Exam 3: Soil Physics
due: 12/12
A recent article (attached) analyzes Darcy’s law from angles not covered in lecture. Summarize
the main point(s) and discuss the practical and theoretical implications of the material presented.
Please type your answers in the following format: double space; with 1" margins on top, bottom,
left and right; and12 point type font.
1
Environ. Sci. Technol. 2004, 38, 5895-5901
Examination of Darcy’s Law for Flow
in Porous Media with Variable
Porosity†
W. G. GRAY* AND C. T. MILLER
Department of Environmental Sciences and
Engineering, University of North Carolina,
Chapel Hill, North Carolina 27599-7431
Henry Darcy’s experimental studies in 1856 of saturated
water flow through a homogeneous porous medium contained
in a vertical column have provided the basis for the
quantitative description of fluid flow in a wide variety of
both natural and engineered porous medium environmental
systems. Extrapolation of Darcy’s original observations
and conclusions has led to several commonly applied
equations used to model flow in porous media. This work
examines this original experimental study, summarizes
the appropriate mathematical expressions that ensue directly
from the data, and indicates expressions in common use
that are suggested, but not actually supported, by the data.
The paradoxes that exist in the common approaches for
the case of a porous medium with a spatially variable porosity
are illustrated. A modified form of Darcy’s law, and also
of the Hubbert potential, is derived based upon fundamental
notions of averaging. The modified form of Darcy’s law
derived here reduces to the conventional form for a
homogeneous porous medium.
Darcy’s Experiments
The study of flow in porous media is important in a wide
range of applications including groundwater flow, exploitation of petroleum reservoirs, filter design, manufacture of
composites, capillary circulation, chemical reactors, etc.
Efforts to quantitatively describe flow and transport processes
in porous media date from the experimental studies of Henry
Darcy in 1856 (1, 2). Darcy’s studies involved packing a vertical
column with sand as depicted in Figure 1. Four different
packings were used, each of different height, with the sand
used in each packing differing primarily by the degree of
washing employed. Manometers tapped reservoirs at each
end of the column, and water was allowed to flow through
the packing. The experiments involved steady, single-fluidphase (hereafter designated as single-phase) flow with water
assumed to fill completely the pore space. Despite difficulties
in obtaining a constant water source, the experiments
demonstrated that the volumetric flow rate through the
porous medium (Q) is proportional to the total head loss
across the sand column (h2 - h1) and the cross-sectional
flow area (A) and inversely proportional to the packed height
of the column (L):
h2 - h1
L
Q ) KA
†
(1)
This paper is part of the Walter J. Weber Jr. tribute issue.
* Corresponding author phone: (919)966-3013; fax: (919)966-7911;
e-mail: graywg@unc.edu.
10.1021/es049728w CCC: $27.50
Published on Web 10/09/2004
 2004 American Chemical Society
In this algebraic expression, K is referred to as the hydraulic
conductivity, and it is a function of both the porous medium
and the fluid properties. Darcy found that K was essentially
constant for a particular packing that he employed.
It is important to note that Darcy’s experiments actually
provide no information about any properties within the
packed column. All data was collected at locations external
to the packed column, and Darcy’s algebraic expression
provides effective information for the column as a whole. In
eq 1, the hydraulic conductivity is characteristic of the column
as a whole and provides no indication of the degree of
homogeneity of the packing in the column. Neither the
volumetric flow rate (Q) nor the volumetric flow rate per
area (Q/A) provides any indication of the speed of the water
flowing within the pores. The area (A) is a property of the
column that was packed and does not necessarily relate to
any effective cross-sectional area of flow. The length parameter (L) is the distance between the sampling points of
the manometers and does not indicate the travel distance
for fluid moving through the porous medium, which is
influenced by the tortuous path created by the medium.
Finally, even the head values (h1 and h2) are obtained from
measurements taken in reservoirs outside of the column.
