VISCO-ELASTIC THEORY OF THE DEFORMATION OF A CONFINED AQUIFER

Yoshiaki FUKUO Disaster Prevention Research Institute, Kyoto University

A B S T R A ~ The author derived the dynamic theory for the deformation of a granular solid

saturated with a liquid, assuming that the liquid filling up the pore space is a Newtonian viscous fluid and the skeleton constituted by solid particles is a linear visco-elastic solid. The theory consists of three fundamental equations, that is. the equations of motion of liquid and skeleton and the equation of continuity between the particles and liquid. In a case where the particle and liquid are taken to be incompressible and the defor- mation of soil is a quasi-static process, these equations are accepted as the theory of three-dimensional consolidation including Terzaghi‘s well-known equation as a special case and are also recognized as the basic equations of motion of confined ground water in a visco-elastic aquifer. A theoretical example will be shown for rheological deformation of a infinite confined aquifer with uniform thickness caused pumping up water at a constant rate.

RESUME L’auteur présente la théorie dynamique de la déformation d’une couche solide granu-

laire saturée par un liquide, en supposant que le liquide remplissant les pores soit newtonien et que le squelette constitué par les particules solides soit un solide linéaire- ment visco-élastique. La théorie consiste en trois équations fondamentales qui sont les équations du mouvement du liquide et du squelette et l’équation de continuité entre les particules et le liquide. Si on suppose les particules et le liquide incompressibles et si on admet que la dbformation du sol constitue un processus quasi statique, ces équations sont acceptées comme constituant celles de la théorie de la consolidation à trois dimen- sions en comprenant l’équation bien connue de Terzaghi comme un cas spécial. Ces équations constituent les équations de base du mouvement de l’eau artésienne dans un aquifer visco-élastique. U n exemple théorique sera montré pour la déformation rhéolo- gique dun aquifer artésien infini d’épaisseur uniforme dans le cas de pompage de l’eau à un taux constant.

INTRODUCTION

It is well known that soil mechanics has made great advances since the conception of pore pressure (Hydrostatische Uberdruck) proposed by K. V. Terzaghi. H e considered that the soil particles, being more or less bound to each other by an attracting force, constitute a skeleton of soil with elastic properties and that the skeleton supports the external burden together with the assistance of the water filling up the pore space between the particles. H e successfully solved the settlement of the soil layer with the idea that a contraction of soil depends on the rate of squeezing out of pore water, which neccessarily brings about the decrease of pore pressure. But he treated only a one-dimensional problem under constant load with a quasi-static method.

In October 1941, M. A. Biot (I) published the theory of three dimensional consolidation and developed the treatment of soil deformation for any arbitrary load variable with time. Recently, in September 1963, M. Mikasa (’) published the useful theory of soft layer consoli- dation showing many suitable examples, especially taking account of finite strain. But they treated only a one dimensional and quasi-static problem in the same way as Terzaghi.

Recently in soil mechanics, much attention has been paid to the correspondence of soil deformation caused by a vibrating agency in connection with the effective performance of engineering construction or safe protection from heavy damage, such as from an earth- quake. It may be clear that the theory of consolidation must be improved and made into a dynamic one for the above requirements. W e shall now derive in this paper the dynamic theory of consolidation, considering the rheological properties of soil.

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Yoshiaki Fukuo

I. DERIVATION OF FUNDAMENTAL EQUATIONS

Soil particles constitute the skeleton of the soil matrix, pushing and rubbing each other’s contact portions against an external burden. Owing to the complexity of its structure, however, one could not expect the direct treatment of forces acting on each particle. In the same situation, it would also be quite impossible to deal quantitatively with the motion of pore water attending to tortuous and irregular pore space. Therefore, we are obliged to consider the representation of motion averaged over a volume element of soil, which is taken to be large enough compared to the size of the pores so that it may be treated as homogeneous and at same time small enough compared to the scale of macroscopic phenomena in which we are interested so that it may be considered as infinitesimal in the mathematical treatment. It will be sufficient in soil mechanics to consider the average conditions over the volume of soil in the above sense.

