GEOS 4430 Lecture Notes: Darcy s Law

GEOS 4430 Lecture Notes: Darcy’s Law

Dr. T. Brikowski

Fall 2013

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file:darcy law.tex,v (1.24), printed October 15, 2013

Introduction

I Motivation: hydrology generally driven by the need for accurate predictions or answers, i.e. quantitative analysis I this requires mathematical description of fluid and

contaminant movement in the subsurface

I for now we’ll examinewater movement, beginning with an

Darcy’s Observations

I Henry Darcy, studying public water supply development in Dijon, France, 1856. Trying to improve sand filters for water purification

I apparatus measured water level at both ends and discharge (rate of flow L_T3) through a vertical column filled with sand (Fig. 1)

Darcy Experimental Apparatus

Figure 1: Simplified view of Darcy’s experimental apparatus. The central shaded area is a square sand-filled tube, in contact with water reservoirs on either side with water levelsh1andh2.

Empirical Darcy Law

I Darcy found the following relationship

Q ∝ A(h1 − h2)

L

or

Q = KAh1 − h2

L (1)

whereK is a proportionality constant termed hydraulic conductivity L_T

Head as Energy

I Head as a measure of energy (1) was used without much further consideration until the 1940’s when M. King Hubbert explored the quantitative meaning of head h, and ultimately derived what he termed force potentialΦ = g·h. He demonstrated that head is a measure of themechanical

energy of a packet of fluid

I mechanical energy of a unit mass of fluid taken to be the sum of kinetic energy, gravitational potential energy and fluid pressure energy (work).

I Energy content found by computing work to get to current state from a standard state (e.g. z =v=p= 0)

Head Energy Components

I Kinetic Energy or “velocity work”. Energy required to accelerate fluid packet from velocityv1 from velocity v2.

Ek = m Z v2 v1 vdv = 1 2mv 2 _(2a)

Remember that the integral sign is really just a fancy “sum”, saying “add up all the tiny increments of change between points 1 and 2”

I Gravitational work. Energy required to raise fluid packet from elevation z1 to elevationz2. W = Z z2 z1 dz = mgz (2b) 7

Head Energy Components (cont.)

I Pressure work. Energy required to raise fluid packet pressure (i.e. squeeze fluid) fromP1 to P2.

P = Z P2 P1 VdP = m Z P2 P1 V m = m Z P2 P1 1 ρ = 1 ρ Z P2 P1 dP = P ρ (2c)

Final Expression for Head

I the sum of (2a)–(2c) is the total mechanical energy for the unit 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 such that h·g is the total energy of a fluid packet. Then

hg = v 2 2 + gz + P ρ (4a) h= v 2 2g + z + P ρg (4b) =z + P ρg (4c) 9

Final Expression for Head (cont.)

where (4c) assumesv is small (true for flow in porous media). I head is a combination ofelevation (gravitational potential)

head and pressurehead (whereP = ρghp, andhp is the pressure head, or pressure from the column of water above the point being considered)

I for a given head gradient (∆_∆xh) discharge flux Q is independent of path (Fig. 4.5, Fetter, 2001).

Head for Variable-Density Fluids

I (4c) indicates that head also depends on fluid density. This is only an issue for saline or hot fluids. For constant density (4c) can be written as

h = z + hp

wherehp 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 meter change in z there is an equal but opposite change inhp).

I when ρ isn’t constant, “fresh-water” head (the head for an equivalent-mass fresh water column, hf = ρ_ρp_fhp) should be used for computing gradients, or in flow models (Pottorff et al., 1987)

Applicability of Darcy’s Law

I Darcy’s Law makes some assumptions, which can limit its applicability

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 (for areas≥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, or if headhis converted to fresh-water equivalent hf

Scale Dependence of Rock Properties

φ, K, k Average

Homogeneous Medium

Porosity, Permeability, or Conductivity

Distance

Heterogeneous Medium

Microscopic Macroscopic

Figure 2: Variation of rock properties with scale. Note heterogeneous media have average properties that vary positively (shown) or negatively (not shown) with distance. See also Bear (Fig. 1.3.2, 1972).

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 per unit area q = Q_A

I poreor seepageor average linear velocity(v traditionally,Vx

in the text) is the velocity at which water actually moves within the pores v = K_φdh_dl

Hydraulic conductivity (

K

)

I the proportionality constant in (1) depends on the properties of 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. From this he found the following form for K:

K = kρg

µ (5)

whereρ is the density of the fluid and µis its dynamic viscosity (_LTM). k is the intrinsic permeabilityof the porous medium Fetter (written as Ki eqn. 3-19, 2001).

