One-Dimensional Consolidation Considering Viscous Soil Behaviour and Water Compressibility-Viscoconsolidation

The aim of this paper is to draw attention to experimental evidence which may contribute to the understanding of the viscous behaviour of soft soils and, above all, present the equation for primary one-dimensional consolidation including the soil viscous resistance and the compressibility of water, with its analytical solution. Initially, a generic equation is presented, in which any constitutive relationship of viscous resistance vs. strain rate may be incorporated. This study, in particular, considers two constitutive relationships: the first, represented by a two-parameter hyperbola with a horizontal asymptote, and the second, represented by a linear relationship which characterizes the soil behaviour as Newtonian. The hyperbolic constitutive relationship was derived from consolidation test results where the viscous resistance was inferred. For this case, the equation was numerically integrated using the software MAPLE 2017. For the linear relationship (Newtonian behaviour), the equation was integrated analytically and its solution presented. Finally, the results of several analyses are presented and compared with results obtained experimentally as well as with classical results of Taylor (1942) and Terzaghi & Frölich (1936).


Introduction
The aim of this paper is to draw attention to experimental evidence that may contribute to the understanding of the viscous behaviour of soft soils (Taylor & Merchant, 1940;Taylor, 1942;Lacerda, 1976;Martins, 1992) and, most of all, to present an equation and its analytical solution for primary one-dimensional consolidation which considers the viscous resistance of the soil and the compressibility of the water in the voids.The drive behind this study surfaced during an experimental research performed at COP-PE-UFRJ with the aim of understanding how the coefficient K 0 varies during secondary consolidation.The one-dimensional consolidation tests performed with a K 0 cell, designed by COPPE showed that at the beginning of each loading step, the pore pressure measured at the base of the sample was very low, increasing gradually with time until it reached a maximum, from which it decreased until finally reaching total dissipation.The solution proposed by Taylor (1942), which takes into account the viscous resistance component of the soil, was used to interpret the results, albeit this solution does not explain the initial pore pressure variation.The initial conjecture at that time was that this behaviour was due to the combined effect of the viscous resistance with the compressibility of water.For this to be true, it was believed that at the beginning of the test, the volumetric strain would lead to an increase in pore pressure with time, since the viscosity of the soil would pre-vent any process to occur instantly.Simultaneously, due to the high strain rates at this stage, the mobilized viscous resistance would be high, just sufficiently to alongside the resistance corresponding to pore pressure and effective frictional stress, satisfy the equilibrium condition.It is worth emphasizing that the component corresponding to the frictional effective stress would be, at this stage, much smaller than the pore pressure, since the stiffness of water is many orders of magnitude superior to that of the soil.With the progress of time, the drainage of the sample leads to a decrease in the pore pressure and an increase in the effective frictional stress.The strain rate, which has dropped monotonically, results in a reduction in the effective viscous stress.Thus, the growth rate of the pore pressure decreases, tending to zero (when the growth rate of the effective frictional stress equals the drop rate of the effective viscous stress) and, from then on, decreasing until the pore pressure is completely dissipated, at the end of the test.
Despite the existence of solutions which consider both the viscous resistance as well as the compressibility of the fluid in the study of one-dimensional consolidation, an analytical solution which considers both these mechanical phenomena combined could not be found within technical literature.
The solution proposed in this paper is able to predict reasonably well the pore pressure variation throughout the test during primary consolidation.