(a) Equation of motion of pore water The motion of water in pores is governed by the hydrodynamic equation of viscous fluid. W e regard the pore water as a Newtonian fluid and denote by V (VI, V2, V3) the particle velocity of pore water. The equation of motion of pore water is expressed by

(1.1) DV 1 - = X - -grad p-(+q-rc) grad û+qV2V, Dt P

where t is time, X is external body force, p andp are the density and pressure of water, respec- tively, 6 is the divergence of water flow, and t,~ and IC are the kinematic viscosity of shear and bulk respectively; the dependences of which on density p are assumed to be slight.

Consider a unit volume of the soil matrix in the sensestated above. Integratingequation (1.1) over the pore space a of the unit volume and using the following notations

we have

DU - = UX - -grad 0 P-(+~-K) l{Ju grad ûdu+q ssf. V2Vdu, Dt P

(1.4)

where U is calIed “Darcy’s velocity” or ‘‘specific flow rate” and P is the “pore pressure” proposed by Terzaghi.

The pore pressure Pis generally taken to be thermodynamic pressure and is determined by the density and temperature of water. In constant temperature, we can write

P Po

(1.5) log - = /3(P-P,), where p is called isothermal compressibility. In the foregoing, po and Po are the density and pressure in some reference state, say a state at rest. In the case where external body force X is gravitational force, it is convenient to introduce the quantity q which is termed the ‘‘ piezometric head ”

P PS

q = - + x g ,

where x3 - axis is taken as positive upward. Appropriate expression to the last term in equation (1.4) is done by refering to Darcy’s

law governing the flow of water in a porous medium. H e postulated the viscous force

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Visco-elastic theory of the deformation of a confined aquifer

acting on water to be proportional to the flow velocity and introduced, as the proportional constant, physical quantity k which is called the coefficient of permeability of the soil, and according to his expression, the viscous force per unit volume of soil is written by

F 3 q JJJu V'Vdv e - JJJu Vdu k

In the next paragraph we shall treat the motion of soil particles together with the flow of pore water, and so it may be reasonable to assume that the viscous force F was propor- tional to the relative motion of water to soil particles, viz. (V- &/at) where u is themean displacement of soil particles as seen latter. From our assumption, it is possible to express its force as follows

Pore water is regarded to be almost incompressible in engineering practice. In this case, inserting the expressions (1.5), (1.6) and (1.7) into equation (1.4) and neglecting the inertia part of acceleration,

Equation (1.8) is, of course, reduced to Darcy's law if the soil particle has no motion and the flow of water is not accelerated.

(b) Equation of motion of soil skeleton W e shall now pay attention to the motion of the soil skeleton. Terzaghi, Biot and Mikasa assumed the elastic isotropy of stress-strain relations for the soil skeleton. While we also accept the isotropy in order to avoid the trouble of mathematical presentation, we had better give up the elastic property of soil with reference to the results of many investi- gations postulating the nonelastic deformation of soil 3). O n the other hand, it may be clear that the non-linear relation between stress and strain make it difficult to analyse the deformation quantitatively. W e now regard it tentatively as a linear viscœlastic relation.

Consider again the volume element in the sense stated previously and take the average over the actual displacement v of soil particle contained in that volume. W e define it as the displacement of skeleton u, that is

Supposing that the difference (v-u) produces only a minor effect on the stress on the skeleton viz. effective stress and assuming the strain to be infinitesimally small, the strain on the skeleton is given by tensor (eij)

(1.10)

Corresponding to this, the effective stress (oij) is exerted on the skeleton. According to the assumption of linear visco-elasticity, stress oij is represented, as positive in compres- sion (4), by

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Yoshiaki Fukuo

where 6ij is Cronecker's notation and 0 is the dilatation of the soil skeleton, that is

au au au ax, ax, ax,

o='+>+>. (1.12)

Introducing the operations with respect to time

a p m a p

n a p n a p

1

I + c ""2 1 + CS,,,

1 + c Y P j p 1 + c Y",, , & = p p = l , (1.13), (1.14) 2 =A-- p = 1

p = 1 p = 1

we can write the stress force per unit cubic element of soil as follows

(1.15)

Furthermore, the skeleton is pushed by the pore pressure of surrounding water. This pressure action f, may not produce any shearing strain by reason of the assumed isotropy and will be expressed by

ao ax ax, -- a% - -(9+A)- - AV2Ul.

gradpdu= -(1-a)gradP (1.16)

per unit volume of soil, where n is the outward unit vector normal to the surface element d a of solid space.

