Kouli_etal_2008_Groundwater modelling_BOOK.pdf - Pantelis ...

Fundamentals of Groundwater Modelling 153
• 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.
3.1. Boundary Value Problem
Mathematical models are all based on the water balance principle. Combining the mass
balance equation and Darcy’s Law produces the governing equation for groundwater flow.
The general equation that governs three-dimensional groundwater steady-flow in
isotropic, homogeneous porous media is:
2 2 2
∂ h ∂ h ∂ h
+ +
2 2 2
∂x
∂y
∂z
= 0
Equation (1)
where h is the groundwater head. This equation is also called the Laplace equation and it has
many applications in physics and hydromechanics. Solving Equation (1) requires knowledge
of boundary conditions to get a unique solution. For this reason, Equation (1) is called a
boundary value problem. So the boundary conditions delineate the area or the domain where
the boundary value problem is valid.
Box 1: Conceptual model: questions to answer
• Is there enough hydrogeological data to describe the geometry of the aquifer/s in
the area of study
• Should the model be one, two or three-dimensional
• Is the aquifer/s homogeneous isotropic
• What are the sources and sinks
• What are the sources of contamination (if applicable)
• Do the boundaries stay the same over time
3.2. 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: ∇h (x,y,z,t)=constant