An experimental investigation
on flow through a permeable
structure
Chen-Yuan Chen1,Cheng-Wu chen2,and Chung-Pan Lee 1
1
Department of Marine Environment and Engineering, National
Sun Yat-sen University Kaohsiung, 80424 Taiwan, R.O.C.
2
Department of Civil Engineering, National
Central University, No. 38, Wu-chuan Li
Chung-li, Taoyuan, Taiwan 320, R.O.C.
ABSTRACT
This paper describes an experimental study in which flow
through permeable structure of porous media under a constant
head permeability device. Porous characteristics such as the
intrinsic permeability and resistance characteristics in porous
media with different grain sizes and shapes have been studied
experimentally in this paper. Two materials used in the study are
well-sorted gravels and uniform polycarbonate spheres with
four D60 (60% passing size through the sieve) of 5.2mm,
7.4mm, 12.2mm, and 31.7mm for gravels and diameters of
13.0mm, 16.0mm, 25.0mm, and 35.0mm for spheres,
respectively. The intrinsic permeability concluded from this
study shows a strong dependence on flow velocity. This is
inconsistent with the traditional definition of intrinsic
permeability that is only a function of grain size which is
determined by linear Darcy Law. On the other hand, if the
resistance force in porous media is quantified by the sum of
linear frictional drag and nonlinear form drag the turbulent
friction coefficient shows a similar trend as that of Shields curve.
The authors analyze the experimental data using a experienced
equation and compare with a newly derived equation. Results
indicate that the intrinsic permeability changes significantly as
the grain size, shapes, and Reynolds number vary and are not
constant value in an approximately uniform flow field. Results
also show that resistance characteristic is a function of the
Reynolds number and it decreases with increases of porosity.
KEY WORDS: porous media, intrinsic permeability, turbulent
coefficient, uniform polycarbonate spheres, convergence,
divergence, and porosity.
1. INTRODUCTION
With the development of international trade and transportation
in recent decades, many harbors around the world have had
vertical concrete quay-walls for ship berthing and shipment

