analytical study of mine closure behaviour in a poro-elastic medium

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Computers and Geotechnics 35 (2008) 645–654

Analytical study of mine closure behaviour in a poro-elastic medium

Henry Wong a,b,*, Mathilde Morvan a,b, F. Deleruyelle c, Chin Jian Leo d

a Universite de Lyon, Lyon, F-69003, Franceb Ecole Nationale des Travaux Publics de l’Etat, CNRS, URA 1652, Departement Genie Civil et Batiment, 3, rue Maurice Audin,

Vaulx-en-Velin, F-69120, Francec Institut de Radio-protection et de Surete Nucleaire (IRSN), BP 17, 92262 Fontenay aux Roses, France

d School of Engineering, University of Western Sydney, Locked Bag 1797 Penrith South DC, Sydney NSW 1797, Australia

Received 3 August 2007; received in revised form 8 November 2007; accepted 8 November 2007Available online 26 December 2007

Abstract

This paper is concerned with the hydro-mechanical behaviour of an underground cavity abandoned at the end of its service life. Dete-rioration of the lining support with time leads to the transfer of the loading from the exterior ground to the interior backfill, both ofwhich are of poro-elastic behaviour and saturated with water. This loading transfer is accompanied by an inward cavity convergenceand an outward water flow, leading to a complex space-time evolution of pore pressures, displacements and stresses. An entirely explicitsolution to this problem is developed, using Laplace transform. A few numerical examples are given to illustrate the hydro-mechanicalbehaviour of the cavity and highlight the influence of key parameters such as backfill stiffness and rate of lining support deterioration.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Poro-elasticity; Laplace transform; Analytical solution; Cavity convergence; Mine closure

1. Introduction

This paper is concerned with the hydro-mechanicalbehaviour of an underground cavity abandoned at theend of its service life, which is taken to be the referencestate. This reference state is the initial state of our analysis(t = 0). To simplify the problem presentation, the mediumsurrounding the cavity is initially in a state of equilibrium,both mechanically and hydraulically, the cavity conver-gence being opposed by an inner lining, in equilibrium withthe surrounding saturated ground. The cavity is initiallyfilled with a saturated backfill whose pore pressure is iden-tical to that of the geological formation. Due to difficulty ofcompaction, the backfill is supposed to have zero initialeffective stress, the difference of inside and outside effectivestress being taken up by the cavity lining. This lining is sup-posed to deteriorate with time thereby inducing cavity con-

0266-352X/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compgeo.2007.11.003

* Corresponding author. Tel.: +33 4 72047268; fax: +33 4 72047156.E-mail address: [email protected] (H. Wong).

vergence, which compresses the backfill and water insidethe cavity and induces outward radial flow. We are inter-ested in the temporal evolution of the cavity convergence,the displacements of the surrounding medium, the porewater pressure and the stresses inside the cavity as well asinside the massif. This problem corresponds closely to thehydro-mechanical behaviour of an abandoned mine cavityat the end of its service life, when its lining progressivelydeteriorates which may cause potential hazards. We pro-pose analytical solutions of this problem which are efficientin parametric studies as part of a decision-making process,besides constituting useful benchmark solutions. We willbegin by recalling the main equations in poro-elasticitywhich will be needed in later paragraphs.

2. Poro-elastic constitutive equations

Poro-elastic behaviour is a classic problem and is well-treated in the literature ([3,6,5,7] or [12]). In the sequel, weonly present the most important equations. Temperature

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646 H. Wong et al. / Computers and Geotechnics 35 (2008) 645–654

effects are neglected in this paper, but can eventually beaccounted for using results from earlier investigators suchas Smith and Booker [11]. Limiting ourselves to small strainsand in the case of linear isotropic behaviour, the poro-elasticconstitutive equation writes (see for example [6,10])

rij ¼ r0ij þ K � 2G

3

� �2 dij þ 2Geij � bðp � p0Þdij;

dij ¼1 if i ¼ j

0 if i 6¼ j

� �; 2¼ ekk ð1Þ

In the above equation, rij and p, are respectively the totalstresses (tensile stress positive) and pore pressure with ini-tial values r0

ij and p0 at t = 0, whereas K and G are the bulkand shear moduli of the porous medium under drainedconditions. The symbol 2 is used to denote the volumetricstrain which will often appear in subsequent developments.The Biot’s coefficient b depends on the bulk modulus of thesolid phase Ks. By supposing incompressible solid phase, qs

remains constant and Ks tends to infinity and Biot’s coeffi-cient then tends to unity (b = 1). To simplify further,incompressibility of the liquid phase is also assumed, sothat its density qf remains constant.