Viscoconsolidation
In perfect analogy with disciplines which incorporate viscous behaviour to classical Solid Mechanics, such as Viscoelasticity and Viscoplasticity, in this paper the term viscoconsolidation is used to define the study of onedimensional primary consolidation where the viscous and hydrodynamic behaviours occur simultaneously, including the effects of the compressibility of the water in the voids.In this case, both the viscous resistance and the hydrodynamic resistance work delaying the strain, making it occur as a function of time during primary consolidation until, at the end of the process, only the frictional resistance takes part in the process and the strain rate cancels out.Naturally, this is not precisely what is actually happening.It is known that the strain rate continues to exist after the end of primary consolidation, during secondary consolidation, until it finally comes to an end.
According to Tsytovich & Zaretsky (1969), the best way to assess the influence of viscous resistance of a soil, also called structural resistance by the authors and plastic structural resistance by Taylor (1942), is through the pore pressure measured in an undisturbed soil sample submitted to a compression test.For this purpose, a coefficient b 0 is defined as b 0 = u 0 /Ds, where u 0 is the initial pore pressure submitted to loading Ds.These authors highlight the consideration of volumetric strain as a result of the unsaturation of the soil for cases with degrees of saturation above 90%.Figures 1 (a), (b) and (c) illustrate some experimental results of pore pressure variation with time in samples with degrees of saturation of 95% and 98%.It may be observed that, in most loading steps, there is an initial growth in pore pressure, up to a maximum value, followed by a decrease leading to total dissipation.Zaretsky (1972) observes that, although this behaviour was present in soils with degrees of saturation smaller than 100%, it was also verified in tests carefully prepared to guarantee total sample saturation.Suklje (1969) presents curves for pore pressure vs. time obtained from consolidation of a normally consolidated lacustrine chalk with degrees of saturation of 96% and 92%, whose shapes resemble those of Tsytovich & Zaretsky (1969).
Due to its extremely low compressibility when compared to the compressibility of the soil structure, water is considered incompressible in most of the analyses of behaviour of saturated soils in Soil Mechanics.The bulk modulus of elasticity of degassed water at 20 °C is 2.15 ´10 3 MPa.It is also known that this modulus varies at different temperatures and pressures.However, when dealing with pore water in layers of soil below the water level, the situation is significantly different as, in these cases, not only may there be microscopic air bubbles, but also dissolved gases in the soil.According to Tsytovich (1987), a degree of saturation just under 100% increases signifi-cantly the compressibility of pore water and the bulk modulus of elasticity of water, K, in this case may be estimated with the equation: where: p a = atmospheric pressure, S = degree of sample saturation.
It is easy to understand that a one-dimensional consolidation test performed on soil with viscous behaviour and compressible water in the voids may be elementarily represented by a rheological model composed of a Maxwell and a Kelvin model connected in parallel, as shown on Fig. 2.
The Kelvin model (a linear spring element and a linear viscous dashpot element connected in parallel) represents the soil with viscous behaviour and the Maxwell model (a linear spring element and a linear viscous dashpot element connected in series) represents the hydrodynamic behaviour of the compressible water under vertical drainage.
It is worth stressing that the time-dependent representation of the hydrodynamic behaviour by a linear function of its velocity, although a simplification, does not invalidate the purpose of this modelling, which is to present the aspect of the progress of the pore pressure with time.
The model's differential equation for a constant loading s 0 obtained by the equilibrium of forces may be written as follows: whose solution for the initial condition e(0) = 0 is: The total resistance components may be represented as: pore pressure: frictional resistance: viscous resistance: Figure 3 shows a result for the variation of pore pressure with time, obtained by Eqs. 3 and 4, for typical values of E, K, h 2 and h 3 .

