Evaluation Of Non-Monotonous Dissipation Test Results.

Evaluation of non-monotonous dissipation test results

E. Imre, P. Q. Trang

Geotechnical Research Group for the Hungarian Academy of Sciences, Budapest, Hungary P. Rózsa

Department of Computer Sc. and Information Theory BME L. Bates, S. Fityus

School of Engineering, University of Newcastle, Australia.

ABSTRACT

A preliminary non-monotonous dissipation test data evaluation procedure is suggested and compared with a monotonous dissipation test evaluation method (Imre and Rózsa, 2004) which is numerically simpler and more robust. In the non-monotonous procedure the initial pore water distribution (including the thickness of the thin interface shear zone with negative initial pore water pressure) is identified. Data of 9 non- monotonous dissipation tests (u2 position, Burns and Mayne, 1998) are used for the comparison in such a way that only the monotonously decreasing final parts are evaluated with the monotonous procedure. Accord- ing to the results, the coefficient of consolidation c identified with the non-monotonous procedure varies within a range and the monotonous procedure gives generally a very good upper bound estimation for it.

Keywords: dissipation test, coupled consolidation, analytical, inverse problem solution

1 INTRODUCTION

The permeability/coefficient of consolidation of the soils can be determined more precisely in situ tests than in laboratory. During the dissipation test the time variation of the pore water pressure is meas- ured (Fig. 1). A model (with initial condition) and an inverse problem solver (including a reliability testing method) are needed for the evaluation of the measured dissipation test data. Neither the testing procedure nor the dissipation test evalua- tion methods have been fully standardized.

The dissipation test pore water pressure data are generally evaluated approximately (Lunne et al, 2002). Either the numerical solution of a 2D un- coupled model or some analytical solutions of 1D models are used (Table 1) are used. “ Dilatory”

pore water pressure data were evaluated first by Burns and Mayne (1998) with an analytical model.

In the case of both models the initial condition was produced with a separate theory using an addi- tional parameter (rigidity index).

In the frame of the ongoing research, two ana- lytical models have been used, the initial condition has been identified and a precise, automatic inverse problem solution method (with reliability testing) has been implemented for “monotonous” data (Imre, 2002; Imre and Rózsa, 2004). It has been

found that the necessary testing time can be con- siderably decreased with the suggested procedure.

The aim of present phase of the ongoing re- search is to investigate whether the foregoing pro- cedure can be applied or extended to “non- monotonous” dissipation test data.

In this paper, the method is extended to the evaluation of non-monotonous dissipation test data in such a way that non-monotonous initial condition shape functions are implemented. The previously published method is applied in such a way that only the monotonously decreasing final parts are evaluated. Results are compared.

2 METHODS 2.1 Models

The system of differential equations of the cou- pled 1 spherical consolidation model (Imre- Rózsa, 2002):

0 r= - u ) v r r( r 1

Eoed r ∂

∂

∂

∂

∂

∂ (1)

0

= ) v r r( r 1 + t r ) r u r( r 1 - k

v ∂

∂

∂

∂

∂

∂

∂

∂

γ (2)

A CB D E F

u u

₁

u

²

3

Figure 1. Some filter positions known from the literature.

a. Teh and Houlsby (1988), b. Kim et al (1997).

Table I. One dimensional linear pile consolidation models Model type Spherical Cylindrical

Uncoupled Torstensson (1975) Soderberg (1962) Coupled 1 Imre-Rózsa (2002) Imre-Rózsa (1998) Coupled 2 Imre-Rózsa (2005) Randolph-Wroth

(1979) Table 2. Thickness of the interface shear zone

Model-

version v [cm] rs =v+d [cm] v/d[-]

a 0.21 1.96 0.12

b 0.42 2.17 0.24

c 0.84 2.59 0.48

d 1.89 3.64 1.08

e 2.94 4.69 1.68

f 3.36 5.11 1.92

Table 3. Tested sites (from Burns and Mayne, 1998)

Site Depth

(m)

Lab cv

(mm²/s) 12 Amherst, MA (crust) 3.0 0.07 - 0.10 13 Brent Cross, U.K. 12.0 0.01 – 0.03 14 Canon’s Park, U.K. 5.7 0.01 – 0.03 15 Cowden, U.K. 17.2 0.05 – 0.19 16 Madingley, U.K. 5.8 0.03 – 0.08 17 Raquette, NY (crust) 3.0 0.03 – 0.70 18 St Lawrence, NY (crust) 6.1 0.07 – 0.80 19 Strong Pit, B.C. 6.7 0.06 – 0.30 20 Taranto, Italy 9.0 0.10 – 0.25 where u [kPa] is pore water pressure, v [m] is dis- placement, r [m] is the space co-ordinate and, t [s] is time, k [m/s] is permeability, γ_v [kN/m³] is unit weight of water, E_oed[kPa] is oedometric modulus.