The fluid mass flux, wf is linked to the pressure gradientvia Darcy’s law and the hydraulic conductivity kh by: wf/qf = kh(�gradp + qfg), while the initial hydraulic equilib-rium (no flow) implies: 0 = kh (�gradp0 + qfg). The differ-ence between the last two equations yields

wf

qf

¼ �khgradðp � p0Þ ð2Þ

Solid phase incompressibility implies that any overall vol-ume increase must come from the change in pore volumedue to fluid influx. Invoking (2), this leads to the followingdiffusion equation of the fluid (see [5] for details)

o 2ot¼ khDðp � p0Þ ð3Þ

The assumptions of incompressible constituents are gener-ally applicable with a good precision in soils and soft rocks.

3. 1D poro-elastic problems in spherical symmetries

The stress tensor must satisfy the equilibrium conditionsat actual (t > 0) and initial time (t = 0)

divrþ qg ¼ 0; divr0 þ q0g ¼ 0 ð4Þwhere q = (1 � /)qs + /qf and q0 = (1 � /0)qs + /0qf arethe overall densities at t > 0 and t = 0. Generally, changeof density is one order of magnitude less than the changein stress, hence we can assume that q � q0, leading to

divðr� r0Þ ¼ 0 ð5ÞFor most soil and rock mechanics problems, the displace-ment field is irrotational, rot(u)=0. On account of the iden-tity Du = grad(divu) � rot(rot(u)), this leads to

grad 2¼ div e ð6Þ

Combining Eqs. (1) and (4), taking into account (6), we get

K þ 4G3

� �grad 2 �gradðp � p0Þ ¼ 0 ð7Þ

In one dimensional problems, the gradient operator is lim-ited to the derivative with respect to the unique space var-iable (see for example [8]). Integration of the aboveequation then gives

K þ 4G3

� �2 �ðp � p0Þ ¼ AðtÞ ð8Þ

where A(t) is the ‘‘constant” of integration independent ofspace. Moreover, in problems of spherical or cylindricalsymmetry involving an infinite medium, the hydro-mechan-ical perturbations are limited to the near field, so thatstrains and pressure variations are negligible far from theorigin, leading to A(t) = 0. Hence

ðK þ 4G=3Þ 2¼ ðp � p0Þ ð9ÞThis last equation is very useful as it relates linearly volu-metric strain to pressure change. It must, however, be re-minded that A(t) is not necessarily null when the domainof analysis does not extend to infinity. Such is actuallythe case for the internal backfill with 0 < r < a analysedin Appendix A. This last relation allows uncoupling ofthe hydraulic diffusion equation (3) from the mechanicalvariables

oðp � p0Þot

¼ khDðp � p0Þ ð10Þ

where the hydraulic diffusion coefficient kh, given by

kh ¼ khðK þ 4G=3Þ ð11Þis only valid for incompressible solid phase. Note also thatthe uncoupling of hydraulic behaviour from the mechanicalbehaviour, leading to the fluid diffusion equation (10) canonly be achieved for irrotational displacement fields. Onedimensional poro-elastic problems obeying spherical sym-metry belong to this category and can be formulated as atwo-fields problem on displacement and pressure basedon the two governing Eqs. (9) and (10), the two indepen-dent variables being (r, t). The compatibility conditionrelating the volumetric strain 2 to the 1D displacementfield depends on the type of symmetry. Concentrating ourattention on spherical symmetry in this paper, we have

2¼ ouorþ 2u

rð12Þ

The problem is to be solved with adequate boundary con-ditions. For linear problems, a very widely used and conve-nient tool is the Laplace transform [4], defined by

Lðf Þ � �f ðsÞ �Z 1

0

f ðtÞe�stdt; f ðtÞ ¼ L�1½�f ðsÞ�

� 1

2pi

Z cþi1

c�i1ets�f ðsÞds ð13Þ

H. Wong et al. / Computers and Geotechnics 35 (2008) 645–654 647

where the inversion integral is to be evaluated along a ver-tical path Re(s) = c, such that all the singularities of �f ðsÞ lieto its left. The Laplace transform of (9) and (10), takinginto account (12), leads to two linear coupled ordinary dif-ferential equations on �u and �p which can be solved simul-taneously taking into account the relevant boundaryconditions in the transformed domain.

4. Mine closure problem in a poro-elastic medium

The life cycle of an underground mining cavity is idea-lised by four stages as shown in Fig. 1. Initially, the groundis in a state of hydro-mechanical equilibrium. The cavity isthen excavated and an internal support is provided tomaintain its stability. Various techniques of support exist.For example, it can be evenly spaced steel bolts or a layerof shotcrete or a combination of them. For modelling pur-poses, this support can be assimilated to a layer of elasticmaterial lining the cavity walls. At the end of its service life,the cavity is backfilled with a poro-elastic material beforebeing abandoned. We are interested in the long term evolu-tion of the hydro-mechanical fields in the surroundingmedium and in the backfill after the its abandon, that is,after stage (c) when the support starts to deteriorate. Theground behaviour is poro-elastic. Necessary symmetryassumptions will be made to render the problem onedimensional so that all field quantities will only dependon a single space variable r, as well as on time t. In partic-ular, the cavity is supposed to be reasonably deep (depthmore than 4 to 5 time its diameter) so that the stress gradi-ent can be neglected when analysing the near field behav-iour. Under these conditions, the cavity is assumed to beperfectly spherical to simplify analysis. This is an approxi-mation to real cavities but its results do constitute useful

(a) initial state (b) excavation & support installation

(c) backfill (d) support deterioration

Fig. 1. Schematic diagram showing the idealisation of the fours stages inthe life cycle of a mining cavity: excavation-support-backfill-abandon.

tools to engineering designs. Analyses of mine cavities asspherical cavities have already been done in the past. Inter-ested readers can consult usefully the work of Berest [2] andBerest and Nguyen-Minh [1].

