Basics of GroundWater Modelling Part 3

Basics of GroundWater Modelling Part 3

*Model Calibration

After the first run of a model, model results may differ from field measurements. This is expected because modelling is just a simplification of reality and approximations and computational errors are inevitable. The process of model calibration is aimed at fine-tuning the model results to match the

measurements in the field. In a groundwater flow models, the resulting groundwater head is forced to match the head at measured points. This process requires changing model parameters (i.e. hydraulic conductivity or groundwater recharge) to achieve the best match. The calibration process is important to make the model predictive and it can also be used for inverse modelling.

*Model Verification and Validation

The term “validation” is not completely true when used in groundwater modelling. Oreskes et. al. (1994) asserted it is impossible to validate a numerical model because modelling is only approximation of reality. Model verification and validation is the next step after calibration. The objective of model validation is to check if the calibrated model works well on any dataset. Because the calibration process involves changing different parameters (i. e. hydraulic conductivity, recharge, pumping rate etc.) different sets of values for these parameters may produce the same solution. Reilly and Harbaugh (2004) concluded that good calibration did not lead to good prediction. The validation process determines if the resulting model is applicable for any dataset. Modellers usually split the available measurement data into two groups; one for calibration and the other for validation.

*Sensitivity Analysis

Sensitivity analysis is important for calibration, optimisation, risk assessment and data collection. In regional groundwater models, there are a large number of uncertain parameter. Coping with these uncertainties is time-consuming and requires considerable effort. Sensitivity analysis indicates which parameter or parameters have greater influence on the output.

Parameters with high influence on model output should get the most attention in the calibration process and data collection. In addition, the design of sampling location, and sensitivity analysis can be used to solve optimisation problems.

The most common method of sensitivity analysis is the use of finite difference approximations to estimate the rate of change in model output as a result of change in a certain parameter. The Parameter Estimation Package “PEST” uses this method (Doherty et. al. 1994).

Some other more efficient methods of sensitivity analysis have been used. Automatic differentiation has been used for sensitivity analysis in groundwater models and it produces precise output compared to finite difference approximations (Baalousha 2007).

*Uncertainty Analysis

Uncertainty in groundwater modelling is inevitable for a number of reasons. One source of uncertainty is the aquifer heterogeneity. Field data has uncertainty. Mathematical modelling implies many assumptions and estimations, which increase the uncertainty of the model output (Baalousha and Köngeter 2006). There are different approaches to incorporate uncertainty in groundwater modelling. The most famous approach is stochastic modelling using the Monte Carlo or Quasi Monte Carlo method (Kunstmanna and Kastensb. 2006: Liou, T. and Der Yeh, H. 1997). The problem with

stochastic models is that they require a lot of computations, and thus they are time consuming. Some modifications have been done on stochastic models to make them more deterministic, which reduces computational and time requirements. Latin Hypercube Sampling is a modified form of Monte Carlo Simulation, which considerably reduces the time requirements (Zhang and Pinder 2003).

*Common Mistakes in Modelling

A major mistake in modelling is conceptualisation. If the conceptual model is incorrect, the model output will be incorrect regardless of data accuracy and modelling approach. A good mathematical model will not resurrect an incorrect conceptual model (Zheng and Bennet, 2002).In all models, it is necessary to identify a certain reference elevation for all head so that the model algorithm can converge to a unique solution (Franke et. al. 1987). Boundary conditions should be treated with care, especially in a steady state simulation. Sometimes boundary conditions change during simulation and become invalid. A model with hydraulic boundary conditions will be invalid if stresses inside or outside the model domain cause the hydraulic boundaries to shift or change. Therefore, boundary conditions should be monitored at all times to ensure they are valid. Model parameterisation is a common mistake in modelling. Theoretical values of hydraulic properties or groundwater recharge should never substitute field data and field investigation. Assumptions like isotropy and homogeneity should not be used without support from field investigation. Selection of the model code is important to obtain a good solution. Different codes involve different mathematical settings that suit a certain problem. The selected code should consider characteristics of the area of interest and the objectives of modelling. Models can be well calibrated and match well with the measured values, but have an

incorrect mass balance. This can be a result of an improper conceptual model.

End of Part 3

Basics of GroundWater Modelling Part 2

Basics of GroundWater Modelling Part 2

*Types of Models

There are different types of models to simulate groundwater movement and contaminant transport. In general, models can be classified into three categories: physical, analogue and mathematical models. The latter type can be classified further depending on the type of solution.