The fact that Darcy’s law is expressed in terms of variables
that may not describe the actual porous medium flow can
be further demonstrated by building an experimental apparatus intrinsically different from that in Figure 1. Consider
the sketch in Figure 2 where the visible, packed column of
Darcy is replaced by an unusually shaped shell that houses
some arbitrary packed, saturated flow system. For example,
this system could be a spiral tube that connects the reservoirs;
it could be a straight column with a small cross-section; it
could be a bent tube or a tube of varying cross-section that
follows some tortuous path between the reservoirs. In fact,
any geometric shape is allowable that can be housed within,
but need not completely fill, the shell depicted. The volumetric flow rate through the system could be measured as
could the heads in the two reservoirs. Then, Darcy’s equation
for saturated flow in this ill-defined region may be expressed
as:
Q ) Keff(h2 - h1)
(2)
Thus the single parameter (Keff) accounts for the effects of
the permeability of the medium and the geometry of the
flow area. Still, no information is obtained about the actual
fluid velocity within the study system. Indeed, a revised
definition of Keff can be used to express eq 2 as:
q ) K′eff(h2 - h1)
(3)
by dividing by some cross-sectional area of the flow. However,
the parameter q with dimensions of length/time provides no
additional insight beyond that of eq 2 into actual flow
velocities. The measured quantities in Darcy’s experiment
are the volumetric flow rate and the difference in the heads
in the reservoirs that bound the study region. These will be
found to be proportional for conditions of slow flow. The
coefficient of proportionality is constant over a range of
specified head differences for a fixed system.
In reporting his experimental results, Darcy was careful
not to overstate their utility and implications. However, the
need for scientists to model flow and transport in porous
medium systems has led to general acceptance of equations
with similarities to Darcy’s eq 1 with the state of the art
reported in references such as refs 3-7. Despite their utility,
VOL. 38, NO. 22, 2004 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
9
5895
The objectives of this work are (i) to examine commonly
used expressions based upon Darcy’s law, (ii) to illustrate
paradoxes inherent in some of these expressions for simple
heterogeneous systems, and (iii) to provide modified forms
of common expressions that apply to both homogeneous
and heterogeneous systems.
Considerations for Hydraulic Conductivity
Because of the geometry of the original apparatus and the
form of the equation put forth by Darcy, it is tempting to
write a differential form in the limit of a very short column
as:
(
)
h 2 - h1
dh
Q
) -K
) q ) - lim K
Lf0
A
L
dL
FIGURE 1. Darcy’s experimental apparatus.
FIGURE 2. Alternative Darcy-like experimental apparatus (1).
these differential equations cannot be justified based on
Darcy’s experimental data. Indeed, if examined carefully,
even the simplest differential form of Darcy’s equation for
single-phase flow in a porous medium is highly restricted.
Subsequent extensions to multiphase flow are even more
problematic (e.g., ref 8). This situation has arisen because
systematic procedures for derivation of equations for flow in
porous media, such as averaging theory, began to develop
approximately 100 years after Darcy’s experiments. The need
to model porous media systems in the intervening years
produced heuristic equations with variables and parameters
that are not precisely related to measurements. However,
our current theoretical understanding makes it possible to
relate experimental measurements to equation variables, thus
facilitating transfer of data between scales.
5896
9
ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 38, NO. 22, 2004
(4)
where a minus sign has been introduced because flow is in
the direction of decreasing head (i.e., negative head gradient).
Based on Darcy’s experimental data, however, there is no
way to determine if the hydraulic conductivity, which is a
characteristic of the full column length, retains its same value
if one looks at a smaller segment of that column. Although
Darcy carefully packed columns of different lengths, he had
no way of examining flow behavior in smaller segments of
a single packed column. His manometers provided information only from the reservoirs at the ends of the column, not
from within the pore space. Based in Figure 2, which depicts
an arbitrary apparatus, it is reasonable to project that some
proportionality relation exists between the head drop and
the flow rate for slow saturated flow. However, the proportionality parameter is expected to vary depending on the
characteristics of the flow region under consideration. Thus,
as has been widely recognized in experimental and mathematical analyses of porous medium systems, the hydraulic
conductivity is generally a function of the length scale of the
system being studied. For a column of constant cross-section
with very uniform packing, K may not vary with the length
of the column until L becomes small, on the order of a few
pore diameters. However, below this length, the heterogeneities in the packing and inherent in the sand will cause the
value of hydraulic conductivity to depend strongly on the
orientation and shape of grains. It will vary so significantly
from point to point and as a function of the length scale that
modeling of averaged system behavior at this scale provides
little information. The concept of a homogeneous differential
length of porous medium is inherently contradictory. For
the more general geometry of Figure 2, similar conceptual
considerations apply.