4.1. Problem resolution

The governing equations on the two-field problem (u,p)have been developed in the last paragraph, which in thecase of spherical symmetry can be written as

oðp � p0Þot

¼ kho2ðp � p0Þ

or2þ 2

roðp � p0Þ

or

� �ð14Þ

K þ 4G3

� �ouorþ 2u

r

� �� ðp � p0Þ ¼ 0 ð15Þ

It remains to specify the initial and boundary conditions.

4.2. Initial conditions at t = 0

The initial state corresponds to stage (c). In reality, theground stresses can be fairly complicated immediately aftersupport installation, depending on the phasing of excava-tion and supporting structures. However, it is thought thata large part of the deviatoric stresses would be dissipatedwith time due to relaxation, so that at stage (c) the stressfield can be taken as isotropic, equal to the geostatic pres-sure. Both the stress and pressure fields are homogeneous.Initial displacements and strains are null. More explicitly

a 6 r <1 : uðr; 0Þ ¼ 0; pðr; 0Þ ¼ p0;

rijðr; 0Þ ¼ �R0dij; eijðr; 0Þ ¼ 0 ð16Þ

Note that such hypotheses can be relaxed as the formula-tion depends on the stress and pressure variation. Nonethe-less they are considered to be reasonable assumptions inthe context of this paper. For the backfill, it is assumedto be very permeable and initially in hydraulic equilibriumwith the surrounding ground and, therefore, shares thesame initial pore pressure. However, since compaction isdifficult, its initial effective stress is taken to be null. Inother words

0 6 r 6 a : uðr; 0Þ ¼ 0; pðr; 0Þ ¼ p0;

rijðr; 0Þ ¼ �p0dij; eijðr; 0Þ ¼ 0 ð17Þ

The difference of total (or effective) stress is supposed to betaken up by the lining support

psðt ¼ 0Þ ¼ ps0 ¼ R0 � p0 ð18ÞThe lining is supposed to be very permeable so that thepore pressures on both sides are at any time identical (ie.no pressure jump across the lining). On the other hand,the effective stress inside the backfill at the interior surfaceis different from that in the massif on the exterior surface.This stress difference is equilibrated by the circumferentialcompressive stress inside the lining.


4.3. Boundary conditions at t > 0

Simplifications, notably the very high hydraulic conduc-tivity of the backfill, allows us to ‘‘eliminate” the fieldquantities in the backfill (0 6 r 6 a) and restrict our analy-sis to the domain exterior to the cavity (a 6 r <1). Detailsof this analysis are reported in Appendix A. We, therefore,need to establish two boundary conditions at the cavitywall, r = a.

For incompressible constituents, the convergence of thecavity wall must be accompanied by the expulsion of anidentical volume of water. On account of (2), this can beformulated as follows

BC-1 : kh

oðp � p0Þor

ða; tÞ ¼ ouotða; tÞ ð19Þ

To formulate the second boundary condition, we need toconsider the equilibrium of the lining support, which im-plies that

�rrrða�; tÞ þ psðtÞ ¼ �rrrðaþ; tÞ ð20ÞThe deterioration of the lining support is simulated by asupport pressure decreasing with time. In the absence ofprecise experimental data, we will assume an exponentialdecay law:

psðtÞ ¼ ðR0 � p0Þ expð�jtÞ ð21ÞThis support-pressure reduction leads to cavity conver-gence, which induces effective stresses and pressure rise inthe backfill and radial out-flow of pore water. The totalstress on the exterior surface of the support, rrr(a

+,t), canbe obtained using the poro-elastic constitutive equation(1) and the appropriate components of the strain tensor.Under spherical symmetry, the latter writes

e ¼ou=or 0 0

0 u=r 0

0 0 u=r

264

375 ð22Þ

Whereas rrr(a�, t) is given by (A6). Combining these results

we get the second boundary condition on the cavity wallr = a

BC-2 : K � 2G3

� �ouorþ 2

ur

� �þ 2G

ouor

� �r¼aþ

¼ 3KR

uða�; tÞa

þ ðR0 � p0Þð1� ejtÞ ð23Þ

4.4. Dimensionless form of the equations

In order to simplify the writings, we will normalise theequations to a dimensionless form by dividing the physicalquantities to their respective characteristic values. The cav-ity radius will be taken as the characteristic length and theclassical characteristic hydraulic diffusion time a2/kh as thecharacteristic time. The initial geostatic overburden R0 willbe used as the characteristic stress and pressure to normal-

ise all quantities having the dimension of stress. The result-ing dimensionless variables will be denoted using anapostrophe (ie. r0 = r/a). Finally the dimensionless deterio-ration rate j0 will be defined so that jt = j0t0. Summarising

r0 ¼ ra

; u0 ¼ ua

; t0 ¼ khta2

;

j0 ¼ a2jkh

; p0 ¼ pR0

; p00 ¼p0

R0

;

