Effects of spatial and temporal variation of soil properties in pore pressure dissipation

by Guillermo Zavala

INTRODUCTION
MODELING APPROACH
RESULTS
CONCLUSIONS
REFERENCES

INTRODUCTION

Purpose of the study

The purpose of this study is to better understand pore water pressure dissipation in low permeability soils when a static load is applied. The simulation will be based on Karl Terzaghi’s one dimensional consolidation theory (Terzaghi et al, 1996), using a finite difference approach in order to simulate different variations to the basic formulation of the theory, and to introduce non-uniformity in the soil. It will also study the effects of the variation in time and space of some parameters that are assumed to be constant in common practice. The importance of this study is to assess the implications of assuming some of the values of soil properties to be constant. Figure 1 shows the basic setup for a consolidation laboratory test.

Figure 1: Basic Experiment Diagram

MODELING APPROACH

Basic Hypotheses: Terzaghi’s one dimensional consolidation theory

This study will be based on Terzaghi’s one dimensional consolidation theory, introducing certain variations. These variations will include changes to some hypotheses of the original theory.

The basic hypotheses for the consolidation theory as it is mostly applied in common practice are the following:

• The soil is saturated (which is assumed in all the cases for this study)
• The water and the soil constituents are incompressible. This is not exactly true but these are relatively incompressible when compared to the soil skeleton (soil viewed as a structure of soil grains)
• The coefficient of hydraulic conductivity k is constant and the soil is an isotropic material. This is a very common assumption in usual practice. For this study, this case will be assumed initially, but the model will then introduce a more realistic case that includes variation of this coefficient (or the other soil parameters that depend on it) with time and space.
• The flow of water in the soil follows Darcy’s law, v=k*i. (where v: flow velocity, k: hydraulic conductivity, and I: hydraulic gradient)
• The time involved in the consolidation process is only due to the soil permeability.
• The soil is confined horizontally, i.e. the soil is only allowed to be deformed in the vertical or “z” direction
• The total and effective stresses are the same for all the points of a horizontal section at all times

Consolidation process

Using the basic hypotheses, some other important facts can be inferred. Considering that the fluid is much more incompressible than the soil skeleton, a static load on the soil (equivalent to an increase in total stress), will initially cause a change in pore water pressure equal to the change in total stress, while the change in effective stress will be equal to zero. As the water starts to drain, the load is slowly transferred to the soil skeleton, constituting a change in effective stress. This water drainage can be considered as a diffusion problem, and the process is described by the equations in the following section.

Equations for the constitutive model

 The most commonly used stress-strain relationship parameter for consolidation of clays in a consolidation laboratory test is the compression index C_c. The compression index relates void ratio change to the change in effective stress in a logarithmic way (empirical relationship), as follows (Holtz, 1980):

                                   (1)

Variation of the coefficient of consolidation

The coefficient of consolidation will be varied for the different stages of the consolidation process. In order to do this, we have to assume certain relationships that will give us the parameters to calculate the Cv. One of these parameters is the coefficient of hydraulic conductivity, which has been related to the void ratio by several researchers using different functions. In this case, one of the very common schemes will be used, and it is a formula of the type:

                                       (5)

where

     k_z = hydraulic conductivity
        e = void ratio
        n = empirically determined parameter
        C = empirically determined parameter
      This relationship was taken from Samarasinghe et al (1982)

Discretization

Equation 2 shows the partial differential equation that governs the pore water dissipation process. The discretization used will be forward difference of first order in time and centered second derivative (second order) in space, which is showed in equation 8:

                    (8)

where:

i is the spatial subindex
j is the temporal subindex

Boundary conditions

 One of the most important aspects of a numerical solution is to adequately consider the boundaries of our model. In this report two cases were considered. The first one includes simple drainage (specimen open to the atmosphere in one end and closed to the atmosphere in the other end) and the second one includes double drainage (specimen open to the atmosphere in both bottom and top). This two conditions can be seen in figure 1.

Numerical solution

 Calculation of pore water pressures vs. time and space was initially based on a constant coefficient of consolidation (C_v). K (coefficient of hydraulic conductivity) was assumed to be constant during the experiment for a certain applied load. All the depths are first initialized (at time zero) to an initial pore water pressure value equal to the applied stress. All the timesteps at the known-value boundaries are filled with the fix boundary value (zero). Then the discretization is applied, calculating the pore water pressure values for all depths at each timestep. In the second stage, the coefficient of consolidation was recalculated for each timestep, rendering more realistic results. This case is much more complicated and computationally expensive because all the parameters necessary to calculate the coefficient of consolidation have to be recalculated for each timestep. These include e, k_z and a_v.

Analytical solution

 The analytical solution was only calculated for the case in which the Cv was considered to be constant. The analytical solution is given in the following form:

                (10)

where

u: excess pore water pressure
Ds’: final change in effective stress, (xload in the program)
z: depth
t: time
h: total height
Cv: coefficient of consolidation
a: (2n+1)*p/2
n: integer, from 0 to infinite

Time factor

The isochron graphs shown refer to a time factor T. This is just a factor that represents a stage of the consolidation of the whole soil layer. It is normalized by the height of the specimen and the coefficient of consolidation, so we can say that a particular time factor value always corresponds a certain degree of average pore water pressure dissipation (e.g. a time factor of 0.197 corresponds to an average of 50% of pore water pressure dissipated).
The expression for time factor is given as (Holtz, 1980):

                                                                (11)

3) RESULTS

The program was run several times using different cases and parameters

In summary, the following models were used:

   A) Assume constant C_v, with single drainage boundary conditions
B) Assume constant C_v  with double drainage boundary conditions
C) Assume variable C_v with double drainage boundary conditions

Single drainage vs. double drainage conditions

Models A and B were compared with the analytical solution, and it was concluded that double drainage boundary conditions gave closer results when compared to the analytical solution when using the same timestep. This occurs because in the single drainage case (Figure 1A) there is one undefined boundary, so it is necessary to make assumptions to be able to model it. In the case of the double drainage, both boundaries are exactly determined. Only results corresponding to the double drainage case are shown in this report. Therefore, we can say  that the single drainage case is more sensitive to the changes in timestep than the double drainage case.