1-Physical Models

Physical models (e.g. sand tanks) depend on building models in the laboratory to study specific problems of groundwater flow or contaminant transport. These models can demonstrate different hydrogeological phenomena like the cone of depression or artesian flow. In addition to flow, contaminant movement can be investigated through physical models. Though they are useful and easy to set-up, physical models cannot handle complicated real problems.

2-Analogue Models

The equation which describe groundwater flow in isotropic homogenous porous media is called the Laplace’s Equation . This equation is very common in many applications in physical mathematics like heat flow, and electricity. Therefore, comparison between groundwater flow and other fields where the Laplace equation is valid, is possible. The most famous analogue model is the flow of electricity. The electric analogue is based on the similarity between Ohm’s law of electric current flow and Darcy’s law of groundwater movement. As electric current moves from high voltage to lower voltage, so does the groundwater, which moves from high head to lower head. Simple analogue models can easily be setup to study the movement of groundwater flow. More detailed information on analogue models is available (Verruijt, 1970, Anderson and Woessner, 1992, Strack 1989; Fetter 2001).

3-Mathematical Models

Mathematical models are based on the conceptualisation of the groundwater system into a set of equations. These equations are formulated based on boundary conditions, initial conditions, and physical properties of the aquifer. Mathematical models allow an easy and rapid manipulation of complex models. Once the mathematical model is set-up, the resulting equations can be solved either analytically, if the model is simple, or numerically.

*Types of Model Solutions

As discussed in the preceding sections, the mathematical models can be solved either

analytically or numerically. Some approaches use a mixture of analytical and numerical

solutions. The following sections briefly discuss the main types of solutions used in

groundwater modelling.

1-Analytical Solutions

Analytical solutions are available only for simplified groundwater and contaminant transport problems. They were developed before the use of numerical models. The advantages of analytical solutions are that they are easy to apply and produce continuous and accurate results for simple problems. Unlike numerical solutions, analytical solutions give a continuous output at any point in the problem domain. However, analytical solutions make many assumptions like isotropy and homogeneity of an aquifer, which are not valid in general. Analytical solutions; therefore, cannot deal with complex groundwater systems. Examples of analytical solutions are the Toth solution (Toth, 1962) and Theis equation (1941). More details on analytical solutions of groundwater problems can be found in Bear (1979) and Walton (1989).

2-Numerical Solutions

Because analytical solutions of partial differential equations (PDE) implies many assumptions, simplifications and estimations that do not exist in reality, they cannot handle complicated real problems. Numerical methods were developed to cope with the complexity of groundwater systems. Numerical models involve numerical solutions of a set of algebraic equations at discrete head values at selected nodal points . The most widely used numerical methods are finite difference and finite element methods. Other methods have been developed, such as the boundary-element method

3-Finite difference method

Finite difference method (FDM) has been widely used in groundwater studies since the early 1960s. FDM was studied by Newton, Gauss, Bessel and Laplace (Pinder and Gray 1977).

This method was first applied in petroleum engineering and then in other fields. The finite difference method depends on the estimation of a function derivative by a finite difference

4-Finite Element Method

The basis of the finite element method is solving integral equations over the model domain. When finite element method is substituted in the partial differential equations, a residual error occurs. The finite element method forces this residual to go to zero. There are different approaches for the finite element method. These are: basis functions, variational principle, Galerkin’s method, and weighted residuals. Detailed description of each method can be found in Pinder and Gray (1970). Finite element method discritises the model domain into elements. These elements can be triangular, rectangular, or prismatic blocks. Mesh design is very important in the finite element method as it significantly affects the convergence and accuracy of the solution. Mesh design in the finite element method is an art more than a science, but there are general rules for better mesh configuration. It is highly recommended to assign nodes at important points like a source or sink, and to refine mesh at areas of interest where variables change rapidly. It is better to keep the mesh configuration as simple as possible. In the case of triangular mesh, a circle intersecting vertices should have its centre in the interior of the triangle.

End of Part 2

Basics of GroundWater Modelling Part 1

Groundwater modelling is a way to represent a system in another form to investigate the response of the system under certain conditions, or to predict the behaviour of the system in the future. Groundwater modelling is a powerful tool for water resources management, groundwater protection and remediation. Decision makers use models to predict the behaviour of a groundwater system prior to implementation of a project or to implement a remediation scheme. Clearly, it is a simple and cheap solution compared to project establishment in reality.