In addition to our inability to handle an arbitrarily small
element of a medium, observe also that if eq 4 were to be
applied to a general system, the value of K would depend on
the location within the apparatus of the medium element
being considered. Only for the case of a perfectly homogeneous packing would one expect K to be a constant for all
equally sized elements of the porous medium. Efforts to pack
columns for porous medium flow experiments have demonstrated the difficulty in obtaining a truly uniform packing.
Although Darcy did use care in packing his columns, the
chance of the medium truly being homogeneous is small.
Thus the values of hydraulic conductivity he obtained are
characteristic of the full columns that he examined, and
reduction to the differential form of eq 4 with the value of
K being independent of position cannot be justified mathematically.
Length Scales
The scale issues mentioned in the preceding discussion have
been considered by porous medium scientists for decades
(e.g., ref 3). The use of common length-scale notions can aid
this discussion. This discussion is concerned with three
different length scales: the microscale, the macroscale, and
the megascale. Since designation of these scales takes on
different meanings in different disciplines, we will expand
on the meanings intended in this study.
First, the microscalesor smallest length scale discusseds
is a length scale over which the details of the pore morphology
and topology are considered. At this scale the fundamental
equations, such as the Navier-Stokes equations or Boltzmann’s equation, may be applied and solved (e.g., ref 9). In
general, the level of detail needed to consider porous medium
flow at the microscale is not available for non-ideal systems.
So while theoretical work is increasingly relying upon
descriptions at such scales, the field of porous medium
science is often concerned with length scales considerably
above that needed to completely define the geometry of the
pore structure.
The length scale at which Darcy’s law is typically written
and applied in a differential form is termed the macroscale.
The macroscale is a scale well above the microscale or pore
scale, say on the order of 10 or more mean particle diameters.
The porous medium field relies upon conservation equations
developed, closed, and applied at this scale in terms of
quantities that are averaged over this length scale. The
minimum length scale for this approach to have meaning is
often referred to as a representative elementary volume (REV)
scale (3).
Darcy’s experiments relied upon measures of flow and
head loss external to the system and indicative of some
averaged properties of the entire system. We will term this
length scale a megascale, which is the largest of the three
length scales discussed. The point made above through
reference to Darcy’s work and the modified apparatus
suggested by Figure 2 is that at the megascale only averaged
quantities characteristic of the entire system are well defined,
and great care must be taken to extend knowledge based
upon observations of such systems to other systems.
Subsequent efforts to extend Darcy’s work and produce
a differential form of Darcy’s law that is applicable within a
domain become paradoxical unless a point is interpreted as
a macroscale average about an REV. Clearly the length scale
of an REV is much larger than the microscale or pore scale,
and the details of flow at the microscale are not resolved
using Darcy’s law. In terms of the notion of an REV, a
differential form of Darcy’s law is an acceptable notion, since
taking a limit as the length scale tends toward zero merely
implies evaluating the difference in averages over an REV as
the distance between the centroids of equal-sized macroscale
neighborhoods tends toward zero.
However, care must be taken in deriving and defining the
averaged quantities over an REV for use in any macroscale
conservation equation. Differential macroscale conservation
equations and their associated closure relations are used to
describe transport phenomena in porous medium systems.
For scientific and mathematical consistency, precise definition and measurability of quantities that appear in those
equations is required. We show below that attention to
achieving this consistency has been lacking historically even
for the simple case of steady single-phase flow. We also derive
a modified Darcy’s law expression that is physically and
mathematically consistent.