Relationship between viscous resistance and strain rate
According to Vyalov (1986), viscosity is a property of fluids (or gases) which causes resistance to the movement of elementary particles relative to one another.Newton (1687) was the first scientist to investigate viscosity.He observed that the resistance offered by a flowing liquid is proportional to its shear velocity.
Newton's law or rheological equation of state of a Newtonian liquid, which relates the shear stress t i with the shear strain rate & g i is given by: The constant h is the viscosity or dynamic viscosity and its SI unit is Pa.s.The reciprocal of viscosity, F = 1/h, is called fluidity.
Although the viscous phenomenon had been originally identified and defined for liquids, it is known to occur, with varying intensities, in virtually all solids found in Nature.
Thus, it may be written: where h c is the coefficient of viscosity for shear and h n is the coefficient of viscosity for compression or tension, often called Trouton factor.Many real solids have a viscous behaviour different from Newton's law.This distinct behaviour, known as anomalous viscosity, is manifested as a variation in the coefficient of viscosity as a function of the magnitude and direction of loading.
The dependency of parameter h on loading is equivalent to the non-linear relationship between the strain rate & e and the stress s.This viscous behaviour is known as non-linear or non-Newtonian.
The rheological equation of state of a non-linear viscous solid may be presented as: & ( ) or ( 10) Ostwald (1926), a pioneer in the study of anomalous viscous media, concluded that those media which have structure present a behaviour pattern distinct from those that are perfectly viscous (Newtonian).The explanation is that the structure changes with strain and, consequently, so does the viscosity.This variable viscosity is also known as structural viscosity.
Consolidation tests, either of one-dimensional or hydrostatic compression, have shown that the behaviour of clayey soils under compression is non-Newtonian.According to Alexandre (2000), the relationship between the effective viscous stress s' v and the rate of void ratio variation & e may be presented as Ostwald's power function, as follows: where n > 1, a = stress increment constant.Santa Maria (2002) proposes adjusting the solution to fitted hyperbolas of two and three parameters, in addition to the power function.Although Santa Maria (2002) observes that the power function leads to better correlations, it is important to note that this was based only on the values of the correlation coeficients of the fitting.A qualitative assesment of the fitting with the three proposed functions indicates that the fitted two-parameter hyperbola presents results which best resemble the behaviour experimentally observed.
Comparing Fig. 4 with the behaviour observed by Santa Maria (2002), Taylor (1942) and Thomasi (2000), it may be stated that the viscous component of the resistance,   either on one-dimensional or hydrostatic compression, does not display characteristics of either a purely viscous fluid (line A), or of a purely plastic one (line B), where the flow does not occur until a certain value of stress is reached.Likewise, it also does not present the behaviour of quasiplastic materials (lines C1 and C2), but complies adequately with the behaviour described by the fitted twoparameter hyperbola, represented by line D.
Figures 5 to 8 regarding one-dimensional consolidation tests (Taylor, 1942;Santa Maria, 2002) and Fig. 9 regarding a hydrostatic consolidation test (Thomasi, 2000), show how the viscous component of the effective stress varies with the absolute value of the void ratio rate and the strain rate.
These figures also show the fitted two-parameter hyperbolas, adjusted visually to the experimental points.The equation for the hyperbolas is as follows: Experimental results from consolidation tests have shown that there is an infinite set of void ratio vs. effective stress curves defining the behaviour of a clayey soil, one for each void ratio rate & e (Bjerrum, 1967;Martins & Lacerda, 1985).Naturally, this set of curves features a limit to the left, characterized by the curve e vs. s' for & e = 0.In this curve, the resistance to strain is exclusively frictional in origin as the viscous component is not mobilized (& e = 0).The new evidence indicates the existence of a limit to the right also,  (Taylor, 1942, after Alexandre, 2000) and fitted two-parameter hyperbolas.³ lim (Fig. 10).This means that after mobilizing all the available resistance (frictional and viscous), any subsequent load increment would result in an incresase in pore pressure, so that the condition of equilibrium is satisfied.

General equation
The general equation for one-dimensional viscous consolidation considering the compressibility of water may be written as (Appendix A): where e = void ratio, function of , e = average void ratio, K = bulk modulus of elasticity of water, a v = coefficient of compressibility, g w = unit weight of water and k = coefficient of permeability.
The following hypotheses are admitted: • soil is homogeneous; • soil particles are incompressible; • vertical soil drainage and strain; • Darcy's law is valid; • small strains; • water is considered as an elastic compressible fluid; • the bulk modulus of elasticity of water is constant for the existing pressure range; • C k and C v are constant for existing stress range and with time; • the viscous resistance s' v is a function of e and ¶ ¶ e t ; • the total applied stress is constant with time.

