promising option for hydro-economic analysis where both regional- and local-scale hydrological processes must be represented adequately in order to predict of field-level groundwater use decisions and the resultant dynamic evolution of the groundwater system to those spatially and temporally distributed choices.

4.2

Methodology

In the following subsections, a description is provided of a hybrid radial-cartesian groundwater model, SPIDERR (Simulating Pumping Boreholes with a Darcy-Forchheimer Regional Radial Flow Model) (Upton et al., 2013), that is used in the analysis presented in this chapter. A spatially-explicit methodology then is proposed to integrate the SPIDERR groundwater flow model with the distributed model of farmers’ individual field-level irrigation decision making that is developed in Chapters 2 and 3, capturing the significant feedbacks that occur between groundwater and agricultural production at both annual and intraseasonal timescales. Finally, a range of simulations are described that will be used to test the ability of the integrated model to predict the long-term co-evolution of agricultural production and groundwater conditions, and to evaluate the effectiveness of policies to improve the sustainability of groundwater-fed irrigation.