Considerations for Hydraulic Head
The differential equation purportedly describing single-phase
flow in a homogeneous column of porous medium is typically
stated as:
dh
dL
Q ) -KA
(5)
with K considered constant. Although this equation integrates
to the Darcy eq 1 at steady state, this differential form contains
an important complexity related to scale. The equation
proposed by Darcy is in terms of quantities at the ends of
the column (megascale) where no solid phase is present,
whereas eq 5 is expressed in terms of quantities that must
be defined within the porous medium (macroscale) (i.e., in
regions where both fluid and solid are present). At the ends
of the column, only fluid is present. However, at some points
within the column, solid grains occupy the space such that
a differential form for the fluid phase at that microscopic
point would not be relevant. Thus, if one is to use a differential
extension of Darcy’s equation at the macroscale, a definition
of h within the porous medium at the macroscale that is
precisely defined, measurable, and mathematically consistent
across all scales is needed.
In natural groundwater flow systems, measurements of
hydraulic head are made in wells rather than in situ. Thus,
as with Darcy’s experiments, the measurement is made
outside the porous medium system such that the actual
average value of h within the porous medium is not directly
measured.
In fact, if all measurements are made externally to the
porous medium, the quantity h that pertains to the macroscale average within the porous medium and which appears
in the differential equation need never be precisely related
to any measured physical quantity. Additionally, the volumetric flow rate (Q) is not actually measured within a porous
medium. It is inferred by the rate at which water is pumped
from or injected into the medium. For the case of the column
flow as described by eq 5, all of the fluid flow forms the
average flux. However in natural systems, the flow within
the pores is actually three-dimensional as it finds a path
through the system.
Despite the difficulties mentioned, the differential form
of Darcy’s law has proven to be quite useful for modeling
single-phase flow in a porous medium. For single-phase,
three-dimensional flow through an isotropic medium, eq 5
is typically expressed as:
k
j - Fg)
q ) - (3p
µ
(6)
where q is the volumetric flow rate per unit area vector, p
j
is the pressure, F is the fluid density, g is the gravity vector,
µ is the dynamic viscosity of the fluid, and k is the intrinsic
permeability, a property of the solid medium with:
K)
kFg
µ
(7)
where g is the magnitude of the gravity vector.
The intrinsic permeability must be determined for each
porous medium system and depends on the distribution of
the sizes, shapes, and orientations of the pores that provide
pathways for flow within the solid system (i.e., the morphology and topology of the pore space). In eq 6, the hydraulic
head has been replaced by the pressure and gravity terms
with:
Fg3h ) 3p
j - Fg
(8)
In fact, the use of head rather than pressure in eq 6 is only
employed when the density is either assumed to be a constant
or dependent only on pressure. As has been shown by
Hubbert (10), eq 8 may be rearranged to an integral
expression:
∫
h(x)
h0
dh′ )
∫
p(x)
p0
dp′
+
gF(p′)
∫
Φ(x)
Φ0
dΦ′
g
(9)
where x is a position vector, Φ is the gravitational potential
VOL. 38, NO. 22, 2004 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
9
5897
FIGURE 3. Homogeneous two-dimensional experimental apparatus.
with g ) -3Φ. As written, the integration in eq 9 can be
performed to obtain the value of h(x). However, if F is a
function of any variables in addition to density, such as
temperature or composition, the integral cannot be evaluated.
In that instance, the more general form of Darcy’s law as
given in eq 6 is employed; and evolution or conservation
equations for the additional variables are also needed to
model the system.
Despite the fact that eq 6 is the equation of choice in most
single-phase flow applications, it is interesting to note that
the pressure appearing in the equation, a pressure existing
within the porous medium, has not been well defined. It is
some average characteristic of the medium associated with
the macroscale point under consideration, but the precise
definition has not been explicitly considered. The fact that
Darcy’s experiments considered only the regions outside the
porous medium for measurement means that those experiments do not provide any indication of the meaning of this
pressure. However, if one is to derive point equations from
a theoretical perspective, it is necessary to have quantities
that appear in the equations be measurable. This problem
will be demonstrated by an example.
Considerations for Pressure
The experimental apparatus depicted in Figure 3 is a
horizontal rectangular column packed with a homogeneous
isotropic sand. At equilibrium, the water levels in the two
reservoirs at the ends of the column will be equal. Equation
1 is satisfied by this situation with Q ) 0 and h1 ) h2.
