Jeffrey Gellis, Andrew Jacobs Professor Andrade REU Summer 2008 September 13, 2008

ConsolidationofaSaturatedPorousMediumExperience

What Exactly, Is Going On? Many people are delighted to see the tourist trap in Italy known as the Leaning Tower of Pisa. The tower was not designed to lean as it does, neither is it clear how much longer it can stand without intervening to help straighten it. What is known for sure though is the phenomenon that has caused it to lean. This phenomenon is known as consolidation. Consolidation is defined as "the pressing of soil particles into a tighter packing in response to an increase in effective stress." The physical result of consolidation known as consolidationsettlement is characterized by a soil that is compressed in conjunction with adjacent structures and objects that have sunk. The Leaning Tower of Pisa is a special case of consolidation settlement in which only some of

the soil has consolidated while other parts have been unaffected by the structure. Thus the Leaning Tower of Pisa has non-uniform soils and a gradient of consolidation which causes it to shift to one side. At this point, we have learned a lot about soils. We have learned about the permeability of soils, the mathematical foundations of water flow through soils, and about the mechanical properties of soils. It may come as no surprise to you that consolidation is the capstone of our 'Experience Series' in that involves every aspect of the past experiments. Consolidation involves permeability, dimensional flow, and mechanical properties of soils.

As seen before, the soil properties we have studied all have real-world application especially in regards to safety. The concept of consolidation is no different; there are endless possibilities in applying this concept. Bridges, buildings, and roads are all structures that can succumb to consolidation settlement unless soils are properly tested.

How Do the Concepts Apply? Consolidation involves all of the properties we have studied so far as talked about earlier: (intrinsic) permeability, dimensional flow, and mechanical

properties. Now however, there are some other concepts we have to understand. Among them are non-steady time dependence, pore pressures, and other dynamic variables. As seen in the 'Water Flow Through Porous Media Experience' flow can have 1, 2, or 3 spatial components. We will be talking about one­dimensionalconsolidation for the remainder of this experience, that is consolidation in one spatial direction. The general equation that describes one-dimensional consolidation was developed by Karl Terzaghi and is defined as follows:

€

∂u∂t

= cv∂2u∂z2

where

u = excess pore water pressure (N/m2)

t = time (sec)

Cv = the coefficient of consolidation

z = vertical distance below the surface of the ground

The one-dimensional consolidation equation or Terzaghi equation is a partial differential equation. It is similar to the equation of two-dimensional flow and to the heat equation except that the Terzaghi equation has time dependence. As with the two-dimensional flow equation, we can also solve Terzaghi's equation using finite differences. This time however, we can unleash the full potential of finite differences by taking a step further and defining our own boundary conditions (as you recall from the 'Water flow experience' our boundary conditions were very complicated and we used a PDE solver to help out). "What's so much different about consolidation?", you might ask. In consolidation

we are combining a saturated porous medium with the stress created by a specific load. In the past we were using an open-cell foam to model soil properties and it wasn't really clear why we couldn't just have used soil. For consolidation however, actual soils demonstrate the concept poorly in a laboratory setting for the purposes of visualization. The times required for consolidation settlements are long and the amount of total deformation that occurs is very small. At this point, the open-cell foam does a remarkable job of simulating what consolidation actually looks like up close in a soil. For visualization purposes, it performs better in experiment than does actual soil. Before we get into solving for how much a soil should theoretically consolidate, we have to learn some things about the process itself. Many different things are happening at once when we place a load on a saturated soil. First of all, we have a stress created by the load that is exerting itself on the specimen. Initially, the stress is not distributed evenly in the soil. When a load is first placed on the soil (time t=0 seconds), all of the stress is on the water in the pores or spaces between soil particles. This stress that is being transferred to the water in the spaces between soil particles results in what is known as the excessporewaterpressure, u. It is a very critical measurement for the soil. After time t=0 (t = 0+ seconds), the stress shifts from the water to the soil

particles themselves. During this period, the soil itself is compressing vertically at an exponential rate. At a specific time, the soil will stop compressing (consolidating) and at this point all of the stress is exerted on the soil particles. When this happens, the excess pore water pressure (u) drops to zero. The water remaining in the soil carries none of the stress and will not drain; this period is known as steadystate. The term steady state is used because it infers that the compression of the soil is held statically in equilibrium and nothing is changing with respect to time. If we look at the Terzaghi equation again:

€

∂u∂t

= cv∂2u∂z2

We notice that Cv is the only quantity that remains constant throughout the

consolidation process. Interestingly enough, Cv is actually a composite of all the

properties we gathered from the previous experiences on 'Permeability' and

'Mechanical Properties' of soils. Cv is defined as:

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Cv =k

µ(Cr + φCf )

where:

Cv = the coefficient of consolidation

k = intrinsic permeability (m2)

µ = the viscosity of fluid (N/m2*s)

Cr = the compression/recompression index

φCf = compressibility of the fluid

As we can see, the equation for Cv involves k and µ, both of which were

determined in prior experiences. Since we are primarily interested in

consolidation involving water, the properties of µ and φCf are known. µ is simply

1.002*10-3 N/m2*s and φCf can be assumed as zero because water for all intents

and purposes is an incompressible fluid. If we have gathered all of the necessary soil properties, we can use the Terzaghi equation to solve for the pore pressure, depth, and time components of consolidation. Once we solve the Terzaghi equation, we can take things a step

further by using the results and applying them to equations that model consolidationsettlement, the physical amount of compression that the soil undergoes over time.

How Do We Model This Computationally? Now comes the actual solving of the Terzaghi equation. As usual there are 2 ways to solve an equation like this: solving analytically to obtain a closed-form solution and solving numerically to obtain an approximate solution. The two methods of solving have their benefits and drawbacks. Solving analytically should produce the exact solution we are looking for however it limits us to using small amounts of data as an input. Solving numerically won't give us the exact solution, however it will offer a high level of refinement and precision and allows us to calculate a lot more data compared to the analytical method. To solve numerically, we would use finite differences again to make approximations. As opposed to the analytical (closed-form) solution, our numerical solution would allow us to see results at finite time and depth intervals---closed-form solutions cannot provide this data and limit the potential for larger scale simulations.

We will look at both methods for solving the Terzaghi equation based on some soil characteristics that will be provided for us. Using both allows us to acquire an exact solution from the analytical method while comparing our result to the larger amount of data from the numerical method; if both match up favorably we know that the simulation was a success. The analytical solution to the Terzaghi equation is a Fourier-series and looks like this:

€

u = Δσ z4

(2N +1)πsin (2N +1)π

2zdrHdr

e

−(2N +1)2 π 2

4CvtHdr

2

N= 0

∞

∑

where:

u = excess pore water pressure (N/m2)

∆σz = change in vertical stress caused by applied load (N/m2)

zdr = vertical distance from specific point to a drainage boundary (m)

Hdr = length of drainage path (length of soil specimen in our case) (m)

Cv = the coefficient of consolidation

t = time after load is applied (seconds) This analytical solution will sum terms until the index produces values that are negligible and convergent. The numerical solution of the Terzaghi equation involves finite differences and a mesh as seen in the 'Water Flow through Porous Media Experience'. This time however, the mesh operates a bit differently. The mesh has different variables involved this time around; mainly time and pore pressure. As opposed to the mesh used in the Water Flow Experience which had the x and y axes comprised of there respective spatial increments, the consolidation mesh uses depth on the

y-axis, time on the x-axis, and records pore pressures at each node. It looks a little something like this:

As for the boundary conditions, it is important to see how they are derived because this is where your basic Calculus comes into play. We can assume that:

€

∂u∂t

=(ui, j+1 − ui, j )

Δt

and that for a second derivative (see Water Flow Experience for reference)

€

∂2u∂z2

=(ui−1, j − 2ui, j + ui+1, j )

Δz2

So if we were to plug these two definitions into the Terzaghi Equation it changes from this:

€

∂u∂t

= cv∂2u∂z2

to this:

€

ui, j+1 = ui, j +CvΔt(Δz)2

(ui−1, j − 2ui, j + ui+1, j )

This equation is fundamental for solving numerically and using in our mesh. However the equation only applies to areas where there is flow. Thus the equation changes in areas where this no flow or when ∂u/∂z = 0; areas of no flow would signify the physical casing holding the sample of soil. The boundary equation in this instance looks like:

€

ui, j+1 = ui, j +CvΔt(Δz)2

(2ui−1, j − 2ui, j )

What we are solving for here is primarily the pore pressure with respect to depth and time. The consolidation settlement can either be calculated from the pore pressures numerically (beyond the scope of this paper) or can be calculated analytically with the following series:

€

settlement =Lkσ z +

8π 2

1(2m −1)2

σ ze−(2m−1)π

L

2

Cvt

m=1

∞

∑

Where: L = length of soil specimen (m)

k = intrinsic permeability (m2)

σz = effective stress (in this case the load that we apply) (N/m2)

Cv = coefficient of consolidation T = time (seconds) One way we can present our data in a meaningful way is to make use of what are called isochrones. An isochrone refers to a trajectory in which all properties of the trajectory are happening at the same time. In the case of consolidation we can use isochrones to show how pore pressure is changing with respect to depth and concurrent times. The result will be a visual diagram displaying the decaying and converging of consolidation as a process. Since we have gone through the basics of programming in MATLAB in the past, we leave it as an exercise to execute and refine the following commented code in the appendix. The m-file codes solutions for excess pore pressure, isochrones, and

consolidation settlement. These solutions were calculated by taking the properties measured in the past 3 experiences (permeability, Cr, etc) and inputting them into the governing consolidation equation solving analytically and numerically. We have produced graphs of the the theoretical or closed-form solution for consolidation settlement and compare with numerical results of the corresponding settlement. Graphs of isochrones and pore pressures were also calculated numerically. Please run the attached m-file within MATLAB to understand and see the results for yourself.