K 0 ¼ KR0

; G0 ¼ GR0

; K 0R ¼KR

R0

ð24Þ

Hence, the dimensionless form of the governing equationswrites

oðp0 � p00Þot

¼ D0ðp0 � p00Þ ð25Þ

2¼ ou0

or0þ 2

u0

r0

� �¼ p0 � p00

x; x ¼ K þ 4G=3

R0

ð26Þ

where D0 stands for the Laplace operator with respect to r0

(ie. D0 = a2D). The dimensionless form of the boundaryconditions write

BC-1 :op0

or0ð1; t0Þ ¼ x

ou0

ot0ð1; t0Þ ð27Þ

BC-2 : K 0 � 2G0

3

� �ou0

or0þ 2

u0

r0

� �þ 2G0

ou0

or0

� �r0¼1þ

¼ 3K 0Ru0ð1�; t0Þ þ ð1� p00Þð1� ejtÞ ð28Þ

4.5. Analytical solution in the Laplace transform domain

The Laplace transform of the governing equations andthe boundary conditions write

o2ðp0 � p00Þor02

þ 2

r0oðp0 � p00Þ

or0� q2ðp0 � p00Þ ¼ 0 ð29Þ

ou0

or0þ 2

u0

r0

� �¼ p0 � p00

xð30Þ

BC-1 :oðp0 � p00Þ

or0ð1; sÞ ¼ xsu0ð1; sÞ ð31Þ

BC-2 : K 0 � 2G0

3

� �ou0

or0þ 2

u0

r0

� �þ 2G0

ou0

or0

� �r0¼1þ

¼ 3K 0Ru0ð1�; sÞ þ ð1� p00Þ1

s� 1

sþ j0

� �ð32Þ

where s is the conjugate variable to t under Laplace trans-form and we have used the notation

q ¼ffiffisp

ð33ÞThe general solution to Eq. (42) is

p0 � p00 ¼ p0 � p00s¼ AðsÞ

r0e�qr0 þ BðsÞ

r0eqr0 ð34Þ

The pressure at infinity being finite, we must have B(s) = 0.Moreover, the constant A(s) can be determined from the firstboundary condition (31). A few steps of calculations lead to

p0 � p00 ¼ p0 � p00s¼ �xse�qðr0�1Þ

r0ð1þ qÞ u0ð1; sÞ ð35Þ


Substitution of the above into the second governing Eq.(30) gives

ou0

or0þ 2

u0

r0¼ � se�qðr0�1Þ

r0ð1þ qÞ u0ð1; sÞ ð36Þ

whose general solution writes

u0ðr0; sÞ ¼ u0ð1; sÞ 1þ qr0

1þ q

� �e�qðr0�1Þ

r02ð37Þ

Here, u0ð1; sÞ appears as an essential unknown. Its determi-nation requires the use of the second boundary condition(32). Detailed calculations lead to

u0ð1; sÞ ¼ � ð1� p00Þx

ð1þ qÞðq2 þ Xqþ XÞ

1

s� 1

sþ j0

� �;

X ¼ 3KR þ 4G=3

K þ 4G=3ð38Þ

Substituting (48) into (35) and (37) yields the pressure anddisplacement fields in the transformed domain

p0ðr0; sÞ � p00 ¼ p0ðr0; sÞ � p00s

¼ ð1� p00Þr0

e�qðr0�1Þ

ðq2 þ Xqþ XÞj0

ðsþ j0Þ ð39Þ

u0ðr0; sÞ ¼ � ð1� p00Þxr02

1þ qr0

q2 þ Xqþ X

� �1

s� 1

sþ j0


ð40Þ

It is interesting to note that the classical theoremlims!0s�f ðsÞ ¼ f ð1Þ applied to (40) gives immediately

u0ðr0;1Þ ¼ lims!0

su0ðr0; sÞ ¼ � 1� p00xX

1

r02ð41Þ

which is consistent with the asymptotic ‘‘drained” limit(B5). The stresses can now be determined in terms of thedisplacement and pressure fields using relations (1)

rrr

R0

¼ � 1

s� 4G0

u0

r0;

rhh

R0

¼ � 1

s� 2G0

p0 � p00x� u0

r0

� �ð42Þ

However, it is the effective stresses which are important inorder to study the failure potential of the ground. Denotingby srr = rrr + p and shh = rhh + p the effective stresses, ands0rr ¼ s0rr=R0 and s0hh ¼ s0hh=R0 their dimensionless counterparts, we deduce from (44)

s0rr ¼ �1

sþ p0 � 4G0

u0

r0;

s0hh ¼ �ð1� p00Þ

sþ K 0 � 2G0=3

x

� �ðp0 � p00Þ þ 2G0

u0

r0ð43Þ

The solutions (38)–(43) can already be exploited by numer-ical inversion using for example the methods of Stehfest[13] or Talbot [14], although it is tempting to invert theseexpressions analytically so that they can be more efficientlyexploited, and can also serve as benchmark examples. In

our case, we have checked the analytical solution usingthe numerical inversion formula due to Stehfest [13]

f ðtÞ ¼ Ln2

t

XN

n¼1

Cn�f n

Ln2

t

� �;