The boundary conditions are as follows:

0 ,

1=

r)| r t

u( _r= (3)

0 y |

r) u(t,

r

=

r ≡

∂

∂

1 (4)

0 v >

r)| t,

v( _r ≡ ₀

0 (5)

0

| r) t,

v( _r ≡

1 (6)

Distance r0 =1.75 cm was used for the radius of the rod (in the following – instead of r0 – some- times d is used). Distance r1 was determined by the strain path method, in every filter position (Imre and Rózsa, 2004).

The transient part of the analytical solution for the pore water pressure u of the spherical coupled 1 model (Imre&Rózsa, 2004) is:

) , (

1

1 _C B_k t r r

= r)

u(t, _k

=

k∑^∞ (7)

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

+

−

⋅ +

⋅

] ) ( )

( [

)]

( )

( [

1 5 . 0 1

5 . 0

5 . 5 0

. 0 , 2

Y r r J

Y r r J

k k k

k k e k

= r) (t

B_k ^- k^cht

µ λ λ

µ λ

γ λ (8)

where Jn and Yn are the Bessel functions of the first and second kind, with the order of n, λk, µk

are the roots of the boundary conditions equations, Ck (k=1...∞) are the Bessel coefficients determinable from the initial condition, c is coefficient of consolidation (c= ch=k Eoed /γv).

The initial pore water pressure u0 is negative due to the interface shear in a thin zone along the shaft (between r0 and rs). The initial pore water pressure is positive everywhere due to the normal stress. To describe these effects, Equations (7) and (8) can be rewritten in vector notation as follows:

) , ( ) 1 (

2 1 C Btr r C

= r)

u(t, + (9)

where C1 and C2 are the Bessel coefficients concerning the initial pore water pressure due to the interface shear and to the normal stress, respectively. Equation (9) indicates that the pore water pressure varies independently due to the two effects.

The shape of the initial pore pressure distribution (r0 ≤r≤ rl) was assumed by specifying one of 50 different initial condition shape functions. The shape functions were described as a combination of 2 parts.

The first part is described by one of 5 linear functions within the shear zone (d=r0 ≤r≤ rs) (Fig 2, I to V). The second part, beyond the shear zone (rs <r≤ r1), is given by one of 10 functions described by equation (10) (Fig 2, 1 to 10)

∞

≠

⎟⎟ ≠

⎟

⎠

⎞

⎜⎜

⎜

⎝

⎛

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜

⎝

⎛ −

− −

F 0;

F e

- e 1 - 1

= r) F

u ^F

r r F

r

r _{1 1 s}

/ ,

0(

(10)

0 5 10 15 20

(r)

0.0 1.0

(u)

I II

IV III

V

1 2

9 10

a, b, c, d, e, f

Figure 2. Initial condition shape functions. I to V parts in the shear zone, 1 to 10 parts outside the shear, a to f thickness of the shear zone (Table 2).

1.0E-1 1.0E+0 1.0E+1 1.0E+2 1.0E+3 1.0E+4 1.0E+5 1.0E+6 1.0E+7

t (min)

-0.5 0.0 0.5 1.0

u (kPa)

II

1

10

Figure 3. The effect of the shape functions 1 to 10.

1.0E-11.0E+01.0E+11.0E+21.0E+31.0E+41.0E+51.0E+61.0E+7

t (min)

-0.5 0.0 0.5 1.0

u (kPa)

I II

III IV V

Figure 4. The effect of negative part of the initial condition.

1.0E-1 1.0E+0 1.0E+1 1.0E+2 1.0E+3 1.0E+4 1.0E+5 1.0E+6

t [min]

0.0 0.2 0.4 0.6 0.8 1.0

u [-]

a c f

Figure 5. The effect of the thickness of the shear zone Hence, the 50 different initial conditions are

described by the possible combinations of func- tions I to V and functions 1 to 10 (Figure 2).

Six similar models - denoted by a to f - were elaborated by assigning 6 different thicknesses for the interface shear zone (varying between 0.1r0

and 2r0) as given in Table 2.

The model-responses (the dissipation curves) are illustrated in Figures 3-5 as a function of the initial condition. The effect of the monotonous part beyond the shear zone is shown in Figure 3.

The effect of the non-monotonous part in the shear zone is shown in Figure 4. The effect of thickness of the shear zone is shown in Figure 5.

It can be seen that (i) after a certain time the ef- fect of the negative part vanishes, the dissipation pattern is controlled by the positive part of the ini- tial pore water pressure distribution, (ii) the adopted thickness of the shear zone may influence the initial part of the dissipation significantly.

