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Darcy’s Law
 •   Last time                                         • Today
     – Groundwater flow is in
       response to gradients of                               – Darcy’s Law
       mechanical energy
     – Three types
         • Potential
                                                              – Hydraulic Conductivity
         • Kinetic
             – Kinetic energy is usually not
               important in groundwater                       – Specific Discharge
         • Elastic (compressional)
     – Fluid Potential, !
             – Energy per unit mass                           – Seepage Velocity
     – Hydraulic Head, h                                            • Effective porosity
             – Energy per unit weight
             – Composed of
                 » Pressure head
                 » Elevation head




                          http://biosystems.okstate.edu/darcy/index.htm



                               Darcy’s Law
                                                                   A FEW CAREER HIGHLIGHTS:
Henry Darcy, a French                                  •    In 1828, Darcy was assigned to a deep well
hydraulic engineer                                          drilling project that found water for the city of
interested in purifying                                     Dijon, in France, but could not provide an
                                                            adequate supply for the town. However, under
water supplies using                                        his own initiative, Henry set out to provide a
sand filters, conducted                                     clean, dependable water supply to the city
                                                            from more conventional spring water sources.
experiments to determine                                    That effort eventually produced a system that
the flow rate of water                                      delivered 8 m3/min from the Rosoir Spring
                                                            through 12.7 km of covered aqueduct.
through the filters.

Published in 1856, his conclusions
have served as the basis for all
modern analysis of ground water
flow
                                                       •    In 1848 he became Chief Director for Water
                                                            and Pavements, Paris. In Paris he carried out
                                                            significant research on the flow and friction
                                                            losses in pipes, which forms the basis for the
                                                            Darcy-Weisbach equation for pipe flow.

                                                       •    He retired to Dijon and, in 1855 and 1856, he
                                                            conducted the column experiments that
                                                            established Darcy's law for flow in sands.
                       Freeze, R. Allen. "Henry Darcy and the Fountains of Dijon." Ground Water 32, no.1(1994): 23–30.




                                                                                                                         1
Darcy’s Law
         Cartoon of a Darcy experiment:
                                     Sand-filled column
        Constant head tanks          with bulk cross-sectional area A
        at each end
                     #
                                                            Q - Rate of
                                     "l                     discharge [L3/T]
   dh
            h1
                                                   #
                                                                   h2
Datum



                     In this experiment,
                 water moves due to gravity.




  What is the relationship between
      discharge (flux) Q and
          other variables?
A plot of Darcy’s actual data:




                                      http://biosystems.okstate.edu/darcy/index.htm




                                                                                      2
What is the relationship between
       discharge (flux) Q and
           other variables?
 Darcy found that:
 1) If he doubled the head difference (dh = h2 – h1),
         then the discharge Q doubled.
 2) If he doubled column length (dl),
         Q halved
 3) If he doubled column area (A),
         Q doubled.                            Sand-filled column
                                        Constant head tanks   with cross-sectional area A
                                        at each end
                                                     #
 Conclusions:                                                 "l
                                                                                Q - Rate of
                                                                                discharge [L3/T]
   Q is proportional to dh    "h
                                            h1
   Q is proportional to 1/dl                                              #
                                                                                      h2
   Q is proportional to A Datum




                  Darcy’s Law
Darcy also found that if he used different kinds
of sands in the column, discharge Q changed,
but for a particular sand, regardless of Q:
                Q dl
                     = a constant
                A dh
This “proportionality constant” is usually called
“hydraulic conductivity” and often is assigned the
symbol, K. Leads to Darcy’s Law:

                   dh
 Q = ! KA                 =conductivity x bulk area x ‘gradient’
                   dl


                                                                                                   3
Darcy’s Law
                                    dh
                    Q = ! KA
                                    dl
            Why is there a minus sign in Darcy’s Law?




                “a little vector calculus”




                                                        (Bradley and Smith, 1995)




                Darcy’s Law
Invert Darcy’s Law to express conductivity in terms of
discharge, area, and gradient:

                        Q & '1 #
                K=
                        A $ dh dl !
                          %       "
This is how we measure conductivity:
     Measure area A; that can be easy.
     Measure gradient, -dh/dl; can be harder*.
     Measure discharge, Q; can also be hard.

                *Imagine trying to measure gradient in a complex geology
                with three-dimensional flow and few observation points.