In addition to above forces, the skeleton tends to be dragged by the flow of pore water in its direction through the reaction of the viscous force acting on the pore water. This drag force fi will be expressed by

'1 = Jj(l-u)surf pnda = - ffJ(l-u,

f 2 - - -.-(.;-u). k (1.17)

W e can thus establish the equation of motion of the soil skeleton, that is

az u at2

(l-a)ps - = (l-o)X+V.o+f,+fz

=(l-a)X+(dp+A')grad

where ps is the density of soil particles and X is an external body force. In almost all cases with which we are concerned, X is a gravitational force. Expressing the vertically upward unit vector by k, we have

aZu at2

(1-n)~~- = -(l-a)psgk+(2+A) grad O + A V z u

(c) The equation of mass continuity Finally, we shall derive the equation of mass continuity per unit volume of soil. Suppose that a skeleton in any volume of soil had porosity ao at an instance of no dilatation 0 =O,

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Visco-elasiic theory of the deformation of a confined aquifer

and that the particles in it had density ps0 at that time. Because the skeleton under consideration is to be framed by the same particles at any instance, the mass of the skeleton must be conserved, that is

p,(1-4(1+0) =Pso(l-o,) 9 (1.20)

since the dilatation O represents the volume increase of soil skeleton per unit initial volume and ps is the density of particles at dilatation 0. The volume of the skeleton is varied with time by external force and consequently, porosity o is also varied. Equation (1.20) gives us the relation between their time rates. Because we are dealing with soil fully saturated with pore water, the change of pore volume results in the flow of pore water into or out of the volume element. This situation is represented by

ô - (op) = - div (opV) = - div (pu) . ôt

(1.21)

where p is the density of water; V is the particle velocity of water and U is the specific ñow rate as defined previously.

Combining equation (1.20) with equation (1.21), we make

a o (‘-‘o)PSo - d,v(pU) -{(i-o)p,+op} = - -- at at{ 1+0 I (1.22)

This is an equation which we expected to derive. Relation of p, to ps0 may be obtained from the consideration of the compression of soil particles due to the effective stress and pore pressure P, although we have little knowledge about it at present. However, in general, the compressibilities of soil particles and water are small. Assuming both den- sities to be constant, we rewrite equation (1.22) as

ÛO (1-~0)-+divU=O

at (1.23)

with good approximation neglecting the small quantity of order 02. Approximating the values o in equations (1.8) and (1.19) to the value o, after equation

(1.23) and summarizing the fundamental equation in the case where the external body force is only gravitational force and the soil particles and pore water are incompressible, we have

a2 u (l--oo)ps0 7 = -(l-oo)(p,o-po)gk+(~++h’)grad O+AV2u

at

!!! at + oog{grad p + t(U-oo $)} = O ao

(1-00) - + div U = O, at

where aUl au au P

@=-+ZJr3 , p=-+xx, ax, ax2 ax, Po 9

Yoshiaki Fukuo

In the remainder, we shall omit a subscript "," from the respective notations. 2. DERNATION OF FUNDAMENTAL EQUATIONS OF SUBSIDENCE

In this section, we shall seek the fundamental equations for the deformation of confined aquifer caused by ground water flow.

