Groundwater Flow in Porous Media
Abdallah Abdel-Hafez
Abstract
Groundwater flow modeling 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
 I   P  O   L   A
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 Error! Reference source not found. simplifies to
 I  O   A
Ground water flows in response to head gradients in accordance with Darcy’s law:
h
q  K
x
where q is the flux and K is the hydraulic conductivity (LT-1).
Incorporating Darcy’s law into flow balance equation, we have


h
h 
h
h 
h
K
K
 xb   Sxy
 K
 yb   K
x x x
x x 
y y  y
y y 
t


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 Txy and by -1

h 
 h
h   h





 x x  x x x    y y  y y y   S h
x
y
T t
We are taking the gradients of the gradients, and if we shrink x and y to differential
size we have:
 2 h  2 h S h


x 2 y 2 T t
The Initial Condition will specify the time 0 heads. So, we can approximate the derivative
∂h/∂t by h/t or
h h hx , y ,t  hx , y ,t t


t t
t
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.
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