Additionally, for this apparatus pressure gages are located,
as indicated, at positions aligned with the vertical center of
the column. The pressure readings of the two gages will be
identical for conditions of no flow. Furthermore, at conditions
of equilibrium, the average pressure calculated for the fluid
within any volume that occupies the entire vertical cross
section of the column will also be equal to the reading at the
gages. This last assertion assumes that the length scale of the
cross section is sufficient to serve as an REV.
Under more general conditions in which flow may occur,
the form of the Darcy equation for this experimental setup
given by eq 6 in the x direction is:
qx ) -
j
k dp
µ dx
(10)
since the gravity vector in the x direction is zero. In this
expression, p
j is the average pressure in the fluid obtained by
averaging the microscale pressure over the volume of fluid
contained within a cross-sectional slice that constitutes a
macroscale REV such that:
p
j)
1
Vf
∫ p dV
Ωf
(11)
where Vf is the volume of fluid, and Ωf represents the
averaging region. Note that p
j can vary with x in that this
volume-averaged pressure can be calculated at any x position
5898
9
ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 38, NO. 22, 2004
FIGURE 4. Heterogeneous two-dimensional experimental apparatus.
along the column. In fact for the case of a homogeneous
medium and assuming a constant density fluid:
p
j (x)
x
B
) h1 + (h2 - h1) Fg
L
2
[
]
(12)
Equation 10 is satisfied at positions within the porous
medium for this volume-averaged pressure. Furthermore,
differentiation of eq 12 with respect to x and elimination of
dp
j /dx between eq 10 and 12 yields:
qx ) -
(
)
Fgk h2 - h1
µ
L
(13)
which is consistent with Darcy’s law. Additionally, the gage
pressures are known as functions of h according to:
h1 )
p1 B
Fg 2
(14a)
h2 )
p2 B
Fg 2
(14b)
and:
Substitution of these expressions into eq 13 yields a flow
equation:
qx ) -
(
)
k p2 - p1
µ
L
(15)
In this form, the difference between the gage pressures is the
driving force for the flow through the column. This equation,
which accurately describes the flow, has a character similar
to Darcy’s expression in that the driving force and the flow
per area describe the column as a whole at steady state but
provides no information of the situation within the column.
Now consider a second experiment using the column
depicted in Figure 4. This column differs from that of the
previous experiment by the presence of a homogeneous layer
of constant thickness that may have porosity and permeability
different from those in the previous experiment. By examination of the flow in the column under these conditions,
some interesting information may be developed. Our discussion will be concerned with a model formulation that is
macroscopic in the direction along the axis of the column
but megascopic in the directions orthogonal to the axis. Thus,
the model will contain gradients in properties along the flow
direction but will be integrated over the flow cross-sectional
area. From this perspective, issues of anisotropy that would
require consideration using a fully macroscale formulation
do not arise.
The equilibrium configuration for this system will be the
same as with the homogeneous packing with h1 ) h2, p1 )
p2, and Q ) 0. For purposes of illustration, we will work with
pressure rather than head. Analysis of experiments performed
on this packed column for slow flows will confirm the
governing equation:
qx ) -
(
)
k* p2 - p1
µ
L
(16)
The medium outside the layer is assumed to be the same
medium that completely fills the domain in Figure 3.
Therefore, k* will be different from k of the original system
when the permeability of the added layer is different from
that of the surrounding medium. For this setup, the value of
k* will not be a function of position along the direction of
flow as each cross-section contains the same fractional area
of each of the two media. Thus, the reasoning that led to eq
10 for the single medium case would seem to apply here as
well with the differential equation taking the form:
qx ) -
j
k* dp
µ dx
(17)
There is no need to employ a tensor form of k*, as would be
appropriate for an anisotropic system, because of the scales
used in formulation of this problem.