Let us ñrst examine a static equilibrium state of the skeleton with a steady flow of pore water. From equations (a), (b) and (c), we have

O = -(i-o)(p,-p)gk+(LZ++)grad O++VZu-pggradcp. (2.1)

O = -(i-a)(p,-p)gk+(LZ++)grad O o l + A V z u O l (2.2)

O = (U++) grad Ooz++V2uoZ-pg grad cp , (2.3)

O,, = div uol, Oo2 = div uoz (2.41, (2.5)

Now, we divide the displacement u into two parts, uol and uoz which satisfy the equations

respectively, where

Equation (2.2) expresses that uol is the displacement caused by the apparent weight of the soil skeleton in water without external load and equation (2.3) means that if uoz is uniform in the entire body of soil, grad cp, consequently U, is zero, say conversely, if pore water flows, the skeleton of the soil must be strained to that extent.

When external load and or piezometric head cp vary with time after the initial state, the strain on the skeleton and the velocity of flow begin to leave the static state. W e proceed to investigate the unsteady motion. Dissolving the quantities u. O, U and cp into the static part u, = u,, + uoz, O, E

O,, + Ooz U, and cp, and respective deviations u', O', U' and cp' from static one, we have for the deviating parts

a2 U' (1-0)~~-

atz

ao' at (1 -o) - + div U' = O.

(2.7)

W e can now derive the equation for only the dilatation O in the following manner. In the remainder, let us omit the prime on each quantity. Taking the divergences of equations (2.6) and (2.7) and time derivative of equation (2.8)

(I-o)p,q = (U+2A)VzO-(1-o)pgVzcp+p at k

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Visco-elastic theory of the deformation of a confined aquifer

azo a at2 at

(1-0)- + -divU=O.

Combining equation (2.10) with equations (2.8) and (2.1 i),

i - d 2 @ i ao cg atz k at

V(p=-- +- -.

Substituting from equations (2.8) and (2.12) into equation (2.9),

(2.10)

(2.11)

(2.12)

(2.13) 1-0 )azo pg ao (1 -OAP, + - p 7 + - 7 = (9+2A)V2@. [ ~ a J at- k öt

This is the equation we want to derive.

Neglecting the accelerated terms in equations (2.12) and (2.13), we find As the particular case which interests us, we shall pick out the quasi-static motion.

ao k -=kV2rp=-(T+2d)V2@. at PS

(2.14)

Differentiating the right part of equation (2.14) with respect to time and inserting the left part of equation (2.14) into this, we make

vz - - -(2+2A)V2cp =o. [: p, 1 (2.15)

This may be regarded as the basic equation of three dimensional consolidation. It is worthy of note that if 'pi (t, xi) is a solution of equation (2.15), 'pi (t, xi) +F1(xi)+Fz(t), where VzF, = O, is also the solution within the limits of quasi-static transition, in physical words, the progress of consolidation is not affected by the non-divergent flow of pore water and/or the gradual change of the water head in the entire region of the soil medium. When the non-divergent flow has already been included in the static part of piezometric head considered previously, equation (2.1 5) is reduced to

(2.16)

In the case of the elastic skeleton, operations 2 and A? are reduced to Lame's constant A and p, respectively. Let us examine the special case of a column of soil supporting a load and confined in a rigid cylinder so that no lateral expansion can occur. Equation (2.16) is then rewritten as

By comparing this with Terzaghi's well-known equation, we can see the relation of operations 2 and d to the value a which is termed the coefficient of compressibility in soil mechanics, (5)

Y+2& = (i+e)/a = i/(i-o)a.

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Yoshiaki Fukuo

Equation (2.14) is taken as the basic equation for subsidence related closely to the flow of confined ground water through the visco-elastic aquifer. W e shall demonstrate it here in a simple example.

3. EXAMPLE W e consider the deformation of a confined aquifer caused by pumping up the ground water at a constant rate in a laterally infinite aquifer with a uniform thickness, as seen in figure 1.6).

Assuming that, at the initial state, the confined water had no flow and the aquifer was in equilibrium under the overburden load in gravitational field, we have the equation for

* -, Depression cone -''-iDrawdom / '\

Confining stratum

\ -\

I -- \

FIGURE 1. Radial flow to a well completely penetrating an infinite confined aquifer with a uniform depth b

the quasi-static motions of soil skeleton and ground water

ao k at PS - = -(9+22)V20 = kV2cp.

