Numerical Model with Finite Differences Approach for CRS Consolidation Test

The Constant Rate of Strain (CRS) consolidation test is extensively used in last time to estimate the settlement of clayey soils in many geotechnical laboratories. Different theoretical solutions and numerical models have been developed to estimate consolidation parameters from CRS consolidation test data, and investigate the strain rate effect on the CRS consolidation results. In this study, a new numerical model is developed to simulate CRS consolidation test for small and large strain conditions and for both linear and nonlinear soils. This numerical model is based on the solution of Terzaghi’s classical consolidation equation by finite differences approach, with taking into account the variation of sample height with test time. Results of this numerical model indicate that applied vertical load at the top boundary of sample and excess pore pressure at its base are dependent on the applied strain rate. Evaluation of the consolidation parameters from numerical results of this model with small and large theoretical solutions shows excellent agreement between all methods in small strain level, and when large strain conditions are reached only use of large strain theories can produce good convergence with model results. However, when great strain rates (approximately β ≥ 0.1) are applied, a significant error can be observed in consolidation parameters calculation by using both small and large solutions. Finally, simulation of some experimental CRS tests reported in literature with this numerical model provides comparable consolidation parameters to those evaluated from the experimental CRS tests data.


Introduction
The Constant Rate of Strain consolidation test (CRS test) became in the last decades as an alternative procedure to the standard consolidation test (IL test) to evaluate the consolidation properties of clayey soils in many countries. The CRS consolidation test, comparatively to the standard consolidation test, can be completed in a reduced time (one to two days) for large range of soils, and produces continuous responses in particular the compressibility curve. The CRS consolidation test was developed for the first time by Hamilton and Crawford (1959) [1] to overcome some disadvantages of standard consolidation test. The main CRS consolidation theories were further developed by Smith and Wahls (1969) [2] and Wissa et al (1971) [3], based on assumptions similar to Terzaghi's ordinary consolidation theory with small strain conditions. Subsequently, several studies based on large strain conditions, have been conducted [4][5][6] to take into account the continuous nature of CRS test with important total stain levels [7,8]. Furthermore, many other studies [5,9,10] considered that the small strain theory can be only used to simulate the CRS consolidation for the small strain levels, and it produces significant error for large strain levels. In all cases, consolidation parameters obtained by CRS consolidation tests are dependent on the applied strain rates [2,7,[11][12][13][14][15][16], and many criteria have been proposed by authors to select adequate strain rates for CRS tests [2,3,5,17].
In this paper, a new numerical model based on finite differences approach is built to simulate the CRS consolidation test by using iteratively, during successive Differences Approach for CRS Consolidation Test small time steps, the Terzaghi's linear one-dimensional consolidation theory. The coefficient of consolidation c v is assumed to be constant during all the consolidation process, but the height of sample h, effective stress σ v ' , and void ratio e, are varying with time. The creep and self-weight effects are ignored. To take into account the effect of compressibility, both linear and nonlinear soils are considered. The results of model are shown and then used to evaluate the consolidation parameters (c v and k v ) by using small and large strain theories, which permits to check the convergence between model results and the large strain theories. The results of some experimental works are also used to check the ability of this numerical model to produce consolidation parameters (c v and k v ) comparable to those evaluated during experimental CRS tests.