Define ∆ ) 2 - 1 where 1 is the porosity outside the
stripe and 2 is the porosity within the stripe. Also, let B be
the height of the flow domain and b be the height of the
stripe. Then at any position x along the column with p
j defined
by eq 11 for the constant density fluid case and assuming
hydrostatic conditions, p
j can be calculated as:
[
]
b
b
2x
11∆
p
j
B
B
L
x
B
) h1 + (h2 - h1) - 1 Fg
L
2
b
1 + ∆
B
(18)
[
]
(
)(
(
)
)
This equation is the same as eq 12 with the addition of
a correction term that accounts for a different average
pressure due to the porosity of the stripe being different
from that of the surrounding medium. The reason for this
correction term is that the nonconstant porosity assigns
different weightings to the pressure variation in the vertical
direction when calculating the average pressure over the fluid
volume. Differentiation of eq 18 with respect to x and
substitution into eq 17 yields:
{
Fgk*
qx ) µ
[ ]}
b
b
1h 2 - h1
B
B B∆
L
L
b
1 + ∆
B
(
) ( )
(19)
This equation differs from Darcy’s law as given by eq 13
by the presence of the last term that accounts for the
difference in porosities of the two solid materials. We can
observe that if the layer has only a different permeability
from the original material but the same porosity, the form
of Darcy’s law is preserved as ∆ ) 0. However, a material
with a different porosity causes the equation to take a different
form. In fact, we know that if h1 ) h2, there will be no flow
in the system. However, eq 19 erroneously indicates that
there will be a flow due to a nonconstant porosity. Note that
this result is obtained even though the porosity of the medium
obtained by integration over its cross section is not a function
of position along the direction of flow.
It is important to examine the factors that contributed to
the nonphysical result of eq 19. The most obvious reaction
is to state that the average pressure as provided by eq 18 is
incorrect. Indeed, if the fact that the fluid occupies only a
portion of the porous medium space is neglected and the
average pressure is calculated by integration without regard
to a porosity distribution, the correction term in the definition
of the average porosity will disappear and eq 19 will reduce
to the expected Darcian form. However, there are at least
two problems with this approach. First, it is important to
work with averaged quantities that are physically based. It
is useful to be able to average physical measurements over
regions of porous media to arrive at new quantities at a larger
scale. Averaging while “pretending” that the solid phase is
not present precludes such a systematic approach. Second,
in the present example, not considering the presence of the
solid phase in doing the averaging leads to the expected result
because of the hydrostatic distribution in the vertical. For a
general fully three-dimensional flow problem, such an
assumption would not be appropriate. Thus, the mathematical formulation of the average pressure, as proposed
in eq 11, is reasonable on physical and computational
grounds. The use of this average pressure in a differential
form of Darcy’s law is where the problem arises. Recall, that
Darcy’s experiment and reported results involved measurements of quantities made outside the porous medium, at the
megascale. These measurements do not capture information
about the pressure distribution within the system. Recognition of the differences between measurements and mathematical forms leads to the resolution of the present paradox.
Properly Formulated Equation
When flow is in the horizontal direction, as in the examples
considered here, it certainly seems reasonable to discard the
gravitational term. If eq 6 is the equation that governs flow
in a porous medium, then the vector component of this
expression in the horizontal direction will not have the
gravitational term. This leads to the conundrum found above
for an inhomogeneous medium. In addition, for any porous
medium, if we average Fg over the fluid volume, for the case
when F is constant, the average will be equal to the point
value, since g is also constant, regardless of the distribution
of porosity. Thus, direct incorporation of gravity into the
flow equation does not solve the problem.
Consider the implications of using the gravitational
potential, 3Φ, in the formulation instead of the gravitational
vector. At the microscale, g ) -3Φ; but after transformation
to the macroscale, it is important to make use of some
averaged potential. The microscale gravitational potential
is:
Φ ) Φ0 + gz
(20)
where Φ0 is a reference constant gravitational potential, and
z is the spatial vertical coordinate. Since the only spatial
dependence of this microscale function is on the vertical
coordinate, the derivative of this function in the net direction
of flow in the example problem is zero:
dΦ
)0
dx
(21)
Following a similar procedure as that used to derive eq
18, we obtain the volume average of Φ over the fluid at a
position along the x direction as:
[
]
b
b
2x
11∆
gB
B
B
L
Φ
h ) Φ0 +
12
b
1 + ∆
B
(
)(
)
(22)
When the porosity is not constant, a correction term appears
that accounts for the unequal weighting of the gravitational
potential through the vertical dimension. Furthermore, the
VOL. 38, NO. 22, 2004 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
9
5899
gradient of this average potential has a nonzero component
in the direction of flow with:
[ ]
b
b
1 - ∆
B
gB B
dΦ
h
)
b
dx
L
1 + ∆
B
(
)
(
(23)
)
Although the microscale gravitational potential does not vary
in the direction of flow, the macroscale potential will vary
if the porosity is not constant. The correspondence between
eq 23 and the correction term in eq 19 is directly observable.
