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Hydrodynamics in Hydrodynamics in Porous MediaPorous Media

We will cover:How fluids respond to local potential

gradients (Darcy’s Law)

Add the conservation of mass to obtain Richard’s equation

We will cover:How fluids respond to local potential

gradients (Darcy’s Law)

Add the conservation of mass to obtain Richard’s equation

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Darcy’s Law for saturated mediaDarcy’s Law for saturated mediaIn 1856 Darcy hired to figure out the water supply to the

town’s central fountain. Experimentally found that flux of water porous media

could be expressed as the product of the resistance to flow which characterized the media, and forces acting to “push” the fluid through the media.

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

hi - The fluid potential in the media at position i, measured in standing head equivalent. Under saturated conditions this is composed 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).

In 1856 Darcy hired to figure out the water supply to the town’s central fountain.

Experimentally found that flux of water porous media could be expressed as the product of the resistance to flow which characterized the media, and forces acting to “push” the fluid through the media.

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

hi - The fluid potential in the media at position i, measured in standing head equivalent. Under saturated conditions this is composed 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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Darcy’s LawDarcy’s LawDarcy then observed that the flow of water in a vertical column was well described by the equation

Darcy’s expression is written in a general form for isotropic media as

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

Darcy then observed that the flow of water in a vertical column was well described by the equation

Darcy’s expression is written in a general form for isotropic media as

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

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Aside on calculus ...Aside on calculus ...

What is this up-side-down triangle all about?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 a length proportional to the slope of the hill.

Later we’ll use •F. The dot just tells us to take the dell and calculate the dot product of that and the function F (which needs to be 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 be

the 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.

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 a length proportional to the slope of the hill.

Later we’ll use •F. The dot just tells us to take the dell and calculate the dot product of that and the function F (which needs to be 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 be

the 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.

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Now, about those parameters...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, or velocity. For vertical flow: speed at which the height of a pond of 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 of water, has units of length.

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, or velocity. For vertical flow: speed at which the height of a pond of 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 of water, has units of length.

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About those vectors...About those vectors...

Is the right side of Darcy’s law indeed a vector? h is a scalar, but H is a vectorSince K is a tensor (yikes), KH is a vector

So all is well on the right hand side

Notes on K: we could also obtain a vector on the right hand side by

selecting K to be a scalar, which is often done (i.e., assuming that conductivity is independent of direction).

Is the right side of Darcy’s law indeed a vector? h is a scalar, but H is a vectorSince K is a tensor (yikes), KH is a vector

So all is well on the right hand side

Notes on K: we could also obtain a vector on the right hand side by

selecting K to be a scalar, which is often done (i.e., assuming that conductivity is independent of direction).

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A few words about the K tensorA few words about the K tensor

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

In anisotropic media, gradients not aligned with bedding give flux not parallel with potential gradients. If the coordinate system is aligned with directions of anisotropy 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 the effects of anisotropy.

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

In anisotropic media, gradients not aligned with bedding give flux not parallel with potential gradients. If the coordinate system is aligned with directions of anisotropy 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 the effects of anisotropy.

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

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Looking holisticallyLooking holisticallyCheck out the intuitively aspects of Darcy’s 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., flow

through twice the length with the same potential drop gives half the flux);

Directly related to the potential energy drop across the system (e.g., double the energy expended to obtain twice the flow).

The expression is patently linear; all properties scale linearly with changes in system forces and dimensions.

Check out the intuitively aspects of Darcy’s 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., flow

through twice the length with the same potential drop gives half the flux);

Directly related to the potential energy drop across the system (e.g., double the energy expended to obtain twice the flow).

The expression is patently linear; all properties scale linearly with changes in system forces and dimensions.

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Why is Darcy Linear?Why is Darcy Linear?Because non-turbulent?

No.Far before turbulence, there will be large local accelerations: 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 with velocities u, v, and w in the x, y, and z (vertical) directions, may be written

Because non-turbulent? No.

Far before turbulence, there will be large local accelerations: 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 with velocities u, v, and w in the x, y, and z (vertical) directions, may be written

ut + u

ux+ v

uy+ w

uz =

-1

Px - g

zx +

u

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Creeping flowCreeping flowNow impose the conditions needed for which Darcy’s Law“Creeping flow”; acceleration (du/dx) terms small compared 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, proportional to velocity and viscosity on right (same as Darcy).

Proof of Darcy’s Law? No! Shows that the creeping flow assumption is sufficient to obtain correct form.

Now impose the conditions needed for which Darcy’s Law“Creeping flow”; acceleration (du/dx) terms small compared 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, proportional to velocity and viscosity on right (same as Darcy).