Numerical solution and experimental results
Admitting that the viscous resistance s' v was defined by the fitted two-parameter hyperbola, expressed by Eq. 14 and illustrated in Fig. 11, Eq. 13 is numerically integrated using the software MAPLE 2017.The solution requires two boundary conditions in z and two initial conditions in t: (1) z t = " 0, -solution of equation:  Santa Maria (2002) performed one-dimensional consolidation tests with remolded clay samples from the Sarapuí river using a K 0 cell designed by COPPE-UFRJ, illustrated in Fig. 12.The instrumentation comprises three load cells (one for each of the two lateral windows, to study the K 0 coefficient, and a third at the base, to determine the friction on the walls), three LVDT's, one for each window and one to measure the vertical displacement of the top cap, and a pore pressure transducer at the base.
The results presented in this paper refer to the incremental test K 0 -13. Figure 13 shows the variation of the pore pressure from the beginning of the test up to 2000 min, at the last two loading steps ( 4th and 5 th steps), alongside the values obtained from the integration of Eq. 13, performed using MAPLE 2017.The data and parameters regarding these two steps, obtained by Santa Maria (2002), are presented in Table 1.As mentioned before, k, a v , a e and b e represent mean values for each step.
It may be seen that from minute 0.6 onwards the theoretical curve is very representative of the behaviour obtained experimentally.Nevertheless, before this moment, there is a significant discrepancy owing to the initial conditions of the theoretical solution where, for time t = 0 + , the vertical strains and, consequently, the pore pressures are null.On the 5 th step of the test, for instance, the first pore pressure measurement occurs at 0.001 min, with a value of 28.5 kPa.Thus, it is thought that the best explanation for this high value of pore pressure at such a small fraction of time may be the inertial effect resultant from the load increment applied with weights on the yoke at the centre of the top cap of the oedometric cell.Naturally, this loading was performed with the utmost care to minimize the effects which are inherently associated with the manual loading process.

Equation for linear relationship between s' v and
¶ ¶

e t
When the functional relationship between s' v and ¶ ¶ e t is linear, it may be written as: For this condition, Eq. 13 becomes (Appendix A):  where h = coefficient of viscosity Eq. 16 is a third-order linear partial differential equation with t and z as independent variables.
To derive this equation, the following assumptions were considered: • soil is homogeneous; • soil particles are incompressible; • vertical soil drainage and strain; • Darcy's law is valid; • small strains; • water is considered as an elastic compressible fluid; • the bulk modulus of elasticity of water is constant for the existing pressure range; • C k1 , C kt , C v and C vt are constant for the existing stress ranges and with time; • the total applied stress is constant with time.

