THE 4th INTERNATIONAL CONFERENCE ON THEORETICAL AND APPLIED PHYSICS (ICTAP-2014)
16-17 October 2014, Denpasar-Bali, Indonesia
SPECIFIC SOLUTIONS GROUNDWATER FLOW EQUATION
Muhammad Hamzah Syahruddin
Geophysics, Physics Department, Hasanuddin University
Jl. Perintis Kemerdekaan Km. 10 Makassar, Indonesia.
e-mail : hamzah@fmipa.unhas.ac.id
Abstract. Groundwater flow under surface, its usually slow moving, so that in laminer flow
condition can find analisys using the Darcy’s law. The combination between Darcy law and
continuity equation can find differential Laplace equation as general equation groundwater flow
in sub surface. Based on Differential Laplace Equation is the equation that can be used to
describe hydraulic head and velocity flow distribution in porous media as groundwater. In the
modeling Laplace equation specific solution using the separation variable method (SVM).
Therefore, distribution hydrolic head and distribution velovity flow modeling in porous media
can showed using the SVM.
Key word : groundwater velocity,hydraulic head, SVM
INTRODUCTION
In mathematics, a partial differential equation
(PDE) is a differential equation that contains unknown
multivariable functions and their partial derivatives.
(This is in contrast to ordinary differential equations,
which deal with functions of a single variable and their
derivatives.) PDEs are used to formulate problems
involving functions of several variables, and are either
solved by hand, or used to create a relevant computer
model.
PDEs can be used to describe a wide variety of
phenomena such as sound, heat, electrostatics,
electrodynamics, fluid flow, elasticity, or quantum
mechanics. This seemingly distinct physical
phenomena can be formalised similarly in terms of
PDEs. Just as ordinary differential equations often
model one-dimensional dynamical systems, partial
differential equations often model multidimensional
systems. PDEs find their generalisation in stochastic
partial differential equations
Empirically flow rate of water in the soil has been
formulated by Henri Darcy in 1856 From Darcy's law
it can be seen the rate of water flow in porous media
such as water infiltration in soil, rock, and others. The
rate of water flow in porous media is proportional to
the constant head permeability and hydraulic gradient.
Darcy's law equation can be seen in the Wesley (2005)
and Tod (1980).
Large flow rate of water in porous media is
determined by the permeability constant. Specifically,
permeability refers to the ability of water to pass
through materials such as silt, sand and clay. The rate
is fastest with sand, the which drains Easily and does
not retain moisture. Drainage is slower and retention is
higher with clay, peat and silt. The structure and
texture as well as other organic elements take part in
raising the rate of soil permeability. Soil with high
permeability increase the rate of water infiltration into
the soil and thus, can reduce the rate of runoff.
In this paper is also an example of modeling that
was conducted to determine how the distribution of
water pressure in the soil of the hydraulic head. And
modeling were carried out to determine how the
velocity distribution of water seepage into the ground
to a homogeneous isotropic medium..
SEPARATION VARIABLE METHOD
(SVM)
Equation flow of water in the soil can be
approximated using the Laplace differential equation
(LDE). The equations is derived from Darcy's law
equation and the law of conservation of mass or
continuity principle. The derived of the Laplace
differential equation Darcy's law and the principle of
continuity can be seen in Syahruddin (2014).
This paper will discuss specific solutions
analytically from the groundwater flow equation is
expressed in the Laplace differential equation (LDE).
THE 4th INTERNATIONAL CONFERENCE ON THEORETICAL AND APPLIED PHYSICS (ICTAP-2014)
16-17 October 2014, Denpasar-Bali, Indonesia
Specific solution of the LDE can be used
analytically the separation of variables method (SVM).
The LDE-specific solutions, obtained by applying a
certain boundary condition on the observed geometry.
Therefore, the analytic solution obtained from the
application of certain boundary condition applies only
to the boundary conditions and not valid for other
boundary conditions .. Thus, any change in the
boundary conditions, the analytical solution was
changed (Kreyzig, 1980).
Simple geometric shapes for review is rectangular
shaped. Geometry terms have certain boundary
conditions. This boundary condition, should be
adapted to the physical conditions approach the real
situation. Boundary condition used is the value of u (x,
y) on the boundary of the terms. Boundary condition is
known as the Dirichlet boundary condition. Geometry
terms can be seen in Figure 1.
is equal to X (x) function multiplied by Y (y) function.
So that the Laplace differential equation can be written
as,
u ( x, y)  X ( x).Y ( y)
u xx  u yy  X "Y  XY "
.................(2)
Equation 2 can be separated variables respectively
to the variables x and y variables. The results of the
separation of each of the variables x and y variables
obtained two equations differensia ordinary. The
results of separation of variables x and y can be seen in
equation 3.
X "X  0
Y "  Y  0
..................(3)
The solution of equation (3) obtained by each of
the functions X (x) and the function Y (y). The
function X (x) and the function Y (y) is obtained by
applying the boundary conditions as in Figure 1 ..
When the function X (x) and the function Y (y) is
substituted into equation (2) the obtained solution is
unique Laplace differential equations. Solution
Laplace differential equations, can be seen in equation
(4)
2 sin nx sin ny 1
0 sin nx f ( x)dx
sinh n
n 1