2.2 Minimisation

The merit function was defined as follows:

)) t ( u max( N

p p

i i m N

i um(ti) u(ti, ) )

( F

∑= −

= ¹

(11)

where N is number of the sampling times, m is measured, p is the parameter vector consisting of c moreover of some linearly and non-linearly de- pendent initial condition parameters (e.g. the inte- ger type serial number for the shape functions (k=1..50)).

The solution of the inverse problem (i.e. the vector pmin related to the global minimum of the merit function) is reliable if it is a unique solution and, if its error is within the range of the parameters.

Geometrically, the solution is unique if the merit function has a “nice” or distinctly determinable single global minimum point. The parameter error can geometrically be interpreted as follows.

The merit function is called noise-free if simu- lated data are used. In Figure 6 the two merit functions are nearly the same everywhere except in the vicinity of the global minimum where the real-life merit function is “filled up” with noise.

The (generalized) standard deviation is a measure of the difference of the two merit functions along the so-called minimal section (Imre, 1996, Fig 6).

The standard deviation of parameter σ(pi) can analytically be determined using the linearized form of the model using some derivatives (Press et al 1986), and the outcomes of this procedure also indicates the uniqueness of the solution.

p

0.01 0.10 1.00 10.00

F [%]

i

~ 2 σ

noise-polluted merit function noise-free merit function

i

Figure 6. Geometrical concept of the standard deviation of parameter pi.

0.40 0.80 1.20 1.60 2.00

v/d [-]

4 8 12 16 20

F

13

17

16

15 12 18

19 20 14

Figure 7. The fitting error F in the function of the thick- ness of the smeared zone v/d.

0.40 0.80 1.20 1.60 2.00

v/d [-]

1E-4 1E-3 1E-2 1E-1 1E+0 1E+1

c [mm2/s]

13 17 16 15

12

18 19

20

14

Figure 8. The identified coefficient of consolidation c in the function of the the thickness of the smeared zone v/d.

In this work both analytical method and a geomet- rical methods were adopted (approximately) to determine every important minimum including the global minimum (Imre, 1996). The minimal section of the merit function concerning the non-linearly dependent parameters was constructed. The reli- ability was tested by representing these sections, and, by computing the standard deviation, also.

2.3 Measurements, model fitting problems

Nine non-monotonous dissipation test results (Ta- ble 3) were used taken from Burns and Mayne

(1998). The notation 12 to 20 indicates the Figure number in the original work.

Four types of inverse problems were solved in this work (Table 4). The dilatory data were evalu- ated with the non-monotonous models elaborated here in problems А(a to f) where a to f are de- scribed in Table 2. If only the monotonous parts of the dilatory data were evaluated with the same models then the problems were denoted by A(a)mon to A(f)mon. If the simpler monotonous model of (Imre and Rózsa, 2004) was used, then the problem was denoted by C. D was used for the results presented in the original work of Burns and Mayne (1998).

Table 4. The tested inverse problems

Sign Model Data

A(a-f) In this work Dilatory A(a-f)mon In this work Monotonous part C Imre – Rózsa (2004) Monotonous part D Burns-Mayne (1998) Dilatory

Table 5. The coefficient of consolidation c (mm²/s) iden- tified in the various problems

Non-monotonous

Data Monotonous part of Data

D A(a-f) A(b )mon C

chD chmin chmax chmon chC

12 0,40 0,25 2,10 0,42 48,39 14 0,40 0,08 0,25 0,03 0,34 15 0,25 0,63 2,10 1,05 2,52 16 0,20 0,13 0,30 0,13 0,25 18 0,50 3,37 8,42 3,37 8,41 19 0,30 0,34 2,10 0,84 2,52 20 0,20 0,63 3,79 0,63 4,21

3 RESULTS

The fitting error F and identified coefficient of consolidation c are shown as a function of the thickness of the shear zone v/d in Figures 7 and 8.

A large fitting error was found in test 13 where only the increasing period was measured. In the shallow (3m) crust tests (tests 12 and 17) F was decreasing with increasing v/d. In other (“nor- mal”) situations (tests 14, 15, 16, 18, 19 20) F was decreasing with decreasing v/d.

The coefficient of consolidation c was gener- ally decreasing with v/d, the thickness of the smeared zone. The decrease was slight for “normal”

soils and was larger for the shallow crust soils.

a. ^{0.00 0.01 0.10 1.00 10.00} 100.00 1000.00

Time [min]

400 600 800 1000

u-u [kPa]

15

measured computed

0

c=0.631-2.1 mm2/s

b.

