# Consolidation

### Introduction and literature review

Consolidation is a time dependent process that occurs in low permeability soils and involves the rearranging of soil particles in order to dissipate excess pore pressures. Many theoretical models were developed during the last century in order to describe and simulate accurately enough the consolidation process in various soil types and conditions.

The consolidation models are generally separated into two categories: the first describing large strain consolidation and the second describing infinitesimal strain consolidation. The infinitesimal strain consolidation models share the restrictions that the decrease in height of the soil is relatively small compared to the soil's initial height and change in void ratio over the consolidation process is infinitesimal compared to the initial one. The large strain consolidation models remove those restrictions to enable analysis of much softer and looser soils without the amount of error produced by those restrictions.

The first who derived a consolidation theory was Karl Terzaghi (1943) marking the beginning of the modern soil mechanics. His model is still the most commonly used for infinitedecimal strain consolidation process in soils. It includes the introduction of the consolidation coefficient (cv) and his assumption that all over the consolidation process it remains constant. The leading consolidation parameters (soil permeability, soil stiffness and drainage path) in real life are not constants as the Terzaghi's consolidation theory assumes. As strain increases the accuracy of Terzaghi's model decreases. Many scientists tried evolving Terzaghi's model by taking into account variations of permeability and compressibility within the soil. An example was a model developed by Davis and Raymond (1965) who assumed that permeability and compressibility of the soil are not constant but decrease as pressure is increasing. Some scientists tried proposing a totally different approach to the consolidation process. An example is Kynch (1952) who proposed a method for estimation of the height of sedimenting soil by only using the assumption of mass concervation within the consolidating soil, that relative velocity of soil particles depends on the concentration of particles in its close neighbour and that all the soil's particles have the same shape and size. This theory was quite interesting at that time period as it doesnt take into account the leading parameters of consolidation like soil permeability or soil stiffness but relates the rate of consolidation just on the velocity of soil particles. The drawback of that theory is that, although it requires only one input parameter; that parameter is difficult to be estimated accurately enough as in reality concentration of consolidating soils varies non- linearly and discontinuities in the consolidation rate of the soil are created. This was noticed by Kynch (1952) and he tried to identify the different discontinuity types but still his theory is very difficult to be used in real life situations.

The first who derived an equation without the limitation of the infinitesimal strain in the normal consolidation theory were Gibson et al (1967) and Mikasa (1965). Gibson et al (1967) derived a general equation describing the consolidation process with the least possible limitations. The only assumptions they made were that movement of fluids within the soil structure follow Darcy's law (relative velocity of fluids in soil is related with the excess pore pressure gradient) and that both pore fluids and solids are incompressible. Moreover they introduced a consolidation coefficient (CF) that can be assumed constant for linear cases. The consolidation coefficient Gibson et al (1967) used was similar to the consolidation coefficient (cv) Karl Terzaghi (1943) introduced in his model. The difference between the two was that Terzaghi's consolidation coefficient (cv) depends only on the current void ratio while Gibson et al (1967) take into account initial void ratio as well. Gibson et al (1967) theoretical model although it has no limitations, it's very difficult to be used in its full potential in reality. This is because in reality it's extremely difficult to derive the exact non-linear relationships between the leading consolidation parameters and obtain easy enough numerical results. As a result many analytical and numerical solutions have been developed for one dimensional finite strain consolidation by introducing additional assumptions into Gibson's et al (1967) model. The additional assumptions are based on the experimental results of insitu conditions for which solutions will be obtained. Such assumptions were made by K.Lee & G.C.Sills (1981), who derived an equation for self-weight consolidating soil by assuming linearity of the permeability-void ratio as well as void ratio-effective stress relationship and coefficient of consolidation is constant. Using the additional assumptions, predictions were made and evaluated by comparing the numerical results with the actual experimental results(K.BEEN and G.C.Sills, 1981). Comparison of the experimental with the theoretical values obtained concluded that in the early consolidation rate is underestimated and later consolidation rate is overestimated but the final soil height values are resonably accurate. It was suggested that large error in the early consolidation is due to the assumption made that permeability varies linearly with void ratio as permeability value affects calculations more than it was expected. This theory though is accurate enough only when the self weight of soil is an important factor in the sedimentation and consolidation process.

Others that tried using Gibson's et al (1967) theoretical model to obtain some numerical solutions were S.D.Koppula & N.R.Morgenstern (1982) who used as the dependent variable excess pore pressure. By interpollating some experimental results of rivers sedimenting with high rate they concluded that linear relationships between void ratiowith logarithm of effective stress and log[k1+e] with logarithm of effective stress are accurate enough approximations. Thus excess pore pressures and water content variation with depth can be predicted accurately enough by the equations they derived. The drawback of that model is that the model is accurate enough only to high sedimenting rate. Moreover the linear relationships that were used are accurate enough with the particular soil conditions, with other soil conditions a slightly different linear relationship maybe more appropriate.