Model Structure
In the CRS consolidation test, a soil sample with an initial thickness of H 0 , is contained within a consolidation cell and deformed at constant strain rate r. A vertical load ∆σ v is applied at the drained top side of sample, and the excess pore pressure ∆u H is measured at the undrained bottom side.
In this model, at any time t of CRS test, the height of sample h is divided into n soil elements with equal thicknesses ∆z. The vertical depth z(t), and the number j of each boundary between two successive elements are defined positive downward from the top side. At each time t, the height of sample h(t) is: And the thickness of each soil element ∆z(t) is: With n is the number of soil sample elements. The Terzaghi's linear one-dimensional consolidation equation is used iteratively during successive time steps ∆t, to simulate the continuous loading of CRS consolidation test. The parameters necessary to calculations are updated at the beginning of each time step. The Terzaghi's consolidation theory assumes essentially the soil sample is homogeneous and saturated; theDarcy's law is valid and the coefficient of consolidation c v is constant during all the consolidation process.
The Terzaghi's linear one-dimensional consolidation equation, for continuous loading, is written as: With: c v is the coefficient of consolidation of soil. u is the excess of pore pressure at vertical depth z and time t.
∆σ v is the applied vertical load. During CRS test, the sample thickness is variable, and use of Terzaghi's consolidation equation leads at every time step ∆t, to neglect the excess pore pressure values of soil part deformed between two successive times t and t+∆t (figure1a). However, although the excess pore water pressure is very small in the neglected part of soil near the drained face, the use of Terzaghi's consolidation equation with normalized parameters permits to take in consideration the totality of sample thickness in calculations (figure1b). Let u r , t r and z r be any arbitrary reference excess pore water pressure, time, and vertical depth, respectively. From these, the following dimensionless terms can be defined [18]: Dimensionless excess pore water pressure / r u u u = Dimensionless applied vertical load / The Terzaghi's linear one-dimensional consolidation equation becomes: The arbitrary reference time can be considered equal to: 2 / r r v t z c = , then equation (4) will be of the form: The left-hand and right-hand sides of equation (5) can be written as: Where , z t t u +∆ and , z t u are the dimensionless pore water pressures at dimensionless depth z and at dimensionless times t t + ∆ and t respectively.
are the dimensionless applied vertical load values at times t t + ∆ and t respectively.
Substituting equations (6), (7) and (8) in equation (5), gives: For equation (9) converges, t ∆ and z ∆ must be chosen such that: To take into account the sample height at every time t of CRS test, the arbitrary reference depth z r is taken equal to: For undrained face of specimen (z=H): and equation (9) becomes: The pore water pressure at any dimensionless depth z and at dimensionless time t t + ∆ is: The value of ∆t is evaluated versus a preset final deformation of sample f ε . If pore water pressure converges for ∆t and ∆z chosen from f ε , it converges automatically during all the test duration ( ε < f ε ).
The applied vertical load is evaluated at any time during CRS test from the specimen deformation (or void ratio diminution): With e 0 is the initial void ratio of sample, and e(t) is the void ratio value e at time t.
For linear soil (a v =constant): With ' 0 v σ is the initial effective vertical stress that considered at equilibrium with initial void ratio e 0 , and ( ) With c c is the compression index that considered constant for nonlinear soil.
At every time during CRS test, effective stress hypothesis of Terzaghi is considered valid: For linear soil (a v =constant): For nonlinear soil (c c =constant): The average pore water pressure excess through the specimen at time t is equal to: Equation (24) is numerically evaluated by the trapezoidal integration method.
To evaluate the applied vertical load at any time t+∆t, and because the value of ( ) at the beginning of each time step can start with the known value of ( ) 19) and then a primary value of ( ) 13), and this process is repeated during each iteration until difference between two values of ( ) v t t σ ∆ + ∆ is less than a tolerance ( figure 2). For this model, it is assumed that any external hydraulic gradient is present; car if it is present the effect of associated seepage forces must be considered [10]. The seepage forces have an effect that reduces the effective stress excess and then increases the pore water pressure. Moreover, the selfweight effect is ignored because the thickness of sample is very small, and its weight is neglected comparatively to the vertical applied load.
The global structure of this numerical model can be illustrated by the flow chart shown in figure 2.