Now consider conditions in the reservoirs at the ends of
the flow region. In a reservoir, the hydraulic head does not
vary with depth. Furthermore, the hydraulic head may be
expressed in terms of the pressure and the gravitational
potential. In the reservoir:
p ) Fg[h - (z - zb)]
(24)
Φ ) g (z - zb)
(25)
within the porous medium is speculative. Megascale data
collected at the exterior of the system is being used to project
macroscale behavior within the system.
The generally employed differential extension of Darcy’s
law given by eq 6 is limited in utility unless the averaged
pressure can be defined. In cases where the porous medium
is homogeneous in porosity, p
j in eq 6 is simply the microscale
pressure averaged over the fluid volume within the averaging
region. In cases where the volume fraction of the fluid within
the averaging region is not constant, inconsistencies may
arise if this volume-averaged pressure is used in eq 6. When
multiphase flow is considered, the distribution of volume
fractions within an averaging volume could cause complications in presenting and applying a consistent flow equation.
The calculations presented to this point have involved
the case of a constant microscale density. If the density is
not constant within the averaging volume, then the average
gravitational potential needed is a mass weighted quantity
with:
where zb is a vertical reference location at the bottom of the
domain. The head is then obtained as:
h)
1
(p + FΦ)
Fg
(26)
This expression for head is for the reservoirs. The appropriate
difference form in terms of pressure that corresponds to
Darcy’s equation, which was written in terms of head, is
then:
(
)
k* (p + FΦ)2 - (p + FΦ)1
qx ) µ
L
(27)
as opposed to eq 15. Thus, the differential form written in
terms of average quantities within the porous medium system
will be:
j + FΦ
h)
k* d(p
qx ) µ
dx
(
)
Fgk* h2 - h1
µ
L
(31)
and:
1
Vf
Fj )
Vf
With these definitions, the extension to the differential form
of Darcy’s law is:
k
q ) - (3p
j + Fj3Φ
h)
µ
(32)
Hubbert (10) derived the relation describing flow in a
porous medium as:
Fk
3H
µ
(33)
where the Hubbert potential (H) is defined as:
(29)
Extensions
The derivation here is intended to indicate the care that must
be exercised in employing the differential form of Darcy’s
law to model flow in porous media. There are several
important observations that can be made.
The experimental work of Darcy involving a porous
medium made use only of data collected outside the column
or at the megascale. Darcy’s law is an algebraic expression
that provides a correlation between flow through the system
and the measured heads at the ends of the system under
study. The actual measured results of Darcy’s experiment
are expressed by eq 2. The extension of this form to eq 1
followed by differentiation to obtain an expression at a point
9
∫ F dV
Vf
(28)
This equation applies at each x location within the porous
medium and is consistent with the equation obtained for
the system as a whole based on measurements outside the
porous medium. Thus, the importance of properly incorporating the averaged gravitational potential into the governing equations is demonstrated.