                                                                                    4
Darcy’s Law
What are the dimensions of K? Dimensional analysis:



           Q dl & ( L3T '1 )( L) # & L #
      K ='     =                  =
           A dh $ ( L2 )( L) ! $ T !
                 %               " % "

                                        = the dimensions of speed or velocity

                                        e.g, m/s, meters per second, in SI units




                    Darcy’s Law
                                         dh
                        Q = ! KA
                                         dl
   This expresses Darcy’s Law in terms of discharge.
   We can also express it in terms of “Darcy velocity”
   or “specific discharge”, that is, discharge per unit
   bulk area, A:

             Q     dh                      &L#
       q=      ='K    = specific discharge $ !
             A     dl                      %T "

   This ‘flux density’ expression is the more common
   and convenient way to express Darcy’s law. For
   example, it applies even when area is varying, and
   with a little generalization also when gradient or conductivity is varying.




                                                                                   5
Darcy’s Law

                    We will look at three major
                    topics important to Darcy’s
                    Law:
                     • Hydraulic Head
            dh         Gradient
 Q = ! KA
            dl       • Bulk cross-sectional
                        area of flow
                     • Hydraulic Conductivity
                       (next time)




            Hydraulic Head
• Head is a measure of the total mechanical
  energy per unit weight.
• If K, Q and A don’t change with distance,
  then
  – hydraulic head varies linearly with distance
                 Q
         dh = $     dl " dh ! dl
                        "#
                 KA
   #
                           In this experiment,
                           with constant K and A, the head
                           drops linearly with distance, and
               #           the specific discharge is constant.
                                       dh
                               q=!K       = constant
                                       dl




                                                                 6
Hydraulic Head
• In Darcy’s experiment, do the drops falling from the
  constant head tanks have constant velocity?




                                         Sand-filled column
                   Constant head tanks   with cross-sectional area A
                   at each end
                                #
                                                              Q - Rate of
                                         "l                   discharge [L3/T]
              "h
                       h1
                                                     #
                                                                    h2
           Datum




                   Hydraulic Head
• In Darcy’s experiment, do the drops falling from the
  constant head tanks have constant velocity?




• No! Water droplets accelerate at 9.8 m/s2.
• Assuming no air resistance, the falling water drop has its
  potential energy converted to kinetic energy as it falls:
                    1 2
            mgz +      mv = total energy = constant,
                    2
             that is, its a conservative energy field.

But water through our column has constant velocity—why?




                                                                                 7
Hydraulic Head
 But water through our column has constant velocity—why?
    #
                                      In this experiment,
                     head level
                                      with constant K, Q and A, the head
                                      drops linearly with distance, and
                     #                the specific discharge is constant.
                                                  dh
                                          q=!K       = constant
                                                  dl
 In a porous medium, the tendency of the fluid to accelerate is opposed
 by friction against the grains.
                          1
              mgz + pV + mv 2 = total energy ! constant
                          2 ~0
It’s a dissipative energy field. Where does the energy go?

  Friction ! Heat; Mechanical Energy ! Thermal Energy




                     Darcy’s Law
• Could we use Darcy’s Law to
  model the falling drops?

• Darcy’s Law works because the driving forces (gravity
  and pressure) in the fluid are balanced by the viscous
  resistance of the medium.
   – Head drop with distance is therefore linear in our simple system.
   – If inertial forces become important, the head drop is no longer
     linear.

• In short, if the driving forces of gravity & pressure are not
  balanced by viscous forces, Darcy’s Law does not apply.




                                                                            8
Darcy’s Law
• What happens if the head gradient is too steep?
  – The fluid will have enough energy to accelerate in
    spite of the resistance of the grains, and inertial forces
    become important.
  – In this case potential energy (head) is not dissipated
    linearly with distance and Darcy’s Law does not apply.
• How can we tell when this occurs?




            Reynolds Number
    •How can we tell when this occurs?
    We examine a measure of the relative importance
    to inertial and viscous forces:

    The Reynolds Number, Re —
    the ratio of inertial and viscous forces


                      inertial forces
             Re =
                      viscous forces




                                                                 9
Conceptual Model

      pore     sand grain

• There are two standard, simple models
  used to explain Darcy’s Law, and thus to
  explore the Reynolds number:
  – Flow in a tube,             Parabolic velocity profile

                                                             R
                                U                                 mean flow


  – Flow around an object, usually a cylinder or
    sphere,                   R
                            U                                     mean flow


      • say, representing a grain of sand.