I (5) demonstrates that hydraulic conductivity depends on the properties of both the fluid and the rock

1

http://en.wikipedia.org/wiki/Navier-stokes 15

Permeability (

k

)

I k (units L2) depends on the size and connectivity of the pore space, and can be expressed ask = Cd2, whereC is the tortuosity of the medium (depends on grain size distribution, packing, etc.; “unmeasurable”), andd is the mean grain diameter (a proxy for the mean pore diameter). See also Fetter (sec. 3.4.3, 2001).

I standard units are the darcy, (1darcy = 9.87x10−9cm2), which is defined as the permeability that produces unit flux given unit viscosity, head gradient, and cross-sectional area (Sec. 3.4.2, Fetter, 2001)

Measuring Rock Hydraulic Properties

I Porosity is measured in the lab with aporosimeter

I sample is dried and weighed

I then resaturate sample with water (or mercury), very difficult to do completely

I measure volume or mass of liquid absorbed by sample

I yieldseffective porosity

I Permeability/Hydraulic Conductivity are measured in the lab using permeameters (Fig. 3), which apply Darcy’s Law to determine K

I Constant Headpermeameter best used for high conductivity, indurated samples (rocks)

I Falling Headpermeameter best used for low-permeability samples or soils

Permeameters

(a)Constant Head Permeameter: K = VL

Ath

(b)Falling Head Permeameter:

K = d2t d2 c L t ln ho h

Figure 3: Constant and Falling-Head Permeameters.a)best for consolidated, high-permeability samples;b)

best for unconsolidated and low-permeability samples.tis the duration of the experiment,V=Qtis the total volume discharged, andAis the cross-sectional area of the apparatus. After Fetter (Fig. 3.16-3.17, 2001), see also Todd and Mays (2005, p. 95-96).

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

The Flow Equation (Continuity Eqn.)

I Approach: use “control” volume, itemize flow in and out, and content of volume (sec. 4.7, Fetter, 2001), see Fig. 4

I Simplifications: assume steady-state, constant fluid properties I Mass flux

I must describe movement of mass across the sides of the control volume, i.e. determine _area·timemass or _tm_·l3

I known fluid parameters: densityρ m_l3

, velocity~q _tl (really specific discharge, or velocity averaged over a cross-sectional area; see references such as Bear (1972); Schlichting (1979); Slattery (1972) for discussion of volume-averaging and REV’s)

Representative Control Volume

Figure 4: Mass fluxes for control volume in uniform flow field.

Control Volume Mass Balance

I Content of control volume (storage)

I Steady-state⇒no change in content with time

I then sum of in and outflows must be 0 (otherwise content would change)

I Sum of flows for control volume in uniform flow I from above know that form will be:

rate of mass in − rate of mass out = 0 (6) I mass flux in rate of mass in = ρ~q_x |{z} mass flux per unit area

· ∆y∆z

| {z }

area of cube face (7)

Control Volume Mass Balance (cont.)

I express change asdistance·(rate of change) and rate of change as ∆qx

∆x

I to be most accurate, determine rate of change over a very small distance, i.e. take limit as ∆x→0

I that’s a derivative lim ∆x→0 ∆qx ∆x = dq dx

I sinceq~can vary as a function ofy andzas well, we write it as a partial derivative, holding other variables constant

d~q dx y,z constant = ∂~q ∂x

I _{then in the form required for (6)} rate of mass out = ρ qx + ∆x ∂qx ∂x ∆y∆z

I net flux inx-direction: −ρ ∂qx

∂x ∆x∆y∆z = −ρ ∂qx

∂x ∆V

Continuity Equation

I sum of flows for general case (y andz sums are same form as for 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 of the 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 and variables are continuous

Flow Equation

I Approach: governing equations (8) and (1) contain similar variables, can we simplify?

I Combined equation

I _{substituting Darcy’s Law (1) for}~qin (8): ∇ ·~q=∇ ·(−K~∇h) =∇2_·h = _∂2_h ∂x2 + ∂2_h ∂y2 + ∂2_h ∂z2 = 0

I This is our governing equation for groundwater flow

I Assumptions: many were made above I steady-state: no time variation

I continuum: fluid and rock properties and variables continuous everywhere in problem domain

I constant density: incompressible fluid, no compositional change, no temperature change

I no viscous/inertial effects: low flow velocities

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 of underground fluids. AIME Transact. 207, 222–239 (1956) Pottorff, E.J., Erikson, S.J., Campana, M.E.: Hydrologic utility of

borehole temperatures in Areas 19 and 20, Pahute Mesa, Nevada Test Site. Report DRI-45060, DOE/NV/10384-19, Desert Research Institute, Reno, NV (1987)

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