0.01 0.10 1.00 10.00 100.00

Time [min]

200 400 600 800 1000

u-u [kPa]

15 measured

computed c=2.52 mm2/s

Figure 9. Test 15, measured and fitted curves. a. dilatory data b. monotonous part of data

a. ^{0.01 0.10 1.00Time [min]10.00 100.00 1000.00}

100 200 300 400 500

u-u [kPa]

16

measured computed

0

c=0.126-0.295 mm2/s

b. ^{1.00 10.00Time [min]100.00 1000.00}

100 200 300 400

u-u [kPa]

16

measured computed

c=0.337 mm2/s

Figure 10. Test 16, measured and fitted curves. a. dilatory data b. monotonous part of data

a. ^{0.00 0.01 0.10Time [min]1.00 10.00 100.00}

0 100 200 300

u-u [kPa]

18 measured

computed

0

c=3.37-8.42 mm2/s

b. ^{0.10 1.00Time [min]10.00 100.00}

100 200 300 400

u-u [kPa]

18

measured computed

c=4.21 mm2/s

Figure 11. Test 18, measured and fitted curves a. dilatory data b. monotonous part of data Table 6. The ratio of the coefficient of consolidation ch

(mm²/s) identified in the various problems (Table 5) chC /

chmaxA

chC / chD

chmaxA / chmin

chminA / chD

chC / chA(b)

14 1,34 0,84 3,00 0,21 10,0

15 1,20 10,10 3,33 2,52 2,40

16 0,86 1,26 2,34 0,63 2,00

18 1,00 16,83 2,50 6,74 2,50

19 1,20 8,41 6,23 1,12 3,00

20 1,11 21,04 6,01 3,16 6,67

mean 1,12 9,75 3,90 2,40 4,43

Sd 0,17 8,13 1,76 2,41 3,22

According to Figures 9 to 11, the difference of the measured and computed data reflects both some measurement or digitalization error and, some qualitative differences at the initial part.

According to Figures 12 and 13, the shape of the merit function concerning the non-

monotonous, extended model is more complicated than for the monotonous model. Several undesired local minima appear on the minimal section for parameter c and the global minimum may vary among these.

The minimum and the maximum coefficients of consolidation c identified throughout problems A(a) to A(f) may differ in a factor of about 4 for

“normal” soils (Tables 5 and 6). This uncertainty is not reflected by the standard deviation values σ(c) computed with the linearized models being about the same in both problems (Table 7). The c identified in problem C was about the same as the maximum c throughout problems A(a) to A(f).

According to Figure 14, the minimal section for the initial condition serial number shows that generally only the first linear part (i.e. I to V, Fig 2) can reliably be identified which is controlled by the negative pore water pressure developed in the interface shear zone.

Table 7. The fitting error F and the coefficient of varia- tion σ(c)/c for inverse problems A(b) and C

Problem Variable

A(b)

F [%] A(b)

σ(c)/c [%] C

F [%] C σ(c)/c [%]

12 7,25 1,6 0,402 0,06 14 2,61 7,57 0,126 0,34 15 2,25 3,76 0,171 0,22 16 4,45 7,26 0,43 0,42 17 7,35 3,03 0,423 0,10 18 2,93 5,01 0,162 0,25 19 1,52 4,16 0,108 0,24 20 1,95 6,11 0,144 0,31

Mean 3,78 4,81 0,24 0,24

Sd 2,3 2,08 0,14 0,12

4 DISCUSSIONS AND CONCLUSIONS

The model used in this work was based on a dimen- sional, point-symmetric consolidation model (Imre- Rózsa, 2002). This model employing a single mo- notonous initial condition shape function (Imre- Rózsa, 2004) was fitted on the monotonous part of the dilatory data sets.

Six model-version with non-monotonous initial condition were elaborated differing in the thick- ness of the interface shear zone. (The simulation results showed that the effect of the negative part of the initial condition vanished soon). These were fitted on the dilatory data sets.

Using the monotonous model, the merit func- tion had a single minimum. Using the non- monotonous model-versions, the shape of the merit function was complicated with several local minima and, the identified coefficient of consolida- tion c varied within a range. The so identified coef- ficients of consolidation c were about 2/9 times larger than the value determined by Burns and Mayne (1998). All the identified c were considera- bly larger than the c values determined with the laboratory tests.

1E-2 1E-1 1E+0 1E+1

c [mm2/s]

0.01 0.10 1.00 10.00 100.00

F [%]

12

Figure 15. The shape of some minimal sections concern- ing c of the real-life and the simulated merit functions in problem A (test 12).

1E-1 1E+0 1E+1

c [mm2/s]

0.01 0.10 1.00 10.00 100.00

F [%]

18

Figure 16. Comparing the minimal sections concerning c in non-monotonous problem A(a) - thin lines, and mo- notonous problem C - thick dashed line (test 18).

0 10 20 30 40 50

Initial condition number [-]

0.01 0.10 1.00 10.00 100.00

F [%]

13

I II III IV V

Figure 17. Minimal section for the initial condition (serial number) parameter in problem A (Test 13).

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