 Flow in the vicinity of a sphere
               y



  U                R
                            x
                                                                       mean flow


                                                 Flow visualization (see slide 32)




                                                                                     10
Flow in the vicinity of a sphere
                                y

                                                     U                   R
• Inertial force per unit "u !u                                      3
                                                                                   x

  volume at any location:    !x                   " = density [M/L ]
                                                  u = local fluid velocity [L/T]
   Rate of change of momentum
                                                  U = mean approach velocity [L/T]
• Viscous force per                  ! 2u         µ = fluid dynamic viscosity [M/LT]
                                    µ 2           ! = fluid kinematic viscosity [L2 /T]
  unit volume:                       !y           x, y = Cartesian coordinates [L],
                                                     x in the direction of free
                                                       stream velocity,U
                                      inertial forces
• Dynamic similarity: Re =
                                      viscous forces
                  "u (!u !x )
         Re =                    = constant for similitude
                 µ (! 2 u !y 2 )                  Don’t worry about where the
                                                           expressions for forces come from.
                                                           H503 students, see Furbish, p. 126




   Flow in the vicinity of a sphere
                                y
 Using a dimensional                                U                    R
                                                                                   x
 analysis approach,
                                                 " = density [M/L3 ]
 assume that                                     u = local fluid velocity [L/T]
                                                 U = mean approach velocity [L/T]
  functions vary with characteristic             µ = fluid dynamic viscosity [M/LT]
  quantities U and R, thus                       ! = fluid kinematic viscosity [L2 /T]
                                                 x, y = Cartesian coordinates [L],
  u !U                                              x in the direction of free
  "u U                                                stream velocity,U
      !
  "x R
  " 2u U
       !
  "y 2 R 2




                                                                                                11
Flow in the vicinity of a sphere

• Inertial force per unit    #u    U2
                          !u    "!
  volume at any location:    #x    R

• Viscous force per unit         " 2u U
                                µ 2 !µ 2
  volume:                        "y   R


• Dynamic similarity:
                U2
              "
         Re =     R = "UR = UR = constant
                 U     µ    !
              µ
                R2




            Reynolds Number
 • For a fluid flow past a sphere
                 "UR UR
            Re =      =       U                        R
                   µ     !
 • For a flow in porous media ?
                  "qL qL
            Re =     =         q
                   µ    !
    – where characteristic velocity and length are:
        q = specific discharge
        L = characteristic pore dimension
           For sand L usually taken as the mean grain size, d50




                                                                  12
Reynolds Number
       • When does Darcy’s Law apply in a porous media?
         – For Re < 1 to 10,
                • flow is laminar and linear,
                • Darcy’s Law applies
          – For Re > 1 to 10,
                                                                               Let’s revisit
                • flow may still be laminar but no longer linear
                                                                               flow around a
                • inertial forces becoming important                           sphere to see why
                    – (e.g., flow separation & eddies)                         this non-linear
                • linear Darcy’s Law no longer applies                         flow happens.



                                      "qd 50 qd 50
                            Re =            =
                                        µ     !



           Flow in the vicinity of a sphere
                        Very low Reynolds Number




                                                                                 Laminar,
                                                                                 linear flow
       mean flow




       "UR UR
Re =      =
        µ   !                                            Source: Van Dyke, 1982, An Album of Fluid Motion




                                                                                                            13
Flow in the vicinity of a sphere
                                  Low Reynolds Number




                                       mean flow




“In these examples of a flow past a cylinder at low flow velocities a      Laminar but non-linear
small zone of reversed flow -a separation bubble- is created on the lee    flow. Non-linear because of
side as shown (flow comes from the left).”                                 the separation & eddies
       "UR UR The numerical values of Re for cylinders, are
Re =      =   ;                                                                             Source: Van Dyke, 1982,
        µ   !   not necessarily comparible to Re for porous media.                          An Album of Fluid Motion




              Flow in the vicinity of a sphere
         High Reynolds Number ! transition to turbulence




   mean flow




       "UR UR
Re =      =
        µ   !                                                        Source: Van Dyke, 1982, An Album of Fluid Motion




