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    Hydrodynamics inPorous Media

    We will cover:

    How fluids respond to local potential

    gradients (Darcys Law)

    Add the conservation of mass to

    obtain Richards equation

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    Darcys Law for saturated mediaIn 1856 Darcy hired to figure out the water supply to the

    towns central fountain.Experimentally found that flux of water porous media could

    be expressed as the product of the resistance to flowwhich characterized the media, and forces acting to pushthe fluid through the media.

    Q - The rate of flow (L3/T) as the volume of water passed through acolumn per unit time.

    hi - The fluid potential in the media at position i, measured instanding head equivalent. Under saturated conditions this iscomposed of gravitational potential (elevation), and static pressure

    potential (L: force per unit area divided by g).K - The hydraulic conductivity of the media. The proportionality

    between specific flux and imposed gradient for a given medium (L/T).

    L - The length of media through which flow passes (L).

    A - The cross-sectional area of the column (L2).

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    Darcys Law

    Darcy then observed that the flow of water in a vertical

    column was well described by the equation

    Darcys expression is written in a general form forisotropic media as

    q is the specific flux vector (L/T; volume of water perunit area per unit time),

    K is the saturated hydraulic conductivity tensor(second rank) of the media (L/T), and

    H is the gradient in hydraulic head (dimensionless)

    Q= K )H(HL

    A0-1 [2.68

    q = -K H [2.69]

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    Aside on calculus ...

    What is this up-side-down triangle all about?

    The dell operator: short hand for 3-d derivative

    The result of operating on a scalar function (like potential) with is the slope of the function

    F points directly towards the steepest direction of up hill with alength proportional to the slope of the hill.

    Later well use F. The dot just tells us to take the dell andcalculate the dot product of that and the function F (which needs tobe a vector for this to make sense).

    dell-dot-F is the divergence of F.

    If F were local flux (with magnitude and direction), F would bethe amount of water leaving the point x,y,z. This is a scalar result!

    F takes a scalar function F and gives a vector slope

    F uses a vector function F and gives a scalar result.

    xi,y

    j,z

    k

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    Now, about those parameters...

    Gradient in head is dimensionless, being length per length

    Q = Aq Q has units volume per unit time

    Specific flux, q, has units of length per time, orvelocity.

    For vertical flow: speed at which the height of a pondof fluid would drop

    CAREFUL: q is not the velocity of particles of water

    The specific flux is a vector (magnitude and direction).

    Potential expressed as the height of a column ofwater, has units of length.

    LHH=H 01 [2.70]

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    About those vectors...

    Is the right side of Darcys law indeed a vector?

    h is a scalar, but H is a vector

    Since K is a tensor (yikes), KH is a vectorSo all is well on the right hand side

    Notes on K:we could also obtain a vector on the right handside by selecting K to be a scalar, which is oftendone (i.e., assuming that conductivity is

    independent of direction).

    q = -KH [2.70]

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    A few words about the K tensor

    Kab relates gradients in potential in the b-direction toflux that results in the a-direction.

    In anisotropic media, gradients not aligned withbedding give flux not parallel with potential gradients. If

    the coordinate system is aligned with directions ofanisotropy the "off diagonal terms will be zero (i.e.,Kab=0 where ab). If, in addition, these are all equal,then the tensor collapses to a scalar.

    The reason to use the tensor form is to capture theeffects of anisotropy.

    q = -

    Kxx Kxy Kxz

    Kyx Kyy KyzKzx Kzy Kzz

    h

    x h

    y h

    z

    = -

    Kxxhx

    +K xy hy

    +K xzhz

    ;K yxhx

    +K yy hy

    +K yzhz

    ;K zxhx

    +K zy hy

    +K zzhz

    flux in x-direction flux in y-direction flux in z-direction

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    Looking holisticallyCheck out the intuitively aspects of Darcys result.

    The rate of flow is:Directly related to the area of flow (e.g., put two

    columns in parallel and you get twice the flow);

    Inversely related to the length of flow (e.g., flowthrough twice the length with the same potentialdrop gives half the flux);

    Directly related to the potential energy dropacross the system (e.g., double the energyexpended to obtain twice the flow).

    The expression is patently linear; all propertiesscale linearly with changes in system forces and

    dimensions.

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    Why is Darcy Linear?Because non-turbulent?

    No.Far before turbulence, there will be large localaccelerations: it is the lack of local acceleration which

    makes the relationship linear.Consider the Navier Stokes Equation for fluid flow.The x-component of flow in a velocity field withvelocities u, v, and w in the x, y, and z (vertical)

    directions, may be written

    ut

    + u ux

    + vuy

    + wuz

    =-1

    Px

    - gzx

    +

    u

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    Creeping flowNow impose the conditions needed for which Darcys Law

    Creeping flow; acceleration (du/dx) terms smallcompared to the viscous and gravitational terms

    Similarly changes in velocity with time are small

    so N-S is:

    Linear in gradient of hydraulic potential on left, proportionalto velocity and viscosity on right (same as Darcy).

    Proof of Darcys Law? No! Shows that the creeping flowassumption is sufficient to obtain correct form.

    uxuyuz0 [2.69]

    ut0 [2.70]

    xPgz

    2u [2.71]

    u

    t+ u

    u

    x+ v

    u

    y+ w

    u

    z=

    -1

    P

    x- g

    z

    x+

    u

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    Capillary tube model for flowWidely used model for flow through porous media is a group

    of cylindrical capillary tubes (e.g.,. Green and Ampt, 1911and many more).

