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Curved Capillary Curved Capillary BarriersBarriers

Making better use of gravityMaking better use of gravityMaking better use of gravityMaking better use of gravity

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The Idea: Use the gradient optimallyThe Idea: Use the gradient optimallyA capillary barrier under uniform application, the A capillary barrier under uniform application, the

diverted flux increases linearly horizontally. diverted flux increases linearly horizontally. In the case of a barrier where the slope in the down dip In the case of a barrier where the slope in the down dip

direction is constant, first failure will always be near the direction is constant, first failure will always be near the down-dip end of the interface, while the upper portion down-dip end of the interface, while the upper portion of the interface could handle much more flow without of the interface could handle much more flow without local penetration. local penetration.

if the slope were greater toward the down-dip portion, if the slope were greater toward the down-dip portion, the diversion could be increased prior to breakthrough. the diversion could be increased prior to breakthrough.

It would appear logical to increase the slope of the It would appear logical to increase the slope of the barrier in proportion to the expected local flux, which barrier in proportion to the expected local flux, which increases linearly with down-slope distance. increases linearly with down-slope distance.

A capillary barrier under uniform application, the A capillary barrier under uniform application, the diverted flux increases linearly horizontally. diverted flux increases linearly horizontally.

In the case of a barrier where the slope in the down dip In the case of a barrier where the slope in the down dip direction is constant, first failure will always be near the direction is constant, first failure will always be near the down-dip end of the interface, while the upper portion down-dip end of the interface, while the upper portion of the interface could handle much more flow without of the interface could handle much more flow without local penetration. local penetration.

if the slope were greater toward the down-dip portion, if the slope were greater toward the down-dip portion, the diversion could be increased prior to breakthrough. the diversion could be increased prior to breakthrough.

It would appear logical to increase the slope of the It would appear logical to increase the slope of the barrier in proportion to the expected local flux, which barrier in proportion to the expected local flux, which increases linearly with down-slope distance. increases linearly with down-slope distance.

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More on optimal use...More on optimal use...optimal use of the total fall in elevation is achieved optimal use of the total fall in elevation is achieved

when the break-through flux through the interface is when the break-through flux through the interface is constant along the entire length of the interface. constant along the entire length of the interface.

If we posit to have selected a barrier geometry such If we posit to have selected a barrier geometry such that this flux penetrates the barrier at all points, thenthat this flux penetrates the barrier at all points, then the pressure is also uniform along the textural interface. the pressure is also uniform along the textural interface. the vertical moisture profile in the upper media is uniform the vertical moisture profile in the upper media is uniform there will be no gradients in pressure in directions tangent there will be no gradients in pressure in directions tangent

to the interface. to the interface.

Barrier parallel flux will be driven by gravitational Barrier parallel flux will be driven by gravitational potential, equal to the local slope of the interface. potential, equal to the local slope of the interface.

optimal use of the total fall in elevation is achieved optimal use of the total fall in elevation is achieved when the break-through flux through the interface is when the break-through flux through the interface is constant along the entire length of the interface. constant along the entire length of the interface.

If we posit to have selected a barrier geometry such If we posit to have selected a barrier geometry such that this flux penetrates the barrier at all points, thenthat this flux penetrates the barrier at all points, then the pressure is also uniform along the textural interface. the pressure is also uniform along the textural interface. the vertical moisture profile in the upper media is uniform the vertical moisture profile in the upper media is uniform there will be no gradients in pressure in directions tangent there will be no gradients in pressure in directions tangent

to the interface. to the interface.

Barrier parallel flux will be driven by gravitational Barrier parallel flux will be driven by gravitational potential, equal to the local slope of the interface. potential, equal to the local slope of the interface.

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For a given location, the lateral rate of flow, Q, is For a given location, the lateral rate of flow, Q, is given by:given by:

T is the thickness of the interface, T is the thickness of the interface, K(z) is the hydraulic conductivity a distance z K(z) is the hydraulic conductivity a distance z

above the interface, and above the interface, and S is the local slope of the interface. S is the local slope of the interface. Since the vertical moisture profile is identical at all Since the vertical moisture profile is identical at all

locations, the integral is constant locations, the integral is constant

For a given location, the lateral rate of flow, Q, is For a given location, the lateral rate of flow, Q, is given by:given by:

T is the thickness of the interface, T is the thickness of the interface, K(z) is the hydraulic conductivity a distance z K(z) is the hydraulic conductivity a distance z

above the interface, and above the interface, and S is the local slope of the interface. S is the local slope of the interface. Since the vertical moisture profile is identical at all Since the vertical moisture profile is identical at all

locations, the integral is constant locations, the integral is constant

So let’s do some math...So let’s do some math...

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Two possible geometriesTwo possible geometries

"Circular-roof" Barrier

Fine

Coarse

"Linear-roof" Barrier

Fine

Coarse

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Linear Roof SolutionLinear Roof SolutionFor a linear-roof design, Q = q x,For a linear-roof design, Q = q x,x = horizontal distance from crestx = horizontal distance from crestq = incident flux. q = incident flux.

But since Q = SC we immediately find thatBut since Q = SC we immediately find that

using the using the

parameterizationparameterizationPutting this into the governing eq.Putting this into the governing eq.

integrating:integrating:

use of a parabolic interface for linear-roof capillary barriers will provide use of a parabolic interface for linear-roof capillary barriers will provide optimal diversion.optimal diversion.

For a linear-roof design, Q = q x,For a linear-roof design, Q = q x,x = horizontal distance from crestx = horizontal distance from crestq = incident flux. q = incident flux.