Cn ¼ ð�1ÞnþN2

Xmin n;N2ð Þ

k¼Int nþ12ð Þ

kN2 ð2kÞ!

N2� k

!k!ðk � 1Þ!ðn� kÞ!ð2k � nÞ!

ð44Þ

where Int(x) means the integer part of x, and N is an eveninteger. It turns out that the analytical and numerical inver-sion give identical results thereby validating the former.Subsequent parametric studies show that pressure andstress variations corresponding to large j values (fast dete-rioration rate) are systematically the most critical.Therefore, in the following section, particular regard is gi-ven to the limiting case where the rate of deterioration ofthe cavity lining is very high compared to the hydraulicdiffusion.

5. Analytical solution in the time domain

To obtain the general expression of u0(r0, t0), we decom-pose Eq. (40) into partial fractions and invert term by term(see for example [9]). Firstly, we expand Eq. (40)

u0ðr0;sÞ

¼ ðp00�1Þxr02

ð1þqr0Þq2þXqþX

� 1

q2� ð1þqr0Þ

q2þXqþX� 1

ðsþj0Þ


ð45Þ

Denoting by X1 and X2 the roots of the quadratic form inthe denominator of Eq. (45), The first term within thebracket [�] can then be decomposed into

1

X1�X2

1

q2ðq�X1Þþ r0

qðq�X1Þ� 1

q2ðq�X2Þ� r0

qðq�X2Þ

� �ð46Þ

where term by term inversion suggests itself. Noting that:sþ j0 ¼ ðqþ i

ffiffiffiffij0pÞðq� i

ffiffiffiffij0pÞ, the second term can also be

decomposed to give the following partial fractions

Aq� X1

þ Bq� X2

þ C

qþ iffiffiffiffij0p þ D

q� iffiffiffiffij0p ð47Þ

with

A ¼ 1þ X1r0

ðX1 � X2Þ j0 þ X21

; B ¼ � 1þ X2r0

ðX1 � X2Þ j0 þ X22

C ¼ i

ffiffiffiffij0p

r0 � 1

2iffiffiffiffij0p

iffiffiffiffij0pþ X1

iffiffiffiffij0pþ X2

;

D ¼ 1þ iffiffiffiffij0p

r0

2iffiffiffiffij0p

iffiffiffiffij0p� X1

iffiffiffiffij0p� X2

ð48Þ


Defining,

gðr0; t0; hÞ ¼ 1

X1 � X2

1

hþ r0

� �expð�hðr0 � 1Þ þ h2t0Þ

�

� erfcr0 � 1

2ffiffiffit0p � h

ffiffiffit0p� �

� 1

herfc

r0 � 1

2ffiffiffit0p

� ��ð49Þ

and,

uðr0; t0; hÞ ¼ L�1 e�q r0�1ð Þ

q� hð Þ

� �

¼ 1ffiffiffiffiffiffipt0p exp �ðr

0 � 1Þ2

4t0

!þ h expð�hðr0 � 1Þ

þ h2t2Þerfcr0 � 1

2ffiffiffit0p � h

ffiffiffit0p� �

ð50Þ

This leads to the solution for radial displacement

u0ðr0; t0Þ ¼ ðp00 � 1Þxr02

hgðr0; t0;X1Þ � gðr0; t0;X2Þ

�Auðr0; t0;X1Þ � Buðr0; t0;X2Þ � Cu r0; t0;�iffiffiffiffij0p� �

�Du r0; t0; iffiffiffiffij0p� �i

ð51Þ

The solution for the pore pressure is also found by decom-posing Eq. (39) into partial fractions and this gives,

�pðr0;sÞ� p00s

¼ 1� p00r0

j0A�

q�X1

þ B�

q�X2

þ C�

qþ iffiffiffiffik0p þ D�

q� iffiffiffiffik0p

( )e�qðr0�1Þ

ð52Þ

Here, A*, B*, C*, D* are the special cases of the coefficientsA, B, C, D in Eq. (48) in which r0 = 0. Thus, after inversionwe have

p0ðr0; t0Þ � p00 ¼1� p00

r0j0 A�uðr0; t0;X1Þ þ B�uðr0; t0;X2Þf

þ C�u r0; t0;�iffiffiffiffij0p� �

þ D�u r0; t0; iffiffiffiffij0p� �o

ð53Þ

The stress fields come directly from (51) and (53)

rrr

R0

¼ �1� 4G0u0

r0;

rhh

R0

¼ �1� 2G0p0 � p00

x� u0

r0

� �ð54Þ

s0rr ¼ �ð1� p00Þ þ ðp0 � p00Þ � 4G0u0

r0;

s0hh ¼ �ð1� p00Þ þx� 2G0

x

� �ðp0 � p00Þ þ 2G0

u0

r0ð55Þ

The first term of the above expressions evidently corre-sponds to the normalised form of the initial effective stress,�(R0 � p0). (54) can be shown to be consistent with (B6) onaccount of (41) and (B1).