    Lets derive the equation for steady flow through a

    capillary of radius ro

    Consider forces on cylindrical control volume shown F = 0 [2.75]

    V

    0

    s

    r

    ro

    s

    Cy lindrica l Control Volum e

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    Force Balance on Control Volume

    end pressures:

    at S = 0 F1 = Pr2at S = S F2 = (P + S dP/dS) r

    2

    shear force: Fs = 2rS

    where is the local shear stress

    Putting these in the force balance gives

    Pr2 - (P + S dP/dS) r2 - 2rS = 0 [2.76]

    where we remember that dP/dS is negative in sign (pressure

    drops along the direction of flow)

    V

    0

    s

    r

    ro

    s

    Cylindrical Control Volume

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    continuing the force balance

    With some algebra, this simplifies to

    dP/dS is constant: shear stress varieslinearly with radius

    From the definition of viscosity

    Using this [2.77] says

    Multiply both sides

    by dr, and integrate

    = -dS

    dPr

    2[2.77

    Pr2 - (P + S dP/dS) r2 - 2rS = 0 [2.76]

    dr

    dv [2.78]

    dr

    dv= r

    2 dPdS

    [2.79]

    )(

    0

    rvv

    v

    dv =

    r

    rr o

    dr

    dS

    dPr

    2

    [2.80]

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    Computing the flux through the pipe...

    Carrying out the integration we find

    which gives the velocity profile in a cylindrical pipe

    To calculate the flux integrate over the area

    in cylindrical coordinates, dA = r d dr, thus

    v(r) = ( )r2-r02

    4 dPdS [2.81]

    Q = Area

    vdA [2.82]

    Q =

    =

    r = 0

    r = r 0

    r 2 - r 02

    4

    d P

    d S r d r d [2 .8 0 ]

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    Rearranging terms...The integral is easy to compute, giving

    (fourth power!!)

    which is the well known Hagen-Poiseuille Equation.

    We are interested in the flow per unit area (flux), forwhich we use the symbol q = Q/r2

    (second power)

    We commonly measure pressure in terms of hydraulichead, so we may substitute gh = P, to obtain

    Q = -

    r04

    8 dPdS [2.84]

    q= -1

    r02

    8 dP

    dS [2.85]

    q= -

    r02

    8

    dh

    dS [2.86]

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    r02/8 is a geometric term: function of the media.

    referred to as the intrinsic permeability, denoted by . is a function of the fluid alone

    NOTICE:

    Recovered Darcys law!See why by pulling out of the hydraulic

    conductivity we obtain an intrinsic property of the solidwhich can be applied to a range of fluids.

    SO if K is the saturated hydraulic conductivity, K= .This way we can calculate the effective conductivity forany fluid. This is very useful when dealing with oil spills

    ... boiling water spills ..... etc.

    q = -r 02

    8

    d h

    d S[2 .83]

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    Darcy's Law at Re# > 1Often noted that Darcy's Law breaks down at Re# > 1.

    Laminar flow holds capillaries for Re < 2000; Hagen-Poiseuille law still valid

    Why does Darcy's law break down so soon?Laminar ends for natural media at Re#>100 due to the

    tortuosity of the flow paths (see Bear, 1972, pg 178).Still far above the value required for the break down of

    Darcy's law.

    Real Reason: due to forces in acceleration of fluidspassing particles at the microscopic level being aslarge as viscous forces: increased resistance to flow,so flux responds less to applied pressure gradients.

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    A few more words about Re#>1Can get a feel for this

    through a simplecalculation of therelative magnitudes ofthe viscous and inertial

    forces.FI Fv when Re# 10.

    Since FI go with v2,

    while Fv goes with v,

    at Re# 1 FI Fv/10,

    a reasonable cut-off forcreeping flow

    approximation

    d2

    d2

    d1

    d1

    Flow

    Isometric View

    v1 v2

    Cross-Section

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    Deviations from Darcys law

    (a) The effect of inertialterms becoming significantat Re>1.

    (b) At very low flow theremay be a threshold

    gradient required to beovercome before any flowoccurs at all due to

    hydrogen bonding of water.

    q

    h

    Darcy'sLa

    w

    0 1 10 100

    Re=0

    Re=1

    Re=10

    Re=100

    K

    1

    q

    h

    Darcy'sLa

    w

    0

    K

    1

    0

    ThresholdPressure

    (b)

    (a)

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    How does this apply to Vadose?Consider typical water flow where v and d are maximized

    Gravity driven flow near saturation in a coarse media.maximum neck diameter will be about 1 mm,

    vertical flux may be as high as 1 cm/min (14 meters/day).

    [2.100]

    Typically Darcy's OK for vadose zone.

    Can have problems around wells

    R = d 1

    v1

    = 1gr/cm3

    x0.1cmx1cm/min0.01gr/cm-sec

    = 0.167

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    What about Soil Vapor Extraction?Does Darcy's law apply?

    Air velocities can exceed 30 m/day (0.035 cm/sec).The Reynolds number for this air flow rate in thecoarse soil used in the example considered above is

    [2.101]

    again, no problem, although flow could be higher

    than the average bulk flow about inlets and outlets

    R =d 1 v1

    =0.001gr/cm 3 x0.1cmx0.035cm/sec

    1.8x10 -4 gr/cm-sec

    = 0.02

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    Summary of Darcy and PoiseuilleFor SATURATED MEDIA

    Flow is linear with permeability and gradientin potential (driving force)

    At high flow rates becomes non-linear due to

    local accelerationPermeability is due to geometric properties of

    the media (intrinsic permeability) and fluidproperties (viscosity and specific density)

    Permeability drops with the square of poresize

    Assumed no slip solid-liquid boundary:

    doesn't work with gas.