Darcy

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Darcys Law and Flow

• Philip B. Bedient
• Civil and Environmental Engineering
• Rice University

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Darcy allows an estimate of

• the velocity or flow rate moving within the
  aquifer
• the average time of travel from the head of the
  aquifer to a point located
  downstream

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Darcys Law

• Darcys law provides an accurate description of
  the flow of ground water in almost all
  hydrogeologic environments.

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Flow in Aquifers
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Darcys Experiment (1856)
Flow rate determined by Head loss dh h1 - h2
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Darcys Law

• Henri Darcy established empirically that the flux
  of water through a permeable formation is
  proportional to the distance between top and
  bottom of the soil column.
• The constant of proportionality is called the
  hydraulic conductivity (K).
• V Q/A, V ? ?h, and V ? 1/?L

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Darcys Law

• V K (?h/?L) and since
• Q VA (A total area)
• Q KA (dh/dL)

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Hydraulic Conductivity

• K represents a measure of the ability for flow
  through porous media
• Gravels - 0.1 to 1 cm/sec
• Sands - 10-2 to 10-3 cm/sec
• Silts - 10-4 to 10-5 cm/sec
• Clays - 10-7 to 10-9 cm/sec

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Conditions

• Darcys Law holds for
• 1. Saturated flow and unsaturated flow 2.
  Steady-state and transient flow 3. Flow in
  aquifers and aquitards 4. Flow in homogeneous
  and heterogeneous systems 5. Flow in
  isotropic or anisotropic media 6. Flow in rocks
  and granular media

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Darcy Velocity

• V is the specific discharge (Darcy velocity).
• () indicates that V occurs in the direction of
  the decreasing head.
• Specific discharge has units of velocity.
• The specific discharge is a macroscopic concept,
  and is easily measured. It should be noted that
  Darcys velocity is different .

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Darcy Velocity

• ...from the microscopic velocities associated
  with the actual paths if individual particles of
  water as they wind their way through the grains
  of sand.
• The microscopic velocities are real, but are
  probably impossible to measure.

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Darcy Seepage Velocity

• Darcy velocity is a fictitious velocity since it
  assumes that flow occurs across the entire
  cross-section of the soil sample. Flow actually
  takes place only through interconnected pore
  channels.

Av voids
A total area
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Darcy Seepage Velocity

• From the Continuity Eqn
• Q A vD AV Vs
• Where Q flow rate A total
  cross-sectional area of        material A
  V area of voids Vs seepage
  velocity VD Darcy velocity

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Darcy Seepage Velocity

• Therefore VS VD ( A/AV)
• Multiplying both sides by the length of the
  medium (L) VS VD ( AL / AVL ) VD ( VT /
  VV )
• Where VT total volume VV void
  volume
• By Definition, Vv / VT n, the soil porosity
• Thus VS VD / n

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Equations of Groundwater Flow

• Description of ground water flow is based
  on Darcys Law Continuity
  Equation - describes conservation of
  fluid mass during flow through a
  porous medium results in a partial
  differential equation of flow.
• Laplaces Eqn - most important in math

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Derivation of 3-D GW Flow Equation from Darcys
Law
z
y

• Mass In - Mass Out Change in Storage

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Derivation of 3-D GW Flow Equation from Darcys
Law

• Replace Vx, Vy, and Vz with Darcy using Kx, Ky,
  and Kz

Divide out constant ?, and assume Kx Ky Kz
K
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Transient Saturated Flow

• A change in h will produce change in ? and n,
  replaced
• with specific storage Ss ?g(? n?).
  Note, ? is the compressibility of aquifer and B
  is comp of water,
• therefore,

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Solutions to GW Flow Eqns.

• Solutions for only a few simple problems can be
  obtained directly - generally need to apply
  numerical methods to address complex boundary
  conditions.

h0
h1
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Transient Saturated Flow

• Simplifying by assuming K constant in all
  dimensions
• And assuming that S Ssb, and that T Kb yields

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Steady State Flow to Well

• Simplifying by assuming K constant in all
  dimensions
• and assuming that Transmissivity T Kb and
• Q flow rate to well at point (x,y) yields

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Example of Darcys Law

• A confined aquifer has a source of recharge.
• K for the aquifer is 50 m/day, and n is 0.2.
• The piezometric head in two wells 1000 m apart is
  55 m and 50 m respectively, from a common datum.
• The average thickness of the aquifer is 30 m, and
  the average width of aquifer is 5 km.

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Compute

• a) the rate of flow through the aquifer
• (b) the average time of travel from the head of
  the aquifer to a point 4
  km downstream
• assume no dispersion or diffusion

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The solution

• Cross-Sectional area 30(5)(1000) 15 x 104
  m2
• Hydraulic gradient (55-50)/1000 5 x 10-3
• Rate of Flow for K 50 m/day
  Q (50 m/day) (75 x 101 m2) 37,500
  m3/day
• Darcy Velocity V Q/A (37,500m3/day)
  / (15 x 104 m2) 0.25m/day

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And

• Seepage Velocity Vs
  V/n (0.25) / (0.2) 1.25 m/day (about 4.1
  ft/day)
• Time to travel 4 km downstream T 4(1000m) /
  (1.25m/day) 3200 days or 8.77 years
• This example shows that water moves very slowly
  underground.

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Limitations of theDarcian Approach

• 1. For Reynolds Number, Re, gt 10 or where the
  flow is turbulent, as in the immediate vicinity
  of pumped wells.

2. Where water flows through extremely
fine-grained materials (colloidal clay)
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Darcys LawExample 2

• A channel runs almost parallel to a river, and
  they are 2000 ft apart.
• The water level in the river is at an elevation
  of 120 ft and 110ft in the channel.
• A pervious formation averaging 30 ft thick and
  with K of 0.25 ft/hr joins them.
• Determine the rate of seepage or flow from the
  river to the channel.

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Confined Aquifer
Confining Layer
Aquifer
30 ft
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Example 2

• Consider a 1-ft length of river (and channel). Q
  KA (h1 h2) / L
• Where A (30 x 1) 30 ft2 K (0.25
  ft/hr) (24 hr/day) 6 ft/day
• Therefore, Q 6 (30) (120 110) /
  2000 0.9 ft3/day/ft length
  0.9 ft2/day

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Permeameters
Constant Head
Falling Head
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Constant head Permeameter

• Apply Darcys Law to find K V/t Q
  KA(h/L) or K (VL) / (Ath)
• Where V volume flowing in time t A
  cross-sectional area of the sample L length of
  sample h constant head
• t time of flow