Proof of Darcy’s Law? No! Shows that the creeping flow assumption is sufficient to obtain correct form.

ux

uy

uz

0 [2.69]

ut

0 [2.70] x

Pgz

2u [2.71]

ut + u

ux+ v

uy+ w

uz =

-1

Px - g

zx +

u

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Capillary tube model for flowCapillary tube model for flowWidely used model for flow through porous media is a group of cylindrical capillary tubes (e.g.,. Green and Ampt, 1911 and many more). Let’s derive the equation for steady flow through a

capillary of radius ro

Consider forces on cylindrical control volume shown

F = 0 [2.75]

Widely used model for flow through porous media is a group of cylindrical capillary tubes (e.g.,. Green and Ampt, 1911 and many more). Let’s derive the equation for steady flow through a

capillary of radius ro

Consider forces on cylindrical control volume shown

F = 0 [2.75]

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Force Balance on Control VolumeForce Balance on Control Volume

end pressures:

at S = 0 F1 = Pr2

at S = S F2 = (P + S dP/dS) r2

shear force: Fs = 2rSwhere 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)

end pressures:

at S = 0 F1 = Pr2

at S = S F2 = (P + S dP/dS) r2

shear force: Fs = 2rSwhere 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)

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continuing the force balancecontinuing the force balance

With some algebra, this simplifies to

dP/dS is constant: shear stress varies linearly with radius

From the definition of viscosity

Using this [2.77] says

Multiply both sides by dr, and integrate

With some algebra, this simplifies to

dP/dS is constant: shear stress varies linearly with radius

From the definition of viscosity

Using this [2.77] says

Multiply both sides by dr, and integrate

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

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Computing the flux through the pipe...Computing the flux through the pipe...Carrying out the integration we find

which gives the velocity profile in a cylindrical pipeTo calculate the flux integrate over the area

in cylindrical coordinates, dA = r d dr, thus

Carrying out the integration we find

which gives the velocity profile in a cylindrical pipeTo calculate the flux integrate over the area

in cylindrical coordinates, dA = r d dr, thus

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Rearranging terms...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), for which we use the symbol q = Q/r2

(second power)

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

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), for which we use the symbol q = Q/r2

(second power)

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

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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 Darcy’s law!See why by pulling out of the hydraulic conductivity we

obtain an intrinsic property of the solid which 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 for any fluid. This is very useful when dealing with oil spills ... boiling water spills ..... etc.

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 Darcy’s law!See why by pulling out of the hydraulic conductivity we

obtain an intrinsic property of the solid which 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 for any fluid. This is very useful when dealing with oil spills ... boiling water spills ..... etc.

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Darcy's Law at Re# > 1Darcy'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 fluids passing particles at the microscopic level being as large as viscous forces: increased resistance to flow, so flux responds less to applied pressure gradients.

Often 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 fluids passing particles at the microscopic level being as large 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#>1A few more words about Re#>1Can get a feel for this through a simple calculation of the relative magnitudes of the 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 for creeping flow approximation

Can get a feel for this through a simple calculation of the relative magnitudes of the 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 for creeping flow approximation

d2

d2

d1

d1

Flow

Isometric View

v1 v2

Cross-Section

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Deviations from Darcy’s lawDeviations from Darcy’s law

(a) The effect of inertial terms becoming significant at Re>1.

(b) At very low flow there may be a threshold gradient required to be overcome before any flow occurs at all due to hydrogen bonding of water.

(a) The effect of inertial terms becoming significant at Re>1.

(b) At very low flow there may be a threshold gradient required to be overcome before any flow occurs at all due to hydrogen bonding of water.

q

h

Darcy's

Law

0 1 10 100

Re=0

Re=1

Re=10

Re=100

K

1

q

h

Darcy's

Law

0

K

1

0

Threshold Pressure

(b)

(a)

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

Consider typical water flow where v and d are maximizedGravity 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

= 1 gr/cm3 x 0.1 cm x 1 cm/min

 0.01 gr/cm-sec

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What about Soil Vapor Extraction?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 the coarse 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

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 the coarse 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

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Summary of Darcy and PoiseuilleSummary of Darcy and PoiseuilleFor SATURATED MEDIAFlow is linear with permeability and gradient in 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 fluid properties (viscosity and specific density)

Permeability drops with the square of pore sizeAssumed no slip solid-liquid boundary: doesn't work with

gas.

For SATURATED MEDIAFlow is linear with permeability and gradient in 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 fluid properties (viscosity and specific density)

Permeability drops with the square of pore sizeAssumed no slip solid-liquid boundary: doesn't work with

gas.