6. Numerical examples

In this section, we will show a few numerical examples toillustrate the applicability of the analytical expressionsderived. The following hydro-mechanical parameters aretaken from an underground waste disposal project inFrance, and will be referred as the ‘‘reference parameters”.

The parameter j can be considered as the inverse of thecharacteristic time of lining deterioration, noted sj, so thatjt = j0t0 = t/sj. Since t0 = t/sh, where sh = a2/kh is the char-acteristic hydro-mechanical diffusion time, and j t = j0t0,we can deduce that: j0 = sh/sj.

To get some intuition on the time scale corresponding tothe above parameters, consider a spherical cavity with aradius of 20 m, then the characteristic hydraulic diffusiontime can be evaluated to be approximately sh � 126 years,in other words, any excess pressure needs around a hun-dred years to be dissipated. The characteristic time forthe lining deterioration is difficult to estimate as the timescale involved is very large and experimental data at thislevel are inexistant. It is conjectured here that this will takea few hundred years. In other words, the characteristictimes of deterioration and hydraulic diffusion are of thesame order, or j0 = O(1).

To start with, let us examine the effect of the deteriora-tion rate, which has been found to have a very importantimpact. Fig. 2 presents the evolution of pore pressure eval-uated using (53) at different radii for different values of j0,the other parameters being those of Table 1.

It can be seen that in general, the pore pressure at anyradius increases with time initially, induced by the hydrau-lic gradient due to the compression of the cavity backfill,reaches a maximum, and then decreases and returns pro-gressively to its initial value. As the radial distanceincreases, the peak pore pressure attained decreases whilethe time needed to reach the peak increases. When the rateof deterioration of the liner support increases (i.e. the char-acteristic time sj decreases, and j0 increases), the effect isthe same as if the radial distance is closer to the cavity,in other words, peak pore pressure increases and the timeto reach it decreases. The limiting case of j0 ?1 leadsto the most critical peak pore pressure. Moreover, thereis in this limiting case a mathematical singularity atr0 = 1, because the pore pressure there jumps instantaneousfrom its initial value p0 to R0 in order to equilibrate the geo-static pressure, then progressively decreases thereafter.

With an increase in pore pressure, it can be expectedthat the effective radial stress will decrease (in terms ofabsolute value), while the trend for the circumferentialstress is not immediately evident. To investigate this point,we plot the variation of the effective stresses in Fig. 3.

Although a reduction of the effective radial stress s0rr

does occur, as expected, the qualitative behaviour actuallydepends on the lining deterioration rate. For small j0 val-ues, the effective radial stress decreases (in absolute value)monotonically, whereas for large j0 values, it may in thefirst stage fall below its asymptotic value before returning

Table 1Typical parameters used in the parametric study

R0 = 12 MPa; p0 = 5 MPa; K = 4200 MPa; G = 1900 MPa(E � 4950 MPa; m � 0.3)

KR = K/50; kh = 10�7 m2s�1; kh � 1.5 � 10�17 m4N�1s�1

Leading to:p00 ¼ 0:417; K0 = 350; G0 = 158; K0R = 7; X = 1.17; x = 561

0 0.5 1 1.5 2 2.5 3

t'

-0.55

-0.5

-0.45

-0.4 kappa'=inf

100

20

5

1

0.2

normalised radial eff. stress at r'=1.5

(a) effective radial stress ( ) 0rr'rr /p Σ+σ=τ

0 0.5 1 1.5 2 2.5 3

t'

-0.64

-0.62

-0.6

-0.58

-0.56

kappa=inf

100 20

5

1

0.2

normalised circum. eff. stress at r'=1.5

(b) effective circumferential stress ( ) 0' /p Σ+σ=τ θθθθ

Fig. 3. Effective stress evolution at r0 = 1.5 for different characteristic timeratios j0 = sh/s.

0 0.5 1 1.5 2 2.5 3

t'

0.5

0.6

0.7

0.8

0.9

1

kappa'=infinity

kappa'=infinity

- - - - 100

20

5

1

0.2

normalised pore pressure at r'=1

(a) r' =1

0 0.5 1 1.5 2 2.5 3t'

0.42

0.44

0.46

0.48

0.5

0.52

10020

5

10.2

normalised pore pressure at r'=1.5

(b) r'=1.5

Fig. 2. Pore pressure evolution at 2 radii (r0 = 1, 1.5) for differentcharacteristic time ratios j0 = sh/sj.


progressively to it. The most extreme variations areobserved for the case of fastest lining deterioration (i.e.maximum j0). Generally, the long term compressive stressis lower than its initial value, since the effective stressexerted by the backfill at the cavity wall, being inducedby radial convergence, is necessarily lower than the initiallining pressure.