Now, supposed that the aquifer deforms only in vertical direction and the flow of water is uniform in vertical cross-section and that the upper and lower boundary surfaces of aquifer are not leaky, we can put on initial and boundary conditions

t = O ; @=O, q = o , w(=u,)=O

8'

a'

1 > O; - 2nbkr-= Q, at r = rs ar

a r @=O, -=O, at r = c o

-- a'-~, w = o , at z=o, -- a' - O at z = b , (3.5), (3.6) az aZ

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Visco-elastic theory of the deformation of a confined aquifer

where b and k are the thickness and permeability of aquifer, respectively, r, is the radius of pumping well, Q is pumping rate of water and the surfaces z = O and b are the lower and upper boundary surfaces of aquifer, respectively.

From the conditions (3.5) and (3.6), we can regard that the quantities 9 and O are inde- pendent of z and that the amount of subsidence is obtained by w (z = b) = bO.

Reducing the equation (3.1) to ordinary differential equation by Laplace transforma- tion,

k PS

pV, = - {Y(p)+2d(p)}V2 V, = kVZI/,

with - Q

a r P r = r,; -2nbkr - - -

where

V, = SV e-P'Odt, V,,, = e-P'qdt. (3.10), (3.11)

Let us now consider the deformation in a Voigt model as a typical visco-elasticity' In this case, the operation Y + 2d is expressed by

and by usual notations shown in figure 2, the quantities A, p, y and c are represented as

E, E2 y=- '' and c = - Y2 1+2p = _ _ _ y

El +E2 El +E2 E2 (3.13)

with notice that l/c < l/y. W e have the solutions of equation (3.7)

(3.14)

4; Q 1 +YP v, = ~

lnvbr, Jp'(1 +cp) K, ($ rmz) where KO and KI are the modified Bessel functions and

(3.15)

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Yoshiaki Fukuo

FIGURE 2. Voigt's model for the rheological character of the aquifer

The solutions cp and O are determined from V, and V, by the use of theinversion theorem for the Laplace transformation

e'PV,dp, O = etPV,dp (3.16), (3.17) 2 d a-ia>

As the functions

1 +cp

have the branch points at p = 0,- l/c, - l/y and - CO, theintegrationsin equations(3.16) and (3.17) are carried out, using the contour of figure 3 with two cuts on the negative real axis so that the integrants are single valued functions of p within and on the contour.

In the limit as the radius of circle ï tends to infinity and the radii of circles 6,, 6, tend to zero, the respective integrals round them can be shown to vanish. O n the circle 6, we can find the limiting value

etPV,dp = - (3.18) 4nvb

as the radius 6, tends to zero, where C = 0.5772 ... is Euler's constant.

and then K,[(,/z/v)rz)], K,[(,/z/v)r,z] are expressed by the formulas Because the argument ofp is n and -n on the lines L,, L, and Li', L,' respectively

n KO( f iz) = - - [ Yo(z) I iJo(z)],

2 z : real, not negative

'II: K,(fiz) = - 2 - [J,(z) T iY,(Z)],

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Visco-elastic theory of the deformation of a confined aquifer

FIGURE 3. The contour of inverse integration for Laplace transformations V, and Ve

we get

Finally, we get the solution for the piezometric head

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Yoshiaki Fukuo

where

r = r & ,JP(~-YP), ,.:, = & r w J ~ ( l - ~ ~ ) V i-cp V 1-cp

O n a similar way, we can find the solution for the dilatation

@=- 4nv2 b

When the radius of the pumping well is very small, we have

2 r 2v 1-cp lim Ji(rp)=O, limr, Yi(+,)= - - A= -- rw-o rw+O

and then

rw+O 4nvb

(3.19)

rw+O 4nv2 b

In the special case where y = O, we have

rw-O 4nvb

- W(pt)Jo - r - + iim ap a (f J1:cJdpl

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Visco-elastic theory of the deformation of a confined aquifer