J.L.Monte & R.J.Krizek (1976) included the term"fluid limit" to Gibson's et al. (1967) theory. This term was defined as the void ratio value at the point where the soil behaves as a heavy fluid. The theory's purpose was the estimation of the sedimentation processes of soil formation. Its difficult thought, to estimate exactly the void ratio at fluid limit of the soil therefore some errors in the estimation of the sedimentation are expected.

The project is centralized into the case of consolidation of soft soil on which additional sediment is continuously deposited on top of it. For the specified soil condition a critical and complete understanding of all the theoretical models that were mentioned above is required to be able to choice the model that will best fit the case under consideration. Up to this particular stage is clear that the best approach to the given situation will be a finite strain theoretical model as finite strain consolidation is expected to occur. Moreover some simplifying assumptions might be required to enable implementation of numerical solution to a general equation (like R.GIBSON's et al.(1967)).

### Aim and objectives of the project

### Aim of the project

The collection, review and critical examination of the different mechanisms of finite strain consolidation process in very soft and loose soils.

The understanding of the differences and alterations in the mechanisms in the case where there will be additional continuous sedimentation (e.g/the situation where a river continiously deposits sediment on the continental shelf).

### Objectives of the project

- Review and critical examination of all the experimental results published in literature of similar cases.
- Review and critical examination of all the theoretical models of finite strain consolidation of soft soils identifying their relative advantages and disadvantages.
- Development of a numerical implementation using appropriate software and method of one or more appropriate theoretical consolidation models.
- Analysis, using the numerical implementation mentioned above, of insitu conditions that may cause deposited sediment to become unstable (e.g. / marine landslides).

### Methodology

There are many journals, articles and books in literature that contain various theoretical consolidation models and experimental results. These will be obtained, reviewed and critically examined using various resources like university library and internet. Analysis of each individual theoretical models will be made as well as of the various experimental results.

The most appropriate theoretical consolidation model will be chosen. The choice of model will be made based on which model satisfies better the situation under consideration which involves as mentioned above the large strain consolidation of soil under continuous sedimentation. A numerical implementation of that numerical model will be made using finite differences method in Matlab software.

Matlab was chosen as its one of the best and more powerful simulation and numerical solution software in the market. Finite differences method was the chosen numerical method as its one of the simplest, faster and accurate enough numerical methods.

Using the methods mentioned above values for the initial soil conditions will be input and qualitative results will be obtained. Using the software's output, time periods and initial soil parameters at which instability in the soil will occur will be obtained.

The accuracy of the predictions will be the accuracy of the numerical method used multiplied with the accuracy of the theoretical numerical model.

### Progress to date

Most of the literature that is required for the current project has been identified and is either acquired or expected to be acquired until the end of Christmas vacations. More than half of the experimental results and theoretical consolidation models have been reviewed and examined. Moreover textbooks for the use of Matlab and finite differences method have been identified.

### References

- J.L.MONTE and R.J.KRIZEK. 1976. One-Dimensional mathematical model for large strain consolidation. Geotechnique. 26(3), pp.495-510.
- K.BEEN and G.C.SILLS. 1981. Self-weight consolidation of soft soils: An experimental and theoretical study. Geotechnique. 31(4), pp.519-535.
- K.LEE and G.C.SILLS. 1981. The consolidation of a soil stratum including self-weight effects and large strains. International Journal for numerical and analytical methods in geomechanics. 5, pp.405-428.
- KYNCH, G. J. 1952. A theory of sedimentation. Trans. Faraday Society. 48, pp.166-176.
- MCNABB, A. 1960. A mathematical treatment of one dimensional soil consolidation. Quarterly of applied mathematics., pp.337-347.
- MIKASA, M. 1965. The consolidation of soft clay, A new consolidation theory and its application., pp.21-26.
- R.GIBSON, G.ENGLAND, and M.J.L.HUSSEY. 1967. The theory of one-dimensional consolidation of saturated clays 1. Finite non-linear consolidation of thin homogeneous layers. Geotechnique. 17, pp.261-273.
- RAYMOND and E.DAVIS. 1965. A non- linear theory of consolidation. Geotechnique., pp.161-173.
- S.D.KOPPULA and N.R.MORGENSTERN. 1982. On the consolidation of sedimenting clays. Canadian geotechnical journal. 19, pp.260-268.
- TERZAGHI, K. 1943. Theoretical soil mechanics. London: Chapman and Hall Ltd.