5900
(30)
q)-
The presence of the averaged potential is a correction to eq
17. Now substitution of the expressions for p
j and Φ
h from eqs
18 and 22 simplifies eq 28 to:
qx ) -
∫ FΦ dV
∫ F dV
Vf
Φ
h )
ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 38, NO. 22, 2004
H)
∫
p(x)
p0
dp′
+ gz
F(p′)
(34)
When F depends on pressure only, this expression may be
differentiated to obtain:
3H )
3p
-g
F
(35)
and a flow equation similar to eq 6 is recovered. However,
if the pressure appearing in this equation is to be an average
explicitly defined in terms of microscale quantities, the
definition of the Hubbert potential should be:
H
h )
∫
p
j (x)
p0
dp′
+Φ
h
Fj(p′)
(36)
This expression makes use of an average pressure as well as
an average gravitational potential, both clearly defined in
terms of microscale counterparts. The form of eq 36 is
equivalent to replacing the microscale elevation coordinate
(z) in eq 34 with the average value of this coordinate within
the fluid in the averaging volume. When the pore space within
the averaging region is distributed uniformly and the density
is constant, the gravitational portions of eqs 34 and 36 are
identical, and the pressure is that obtained from volume
averaging of the microscale pressure.
In many cases of flow in porous media when the variation
in volume fraction is small, the corrections presented here
may also be small. Nevertheless, they are important for
ensuring that the fundamental flow equations being employed are consistent. If one wishes to invoke a simplifying
assumption at some point in the analysis to make use of the
traditional equations, this approximation may be made. In
many instances, the approximation will be a good one.
However, beginning an analysis with unrecognized assumptions built in can lead to faulty and inconsistent models. For
porous medium problems in which change of scale is an
important component of the modeling process, rigorously
defining variables to allow for a consistent change of scales
is an important consideration.
One final disclaimer that may be important is to state
that this analysis has not considered the case where temperature and/or concentration gradients would drive the flow.
Although, in the general form provided as eq 32, Fj may depend
on temperature or concentration, variations in these quantities are not deemed significant enough to provide potentials
for flow. Further examination of this issue awaits a more
detailed analysis than a visitation of the experimental results
of Darcy and the use of his correlation as a starting point for
a more general physical description.
Acknowledgments
The authors are pleased to submit this manuscript in
recognition of the distinguished research career of Walter J.
Weber, Jr. C.T.M. acknowledges the guidance of Professor
Weber early in his career and his encouragement to do
fundamental work; hopefully, this current collaborative work
is some evidence that this message got through. This work
was supported in part by the National Science Foundation
(NSF) Grant DMS-0327896. Partial support of this work was
also provided by NSF through DMS-0112069 to the Statistical
and Applied Mathematical Sciences Institute in Research
Triangle Park, where the initial ideas for this work were
formulated. C.T.M. was also supported in part by Grant P42
ES05948 from The National Institute of Environmental Health
Sciences.
Literature Cited
(1) Darcy, H. Les Fontaines Publiques de la Ville de Dijon; Dalmont:
Paris, 1856.
(2) Darcy, H. Determination of the laws of flow of water through
sand. In Physical Hydrology; Freeze, R. A., Back, W., Eds.;
Hutchinson Ross: Stroudsburg, PA, 1983.
(3) Bear, J. Dynamics of Fluids in Porous Media; Elsevier: New York,
1972.
(4) Bear, J. Hydraulics of Groundwater; McGraw-Hill: New York,
1979.
(5) de Marsily, G. Quantitative Hydrogeology: Groundwater Hydrology for Engineers; Academic Press: San Diego, CA, 1986.
(6) Domenico, P. A.; Schwartz, F. W. Physical and Chemical
Hydrogeology; John Wiley and Sons: New York, 1998.
(7) Freeze, R. A.; Cherry, J. A. Groundwater; Prentice-Hall: Englewood Cliffs, NJ, 1979.
(8) Gray, W. G.; Hassanizadeh, S. M. Paradoxes and realities in
unsaturated flow theory. Water Resour. Res. 1991, 27 (8), 18471854.
(9) Pan, C.; Hilpert, M.; Miller, C. T. Pore-scale modeling of saturated
permeabilities in random sphere packings. Phy. Rev. E 2001, 64
(6), 066702.
(10) Hubbert, M. K. The Theory of Ground-Water Motion and Related
Papers; Hafner Publishing Company: New York, 1969.
Received for review February 20, 2004. Revised manuscript
received July 27, 2004. Accepted September 1, 2004.
ES049728W
VOL. 38, NO. 22, 2004 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
9
5901