Major Assumptions: For this concept to be modeled ideally, we will have to make some assumptions:

1. The Liquid That is Flowing is Inviscid ---> Inviscid means that the liquid flows smoothly. A viscous material would be something like molasses, but since we are dealing with water, we can assume it is inviscid.

2. The foam is adequately saturated ---> The foam must be saturated (with water) to ensure that the flow is undeterred and that it follows a predictable path.

3. Foam exhibits an elastic response--> The foam should be able to return to its initial configuration under its own power after being compressed or

lengthened. 4. Foam exhibits a linear stess-strain relationship for first 1-20% of

deformation We are assuming that the foam will have an initially linear response before it is compressed over 20%, these are relatively general guidelines for most materials.

How Do We Model This Physically? Analytical and numerical simulations are powerful tools, but they are sometimes meaningless without a physical model to compare to. Thus we have developed a consolidation apparatus to demonstrate what is physically happening during the process of consolidation. As mentioned earlier, we are using the same open-cell foam as seen in 2 of the 3 past experiences. What You Will Need: -the foam specimen (should be about 26-29 cm) - an acrylic casing with standpipe tube - a polyethylene puck with drainage holes - metal washers (for loading) and a washer holder - a ruler, a stopwatch, and water

Procedure:

1. Gather the foam specimen, acrylic casing with standpipe tube, vertical standing ruler, and a stopwatch.

2. Place acrylic casing setup on a stand of some sort (blue stand should be somewhere in vicinity), thread outflow tube through the bottom of the stand.

3. Sit the standpipe tube upright and affix to a ring stand or something hanging from the ceiling (whatever is available). Place apparatus in a sink so that drainage isn't as messy.

4. Take foam specimen and slide into the acrylic casing until it hits the bottom of the casing.

5. Fill the acrylic casing with water until the level is about 2 inches from the top.

6. Mercilessly squeeze the foam underwater to saturate it with water. THIS IS A CRUCIAL STEP.

7. Keep squeezing foam to release air bubbles and properly saturate it. Work at this for about 2-3 minutes or until your hand is sufficiently tired...results will not be accurate unless the foam is as saturated as possible.

8. Keep the water level in the casing above the foam sample AT ALL TIMES. As soon as the foam specimen is exposed to air, it will tamper with accurate results.

9. After the foam is sufficiently saturated, leave a small level of water above the specimen.

10. Look at the standpipe tube and try to release any air pockets or air bubbles by squeezing the tube until they are released (we want to make sure the flow through the apparatus is undeterred). Take note of the water level in the standpipe tube.

11. Place the polyethylene puck with drainage holes on top of the saturated foam specimen and measure the level of the foam from a datum point (the base is a good reference) with the vertical standing ruler.

12. Choose a specific load in terms of metal washers to place on the foam specimen; 2, 3, 4, or 5 are good amounts and make sure that the 5-20% deformation rule from the 'Mechanical Properties' experience is followed.

13. Place the chosen amount of washers on the washer holder (simply a plastic cup) so that they look like the figure above. This is done so that the drainage holes of the puck are not impeded by the washers.

14. Take your ruler out and place it on your base or your datum point and let it stand parallel right next to the casing with the centimeter (cm) side next to it and in plain view.

15. Take out your stopwatch. 16. Take the washers and washer holder and place them evenly and centered

on the drainage puck while IMMEDIATELY starting the stopwatch. ***Notice the spike of the water level in the standpipe tube when you do

this and how it decreases afterward. 17. Watch as the foam compresses and use the vertical standing ruler track

its progress. Every 3-5 mm of compression, you should lap or record the time on the stopwatch to get a good amount of data. The initial compression might be large, so estimate a reasonable time for how long it took if this is necessary.

18. Over time, the compression will slow down. Keep taking measurements for about 35-40 minutes or until there is no more compression (consolidation) in the foam.

(Progression of foam compressing/water level increasing in apparatus)

19. Remember that when the foam stops compressing, you have reached steady­stateand all of the stress is being carried by the foam.

20. Repeat with other weight setups (different amounts of washers) or incrementally load with more washers to get a reliable set of data.

