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GEOS 4430 Lecture Notes: Darcy’s Law
Dr. T. Brikowski
Fall 2013
0file:darcy law.tex,v (1.24), printed October 15, 2013
1
Introduction
I Motivation: hydrology generally driven by the need foraccurate predictions or answers, i.e. quantitative analysis
I this requires mathematical description of fluid andcontaminant movement in the subsurface
I for now we’ll examine water movement, beginning with anempirical (qualitative) relation: Darcy’s Law
2
Darcy’s Observations
I Henry Darcy, studying public water supply development inDijon, France, 1856. Trying to improve sand filters for waterpurification
I apparatus measured water level at both ends and discharge(rate of flow L3
T ) through a vertical column filled with sand(Fig. 1)
3
Darcy Experimental Apparatus
L
h
h1
A
Cross−sectional area
Q
2
Figure 1: Simplified view of Darcy’s experimental apparatus. The centralshaded area is a square sand-filled tube, in contact with water reservoirson either side with water levels h1 and h2.
4
Empirical Darcy Law
I Darcy found the following relationship
Q ∝ A(h1 − h2)
L
or
Q = KAh1 − h2
L(1)
where K is a proportionality constant termed hydraulicconductivity L
T
5
Head as Energy
I Head as a measure of energy (1) was used without muchfurther consideration until the 1940’s when M. King Hubbertexplored the quantitative meaning of head h, and ultimatelyderived what he termed force potential Φ = g · h. Hedemonstrated that head is a measure of the mechanicalenergy of a packet of fluid
I mechanical energy of a unit mass of fluid taken to be the sumof kinetic energy, gravitational potential energy and fluidpressure energy (work).
I Energy content found by computing work to get to currentstate from a standard state (e.g. z = v = p = 0)
I energy units are kg m2
sec2 = Nt ·m = joule
6
Head Energy Components
I Kinetic Energy or “velocity work”. Energy required toaccelerate fluid packet from velocity v1 from velocity v2.
Ek = m
∫ v2
v1
vdv =1
2mv2 (2a)
Remember that the integral sign is really just a fancy “sum”,saying “add up all the tiny increments of change betweenpoints 1 and 2”
I Gravitational work. Energy required to raise fluid packet fromelevation z1 to elevation z2.
W =
∫ z2
z1
dz = mgz (2b)
7
Head Energy Components (cont.)
I Pressure work. Energy required to raise fluid packet pressure(i.e. squeeze fluid) from P1 to P2.
P =
∫ P2
P1
VdP = m
∫ P2
P1
V
m= m
∫ P2
P1
1
ρ
=1
ρ
∫ P2
P1
dP =P
ρ(2c)
where (2c) assumes a unit mass of incompressible fluid
8
Final Expression for Head
I the sum of (2a)–(2c) is the total mechanical energy for theunit mass (i.e. m = 1)
Etm =v2
2+ gz +
P
ρ(3)
I in a real setting, energy is lost when flow occurs Fetter (Sec.4.2, 2001), so Etm = constant implies no flow. No flow(Q = 0) in (1) implies constant head (h1 = h2).
I Following the analysis of Hubbert (1940) define head h suchthat h · g is the total energy of a fluid packet. Then
hg =v2
2+ gz +
P
ρ(4a)
h =v2
2g+ z +
P
ρg(4b)
= z +P
ρg(4c)
9
Final Expression for Head (cont.)
where (4c) assumes v is small (true for flow in porous media).
I head is a combination of elevation (gravitational potential)head and pressure head (where P = ρghp, and hp is thepressure head, or pressure from the column of water above thepoint being considered)
I for a given head gradient ( ∆h∆x ) discharge flux Q is
independent of path (Fig. 4.5, Fetter, 2001).
10
Head for Variable-Density Fluids
I (4c) indicates that head also depends on fluid density. This isonly an issue for saline or hot fluids. For constant density (4c)can be written as
h = z + hp
where hp is the height of water above the point of interest(eq. 4.11, Fetter, 2001).
I This implies that head is constant vs. z (for every meterchange in z there is an equal but opposite change in hp).