u ( x, y )  
..............(4)
Equation (4) represent the potential distribution at
coordinates (x, y) in the medium. Equation (4) can be
used to calculate the pressure distribution in the
medium. The pressure distribution in the medium can
be determined by the equation,
  2 sin nx sin ny 1

P( x, y )  g  
sin nx f ( x)dx 

0
sinh n
 n1

Figure 1 Medium seepage of water
...................(5)
To resolve such Dirichlet cases in Figure 1 would
require a two-dimensional differential equations
nLaplace. Two-dimensional Laplace differential
equation is shown in equation 1.
 2u  2u

0
x 2 y 2
................(1)
By applying the method of separation of variables,
the variables x and y can be separated. The trick is, the
solution of differential equations of Laplace, at first
considered that u (x, y) can be separated into, u (x, y)
From equation (4) can be easily derived with
respect to x and y variable. Derivative of equation (4)
of the variable x as a potential gradient, which is the
flow lines in the medium. The derivative with respect
to x variable obtained equation (6) is,
u  2 cos(n x) sinh( n y )

1  cos(n )
x n 1
sinh(n )
................(6)
Equation (6) represent the potential gradient at
coordinates (x, y) in the medium. Equation (6) can be
THE 4th INTERNATIONAL CONFERENCE ON THEORETICAL AND APPLIED PHYSICS (ICTAP-2014)
16-17 October 2014, Denpasar-Bali, Indonesia
used to calculate the distribution of the rate of water
seepage in the medium. Distribution rate of water
seepage in the medium can be determined by
Substituting equation (6) into equation Darcy's law.
Results substituting equation (6) into the Darcy law is,
  2 cos nx sinh ny
1  cos n 
v ( x, y )  K  
sinh
n

 n 1

TABLE 1. Results of Calculation
(x,y)
0,1 , 0,5
0,2 , 0,5
0,3 , 0,5
0,4 . 0,5
0,5 . 0,5
0,6 , 0,5
0,7, 0,5
0,8, 0,5
0,9 , 0,5
u
0,08159
0,15275
0,20634
0,23906
0,25
0,23906
0,20634
0,15275
0,08159
............... (7)
Calculation of potential and potential gradient in
equation (4) and (5) be easily done with a computer
program. The calculation is done using the
programming languages (C ++). The results of the
calculations are presented in Table 1.
du/dx
0,77914
0,63217
0,4344
0,21874
0
-0,21874
-0,4344
-0,63217
-0,77914
du/dy
0,29924
0,54494
0,71267
0,80546
0,83462
0,80546
0,71267
0,54494
0,29924
RESULTS
ACKNOWLEDGMENTS
Seepage of water in the soil can be approximated
by the Laplace differential equation. One of the
solutions that can be used to solve analytically,
Laplace differential equations, is the method of
separation of variables. Analytic solutions, Laplace
differential equations, is a specific solution because it
depends on the boundary conditions used. If the
boundary conditions change then the solution of
Laplace equation change as well.
Potential obtained from the Laplace differential
equations solution can be used to describe the
distribution of water pressure in the soil. While the
derivative of potential as hydraulic head gradient can
be used to obtain the distribution of the flow rate of
water in the soil. A potential gradient flow lines of
water from high pressure to low pressure.
We express our gratitude to the leader Hasanuddin
university for permission and financial assistance to us
in order to present and presentation of papers at the
ICTAP 2014Denpasar Bali Indonesia
.
REFERENCES
Kreyzig, E. (1980) : Advanced Engineering Mathematics,
John Wiley & Sons, 667-668.
Syahruddin, M.H., (2014) : Persamaan Aliran air dalam
Tanah, Prosiding SFN Bali.
Tood, D.K. (1980) : Groudwater Hydrology, John Wiley
and Sons, Inc, New York.
Wesley, D.L., 2005: Mekanika Tanah. Badan Penerbit
Pekerjaan Umum, Jakarta