+ r JO1'' W(pt) 2 J- * J, ($ r J p ) d p ] V ap I-cp I-cp

- - "[-C-iog(Lf)+$rJ: 4nbk 4v2 t W(->Jl($rt)dS], ct2 + 1

where

is well function and t = and also

- -re fi -'"[:W{- }J,($r<)d<]. Y C(Ct2+ 1)

Using the dimension time t* and distance r* -

t - C

t* s-, r*

the amount of subsidence 6 is obtained by i=bO

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Visco-elastic theory of the deformation of a conJned aquifer

* OD (-1)" *n+m

- - 271(A+2p)k PgQ [(l-e-f*)Ko(r*) + n = i 1 (n-l)!n! (-)"X,,(r*) 2 m=om!(n+m)

* 1 n = i (n!)

-e-$* - 2 t*nGrKfl(r*)]

(l-e-'*)K0(r*) + 271 (A + 2p)k

- -

Summarizing the result of the calculation, we have the solutions cp and 5 in a special case where

9 + 2 A = ( L + 2 p ) l+c- ( 3 n-1 *m

Ko(r*)+ (2J { c]] (3.21) rw+û 2 ~ b k n = i n! m = o m!

* lim cp = - - - K,(r*) 1-e-"

where

(3.23)

4. PRACTICAL DETERMINATION OF PROPER CONSTANTS OF AQUIFER IN VOIGT MODEL

W e may have to find the adequate method determining proper constants of the aquifer, that is, elastic factor A+2p, permeability k and retarding time c for practical utility of above solutions (3.21) and (3.22). First approximations will make it possible to us in following way.

Differentiating these with respect to tume t, we have

and could determine the retarding time c from the slope of each line along which the rate values of variation of head cp or subsidence 5 were plotted against time on the semi- logarithmic graphs, respectively. Moreover, from the segments on the ordinate of graphs, respectively. Moreover, from the segments on the ordinate of graphs cut off by plotted lines, log a, and log a(, we obtain the relations

471c r* -a,=-K,(r Q bk

271c -ac = " Ko(r*) = zzKo($z), Q k (1 + 2P)

(4.3)

(4.4)

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Yoshiaki Fukuo

where

So, if we are ready to use the family of curves, depending on the Bessel function KO (dz) with respect to the parameter a e r/Jc as seen in figure 4, we may seek easily the value of argument z in equation (4.4) fitting to calculated value 2ncac/Q, which is regarded as the

1

- 0 o. I o. 2 I

FIGURE 4. Graph for finding the value of argument z of Bessel function fittiag to equation (4.4)

ordinate value in this figure, because the parameter a has been specified from retarding time c and actual distance r from pumping well to observing point. When the argument z was decided, the value of permeability k will be determined by equation (4.3) in use of data for the thickness b of aquifer and subsequently the value of elastic factor (1+2p) will be also determined by equation (4.5).

REFERENCES

1.

2.

3.

4.

5.

6.

BIOT, M. A. (1941) ; General Theory of Tree-Dimensional Consolidation, J. Appl. Phys., Vol. 12, February 1941, pp. 155-164. MIKASA, M. (1963); The Consolidation of Soft Clay-A New Consolidation Theory and Its Application-, (Monograph in Japanese), Inst. of Kashima Inc. Novembre 1963, pp. 1-126. KRAVTCHENKO, J. and SIRIEYS, P. M.: (1966): Rheology and Soil Mechanics, Symposium Grenoble, Apr. 1-8, 1964, Springer Verlag, Berlin. FREDERICKSON, A. G. (1964): Principles and Applications of Rheology, Prentice-Hall, Inc. Englewood Cliffs, N. J. JUMIKIS, A. R. (1962): Soil Mechanics, D. Van Nostrand Co. Inc., Priceton, New Jersey New York. DE WIEST, R. J. M. (1965): Geohydrology, John Wiley & Sons, Inc., New York,

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