                                                                                                                        14
Flow in the vicinity of a sphere
                Higher Reynolds Number ! Turbulence




                    mean flow




       "UR UR
Re =      =
        µ   !                                             Source: Van Dyke, 1982, An Album of Fluid Motion




                                Turbulence
                                                     False-color image of (soap film) grid
                                                     turbulence. Flow direction is from top
                                                     to bottom, the field of view is 36 by 24
                                                     mm. The comb is located immediately
                                                     above the top edge of the image. Flow
                                                     velocity is 1.05 m/s.
                                                     http://me.unm.edu/~kalmoth/fluids/2dturb-
                                                     index.html




            Experimentally observed velocity variation
                         (here in space)
                           Velocity           It is difficult to get turbulence to occur in
                    temporal covariance         a porous media. Velocity fluctuations
                     spatial covariance          are dissipated by viscous interaction
                                                       with ubiquitous pore walls:
                         Highly chaotic
                                                 You may find it in flow through bowling ball
                                                   size sediments, or near the bore of a
                                                    pumping or injection well in gravel.




                                                                                                             15
Reynolds Number
• When does Darcy’s Law apply in a porous media?
  – For Re < 1 to 10,                   "qd 50 qd 50
     • flow is laminar and linear, Re =       =
        • Darcy’s Law applies                                        µ         !
    – For Re > 1 to 10,
        • flow is still laminar but no longer linear             q
        • inertial forces becoming important
            – (e.g., flow separation & eddies)
        • linear Darcy’s Law no longer applies
    – For higher Re > 600?,
        • flow is transitioning to turbulent
        • laminar, linear Darcy’s Law no longer applies

            Notice that Darcy’s Law starts to fail when inertial effects
            are important, even through the flow is still laminar. It is not
            turbulence, but inertia and that leads to non-linearity.
            Turbulence comes at much higher velocities.




            Specific Discharge, q
                                                dh
                            Q = ! KA
                                                dl
Suppose we want to know water “velocity.” Divide Q by A to get the
volumetric flux density, or specific discharge, often called the Darcy
velocity:              3 !1
              Q LT                   L
                = 2              =            ! units of velocity
              A   L                  T

              Q          dh                           By definition, this is the
                = q = !K                              discharge per unit bulk cross-
              A          dl                           sectional area.


But if we put dye in one end of our column and measure the time it
takes for it to come out the other end, it is much faster suggested by
the calculated q – why?




                                                                                       16
Bulk Cross-section Area, A
                                    dh
                    Q = ! KA
                                    dl
A is not the true cross-sectional area of flow. Most of A is
taken up by sand grains. The actual area available for flow is
smaller and is given by nA, where n is porosity:


                        Vvoids
     Porosity = n   =              !1
                        Vtotal




           Effective Porosity, ne
   In some materials,
   some pores may be
   essentially isolated
   and unavailable for
   flow. This brings us
   to the concept of
   effective porosity.


             V of voids available to flow
      n =                                 !n
       e                 V
                                 total




                                                                 17
Average or seepage velocity, v.
• Actual fluid velocity varies throughout the pore space,
  due to the connectivity and geometric complexity of that
  space. This variable velocity can be characterized by its
  mean or average value.
• The average fluid velocity depends on
   – how much of the cross-sectional area A is
     made up of pores, and how the pore
                                                                     q               v’
     space is connected
                                                                           v = spatial
• The typical model for average velocity is:
                                                                         average of local
                                 q
                           v=             >q                                velocity, v’
                                 ne
    v = seepage velocity
       (also known as average linear velocity,
        linear velocity, pore velocity, and interstitial velocity)




                        Darcy’s Law
• Review                                   • Next time
   – Darcy’s Law                                – Hydraulic Conductivity
      • Re restrictions                         – Porosity
                                                – Aquifer, Aquitard, etc
   – Hydraulic Conductivity