This paper is supported in part by the National Science Council of
Taiwan, R.O.C. under projects no’s- NSC 88-2211-E-018 and NSC
91-2211-E-008-038.
handling. These structures are designed based on geo-technical
and structural stability at a specified load level and risk
measurement. Field measurements indicate that this kind of
quay-wall reflects about 85% of the incident wave energy back
to the harbor basin. How to reduce the unnecessary oscillations
in the harbor and improve the tranquility has become a most
desirable tank. The building along coastlines in the harbor of
various permeable structures for the energy dissipation of inner
harbor waves, such as perforated caisson-type seawalls,
crib-type breakwater, piled barriers, and slotted vertical walls,
has been suggested. It is required that reflection from these
structures be minimized. As a result of the requirements for
effective seawall structures, theoretical models involving
wave-energy dissipation have been developed (Evans, 1990;
Evans and Linton, 1991).
The objective of many papers is focused on the experimental
studies of the reflection characteristics of porous seawalls.
Wave reflection and transmission through a permeable structure
has been studied widely, especially after the theoretical study on
a crib-type breakwater by Sollitt and Cross (1972). Seawalls
with porous toe, caissons on rubble foundation, rubble-mound
breakwaters, and submerged porous breakwaters are some types
of typical permeable structures. Permeable structures protect
lee-side wave attack by reflecting and dissipating wave energy
through the viscosity-induced resistance in the porous media.
Submerged breakwaters in addition may trigger the early
breaking of incident waves and dissipate most of the energy.
Because of the submergence of the breakwater, its application
on protecting coastal area may attract more attention for
environmental concern. The reflected waves and the dissipated
wave energy are strongly affected by water depth, wave
properties such as period and height, and structure properties.
The major properties of permeable structures are porosity,
intrinsic permeability, size distribution and shape function of
the components of the porous media.
These factors also determine the stability of the structures,
reflection and/or transmission, wave run-up and run-down, etc,
under wave attack in a given water depth. A study to optimize
the cost of constructing a secure and efficient structure is
usually needed. This requires an understanding and prediction
of the flow inside and outside the structure and the force
distribution on the structure under given wave conditions.
Unfortunately, at the present time, only very simplified structure
geometries (e.g., crib-type breakwaters: Sollitt and Cross, 1972)
can be examined analytically. And most of the numerical
methods have been developed only for specific structures.
Therefore, many design problems concerning characteristics
inside porous media are still solved by empirical equations such
as Hudson equation or by experimental tests.
Owing to the importance of the factors, researches about
characteristics including porosity, grain-size and shape are
demanded immediate attention. At the present time, only very
simplified empirical equations concept suggested by Ward
(1964), Dinoy (1971), and Sollitt and Cross (1972) have been
enormously useful in analyzing both intrinsic permeability and
resistance coefficient. According to previous results, it is shown
that the permeability scales directly proportional to the square
of the length ratio, and the turbulent coefficient are the same in
similar materials (Ward, 1964; Sollitt and Cross, 1972). The
parameter of intrinsic permeability plays a crucial role in many
of the calculations and formula of the hydraulic engineering
field that deal with fluid flow through porous media. Although
this permeability factor and the turbulent coefficient have long
been accepted as a constant for any particular media,
experiments have shown that they do vary and are not constant
as was previously assumed (Cherry and Freeze, 1979).
The study of the convective transport properties of fluid in
porous media is a fundamental issue both theoretically and
technologically. Many different models have been developed to
define a suitable set of geometrical parameters able to describe
the complex interplay between geometry, connectivity and
transport properties. Fluid flow in porous media takes place
within complicated boundaries that makes the rigorous solution
of the Navier-Stokes equations practically impossible. Frisch et
al. (1986) introduced a cellular automaton fluid model to solve
the Navier-Stokes equations numerically. When flow through a
porous medium is in the regime of low Reynolds number,
Darcy's Law is often applicable (Dullien, 1979). This law is
given by Q  ( K /  ) P / L . This linear relationship,
between the volumetric flow rate per unit area Q and the applied
pressure P divided by the length L of the sample, is used
to define the permeability K of the medium, provided that the
dynamic viscosity  of the fluid is known. A property of
porous systems frequently considered is the porosity  that is
the volume fraction of void space. Some effort has been made to
elucidate the nature of the exponent for the power law that
relates the permeability to the porosity, in some simple porous
systems.
2. THEORY
2.1 Deriving a Empirical Equation
Porous media are modeled by the addition of a momentum
source term to the standard fluid flow equations. The source
term is composed of two parts, a viscous loss term (Darcy), and
an inertial loss term
3
3
j 1
j 1
S i   Dij u j  Cij
1
uj uj
2
Si is the source term for the i-th (x, y, or z) momentum
equation, and D and C are prescribed matrices. u j are the
where
factor, simply specify D and C as diagonal matrices
with 1 / K and C2, respectively, on the diagonals (and zero for
the other elements).
In laminar flows through porous media, the pressure drop is
typically proportional to velocity and the constant C2 can be
considered to be zero. Ignoring convective acceleration and
diffusion, the porous media model then reduces to Darcy's Law
given by  p  v / K . Where p is the pressure gradient,
 the fluid viscosity, K the permeability, and v the
superficial velocity. The equation is known as Darcy's law. In
an external force field, Darcy's law may be extended as
 p  f  v / K . Forchheimer (1901) was the first to
point out that the departure of predictions by Darcy's law from
measurements may be due largely to the kinetic effect of fluid
which is not included in the models for small Reynolds number
flows. For this reason, he suggested that a term representing the
u 2 , be included in Darcy's Law
 p  v / K  au 2 . Accordingly, the
kinetic energy of fluid,
equation, i.e.,
parameter a is called the Forchheimer constant or parameter.
Ward (1964) had studied the expression for Forchheimer
constant, and the most widely used expression for it is that
given by a  C E / K , where C E is the so-called Ergun
constant. Although the Ergun constant is dimensionless, it is not
a universal constant and is often found to vary with changes in
porosity and structure of the porous medium. The Ergun
equation is often written in the vector form in the literature as
(Nield and Bejan, 1992)
1/ 2
 p 