I when ρ isn’t constant, “fresh-water” head (the head for anequivalent-mass fresh water column, hf =
ρpρfhp) should be
used for computing gradients, or in flow models (Pottorffet al., 1987)
11
Applicability of Darcy’s Law
I Darcy’s Law makes some assumptions, which can limit itsapplicability
I Assumption: kinetic energy can be ignored. Limitation:laminar flow is required (i.e. Reynold’s number low, R ≤ 10,viscous forces dominate)
I Assumption: average properties control discharge.Limitation: (1) applicable only on macroscopic scale (forareas ≥ 5 to 10 times the average pore cross section, Fig. 2)
I Assumption: fluid properties are constant. Limitation: (1)applicable only for constant temperature or salinity settings, orif head h is converted to fresh-water equivalent hf
12
Scale Dependence of Rock Properties
φ, K, k
Average
Homogeneous Medium
Po
ros
ity
, P
erm
ea
bil
ity
, o
r C
on
du
cti
vit
y
Distance
Heterogeneous M
ediumMicroscopic Macroscopic
Figure 2: Variation of rock properties with scale. Note heterogeneousmedia have average properties that vary positively (shown) or negatively(not shown) with distance. See also Bear (Fig. 1.3.2, 1972).
13
True Fluid Velocity
I (1) gives total discharge through the cross-sectional area A
I specific discharge (q traditionally, v in the text) is the flux perunit area q = Q
A
I pore or seepage or average linear velocity (v traditionally, Vx
in the text) is the velocity at which water actually moveswithin the pores v = K
φdhdl
14
Hydraulic conductivity (K )
I the proportionality constant in (1) depends on the propertiesof the fluid, a fact that wasn’t recognized until Hubbert(1956)
I Hubbert rederived Darcy’s Law from physical principles,obtaining a form similar to the Navier-Stokes equation1. Fromthis he found the following form for K :
K =kρg
µ(5)
where ρ is the density of the fluid and µ is its dynamicviscosity ( M
LT). k is the intrinsic permeability of the porousmedium Fetter (written as Ki eqn. 3-19, 2001).
I (5) demonstrates that hydraulic conductivity depends on theproperties of both the fluid and the rock
1http://en.wikipedia.org/wiki/Navier-stokes15
Permeability (k)
I k (units L2) depends on the size and connectivity of the porespace, and can be expressed as k = Cd2, where C is thetortuosity of the medium (depends on grain size distribution,packing, etc.; “unmeasurable”), and d is the mean graindiameter (a proxy for the mean pore diameter). See alsoFetter (sec. 3.4.3, 2001).
I standard units are the darcy, (1 darcy = 9.87x10−9cm2),which is defined as the permeability that produces unit fluxgiven unit viscosity, head gradient, and cross-sectional area(Sec. 3.4.2, Fetter, 2001)
16
Measuring Rock Hydraulic Properties
I Porosity is measured in the lab with a porosimeterI sample is dried and weighedI then resaturate sample with water (or mercury), very difficult
to do completelyI measure volume or mass of liquid absorbed by sampleI yields effective porosity
I Permeability/Hydraulic Conductivity are measured in the labusing permeameters (Fig. 3), which apply Darcy’s Law todetermine K
I Constant Head permeameter best used for high conductivity,indurated samples (rocks)
I Falling Head permeameter best used for low-permeabilitysamples or soils
17
Permeameters
(a) Constant HeadPermeameter: K = VL
Ath
(b) Falling Head Permeameter:
K =d2t
d2c
Lt
ln(hoh
)Figure 3: Constant and Falling-Head Permeameters. a) best for consolidated, high-permeability samples; b)best for unconsolidated and low-permeability samples. t is the duration of the experiment, V = Qt is the totalvolume discharged, and A is the cross-sectional area of the apparatus. After Fetter (Fig. 3.16-3.17, 2001), see alsoTodd and Mays (2005, p. 95-96).
18
Applications of Darcy’s Law
I find magnitude and direction of discharge through aquifer
I solve for head at heterogeneity boundary in composite(two-material) aquifer
I find total flux through aquitard in a multi-layer system
19
The Flow Equation (Continuity Eqn.)