   – Specific Discharge

   – Seepage Velocity
       • Effective porosity




                                                                                            18

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L2 darcys law

  • 1. Darcy’s Law • Last time • Today – Groundwater flow is in response to gradients of – Darcy’s Law mechanical energy – Three types • Potential – Hydraulic Conductivity • Kinetic – Kinetic energy is usually not important in groundwater – Specific Discharge • Elastic (compressional) – Fluid Potential, ! – Energy per unit mass – Seepage Velocity – Hydraulic Head, h • Effective porosity – Energy per unit weight – Composed of » Pressure head » Elevation head http://biosystems.okstate.edu/darcy/index.htm Darcy’s Law A FEW CAREER HIGHLIGHTS: Henry Darcy, a French • In 1828, Darcy was assigned to a deep well hydraulic engineer drilling project that found water for the city of interested in purifying Dijon, in France, but could not provide an adequate supply for the town. However, under water supplies using his own initiative, Henry set out to provide a sand filters, conducted clean, dependable water supply to the city from more conventional spring water sources. experiments to determine That effort eventually produced a system that the flow rate of water delivered 8 m3/min from the Rosoir Spring through 12.7 km of covered aqueduct. through the filters. Published in 1856, his conclusions have served as the basis for all modern analysis of ground water flow • In 1848 he became Chief Director for Water and Pavements, Paris. In Paris he carried out significant research on the flow and friction losses in pipes, which forms the basis for the Darcy-Weisbach equation for pipe flow. • He retired to Dijon and, in 1855 and 1856, he conducted the column experiments that established Darcy's law for flow in sands. Freeze, R. Allen. "Henry Darcy and the Fountains of Dijon." Ground Water 32, no.1(1994): 23–30. 1
  • 2. Darcy’s Law Cartoon of a Darcy experiment: Sand-filled column Constant head tanks with bulk cross-sectional area A at each end # Q - Rate of "l discharge [L3/T] dh h1 # h2 Datum In this experiment, water moves due to gravity. What is the relationship between discharge (flux) Q and other variables? A plot of Darcy’s actual data: http://biosystems.okstate.edu/darcy/index.htm 2
  • 3. What is the relationship between discharge (flux) Q and other variables? Darcy found that: 1) If he doubled the head difference (dh = h2 – h1), then the discharge Q doubled. 2) If he doubled column length (dl), Q halved 3) If he doubled column area (A), Q doubled. Sand-filled column Constant head tanks with cross-sectional area A at each end # Conclusions: "l Q - Rate of discharge [L3/T] Q is proportional to dh "h h1 Q is proportional to 1/dl # h2 Q is proportional to A Datum Darcy’s Law Darcy also found that if he used different kinds of sands in the column, discharge Q changed, but for a particular sand, regardless of Q: Q dl = a constant A dh This “proportionality constant” is usually called “hydraulic conductivity” and often is assigned the symbol, K. Leads to Darcy’s Law: dh Q = ! KA =conductivity x bulk area x ‘gradient’ dl 3
  • 4. Darcy’s Law dh Q = ! KA dl Why is there a minus sign in Darcy’s Law? “a little vector calculus” (Bradley and Smith, 1995) Darcy’s Law Invert Darcy’s Law to express conductivity in terms of discharge, area, and gradient: Q & '1 # K= A $ dh dl ! % " This is how we measure conductivity: Measure area A; that can be easy. Measure gradient, -dh/dl; can be harder*. Measure discharge, Q; can also be hard. *Imagine trying to measure gradient in a complex geology with three-dimensional flow and few observation points. 4
  • 5. Darcy’s Law What are the dimensions of K? Dimensional analysis: Q dl & ( L3T '1 )( L) # & L # K =' = = A dh $ ( L2 )( L) ! $ T ! % " % " = the dimensions of speed or velocity e.g, m/s, meters per second, in SI units Darcy’s Law dh Q = ! KA dl This expresses Darcy’s Law in terms of discharge. We can also express it in terms of “Darcy velocity” or “specific discharge”, that is, discharge per unit bulk area, A: Q dh &L# q= ='K = specific discharge $ ! A dl %T " This ‘flux density’ expression is the more common and convenient way to express Darcy’s law. For example, it applies even when area is varying, and with a little generalization also when gradient or conductivity is varying. 5