Analytical solution and experimental results
The solution to the equation requires two boundary conditions in z and two initial conditions in t, as shown below: (1) z t = " 0, -solution for equation:  where s is the total applied stress.
Considering that the solution for the differential equation presented in conditions ( 1) and ( 2) is where It can be easily shown that Eq. 17 becomes Taylor's solution (1942) when K ® ¥.
Figures 14 and 15 present the results of a comparative analysis between the solution proposed in this study and that presented by Taylor for the same parameter values corresponding to the 5 th step of test K 0 -13 (Table 1)   Soils and Rocks, São Paulo, 41(1): 33-48   It becomes clear that, as the bulk modulus of elasticity of water increases, the curves get closer to the curve corresponding to Taylor's solution (1942), as expected.
With the aim of comparing and verifying the consistency between analytical and numerical results, Fig. 16 presents the curves obtained by means of Eq. 17 and through numerical integration of Eq. 16 using MAPLE 2017, for the parameter values corresponding to the 5 th step of test K 0 -13 (Table 1).
As a perfect match was obtained, to distinguish the numerical result from the analytical the former was multiplied by a factor of 0.999.
Figure 17 presents the progress of pore pressure with time for the same parameters presented in Table 1, where the coefficients of viscosity, h, are considered equal to the angular coefficient of the tangent, a e , of the two-parameter hyperbola at the origin.As expected, the match between the theoretical and experimental curves is not as good as the one obtained by the general equation with function s' v represented by a fitted two-parameter hyperbola.
To fully envision the importance of the coefficient of viscosity in the behaviour of a soil during one-dimensional consolidation, the curves for void ratio vs. time and pore pressure vs. time are plotted for various values of h, as well as the curve for the Terzaghi & Frölich's solution (1936), which does not contemplate this effect (Figs. 18 and 19).The values of the remaining parameters are those presented in Table 1.It is worth noting that, for numerical convenience, in this case hour has been adopted as a unit of time.It becomes evident that as h increases, the values of the maximum pore pressure decreases, with the peaks displacing to the right.Additionally, as expected, the more the h value decreases, the more the pore pressure curves approach in aspect to that of Terzaghi & Frölich's solution (1936).With respect to the void ratio, it is observed that the lower the value of h, the more the curve moves away from that of Terzaghi & Frölich's solution (1936) in the initial branch.In the final branch of the curve, exactly the opposite occurs.
As a final analysis, the condition of applying a loading while the drain is closed is considered to predict the progress of pore pressure growth.In this case, except for the coefficient of permeability, all test parameters refer to the 5 th step in Table 1.To simulate the effect of closed drainage, a coefficient of permeability of 1 ´10 -50 m.min -1 is considered.
Figure 20 shows that, as expected, the higher the stiffness of water, the faster the pore pressure reaches its maximum and the closer it gets to the applied load (167.5 kPa).The difference between the maximum pore pressure value and the applied load represents the effective stress resisted by the soil, since it is submitted to the same volumetric strain as the water.

Conclusions
The study that resulted in this paper allows to highlight the following main conclusions: (i) For the soils investigated by Taylor (1942) and Santa Maria (2002) in one-dimensional consolidation tests and by Thomasi (2000) in hydrostatic consolidation tests, the relationship between the mean values of the viscous component of the effective stress and the strain (or void ratio) rate is characterized by a curve which may be satisfactorily represented by a twoparameter hyperbola going through the origin and featuring a horizontal asymptote.
(ii) For these soils, the shape of the curve viscous resistance vs. strain (or void ratio) rate suggests the existence of a limit value, represented theoretically by the ordinate of the fitted two-parameter hyperbola's horizontal asymptote.(iii) For one-dimensional consolidation of a soil with viscous behaviour and compressible water in the voids, both the general Eq. 13, where the viscous resistance vs. void ratio rate is represented by a two-parameter hyperbola (Eq.14), and and Eq. 16, where the soil behaviour is considered Newtonian, can predict the pore pressure variation with time.(iv) As expected, the general Eq. 13 with the viscous resistance defined by Eq. 14 presents better results than Eq. 16, where viscous behaviour is linear (Newtonian).(v) The analytical solution obtained assuming Newtonian behaviour for the soil was succesfully verified (a) by the fact that it becomes Taylor's solution (1942) when K ® ¥ and (b) by the perfect match between their results and those obtained by numerical integration of Eq. 16 using the software MAPLE 2017.(vi) Although the assumptions for both of the solutions presented lead to a condition of null pore pressure at the t = 0 + instant, the initial values measured are approximately 20% of the total applied stress and may be accounted for as a result of dynamic effects inherently associated to the application of the load increment at the beginning of each step.
where & v is the apparent velocity related to the gross crosssectional area of the soil.

Derivation of the equation Adding
The particle-liquid relative velocity must then be considered The Expanded Principle of Effective Stress (EPES) (Martins, 1992)  Differentiating the EPES with respect to t ¶ ¶ ¶s ¶ ¶s ¶ ¶s ¶ For the particular case where the soil behaviour is Newtonian Appendix B -Analytical solution of the one-dimensional consolidation Eq.A.9 The boundary conditions are Substituting the above equations in Eq.A.9, it leads to: which, by simplification, can be written as 3) The boundary conditions for f(z,t) are According to Taylor (1942), the aspect of Eq.B.3 and the boundary conditions above suggest that the solution may be written as Figure 1 -(a)-Pore pressure dissipation for experiments of Sipidin (Cambrian clay, S = 98% and water content = 30%); (b)-Pore pressure dissipation for experiments of Ter-Martirosyan and Tsytovich (Saratov clay of a disturbed structure, S = 98%, with loading step q = 100 kPa, where curves correspond to the consecutive loading steps); (c)-Pore pressure dissipation for experiments of Kogan (Silty loam of an undisturbed structure, S = 95%, water content = 30%, with loading step q = 100 kPa) (after Zaretsky, 1972).