Model Results
To illustrate different results of this numerical model, the Resedimented Boston Blue clay (BBC) is taken as example; it is characterized by the following properties [10]: Specific gravity of solids G s =2.80, liquid limit LL=47.1, compression index c c =0.40, initial void ratio e 0 =1.26, coefficient of consolidation c v =5.016x10 -2 cm 2 /min, initial effective stress ' 0 v σ =68.4kPa, initial coefficient of compressibility a v0 =0.00254/kPa and initial hydraulic permeability k vo =5,526x10 -4 cm/min.
Different simulations with this model are conducted for test samples with initial height of H o =25mm.
The two important results of this numerical model are the vertical load ∆σ v (t)applied at the top boundary of sample, and the excess pore water pressure ∆u H (t) measured at the bottom of sample during CRS consolidation test. These two parameters are used with others parameters to evaluate the consolidation properties of soils such as compressibility curve, coefficient of consolidation c v and hydraulic conductivity k v [3,12,19]. (1) Convergence between linear and nonlinear variations of applied vertical load or base excess pore water pressure is observed only during small stains range. (2) Applied vertical load increases linearly with strain for linear soils and nonlinearly for nonlinear soils. (3) Base excess pore pressure increases linearly until reaches a pick and then decreases with strain for linear soils and increases nonlinearly and continuously for nonlinear soils.
(4) Variations of applied vertical stress or base excess pore pressure have similar trends for all strain rates but with different magnitudes. This indicates that consolidation properties evaluated from CRS test results are strain rate-dependent. The above results of this numerical model deviate from CRS consolidation theories [3] based on small strain conditions and shows Similar findings with numerical and theoretical methods based on large strain conditions [5,10]. For example, the base excess pore pressure for linear soils is assumed to be constant by small strain theories during all duration of CRS test [3], but results of this numerical model indicate that it increases until reaches a pick and then decreases continuously until the end of the test. Furthermore, the use of CRS data of this numerical model to evaluate c v and k v variations by the small and large strain theories, permits to verify which c v and k v values, from small or large strain theories, is closer to the assumed values of c v and k v used to generate CRS data of this model [20]. The small strain theory of Wissa et al [3] assumes in the steady state (T v > 0.5) that the coefficient of consolidation c v and hydraulic permeability k v are defined as For linear soil: For nonlinear soil: Where : ∆σ v (t+∆t ) and ∆σ v (t) are the applied vertical load values at times t+∆t and t respectively.
∆u H (t) is the base excess pore pressure at time t. σ v,avg is the average value of σ v over ∆t. ∆u H,avg is the average value of base excess pore pressure over ∆t.
σ ' v,avg is the average value of effective vertical stress over ∆t.
H 0 is the initial sample height. γ w is the unit weight of water, and ∆t is the time step. The large strain theories of Lee [5] and Sheahan and Watters [12] assume in the steady state (T v > 0.5), that the coefficient of consolidation c v and hydraulic permeability k v are defined as For linear soil: For nonlinear soil: Where h is the current specimen height at time t.
A dimensionless time factor T v was derived for CRS conditions [12], it indicates the degree of transience in the specimen strain distribution, and is evaluated from a function F that at any time equal to Differences Approach for CRS Consolidation Test For linear soil: For non linear soil: For both linear and non linear soils: 3 2 4.78( ) 3.21( ) 1.63 The calculated values of c v and k v by using the small and large strain theories for the linear soil case, are plotted versus average strain in figures 4 and 5 (a, b and c) for the strain rates of (0.1%/h, 1%/h and 10%/h) respectively. During the transient state stage (T v < 0.5), high values of c v and k v are obtained by using both small and large strain theories for all strain rates. Then, at the start of steady state stage, a good accuracy is observed between c v and k v values obtained by all methods for the two strain rates of (0.1%/h and 1%/h). Subsequently, as the strains increase, the c v and k v variations obtained by the large stain theory, closely correspond to the numerical model values for the two strain rates of (0.1%/h and 1%/h), but the small stain theory overestimates c v and k v values comparatively to the c v and k v model values. For the highest strain rate (10%/h), the values of c v and k v estimated by both small and the large strain theories are largely different to the constant values of c v and k v assumed by this model. This deviation is justified by the strain rate effect on the consolidation parameters of CRS consolidation test [20]. Results of this model indicates also that the obtained c v value by using the large strain theory is constant during all steady state duration, which is similar to the Wissa et al assumption [3]. It can be also observed that for the BBC soil case, the strain rate effect on the CRS results is considerable only when the standardized strain rate ≥ which is in good agreement with several others recommendations proposed to select proper strain rates for CRS consolidation test [5,21,22]. Results of others simulations (not shown) indicate the same trends for the nonlinear soil case.

Experimental Verification
Results of two experimental works [12,23] have been used to check the performance of this numerical model to produce results that are convenient with large strain conditions of CRS consolidation test. Sheahan and Watters (1997) [12] conducted three incremental tests and nine CRS consolidation tests with three different strain rates (0.1%/h, 1%/h and 3%/h) on resedimented Boston Blue Clay (BBC).