Analytical vs. Numerical Solution

The comparison between the analytical and numerical model for model B, using Cv=0.3x10^-3 cm²/s, is shown. Figure 2 shows the pore water pressure dissipation for the analytical and numerical solution at the middle depth. Figure 3 shows the analytical solution isochron graphs and figure 4 shows the numerical solution for the same case. The analytical solution and the numerical solution match very well when an appropriate timestep and spatial step are selected. Other spatial and timesteps were also tested in the program but the results are not shown.

Figure 2: Analytical vs. numerical solution at middle depth

Figure 3: Analytical Solution

Figure 4: Numerical Solution, constant Cv=0.3x10^-3 cm²/s

Constant C_v vs. Variable C_v

Models B and C were run three times changing different parameters in order to assess the differences that between the constant and variable C_v assumption. Variation of C_v with time and space was assumed. In order to obtain the values of C_v, relationships described above where used.The program was run using experimental data from Samarasinghe et al (1982). The following soil properties were assumed in a first case:
    eo = 0.57
    Cc = 0.16

seating load = 50 kPa

The parameters for the calculation of k_z (hydraulic conductivity) were also taken from Samarasinghe et al (1982), for different cases:

Case 1

n = 5.2
   C = 7.7 x 10^-7 cm/s

Figures 4 and 5 show the isochrons for the constant Cv model and for the variable Cv model, respectively. We can see that the constant Cv model predicts slower pore water pressure dissipation than the changing Cv model. Figure 6 shows the variation of e, k_z, a_v and Cv parameters at the depth. We can see that the coefficient of consolidation Cv (diffusion constant) increases with time (faster dissipation), which is the explanation of why the constant Cv assumption indicates that more time was needed for consolidation to occur.

Figure 5: Numerical Solution, variable Cv, n=5.2, C=7.7x10^-7 cm/s

Figure 6: Variation of soil parameters vs time, variable Cv, n=5.2, C=7.7x10^-7 cm/s

Case 2

This case considered
  n = 7
  C = 15.4 x 10^-7 cm/s

In this case, something similar to the first case could be seen. However, the parameters that define the hydraulic conductivity were varied in a way that hydraulic conductivity had a larger decrease with a decrease of the value of void ratio (e). We can see in this case that C_v remains almost constant (the variation is very small), so it can be seen in the isochrons plot that both predictions match very closely. Figures 7, 8 and 9 show the correspondent graphs for this case (constant Cv isochrons, variable Cv isochrons, and the variation of the other parameters with time, respectively)

Figure 7: Numerical Solution, constant Cv=0.221x10^-3 cm²/s

Figure 8: Numerical Solution, variable Cv, n=7, C=15.4x10^-7 cm/s

Figure 9: Variation of soil parameters vs time, variable Cv, n=7, C=15.4x10^-7 cm/s

Case 3
This case considered:
       n = 8
      C = 23.1 x 10^-7 cm/s

In this case the variation of k_z was selected such as the value of C_v will decrease with time (this can be seen in figure 12). Then, comparing figure10 and 11 it can be seen that this case is opposite to the other ones. Constant C_v assumptions predict a faster rate of pore water pressure dissipation.

Figure 10: Numerical Solution, constant Cv=0.189x10^-3 cm²/s

Figure 11: Numerical Solution, variable Cv, n=8, C=23.1x10^-7

Figure 12: Variation of soil parameters vs time, variable Cv, n=8, C=23.1x10^-7 cm/s

In summary, all these different cases consider different trends of variation of the C_v coefficient vs. void ratio. Then, results obtained using this parameters are compared with the case of constant C_v.

More analyses should be done using input parameters for different kinds of soils to test the sensitivity of the parameters in a wide variety of soil types.

4) CONCLUSIONS

• Double drainage conditions are easier to model because both boundaries are open to the atmosphere and therefore they have a constant value of pressure equal to zero.
• The differences between the consideration of constant C_v and variable C_v depend largely on the soil parameters. From equation 3 it can be concluded that C_v depends on the variation of k_z and a_v with void ratio. Both of these parameters decrease with void ratio, but their relative decrease determines if the value of C_v increases or decreases with void ratio. This relative decrease (or increase) will directly affect the error in the constant C_v assumptions.
• The assumption of C_v constant assumes that the variation of a_v and kz is similar, which will render an approximate constant value of C_v. This can be true in several cases, but can also be different for others.
• More studies need to be made using different data for other soils in order to assess the impact of the constant C_v assumption

  REFERENCES

 1.      TERZAGHI, K., PECK R., MESRI, G. (1996), “Soil Mechanics in Engineering Practice. Third Edition”, John Wiley & Sons Inc.

2.      MITCHELL, J. (1993),  “Fundamentals of Soil Behavior”, John Wiley & Sons Inc.

3.      HOLTZ, R.D., KOVACS, W.D. (1980), “An Introduction to Geotechnical Engineering”, Prentice Hall, NJ.

4.      DAS, B.M., (1998), “Principles of Geotechnical Engineering”, International Thomson Publishing.

5.      SAMARASINGHE, A.M., HUANG, Y.H., DRNEVICH, V.P. (1982), “Permeability and Consolidation of Normally Consolidated Soils”, Journal of Geotechnical Engineering Division,  Vol. 108, Num. GT6, pp. 835-850.