CWR 6537 Subsurface Contaminant Hydrology
Lecture 3
WATER FLOW IN POROUS MEDIA
A.
Historical Highlights:
1805
1822
1822
1827
1842
1845
1856
1907
1931
P. Laplace
C. Fourier
L. Navier
G. Ohm
J. L. Poiseuille
G. G. Stokes
H. Darcy
E. Buckingham
L. A. Richards
Differential equation operator
Heat Flow
Viscous Flow of Fluids
Electricity
Fluid flow in a capillary
Generalized Navier-Stokes law
Saturated water flow in porous media
Capillary potential, Darcy equation for unsaturated flow
Transient water flow in porous media
Textbooks
1972 Jacob Bear
1979 A. Freeze and J. Cherry
1980 C.W. Fetter
1993 C.W. Fetter
1990 Domenico and Schwartz
2000 R. Charbeneau
Dynamics of Fluids in Porous Media
Groundwater
Applied Hydrogeology
Contaminant Hydrogeology
Physical and Chemical Hydrogeology
Groundwater Hydraulics and Pollutant Transport
Stochastic Texts
1989 Gedeon Dagan
1990 W. Jury and K. Roth
1993 Lynn Gelhar
2003 Yoram Rubin
Flow and Transport in Porous Formations
Transfer Functions and Solute Movement through Soil
Stochastic Subsurface Hydrology
[Eulerian]
Applied Stochastic Hydrogeology
[Lagrangian]
(1)
A series of linear flux laws were proposed over a period of four decades (1820-1860), all of
which state that the flux (of heat, electricity, fluids, or mass) is proportional to the driving
force, i.e., the gradient in potential (temperature, voltage, water potential, or chemical
potential).
(2)
Nonlinear flux laws, where the proportionality constant is a strong, nonlinear function of the
state of the system are needed to describe transient water flow in porous media; e.g., soil
hydraulic conductivity as a function of soil-water matric potential (R).
Water Flow
Saturated
Unsaturated
- steady
- steady
- unsteady
- transient
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CWR 6537 Subsurface Contaminant Hydrology
Lecture 3
Buckingham (1907) extended the applicability of Darcy's law to unsaturated flow conditions,
with conductivity [K(R)] as a function of R (or 2). Richards (1931) combined DarcyBuckingham equation with the continuity expression to obtain the governing equation for
transient, unsaturated flow.
(3)
The study of the hydraulics of groundwater flow began to be integrated with geology and
geochemistry in the 1970s. At the same time, stochastic approaches emerged.
B. Soil-Water Potential
The amount of work that must be done per unit quantity of water in order to transport irreversibly
and isothermally an infinitesimal quantity of water from a pool of pure water (chosen as the
reference) at a specified elevation and atmospheric pressure, to specified position in soil.
Potential Energy: Energy = Force x distance
Specific Potential Energy: energy/volume or energy/weight
Energy = Force x length = Force
volume
volume
area
6 PRESSURE
Energy = Force x length = mass x acceleration x length 6 LENGTH
weight
weight
mass x gravity
NOTE: matric potential is negative in unsaturated soils and equal to zero in saturated soils.
Total Soil-Water Potential (N)
N = 3 Ni
Here, we only consider matric or pressure potential (R) and gravitational potential (z)
N = R + z ; R # 0 for unsaturated soils
R $ 0 for saturated soils
z < 0 with z = 0 at ground surface
Note that other major potential components ignored include: osmotic, pneumatic, and over-burden.
Also, z is taken as positive upward (or negative down) with ground surface as the datum (z = 0).
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C.
CWR 6537 Subsurface Contaminant Hydrology
Lecture 3
Conservation of Mass (Continuity Equation)
qz
Consider a unit volume element )x )y )z in
which flux density (Dq) in each direction is
specified.
∆z
qx
D = liquid density
q = Darcy flux
z
qy
x
∆y
∆x
y
S)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
Flow
Direction
Mass Inflow Rate
Mass Outflow Rate
S)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
S)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))Q
Net flow rate = Inflow Rate-Outflow Rate = change in mass storage rate
(1)
where, 2 is volumetric water content
For incompressible fluids (D constant in time) and homogeneous fluid density (D constant in space)
(2)
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CWR 6537 Subsurface Contaminant Hydrology
Lecture 3
D. The Storage Term:
By conservation of fluid mass:
(3)
r
where D = fluid density [MAL-3]; 2 = volumetric water content; and q = specific discharge vector,
[LAT-1]. The fraction of water filled porosity is defined by:
(4)
where Sw = the saturated fraction and n = porosity. Eq. (4) can be substituted into the storage term
[the left-hand-side of Eq(3)] to obtain:
(5)
It may be recalled that
(6)
where N = hydraulic head, piezometric head, or potential [L]; z = elevation or gravitational head [L];
and R = pressure head [L].
Using the chain rule, Eq. (5) can be cast in terms of measurable pressure heads, R.
(7)
The first term of Eq. (7) accounts for the change in fluid storage due to a change in volumetric water
content (i.e., in the vadose zone where Sw < 1. and n is usually assumed constant). From the slope
of a moisture retention curve the following relationship is obtained:
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CWR 6537 Subsurface Contaminant Hydrology
Lecture 3
(8)
in which C(R) = specific moisture capacity [L-1].
The second term in Eq. (7) accounts for change in fluid storage due to compressibility of solid
matrix.
(9)
where " = solid matrix compressibility [LAT2AM-1] and g = acceleration of gravity [LAT-2].