  • 6. Darcy’s Law We will look at three major topics important to Darcy’s Law: • Hydraulic Head dh Gradient Q = ! KA dl • Bulk cross-sectional area of flow • Hydraulic Conductivity (next time) Hydraulic Head • Head is a measure of the total mechanical energy per unit weight. • If K, Q and A don’t change with distance, then – hydraulic head varies linearly with distance Q dh = $ dl " dh ! dl "# KA # In this experiment, with constant K and A, the head drops linearly with distance, and # the specific discharge is constant. dh q=!K = constant dl 6
  • 7. Hydraulic Head • In Darcy’s experiment, do the drops falling from the constant head tanks have constant velocity? Sand-filled column Constant head tanks with cross-sectional area A at each end # Q - Rate of "l discharge [L3/T] "h h1 # h2 Datum Hydraulic Head • In Darcy’s experiment, do the drops falling from the constant head tanks have constant velocity? • No! Water droplets accelerate at 9.8 m/s2. • Assuming no air resistance, the falling water drop has its potential energy converted to kinetic energy as it falls: 1 2 mgz + mv = total energy = constant, 2 that is, its a conservative energy field. But water through our column has constant velocity—why? 7
  • 8. Hydraulic Head But water through our column has constant velocity—why? # In this experiment, head level with constant K, Q and A, the head drops linearly with distance, and # the specific discharge is constant. dh q=!K = constant dl In a porous medium, the tendency of the fluid to accelerate is opposed by friction against the grains. 1 mgz + pV + mv 2 = total energy ! constant 2 ~0 It’s a dissipative energy field. Where does the energy go? Friction ! Heat; Mechanical Energy ! Thermal Energy Darcy’s Law • Could we use Darcy’s Law to model the falling drops? • Darcy’s Law works because the driving forces (gravity and pressure) in the fluid are balanced by the viscous resistance of the medium. – Head drop with distance is therefore linear in our simple system. – If inertial forces become important, the head drop is no longer linear. • In short, if the driving forces of gravity & pressure are not balanced by viscous forces, Darcy’s Law does not apply. 8
  • 9. Darcy’s Law • What happens if the head gradient is too steep? – The fluid will have enough energy to accelerate in spite of the resistance of the grains, and inertial forces become important. – In this case potential energy (head) is not dissipated linearly with distance and Darcy’s Law does not apply. • How can we tell when this occurs? Reynolds Number •How can we tell when this occurs? We examine a measure of the relative importance to inertial and viscous forces: The Reynolds Number, Re — the ratio of inertial and viscous forces inertial forces Re = viscous forces 9
  • 10. Conceptual Model pore sand grain • There are two standard, simple models used to explain Darcy’s Law, and thus to explore the Reynolds number: – Flow in a tube, Parabolic velocity profile R U mean flow – Flow around an object, usually a cylinder or sphere, R U mean flow • say, representing a grain of sand. Flow in the vicinity of a sphere y U R x mean flow Flow visualization (see slide 32) 10
  • 11. Flow in the vicinity of a sphere y U R • Inertial force per unit "u !u 3 x volume at any location: !x " = density [M/L ] u = local fluid velocity [L/T] Rate of change of momentum U = mean approach velocity [L/T] • Viscous force per ! 2u µ = fluid dynamic viscosity [M/LT] µ 2 ! = fluid kinematic viscosity [L2 /T] unit volume: !y x, y = Cartesian coordinates [L], x in the direction of free stream velocity,U inertial forces • Dynamic similarity: Re = viscous forces "u (!u !x ) Re = = constant for similitude µ (! 2 u !y 2 ) Don’t worry about where the expressions for forces come from. H503 students, see Furbish, p. 126 Flow in the vicinity of a sphere y Using a dimensional U R x analysis approach, " = density [M/L3 ] assume that u = local fluid velocity [L/T] U = mean approach velocity [L/T] functions vary with characteristic µ = fluid dynamic viscosity [M/LT] quantities U and R, thus ! = fluid kinematic viscosity [L2 /T] x, y = Cartesian coordinates [L], u !U x in the direction of free "u U stream velocity,U ! "x R " 2u U ! "y 2 R 2 11
  • 12. Flow in the vicinity of a sphere • Inertial force per unit #u U2 !u "! volume at any location: #x R • Viscous force per unit " 2u U µ 2 !µ 2 volume: "y R • Dynamic similarity: U2 " Re = R = "UR = UR = constant U µ ! µ R2 Reynolds Number • For a fluid flow past a sphere "UR UR Re = = U R µ ! • For a flow in porous media ? "qL qL Re = = q µ ! – where characteristic velocity and length are: q = specific discharge L = characteristic pore dimension For sand L usually taken as the mean grain size, d50 12