Major Assumptions: For this concept to be modeled ideally, we will have to make some assumptions

1. The Liquid That is Flowing is Inviscid ---> Inviscid means that the liquid flows smoothly. A viscous material would be something like molasses, but since we are dealing with water, we can assume it is inviscid.

2. The foam is adequately saturated ---> The foam must be saturated (with water) to ensure that the flow is undeterred and that it follows a predictable path.

3. Foam exhibits an elastic response--> The foam should be able to return to

its initial configuration under its own power after being compressed or lengthened.

4. Foam exhibits a linear stess-strain relationship for first 1-20% of deformation--> We are assuming that the foam will have an initially linear response before it is compressed over 20%, these are relatively general guidelines for most materials.

What Are the Results? If you have completed the prior 3 experiences, your results will likely look different than those obtained by us. However, they should be in the ballpark if everything happened correctly. After dozens of trials we obtained these values: intrinsicpermeability, k = 3.8295*10^-12 m2 compressionindex, Cr for 2 washer load = 3.5842*10^-5 compressionindex, Cr for 3 washer load = 3.2825*10^-5 compressionindex, Cr for 4 washer load = 4.0770*10^-5

With these values and the constants involved in the Terzaghi equation we were able to obtain these solutions for consolidation:

*****The tonal contour graphs show the pore pressures with respect to time and depth. The red signifies very high pore pressures while the blue signifies very small pressures....it is clear how the pressure decreases over time as the stress is being transferred to soil particles. (The above is an example of the solution for a 3 washer or 1069 pascal load) *******The isochrone graphs display pore pressures across concurrent times and depths as well, but are presented in a slightly different way. We can track the pore pressure at a specific time across the stratum of the soil with these

isochrone graphs. (The above is an example of the solution for a 3 washer or 1069 pascal load) *******Lastly, the consolidation settlement graphs display the theoretical compression of the soil (based of the analytical/closed-form solution) as well as 3 physical trials of consolidation using the apparatus that was built. We can see from the graphs that during our trials, consolidation settlement occurred rather quickly compared to the theoretical result. It is unclear the exact reason for this, but it is the author's assumption that difficulty in centric loading and friction from the drainage puck are factors at play here. Although consolidation settlement happens more quickly than desired, the ultimate consolidation settlement (degree of compression at steady-state) is remarkably close across the board. Percent error ranges from 3-19% on this measurement. This is a pretty good result considering that the apparatus that was constructed for consolidation was created and machined by a college and high school student. (The above is an example of the solution for a 3 washer or 1069 pascal load) Foramoredetailedlookatthecodingandlogicbehindthesolutions,please

referencethem­fileincludedwiththisdocument.

What's Going to Happen in the Real World? Let's go back to our initial case study of the Leaning Tower of Pisa. Based on what we have learned, we could theoretically obtain a sample of soil that the tower was built upon and then run tests on it for certain key properties. We would want to determine the soil's intrinsic permeability, dimensional flow characteristics, and it's compression index (Cr). We then know that we could take these properties and use them in the calculation of excess pore pressures in the soil and for the consolidation settlement. With this information, we could make an estimate of how far the Leaning Tower of Pisa will sink and tip over in the years

to come. Being able to simulate this in the laboratory is much more useful than measuring soil characteristics in the field. It is also much better to determine what essential structural improvements must be made to the Tower based off of lab results. Paying exorbitant sums on unnecessary retrofitting or on the other extreme, waiting complacently until the Leaning Tower of Pisa collapses are two scenarios that can be prevented with proper laboratory testing of soils from the Tower. The Leaning Tower of Pisa is not a case on its own. The same methods can be applied to most structures built upon questionable soil foundations. That's what makes the Terzaghi equation so powerful.

What Have We Learned? The point of this experience was to show you how many concepts can be interrelated and dependent on each other at once. Consolidation is the culmination of many factors that are sometimes thought of as trivial by those unfamiliar with them; however we have learned how important many soil characteristics are in ensuring the safety of people and structures. It is our hope

that you appreciate the fundamentals you have learned throughout your education in physics and calculus. It is these fundamental concepts that form the basis of geotechnical engineering; a field that is continually being refined and advanced for the future of structural development. The more we try to understand soils and their attributes, the further we can push the limits of structural technology, preservation, and safety in the years to come.

REFERENCES 1. Budhu, Muni, (2007). Soil: Mechanics and Foundations. 2nd Edition. ©2007 John Wiley & Sons, Inc. 2. Coduto, Donald P., (1999). Geotechnical Engineering: Principles and Practices. 1st Edition. ©1999 Prentice-Hall, Inc. 3. Professor Jose Andrade, Northwestern University, Evanston, IL.