Finally the third term accounts for change in fluid storage due to fluid compressibility.
(10)
where $ is fluid compressibility [LAT2AM-1]. Eqs. (7) through (9) can be combined to give:
(11)
in which Ss = the specific storage [L-1], and is defined by:
(12)
Specific storage represents the volume of water produced per unit volume aquifer per unit decline
in head.
Notes:
C(R) - large in the unsaturated zone, zero in saturated zone. However, in unconfined saturated
systems (e.g., aquifers), water is also produced (or stored) as the water level changes. This
is quantified by the specific yield, Sy.
- small term typically neglected in unsaturated zone. Can be important in confined aquifers
Ss
(for example in the analysis of well hydraulics).
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CWR 6537 Subsurface Contaminant Hydrology
Lecture 3
E. Richards Equation
Now consider the flux term of the Continuity Equation. We can substitute the Darcy-Buckingham
Equation for q:
(13)
If we substitute Equations 11 and 13 back into Equation 3 (the continuity equation), we obtain the
Richards Equation:
(14)
If we further neglect variations in water density and ignore aquifer and fluid compressibility effects,
14 may be alternatively expressed as:
(15)
which is equivalent to the following (sometimes referred to as the mixed-form of the Richards
Equation)
(16)
The 2-based form is
(17)
where soil-water diffusivity is defined as D(2) = K(R)/C(R).
Note:
@ C(R) and K(R) are both highly non-linear
@ The Richards equation is valid for saturated and unsaturated flow. In the saturated zone
C(R) = 0 ; K = Ksat = Constant ; Sw = 1 ; R $ 0
In the unsaturated zone
C(R) … 0 C(R) >> Sw Ss ; R # 0 ; Swr # Sw # 1
K(R) $ 0 Sw > Swr = 0 S = Swr
@ In general K is a 3x3 tensor. Kx is a fairly straightforward concept. Kxy is related to the
flow induced in the x-direction by a gradient in the y-direction. When the coordinate system
is aligned to the principal axes for K, the second-order tensor is reduced to a diagonal tensor
(i.e., a vector of Kx, Ky, and Kz).
@ These equations assume rigid media and incompressible fluid, i.e., Ss=0.
@ R-based and mixed equations are valid for both saturated and unsaturated zone, but 2-based
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CWR 6537 Subsurface Contaminant Hydrology
Lecture 3
equation is not valid in saturated zone because D(2) "blows up" as it approaches saturation
and C(2)=0. Also, R is a continuous function as it moves through layered media - 2 is not.
@ 2-based equations are very useful for fully unsaturated flow 6 traditionally favored by soil
scientists; looks like advective-dispersive transport equation, but highly non-linear.
Starting with the mixed-form, Equation (16), consider the following cases:
(1) Homogeneous, isotropic media [LK = 0; and Kx = Ky = Kz = K]
For saturated conditions, M2/Mt = 0 and
which is known as the Laplace Equation.
(2) Heterogeneous, isotropic media [LK … 0; and Kx = Ky = Kz = K]
(3) Homogeneous, anisotropic media [LK = 0; and Kx … Ky … Kz]
Finally, an example simplification is provided for the case of one-dimensional flow in the xdirection:
The 2-based form is then
F. Soil Hydraulic Properties
From the above analysis, we need the following hydraulic properties to describe water flow
(saturated and unsaturated) in porous media:
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CWR 6537 Subsurface Contaminant Hydrology
Lecture 3
2(R) or R(2) 6 soil-water characteristic function
K(R) or K(2) 6 soil hydraulic conductivity function
C(R) or C(2) 6 specific soil-water moisture capacity function
D(2)
6 soil-water diffusivity function
G. Considerations for Solving Equations
For any solution to these equations, we must consider the initial and boundary conditions. Some
example boundary conditions are discussed here.
1.
prescribed water content (2 = 21 or R = R1) on boundaries
Examples:
1-D: flood irrigation, or post-ponding infiltration
2-D: furrow irrigation, leakage from a ditch
3-D: flow from a puddle, post-ponding drip irrigation
Usually water assumed to be supplied as "free water" at atmospheric pressure ˆ R = 0 and
2 = 2s. If water depth over supply surface is not negligibly small have R1>0 and must use
R-based equation.
2.
Prescribed flux
Examples
NOTE:
1-D: sprinkler irrigation, rainfall pre-ponding
2-D: injection from buried pipe
3-D: drip irrigation pre-ponding
Soil must be able to take up all arriving water or else reverts to prescribed water
content (or prescribed head) condition.
This type B.C. is more difficult to deal with since 2 and R at boundary change with time.
Eventually will get runoff for continuous flux if soil is unable to transmit infiltrating water fast
enough.
Techniques for Solving Equations
1.
Analytical solutions - require restrictive assumptions, but are useful for making general
inferences about system. Require homogeneous soil properties, temporally constant B.C.,
uniform I.C.
2.
Quasi-analytic solutions - mathematical analysis establishes the form of the solution, but
some coefficients and parameters require approximation or solution by numerical means.
3.
Numerical solution - most general; will deal with wide variety of flow geometries and
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CWR 6537 Subsurface Contaminant Hydrology
Lecture 3
boundary and initial conditions, and can handle both saturated and unsaturated systems
simultaneously. Issues of computational accuracy, efficiency, and stability become
important. Particularly important for unsaturated flow and transport problems which are
inherently non-linear.
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