# Groundwater flow mathematical modeling

Kimmo Hentinen, Geological Survey of Finland, P.O. Box 1237, FI-70211 KUOPIO, FINLANDkimmo.hentinen[at]gtk.fi

## Introduction

Mathematical modelling can be used for studying groundwater management in mine closure procedure. Mathematical models can be created e.g. for fluid flow, mass and heat transport. Various processes can be modelled mathematically and with modelling, problems can be identified and avoided beforehand. For a certain process a mathematical model is derived and then solved to gain information about the area of interest. In groundwater flow modelling very simplified scenarios can be solved analytically but usually the mathematical model is very hard or impossible to solve exactly. Instead an approximative solution is achieved by numerical methods (Wang & Anderson 1982). The solution of the model can be then analyzed and visualized.

## Description of the method

Mathematical models for flow, heat and mass transport are derived using the principle of conservation. For example mass doesn’t disappear nor is it created out of nothing. The same applies for thermal energy and momentum. These equations derived from the principle of conservation are also called continuity laws (or balance laws) (Wang & Anderson 1982, Hämäläinen & Järvinen 2006). Flow, mass and heat transfer can be studied in time independent steady state or in time dependent transient situations.

Continuity laws are usually in a form of partial differential equations (PDE). In some very simple cases it is possible to solve these equations analytically and to achieve an exact solution which describes e.g. flow in porous media. Usually the solution can’t be obtained analytically and numerical methods must be applied in order to achieve approximative solution. In order to obtain an unique solution for PDE boundary conditions are required (Wang & Anderson 1982). In groundwater modelling boundary condition can be e.g. fixed value for hydraulic head at the domain boundary.

Numerical methods for PDE solving are e.g. finite difference method (FDM), finite element method (FEM) and finite volume method (FVM) of which FEM and FVM are probably the most common (Hämäläinen & Järvinen 2006). FDM is, however, widely used in the groundwater flow modelling. In all of these methods the area of interest is divided into finite number of elements or nodes and the governing continuity laws are then applied to each of them. In this way a system of linear equations is created which is written in matrix form and solved with a computer. As a result, an approximative solution of the PDE is produced which is also the approximative solution of the mathematical model.

Most commonly used groundwater flow modelling programs are MODFLOW (and its commercial graphical user interface GMS), MODFLOW SURFACT and FEFLOW. MODFLOW and MODFLOW SURFACT use FDMand FEFLOW uses FEM. Multiple reactive solute transport modelling can be made with MT3DMS program coupled with MODFLOW. PHREEQC is code for geochemical reaction and transport modelling.

## Appropriate applications

In order to use mathematical modelling, simplifications are needed. The continuity laws themselves are only theories describing the ‘reality’. In addition, spatial and temporal discretizations are required for numerical methods and the properties of the modelled area have to be averaged because it is impossible to know the accurate properties in every point of the domain. Some simplifications have to be made but usually simplifications are made in order to achieve a compromise between the accuracy of the model and the computational cost (Diersch 2014).

Mathematical modelling is also quite inexpensive method to study groundwater flow. Mathematical modelling itself can be quite inexpensive but without any or limited data (hydraulic conductivity, groundwater level, etc.) about the mine site, the modelling is useless. If enough data isn’t available from the mine site it can be expensive to gather this data with measurements.

In groundwater modelling some information for example about the boundary conditions and the properties of the model domain is required. Necessary information is applied to the mathematical model of groundwater flow and mass transport. As a result e.g. hydraulic head levels, flow fluxes, distribution of potential contaminants and water balance can be computed. With this information it’s possible to evaluate for example the environmental risks in certain areas.

In the case presented by Wels et al. (2012), groundwater flow modelling was used to simulate discharge into the horizontal underground mine, groundwater flow and quality during the mining and after the closure. Drinking water sources and fish-bearing habitat were near the mine site and thus groundwater flow field and transport modelling were essential for this site. Modelling scope was 10 years of mine life and 1000 years post-mining. MODFLOW, MODPATH and MT3DMS were used as modelling codes. After the closure transient modelling simulations were used for assessing the infilling rate of the mine and changes in flow field as a result of the mine flooding. Transport modelling of possible contaminants was also transient. Simulation results were used to evaluate effects of mining and closure to the water levels and concentrations of contaminants in the supply wells and surface water.

## Performance

With the assumption that commercial groundwater modelling tools are used it is fairly simple for experienced user to implement model geometry, material properties and other parameters into the model. Commercial programming tools are also restrictive due to usually poor customization options for built in models. Modern modelling tools are quite extensive and versatile thus built in models are usually suitable for modelling many situations. Accuracy of the mathematical model depends on the spatial and temporal discretization and the complexity of the model. These properties directly affect the computational cost. The more computational capacity the tool has, the greater the accuracy in shorter time.

## Design requirements

In order to use mathematical modelling, information is needed about the mine site. For example in groundwater flow modelling domain geometry, boundary conditions and domain material properties (e.g. hydraulic conductivity) are needed for modelling. At least a powerful PC-computer and usually a commercial modelling program for a certain field are essential for modelling. A modelling professional is also needed. The use of modelling tools would be too challenging for non experts. Groundwater flow mathematical modelling is technically very mature. FEM has been used since 1950’s for various purposes and FDM even before that (Diersch 2014). Also multiple commercial FEM and other numerical method modelling tools for different fields prove the maturity of mathematical modelling.

## References

Wang, H.F. & Anderson M.P. 1982. Introduction to groundwater modeling, Academic Press. 237 pages.

Diersch, H-J.G. 2014. FEFLOW Finite Element Modeling of Flow, Mass and Heat Transport in Porous and Fractured Media, Springer. 996 pages.

Hämäläinen, J. & Järvinen, J. 2006. Elementtimenetelmä virtauslaskennassa, CSC – Tieteellinen laskenta Oy. 215 pages.

Wels, C., Mackie D. & Scibek, J. 2012. Guidelines for Groundwater Modelling to Assess Impacts of Proposed Natural Resource Development Activities, Report no. 194001 for the British Columbia Ministry of Environment Water Protection & Sustainability Branch, Robertson GeoConsultants Inc. & SRK Consulting Inc. Canada. 385 pages. http://www.env.gov.bc.ca/wsd/plan_protect_sustain/groundwater/groundwater_modelling_guidelines_final-2012.pdf, 11.2.2015