Groundwater Flow in Porous Media
Abdallah Abdel-Hafez
Abstract
Groundwater flowmodeling is one of the important topics in engineering and geosciences. Groundwater flow can be modeled using partial differential equations that can be solved in different numerical solutions; among them is finite element method (FEM).
In this study, a simple groundwater flow model has been introduced using COMSOL Multiphysics (FEMLAB), which indicated that it is very useful to solve such a problem.
I. Introduction
A Model is the tool for understanding a system and its behavior and for predicting its future response. Mathematical model is the model that takes the form of a set of mathematical equations, involving one or more partial differential equations.
Mathematical models are frequently used in studying groundwater systems, it used to simulate (or to predict) the groundwater flow in porous media.
Groundwater is considered one of the most valuable recourses of fresh water in many countries, figure1 shows the components of the hydrological cycle, in which the groundwater is one of the main parts. But unfortunately, this precious source is subjected to many environmental problems, affecting both the quantity and quality. Groundwater can be extracted from aquifers through pumping wells and supplied for domestic use, industry and agriculture.
With increased withdrawal of groundwater, the quality of groundwater will be continuously deteriorating and the aquifers will be depleted.
Studying the process of groundwater flow in subsurface systems using numerical simulation has become a widely practiced nowadays, among these is COMSOL Multiphysics (FEMLAB) which is a finite element analysis and solver software package for various physics and engineering applications.
Figure 1. The hydrological cycle and groundwater flow (After Winter 2001)
II. Governing Equations
The fundamental mass balance equation is
where: I = inputs, P = production, O = outputs, L = losses, A = accumulation
A non-steady state (or transient) ground water system has changes in water storage with time (water accumulation or removal); hence A ≠ 0. Let’s ignore Production/Loss.
So ( 1 ) simplifies to
Ground water flows in response to head gradients in accordance with Darcy’s law:
where q is the flux and K is the hydraulic conductivity (LT-1).
Incorporating Darcy’s law into flow balance equation, we have
The product of the hydraulic conductivity K and the aquifer thickness b is known as the transmissivity T and has units of L2T-1. Both K and b are considered constant here. Incorporating T, dividing both sides by Txy and by -1
We are taking the gradients of the gradients, and if we shrink x and y to differential size we have:
The Initial Condition will specify the time 0 heads. So, we can approximate the derivative ∂h/∂t by h/t or
III.Discussion (model formulation,solution, parametric study and validation)
Femlab was used to solve a simple problem of groundwater flow in porous media, using Darcy law and Diffusion mode.
The Boundary conditions were set as the following:
Insulation (No flow)
Insulation (No flow)
The subdomain setting was as follows:
The output and the post processing of the Flux is shown in the following figures, which represent the flux distribution in the geometry, which decreases with time.
Changing the boundary conditions to be inflow from the left and no flow for the rest, the output shown as below:
Also changing the subdomain setting of the Hydraulic conductivity parameter (k) resulted in changing the flow patterns as shown below.
Model validation aims to determine how well the mathematical representation of the processes describes the actual system behavior in terms of the degree of correlation between model calculations and actual measured data.
The best way to validate any model is to compare the calculated data obtained by the model with actual experimental data.
VI. Conclusions
Flow may go from lower to higher pressure or from lower to higher elevation: but it always goes from higher to lower potential.
FEMLAB represents a good tool for modeling the groundwater movement and flow.
Groundwater flow models can be used to:
- Determine the total volume that may be withdrawn annually from the aquifer.
- Make Decisions related to groundwater quality.
- Study the effect of Agricultural activities such as the use of fertilizers and pesticides
- Identify the best potential locations for future well installations and show how the pumping of one or more wells will affect the other wells in the aquifer and surface water bodies.
V. References
Domenico, P., Scwartz, F. Physical and chemical hydrogeology, (1998), 2nd Edition, Wiley.
P. P.Wang and Z. Chunmaio, An efficient approach for successively perturbated groundwater models, Adv Water Res (1998) 21 499-508.
M. Bakker, Simulating groundwater flow in multiaquifer systems with analytical and numerical Dupuitmodels, J Hydrol (1999) 222, 55-64.
P. W. France, Finite element analysis of threedimensional groundwater flow problems, J Hydrol, (1974) 21, 381-398.
Anderson, M. P., and W. W. Woessner, Applied Groundwater Modeling, Simulation of Flow and Convective Transport, Academic (1991)San Diego, Calif.
H. Karahan and M. T. Ayvaz, Transient groundwater modeling using spreadsheets, Adv Eng Software (2005) 36, 374-384.
FemLab User Guide. ComSol. (Available online:
Abdulla FA, Al-Khatib MA, Al-Ghazzawi ZD, Development of groundwater modeling for the AZROQ basin. Environ Geol (2000) 40(1/2):11–18
H. Karahan and M. T. Ayvaz ,Time-Dependent Groundwater Modeling Using Spreadsheet, Computer applications in engineering education 13 (2005) ,192 -199
Winter TC, The concept of hydrologic landscapes. J Am Water Resources Assoc (2001), 37:335–349
Heinl M, Brinkmann PJ, A groundwater model of the Nubian aquifer system. Hydrol Sci J (1989), 34:425–447