I Approach: use “control” volume, itemize flow in and out, andcontent of volume (sec. 4.7, Fetter, 2001), see Fig. 4
I Simplifications: assume steady-state, constant fluid properties
I Mass fluxI must describe movement of mass across the sides of the
control volume, i.e. determine massarea·time or m
t·l3I known fluid parameters: density ρ
(ml3
), velocity ~q
(lt
)(really
specific discharge, or velocity averaged over a cross-sectionalarea; see references such as Bear (1972); Schlichting (1979);Slattery (1972) for discussion of volume-averaging and REV’s)
I Then mass flux = ρ · ~q
20
Representative Control Volume
q xq x x+
∆ ∆ ∆
∆ z
∆ y
∆ x
ρ ρ∆ x
(x,y,z)
(x+ x, y+ y, z+ z)
Figure 4: Mass fluxes for control volume in uniform flow field.
21
Control Volume Mass Balance
I Content of control volume (storage)I Steady-state ⇒ no change in content with timeI then sum of in and outflows must be 0 (otherwise content
would change)
I Sum of flows for control volume in uniform flowI from above know that form will be:{
rate ofmass in
}−{
rate ofmass out
}= 0 (6)
I mass flux in{rate ofmass in
}= ρ~qx︸︷︷︸
mass flux per unit area
· ∆y∆z︸ ︷︷ ︸area of cube face
(7)
I mass flux outI depends on ~qout , let ~qout = ~qin + change
22
Control Volume Mass Balance (cont.)
I express change as distance · (rate of change) and rate ofchange as ∆qx
∆xI to be most accurate, determine rate of change over a very
small distance, i.e. take limit as ∆x → 0I that’s a derivative
lim∆x→0
∆qx∆x
= dqdx
I since ~q can vary as a function of y and z as well, we write itas a partial derivative, holding other variables constant
d~qdx
∣∣∣y,z constant
=∂~q
∂x
I then in the form required for (6){rate ofmass out
}= ρ
(qx + ∆x
∂qx∂x
)∆y∆z
I net flux in x-direction: −ρ ∂qx∂x ∆x∆y∆z = −ρ ∂qx
∂x ∆V
23
Continuity Equation
I sum of flows for general case (y and z sums are same form asfor x)
−ρ∆V
(∂qx∂x
+∂qy∂y
+∂qz∂z
)= 0(
∂qx∂x
+∂qy∂y
+∂qz∂z
)= 0
∇ · ~q = 0 (8)
I this is the divergence of the specific discharge, a measure ofthe fluids to diverge from or converge to the control volume
I assume this equation applies everywhere in problem domain,i.e. that fluid is everywhere, and fluid/rock properties andvariables are continuous
I then rock and fluid are overlapping continua
24
Flow Equation
I Approach: governing equations (8) and (1) contain similarvariables, can we simplify?
I Combined equationI substituting Darcy’s Law (1) for ~q in (8):
∇ · ~q = ∇ · ~(−K∇h)
= ∇2 · h
=
(∂2h
∂x2+
∂2h
∂y2+
∂2h
∂z2
)= 0
I This is our governing equation for groundwater flowI Assumptions: many were made above
I steady-state: no time variationI continuum: fluid and rock properties and variables continuous
everywhere in problem domainI constant density: incompressible fluid, no compositional
change, no temperature changeI no viscous/inertial effects: low flow velocities
25
References
Bear, J.: Dynamics of Fluids in Porous Media. Elsevier, New York, NY(1972)
Fetter, C.W.: Applied Hydrogeology. Prentice Hall, Upper Saddle River,NJ, 4th edn. (2001), http://vig.prenhall.com/catalog/academic/product/0,1144,0130882399,00.html
Hubbert, M.K.: The theory of ground-water motion. J. Geol. 48,785–944 (1940)
Hubbert, M.K.: Darcy’s law and the field equations of the flow ofunderground fluids. AIME Transact. 207, 222–239 (1956)
Pottorff, E.J., Erikson, S.J., Campana, M.E.: Hydrologic utility ofborehole temperatures in Areas 19 and 20, Pahute Mesa, Nevada TestSite. Report DRI-45060, DOE/NV/10384-19, Desert ResearchInstitute, Reno, NV (1987)
Schlichting, H.: Boundary-Layer Theory. McGraw-Hill, New York (1979)
Slattery, J.C.: Momentum, Energy, and Mass Transfer in Continua.McGraw-Hill, New York (1972)
26
References (cont.)
Todd, D.K., Mays, L.W.: Groundwater Hydrology. John Wiley & Sons,Hoboken, NJ, 3rd edn. (2005), http://www.wiley.com/WileyCDA/WileyTitle/productCd-EHEP000351.html
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