Figure 3 -
Figure 3 -Progress of pore pressure with time obtained from a rheological model (Maxwell and Kelvin in parallel) of a one-dimensional consolidation test on viscous soil with compressible water in the voids.

Figure 2 -
Figure 2 -Rheological model representing a one-dimensional consolidation test on viscous soil with compressible water in the voids.

Figure 4 -
Figure 4 -Relationship between viscous resistance and strain rate.A = Newtonian viscous liquid; B = plastic viscous material; C1 and C2 = quasi-plastic viscous material; D = saturated clay under one-dimensional consolidation test.
and a e represent the angular coefficient of the tangent at the origin while b e and b e represent the ordinate of the horizontal asymptote.Figures 6 to 8 concern one-dimensional consolidation tests.It is worth drawing attention to the fact that the results presented were obtained for mean values of & e and s' v .As may be observed, the fitted two-parameter hyperbola represents reasonably well the viscous resistance variation with strain rate & e, volumetric strain rate & e v , and void ratio rate & e. Figures 5 to 9 clearly indicate a trend for the viscous resistance to reach a maximum value when the strain rate increases.This evidence has an important implication.
Figure5-Viscous component of the effective stress vs. void ratio rate -actual data from one-dimensional consolidation on a Boston Blue Clay sample(Taylor, 1942, after Alexandre, 2000)  and fitted two-parameter hyperbolas.
Figure 8 -Normalized viscous resistance vs. void ratio rate -fitted two-parameter hyperbola compared with actual points obtained from one-dimensional consolidation test K 0 -13 / all loading steps (Santa Maria, 2002).

Figure 9 -
Figure 9 -Viscous component of the effective stress vs. volumetric strain rate during a hydrostatic consolidation test -actual data (Thomasi 2000) and fitted two-parameter hyperbola.

Figure 10 -
Figure 10 -Schematic representation of the two boundaries of the region where the pattern of behaviour is possible.
the total stress applied.Eq. 15 is obtained by equilibrium and allows the evaluation of the pore pressure.

Figure 11 -
Figure 11 -Definition of limit viscous resistance (b e ) and initial coefficient of viscosity (a e ) for one-dimensional consolidation tests.

Figure 12 -
Figure 12 -Detail of the K 0 cell with the LVDTs holder.
Eq. 16 for the above conditions leads to the following solution (Appendix B): Figure13-Comparison between pore pressures predicted from Eq. 13 and obtained experimentally (SantaMaria, 2002).

Figure 14 -
Figure14-Comparison of void ratio values determined by Eq. 17 for several values of K and byTaylor's solution (1942).

Figure 15 -
Figure 15 -Comparison of pore pressure values determined by Eq. 18 for several values of K and by Taylor's solution (1942).

Figure 16 -
Figure16-Comparison between numerical and analytical solutions of Eq. 16, suggesting a perfect match (an offset was introduced in the curves, otherwise only one line would be seen).

Figure 18 -
Figure18-Void ratio vs. time comparison between values determined by Eq. 17 for various values of h and byTerzaghi &  Frölich's solution (1936).

Figure 19 -
Figure 19 -Pore pressure vs. time comparison between values determined by Eq. 18 for various values of h and by Terzaghi & Frölich's solution (1936).
Figure 20 -Pore pressure growth with time in an undrained soil sample with viscous behaviour considering several values of K.
by simplification that e e = (mean value of e), then ¶

Table 1 -
Data and parameters regarding the 4