But since Q = SC we immediately find thatBut since Q = SC we immediately find that

using the using the

parameterizationparameterizationPutting this into the governing eq.Putting this into the governing eq.

integrating:integrating:

use of a parabolic interface for linear-roof capillary barriers will provide use of a parabolic interface for linear-roof capillary barriers will provide optimal diversion.optimal diversion.

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What does this buy us?What does this buy us?To estimate the improvement in barrier performance To estimate the improvement in barrier performance

obtained using the curved interface, consider a barrier obtained using the curved interface, consider a barrier which has a total fall of H over a horizontal distance L. which has a total fall of H over a horizontal distance L.

For the straight barrier, we get a diversion capacity of CL/HFor the straight barrier, we get a diversion capacity of CL/H

For a parabolic system we get a diversion of 2CL/HFor a parabolic system we get a diversion of 2CL/H

The parabolic interface provides a factor of two increase in The parabolic interface provides a factor of two increase in diversion capacity for a given total elevation loss.diversion capacity for a given total elevation loss.

To estimate the improvement in barrier performance To estimate the improvement in barrier performance obtained using the curved interface, consider a barrier obtained using the curved interface, consider a barrier which has a total fall of H over a horizontal distance L. which has a total fall of H over a horizontal distance L.

For the straight barrier, we get a diversion capacity of CL/HFor the straight barrier, we get a diversion capacity of CL/H

For a parabolic system we get a diversion of 2CL/HFor a parabolic system we get a diversion of 2CL/H

The parabolic interface provides a factor of two increase in The parabolic interface provides a factor of two increase in diversion capacity for a given total elevation loss.diversion capacity for a given total elevation loss.

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How about a circular cover?How about a circular cover?It is straightforward to compute the optimum shape of a It is straightforward to compute the optimum shape of a

circular-roof diversion. circular-roof diversion. For any radial distance r from the peak, the total flux For any radial distance r from the peak, the total flux

intercepted by the barrier will be intercepted by the barrier will be r r22 q q

which transects a circumference with length which transects a circumference with length 2 2 r. r.

The local lateral flow will be qr/2, which by comparison to The local lateral flow will be qr/2, which by comparison to the linear-roof, implies an optimal interface parameterized the linear-roof, implies an optimal interface parameterized by z = rby z = r22q/C. q/C.

The slope should be half that of a linear-roof for any given The slope should be half that of a linear-roof for any given distance from the crest. distance from the crest.

It is straightforward to compute the optimum shape of a It is straightforward to compute the optimum shape of a circular-roof diversion. circular-roof diversion.

For any radial distance r from the peak, the total flux For any radial distance r from the peak, the total flux intercepted by the barrier will be intercepted by the barrier will be

r r22 q q which transects a circumference with length which transects a circumference with length

2 2 r. r. The local lateral flow will be qr/2, which by comparison to The local lateral flow will be qr/2, which by comparison to

the linear-roof, implies an optimal interface parameterized the linear-roof, implies an optimal interface parameterized by z = rby z = r22q/C. q/C.

The slope should be half that of a linear-roof for any given The slope should be half that of a linear-roof for any given distance from the crest. distance from the crest.

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Segmented changing slopeSegmented changing slopeconstruction of a smoothly changing slope presents construction of a smoothly changing slope presents

logistical difficulties. logistical difficulties. Approximate by a series of n straight segments of Approximate by a series of n straight segments of

equal horizontal extent and linearly increasing slope. equal horizontal extent and linearly increasing slope. For a barrier with total horizontal extent H and total For a barrier with total horizontal extent H and total

fall L, the slope of the most down-dip segment is fall L, the slope of the most down-dip segment is

construction of a smoothly changing slope presents construction of a smoothly changing slope presents logistical difficulties. logistical difficulties.

Approximate by a series of n straight segments of Approximate by a series of n straight segments of equal horizontal extent and linearly increasing slope. equal horizontal extent and linearly increasing slope.

For a barrier with total horizontal extent H and total For a barrier with total horizontal extent H and total fall L, the slope of the most down-dip segment is fall L, the slope of the most down-dip segment is

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Expected Performance Expected Performance

In a segmented system, first leakage would occur at In a segmented system, first leakage would occur at the lowest end of each segment simultaneously, the lowest end of each segment simultaneously, since the diverted flux/slope ratio will be minimal and since the diverted flux/slope ratio will be minimal and equal for all segments at these points. equal for all segments at these points.

Using 3 and 5 straight segments the diversion Using 3 and 5 straight segments the diversion capacity is predicted to be capacity is predicted to be at leastat least 75% and 83% 75% and 83% respectively of the diversion of a smoothly curved respectively of the diversion of a smoothly curved interface, demonstrating a rapid convergence to the interface, demonstrating a rapid convergence to the optimal performance. optimal performance.

In a segmented system, first leakage would occur at In a segmented system, first leakage would occur at the lowest end of each segment simultaneously, the lowest end of each segment simultaneously, since the diverted flux/slope ratio will be minimal and since the diverted flux/slope ratio will be minimal and equal for all segments at these points. equal for all segments at these points.

Using 3 and 5 straight segments the diversion Using 3 and 5 straight segments the diversion capacity is predicted to be capacity is predicted to be at leastat least 75% and 83% 75% and 83% respectively of the diversion of a smoothly curved respectively of the diversion of a smoothly curved interface, demonstrating a rapid convergence to the interface, demonstrating a rapid convergence to the optimal performance. optimal performance.