  • 13. Reynolds Number • When does Darcy’s Law apply in a porous media? – For Re < 1 to 10, • flow is laminar and linear, • Darcy’s Law applies – For Re > 1 to 10, Let’s revisit • flow may still be laminar but no longer linear flow around a • inertial forces becoming important sphere to see why – (e.g., flow separation & eddies) this non-linear • linear Darcy’s Law no longer applies flow happens. "qd 50 qd 50 Re = = µ ! Flow in the vicinity of a sphere Very low Reynolds Number Laminar, linear flow mean flow "UR UR Re = = µ ! Source: Van Dyke, 1982, An Album of Fluid Motion 13
  • 14. Flow in the vicinity of a sphere Low Reynolds Number mean flow “In these examples of a flow past a cylinder at low flow velocities a Laminar but non-linear small zone of reversed flow -a separation bubble- is created on the lee flow. Non-linear because of side as shown (flow comes from the left).” the separation & eddies "UR UR The numerical values of Re for cylinders, are Re = = ; Source: Van Dyke, 1982, µ ! not necessarily comparible to Re for porous media. An Album of Fluid Motion Flow in the vicinity of a sphere High Reynolds Number ! transition to turbulence mean flow "UR UR Re = = µ ! Source: Van Dyke, 1982, An Album of Fluid Motion 14
  • 15. Flow in the vicinity of a sphere Higher Reynolds Number ! Turbulence mean flow "UR UR Re = = µ ! Source: Van Dyke, 1982, An Album of Fluid Motion Turbulence False-color image of (soap film) grid turbulence. Flow direction is from top to bottom, the field of view is 36 by 24 mm. The comb is located immediately above the top edge of the image. Flow velocity is 1.05 m/s. http://me.unm.edu/~kalmoth/fluids/2dturb- index.html Experimentally observed velocity variation (here in space) Velocity It is difficult to get turbulence to occur in temporal covariance a porous media. Velocity fluctuations spatial covariance are dissipated by viscous interaction with ubiquitous pore walls: Highly chaotic You may find it in flow through bowling ball size sediments, or near the bore of a pumping or injection well in gravel. 15
  • 16. Reynolds Number • When does Darcy’s Law apply in a porous media? – For Re < 1 to 10, "qd 50 qd 50 • flow is laminar and linear, Re = = • Darcy’s Law applies µ ! – For Re > 1 to 10, • flow is still laminar but no longer linear q • inertial forces becoming important – (e.g., flow separation & eddies) • linear Darcy’s Law no longer applies – For higher Re > 600?, • flow is transitioning to turbulent • laminar, linear Darcy’s Law no longer applies Notice that Darcy’s Law starts to fail when inertial effects are important, even through the flow is still laminar. It is not turbulence, but inertia and that leads to non-linearity. Turbulence comes at much higher velocities. Specific Discharge, q dh Q = ! KA dl Suppose we want to know water “velocity.” Divide Q by A to get the volumetric flux density, or specific discharge, often called the Darcy velocity: 3 !1 Q LT L = 2 = ! units of velocity A L T Q dh By definition, this is the = q = !K discharge per unit bulk cross- A dl sectional area. But if we put dye in one end of our column and measure the time it takes for it to come out the other end, it is much faster suggested by the calculated q – why? 16
  • 17. Bulk Cross-section Area, A dh Q = ! KA dl A is not the true cross-sectional area of flow. Most of A is taken up by sand grains. The actual area available for flow is smaller and is given by nA, where n is porosity: Vvoids Porosity = n = !1 Vtotal Effective Porosity, ne In some materials, some pores may be essentially isolated and unavailable for flow. This brings us to the concept of effective porosity. V of voids available to flow n = !n e V total 17
  • 18. Average or seepage velocity, v. • Actual fluid velocity varies throughout the pore space, due to the connectivity and geometric complexity of that space. This variable velocity can be characterized by its mean or average value. • The average fluid velocity depends on – how much of the cross-sectional area A is made up of pores, and how the pore q v’ space is connected v = spatial • The typical model for average velocity is: average of local q v= >q velocity, v’ ne v = seepage velocity (also known as average linear velocity, linear velocity, pore velocity, and interstitial velocity) Darcy’s Law • Review • Next time – Darcy’s Law – Hydraulic Conductivity • Re restrictions – Porosity – Aquifer, Aquitard, etc – Hydraulic Conductivity – Specific Discharge – Seepage Velocity • Effective porosity 18