Similar comments also applies on the influence of j0 onthe temporal evolution of the effective circumferentialstress s0hh. At large time, s0hh is generally more compressivethan at the initial time due to the reduction of the effectivepressure at the cavity wall. Again, for large j0 values, s0hh

may at the first stage pass through a point of minimum

compression, induced by the sharp increase in pore pres-sure. Since failure depends much on the deviatoric stresss0rr � s0hh, whether changes in deviatoric stress would leadto problems would be a point of interest. For this purpose,we have plotted the variation of the deviatoric stresss0rr � s0hh in Fig. 4.

The variation of the deviatoric stress is seen to be lesscomplex than the individual stresses. It increases monoton-ically from the zero initial value to its asymptotic value, lar-ger values of j0 correspond to faster rates of increase.

In conclusion, it can be observed that larger values of j0

correspond to sharper increase of pore pressures and stres-ses. The limiting case j0 ?1 should, therefore, lead topessimistic estimates. On account of this, we propose arestricted parametric study in the following on this limitingcase.

Fig. 5 shows the space-time evolution of the pore pres-sure. It completes the information given in Fig. 1. In partic-ular, Fig. 5b shows how the initial step-function porepressure (equal to R0 inside the backfill and zero in themedium) gradually dissipates with time.

0 0.5 1 1.5 2 2.5 3t'

0.5

0.6

0.7

0.8

0.9

1

r' = 1

1.1

1.2

1.4

2

1.25 1.5 1.75 2 2.25 2.5 2.75 3r'

0.5

0.6

0.7

0.8

0.9

1

t' = 0.005

0.1

0.4

1

3

normalised pore pressure

normalised pore pressurea

b

Fig. 5. (a) Pore pressure evolution with time at different radii and (b) porepressure profiles at different instants.

0 0.5 1 1.5 2 2.5 3

t'

0.05

0.1

0.15

0.2

0.25

kappa=100,inf

20

5

1 0.2

normalised dev. stress atr'=1.5

Fig. 4. Deviatoric stress evolution at r0 = 1.5 for different characteristictime ratios j0 = sh/sj.

1 2 3 4 5 6 7 8t'

-0.8

-0.6

-0.4

-0.2

0

r' = 1

1.2

1.5

2

4

1.5 2 2.5 3 3.5 4r'

-0.8

-0.6

-0.4

-0.2

0t'=0.02

0.3 1 5 inf

normalised convergence

normalised convergencea

b

Fig. 6. (a) Temporal evolution of normalised displacementu

u1¼ uðR0�p0Þ=ð3KRþ4GÞ at different radii and (b) normalised displacement

profiles at different instants.


In Fig. 6 we have shown the space-time variation of thenormalised displacement

uðr0; t0Þuð1;1Þ ¼

uðr0; t0ÞðR0 � p0Þ=ð3KR þ 4GÞ ð56Þ

The normalisation is such that the value of the displace-ment must lie inside the interval (�1,0), where the extremevalue �1 can only be attained for the cavity wall r0 = 1. Inkeeping with expectation, the radial convergence takesplace progressively as pore pressure dissipates. The asymp-totic state is practically reached at 5 times the characteristictime of hydro-mechanical diffusion sh = a2/kh.

To illustrate the influence of the backfill stiffness KR, alimited parametric study is performed on the pore pressureand the deviatoric stress. Fig. 7 shows the pore pressureevolution at an arbitrary radius r0 = 1.1 for six stiffnessratios (K/KR = 0.2, 1, 2, 5, 10, 1) which reflects in six dif-ferent values of X (X = 10, 3, 2.06, 1.50, 1.32, 1.13), the lastcase corresponding to the absence of backfill (KR = 0).

It could be seen that a stiff backfill helps to limit the porepressure increase, although compaction is difficult espe-cially when one approaches the ceiling of the cavity dueto increasing confinement of space. To complete the infor-mation, the temporal evolution of the normalised deviator-ic stress ðs0rr � s0hhÞ is plotted in Fig. 8 for a particular radiusr0 = 1.1, for six different stiffness ratios (K/KR = 0.2, 1, 2, 5,10, 1) as above.

0 0.5 1 1.5 2t'

0.1

0.2

0.3

0.4

0.5

0.6

K KR=0.2

1

2

510

inf.-->KR=0

normalised dev.stress

Fig. 8. Influence of the backfill stiffness on the normalised deviatoricstress.

0 0.5 1 1.5 2t'

0.45

0.5

0.55

0.6

0.65

0.7

0.75

K KR=0.2

12

510

inf.--> KR=0

normalised pore pressure at r'=1.1

Fig. 7. Influence of the backfill stiffness KR on the pressure field.