K
uj 
CE
 uj uj
K1/ 2
Empirically, skin friction is proportional to u j , and the form
drag is proportional to u j u j . The equation was established by
Ward (1964) for large porous media in steady flows. The virtual
force is proportional to du j / dt . Thus, a permeable structure is
considered to be anisotropic but homogeneous. The equation of
motion for a steady, non-convective flow in large grain
(1)
permeable media can be written as (Sollitt and Cross, 1972)
Sj
Cf2
1 


( p  gz )  u j  1 / 2 u j u j
dt
  xj
k
k
du j
velocity components in the x, y, and z directions, and  is the
density of water. This momentum sink contributes to the
pressure gradient in the porous cell, creating a pressure drop that
is proportional to the fluid velocity (or velocity squared) in the
cell.
where j =1,2 in a two dimension flow field, g is acceleration due
to gravity,  is the kinematic viscosity, C f is the
To recover the case of simple homogeneous porous media
called the virtual mass coefficient. The form drag and the virtual
force are similar to those in the Morison equation (Steimer and
(2)
Sollitt, 1978). The viscous drag forces dissipate energy. On the
other hand, the virtual force (also called inertia drag) can be
considered to contribute to the kinetic energy of an added mass
Si 

K
ui  C2
1
 ui ui
2
where K is the permeability and C2 is the inertial resistance
dimensionless turbulent coefficient, k is intrinsic permeability,
x1= x, x2= z where the origin of the coordinate system is at the
still water level, u1=u, u2=w, and p is the pressure. S j is
that does not dissipate energy.
2.2 Deriving a New Equation
Consider flow field within a multi-layered anisotropic but
homogeneous permeable structure, it is shown as Fig. 1.

Force 
Pr essure  d  Area 
Area
 (stagnation pressure) × (frontal area) 
2
A0 u j / 2
C D equal to the factor of proportionality, then leads
2
to FD  C D Au / 2 , where C D is a dimensionless drag
Let
coefficient, which is function of body shape and turbulent state
(Reynolds number). A0 is frontal area, perpendicular to u.
This result in
 LA1    
2
FD  C D u j u j / 2  
  D / 4
3
 D / 6 
3 C D 1   u j u j

LA
4
D

In a flow field of uniform flow speed along the streamlines,
if streamlines diverge, divergence is occurring; if the
streamlines converge, convergence is taking place. Normally,
the convergence and divergence components are combined
inside porous media. Therefore, a fluid moving through a body
shown as figure above experiences a drag force, which is
usually divided into two components: frictional drag
(sometimes called viscous drag) and form drag (sometimes
called pressure drag). The corresponding frictional drag is in the
form as given by Newton’s viscosity law as     dv / ds .
The bulk volume of a permeable structure is the sum of the
volume of grains and pores, and the solid volume is the volume
of grains. Consequently, Bulk Volume U b = Solid Volume U v
+ Volume of Pores, and the number of grains = Solid
Volume U v ÷ volume of each grain (Particle global volume).
They are given as the following
Solid Volume U v = cross sectional area A of the meterial
sample × length L of the pore media through which flow occurs.
3
Vs = D / 6
Particle global volume = spherical volume
The surface force can then be expressed as
Surface force = (Dimensionless sheer coefficient C ) × (Sheer

3. CONCLUSION
This experiment has been discussed before, as the porous
pressure is larger in small-sized grain, which leads to larger
differences in water head, and thereby puts a limit on the
Reynolds number of materials with small-sized particles. This
experiment had overcome the problem of the Reynolds number
and using simple experimental data to find relevant information.
It is hoped that we can further understand the porous material
and the true relationship between intrinsic permeability and the
drag coefficient. However, the real data on two other  , where
porosity  =0.32, 0.49 and 0.12, still await analysis.
Therefore, the authors have continued to step up the pace on
further research and discussion of this problem so that we can
find an appropriate relationship model.
ACKNOWLEDGEMENT
The authors wish to acknowledge the support of National
Science Foundation, R.O.C., under Grant Number
NSC89-2611-E-110-019 and Mr. W.Q. Ker, National Sun
Yat-sen University, for technical advice.
stress) × (Particle number) × (Sphere volume)
 u   LA1    
2
F  C   j   
  D
3
 RH   D / 6 
C 1   u j
 6
LA
RH D
where R H is hydraulic radius. D is particle diameter,  is
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