Experimental Works of Sheahan and Watters (1997)
Resedimented BBC was prepared in batches using a process based on methods used at Massachusetts Institute of Technology. Natural BBC passing a sieve no 40 was oven dried and ground into a powder of which 95% passed the sieve no 100. The slurry was mixed at an initial water content of 100%. The slurry was then drawn into an evacuated cylindrical chamber (25.4cm diameter, 35cm high), which was doubly drained and loaded using a piston. The slurry was incrementally consolidated to a maximum effective stress of 100kPa and rebounded to a final effective stress of 25kPa. After batch removal from the cylinder, pieces were cut, sized approximately for each test type to be performed (CRS consolidation tests, incremental consolidation tests), coated with a paraffin/petroleum jelly mixture, wrapped in cellophane, and stored in a humid room.
The CRS consolidation tests were performed using a computer-automated, hydraulically loaded Rowe cell [24]. During specimens' saturation phase, back pressure is applied through both the base porous element and the top drainage surface. Pore pressures are measured at five points through the specimen depth: at the top surface; at the specimen base; and at three middle depth points. The maximum effective stress reached during CRS tests varied from 320 to 510kPa, and the final stress levels were maintained for 24h to monitor final pore pressure dissipation and secondary compression. For this tests program, no unloading of specimens was performed.
Conventional incremental tests on the same soil were also conducted using procedures recommended in ASTM Standard D2435. The specimens were doubly drained and no back pressure was applied prior to consolidation. A load increment ratio of about 2 was applied, and each load was maintained 3 to 4h prior the application of the next loading. Stress-strain results from the IL tests were evaluated based on end of primary states.
Characteristics During simulations, each sample is divided into 100 elements, and BBC soil is first simulated as linear soil (a v =constant) and then as nonlinear soil (c c =constant). Subsequently, the obtained simulations data are used to evaluate c v and k v variations by using nonlinear method (equations. 32 and 33).  For the 3%/h CRS tests, a good convergence is also obtained between experimental and numerical c v variations estimated by large strain theories for the linear soil case (figure 6b). However, a considerable deviation is observed between experimental and numerical k v variations for both linear and nonlinear soils (figure 7b).

Experimental Works of T. Lok and X. Shi (2008)
T. M. H. Lok and X. Shi (2008) [23] performed IL and CRS consolidation tests on two types of Macau Marine Clays (MMC), the first is reconstituted characterized by LL =65 and e 0 = 1.65, and deformed under a strain rate of 2 % / h during CRS tests, the second is undisturbed characterized by LL = 60 and e 0 = 1.37, and consolidated using a strain rate of 1 % / h during CRS tests.
A consolidometer was used to prepare the reconstituted samples. The material necessary to prepare the reconstituted samples was taken from an excavation site in Macau, and was then mixed into thick slurry and passed through 600µm standard sieve to remove all large soil particles and shells. The slurry was poured into the consolidometer for consolidation, and a pressure of 1bar was applied to the slurry from the bottom cap. The consolidation was stopped when the primary consolidation phase had finished. After finishing the consolidation, the sample was carefully pushed out by hydraulic extruder and was cut into different sizes depending on the tests to be performed. The peripheral part of the specimen was discarded because of disturbance.
Two different sizes of undisturbed samples were obtained with 76 mm stainless steel Shelby piston tube sampler and U100 steel tube sampler with the length of 1m and 0.5m, respectively. The undisturbed clay samples were taken at the depth of 3m to 6m from the first site (Taipa), and at the depth of 6m to 13m from the second site (Cotai). After the sample was taken out from the bore-hole, both ends of the sample tube were sealed with wax in the field.
In incremental tests, the cell of 75mm in diameter was used to explore the consolidation behavior of reconstituted samples; while the cell of 50mm in diameter was used to explore the behavior of undisturbed samples. Both cells have the height of 20mm. Each load increment is maintained constant for 24 hours, and the load was doubled for the next increment. During the consolidation process, the vertical displacements were recorded at time intervals of 0, 0.   For UMMC, a small deviation is also obtained between experimental and numerical variations of c v and k v , but generally no significant difference is observed especially for nonlinear soil (figures 9a and 9b). Finally, from different comparisons it is observed that the results of numerical simulation of BBC linear soil are close to the experimental results than BBC nonlinear soil car probably the BBC soil has a linear behavior. Furthermore, because the numerical simulation of nonlinear soil of MMC provided results that more nearer to the experimental results than the linear soil, the MMC soil has probably a nonlinear behavior. Subsequently, choose of congruent method for analyzing linear or nonlinear soil is an important phase for interpreting the CRS tests [20].

Conclusions
In this study, the Terzaghi's numerical solution with finite differences approach of consolidation for constant loading is used to develop a new numerical model that simulates iteratively the continuous loading of CRS consolidation test. Results of this numerical model lead to the following: 1) Use of variable height of sample during CRS consolidation in this numerical model permits to get results (applied vertical stress and base excess pore pressure) that converge to analytical and numerical results based on large strain conditions. 2) Variations of coefficient of consolidation c v and hydraulic permeability k v estimated by the large strain theories from numerical model data, are very close to those corresponding to the model results.