The results are consistent with Fig. 6: a stiffer backfillalso reduces the deviatoric stresses, and thus reduce thefailure potential, and increases the safety margin.

7. Conclusions

In this paper, an original analytical solution on the mineclosure behaviour inside a poro-elastic medium is pre-sented. This solution gives entirely explicit expressions forthe space-time variation of displacement, pore pressureand stress fields induced by the deterioration of the liningsupport when the cavity is abandoned at the end of its ser-vice life. The presence of a poro-elastic backfill inside thecavity is taken account of. The analytical solution onlyinvolves exponential and error functions depending oncomplex arguments. The computations are fast and effi-cient (using for example Mathematica or Maple), makingthe solution ideally suited to parametric studies. The solu-tion also constitutes a very useful benchmark example tocheck complex numerical computations as well as to vali-date computer codes.

Results of analyses show that the pore pressure andeffective stresses can exhibit non-trivial time variations, giv-ing rise to local maxima and minima. The local ‘‘peaks”

attained during the transient variation can sometimes bemore critical than the long term asymptotic values sincefast lining deterioration cannot be identified by long termanalysis under ‘‘drained” conditions. The most criticalcases are generally found to correspond to fast lining sup-port deterioration and low backfill stiffness. For this rea-son, a hypothetical limiting case of instantaneous liningfailure is considered which gives bounding values to porepressure and stress variations. The present analysis is partof a series of studies on cavity convergence. This work iscurrently being extended to cover more general constitutivebehaviours.

Acknowledgement

The financial contribution from IRSN (Institut deRadio-protection et de Surete Nucleaire) making the pres-ent study possible is gratefully acknowledged.

Appendix A. Hydro-mechanical fields within poro-elastic

backfill

We are interested here in the spherical cavity (0 6 r 6 a)with saturated backfill of zero initial effective stress, butnon zero initial pore pressure p0. To respect the assumptionof spherical symmetry, gravity field will be neglected, sothat p0 is homogeneous (ie. gradp0 = 0). To simplify theanalysis, we will suppose the backfill to be much more per-meable than the massif, so that its hydraulic conductivitycan be taken to be infinite

khR !1 ðA1Þ

The poro-elastic constitutive equations (1)–(25) also applyhere, the only change being the material properties. In par-ticular, the bulk and shear moduli of the massif K and G

are to be replaced by those of the backfill, KR and GR.On account of assumptions gradp0 = 0 and Eqs. (A1)and (9), we have

gradp ¼ 0 ðA2Þ

In other words, the pressure field inside the backfill ishomogeneous, equal to the pore pressure at the cavity wall:p(r, t) = p(a, t). Eq. (17) will then become, for the backfill

grad 2¼ 0 ðA3ÞRecalling the definition (25) of 2, and that the displace-ment at the origin must remain finite, (A3) can be inte-grated to give

uðr; tÞ ¼ uða; tÞ ra

ðA4Þ

On account of (1) and (A2), Eq. (A4) implies that both thestress and strain tensors are isotropic and homogeneous,given by


e ¼ ðuða; tÞ=aÞ1; r ¼ ð3KRuða; tÞ=a� pða; tÞÞ1 ðA5Þwhere 1 is the second order identity tensor. In particular,the total stress on the inner wall of the lining support is

rrrða�; tÞ ¼ 3KRuða; tÞ

a� pðaþ; tÞ ðA6Þ

Appendix B. Asymptotic displacement field at large time

At a very large time, all excess pore pressure will havebeen dissipated so that all fluid flow will cease. Eq. (9)implies

pðr;1Þ ¼ p0 ðB1ÞEq. (19) then yields ou

or þ 2 ur ¼ 0 can then be integrated to

give

uðr;1Þ ¼ uða;1Þ ar

� �2

ðB2Þ

where we have used the continuity of u at r = a and the factthat u(r,1) must remain finite. Substituting (B1) and (B2)into the poro-elastic constitutive equation (1) leads to

rrrðaþ;1Þ ¼ �R0 � 4Guða;1Þ

aðB3Þ

The lining support having totally deteriorated, we haveps(1) = 0. The stress continuity at cavity wall, on accountof (B1), (B3) and (A6) allows the determination of theasymptotic cavity convergence

uða;1Þa

¼ � R0 � p0

3KR þ 4GðB4Þ

From (B2), we also get the asymptotic convergence at anyother radius:

uðr;1Þ ¼ � R0 � p0

3KR þ 4Ga3

r2ðB5Þ

Substitution of (B5) into (1), gives, on account of (B1)

rrrðr;1Þ ¼ �R0 þ 4GR0 � p0

3KR þ 4Gar

� �3

rhhðr;1Þ ¼ �R0 � 2GR0 � p0

3KR þ 4Gar

� �3ðB6Þ

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analytical study of mine closure behaviour in a poro-elastic medium

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