Modelling Approach

Groundwater Models can be simple, like one-dimensional analytical solutions or spreadsheet models (Olsthoorn, 1985), or very sophisticated three-dimensional models. It is always recommended to start with a simple model, as long as the model concept satisfies modelling objectives, and then the model complexity can be increased (Hill 2006). Regardless of the complexity of the model being used, the model development is the same.

The stepwise methodology of groundwater modelling is shown in Figure 1. The first step in modelling is identification of model objectives. Data collection and processing is a key issue in the modelling process. The most essential and fundamental step in modelling, however, is model conceptualization. Calibration, verification and sensitivity analysis can be conducted after model completion and the first run. The following sections explain in detail each step in groundwater modelling.

Objectives of Modelling

Groundwater models are normally used to support a management decision regarding

groundwater quantity or quality. Depending on the objectives of modelling, the model extent, approach and model type may vary.

Groundwater models can be interpretive, predictive or generic. Interpretive models are used to study a certain case and to analyse groundwater flow or contaminant transport.

Predictive models are used to see the change in groundwater head or solute concentration in the future. Generic models are used to analyse different scenarios of water resource management or remediation schemes.

Objectives of groundwater modelling can be listed as:

• Prediction of groundwater flow and groundwater head temporally and spatially.

• Investigating the effect of groundwater abstraction at a well on the flow regime and

predicting the resulting drawdown.

• Investigating the effect of human activities (e.g. wastewater discharge, agricultural

activities, landfills) on groundwater quality.

• Analysis of different management scenarios on groundwater systems, quantitatively

and qualitatively.

Depending on the objectives of study and the intended outcome, selection of model

approach and data requirements can be made to suit the area of study and the objectives. For example, if the objective is a regional groundwater flow assessment, then a coarse model may satisfy this objective, but if the area of study is small then a fine-grid model with high datadensity should be used.

Conceptual Model

A conceptual model is a descriptive representation of a groundwater system that incorporates an interpretation of the geological and hydrological conditions. Information about water balance is also included in the conceptual model. It is the most important part of groundwater modelling and it is the next step in modelling after identification of objectives. Building a conceptual model requires good information on geology, hydrology, boundary conditions, and hydraulic parameters. A good conceptual model should describe reality in a simple way that satisfies modelling objectives and management requirements (Bear and Verruijt 1987). It should summarise our understanding of water flow or contaminant transport in the case of groundwater quality modelling. The key issues that the conceptual model should include are:

• Aquifer geometry and model domain

• Boundary conditions

• Aquifer parameters like hydraulic conductivity, porosity, storativity, etc

• Groundwater recharge

• Sources and sinks identification

• Water balance

Once the conceptual model is built, the mathematical model can be set-up. The

mathematical model represents the conceptual model and the assumptions made in the form of mathematical equations that can be solved either analytically or numerically.

Boundary Conditions

Identification of boundary conditions is the first step in model conceptualisation. Solving of groundwater flow equations (partial differential equations) requires identification of boundary conditions to provide a unique solution. Improper identification of boundary conditions affects the solution and may result in a completely incorrect output. Boundary conditions can be classified into three main types:

• Specified head (also called Dirichlet or type I boundary). It can be expressed in a

mathematical form as: h (x,y,z,t)=constant

• Specified flow (also called a Neumann or type II boundary). In a mathematical form

it is: Ñh (x,y,z,t)=constant

Head-dependent flow (also called a Cauchy or type III boundary). Its mathematical

form is: Ñh (x,y,z,t)+a*h=constant (where “a” is a constant).

In addition to the above-mentioned types there are other sub-types of boundaries. These will be explained later.

In groundwater flow problems, boundary conditions are not only mathematical constraint, they also represent the sources and sinks within the system (Reilly and Harbaugh 2004).

Selection of boundary conditions is critical to the development of an accurate model (Franke et. al. 1987).

It is preferable to use physical boundaries when possible (e.g., impermeable boundaries, lakes, rivers) as the model boundaries because they can be readily identified and

conceptualised. Care should be taken when identifying natural boundaries. For example groundwater divides are hydraulic boundaries and can shift position as conditions change in the field. If water table contours are used to set boundary conditions in a transient model, in general it is better to specify flux rather than head. In transient simulation, if transient effects (e.g. pumping) extend to the boundaries, a specified head acts as an infinite source of water while a specified flux limits the amount of water available. If the groundwater system is heavily stressed, boundary conditions may change over time. For this reason, boundary conditions should be continuously checked during simulation.

End of part 1