1 INTRODUCTION

Several basic processes in geomechanics depend onthe microscopic level dissolution of mineral atstressed intergranular contact. They include: structu-ration, soil aging in laboratory tests and pressuresolution in oil bearing sediments. The mechanisms ofthe removal include: dissolution of mineral at the in-terface between grains, plastic deformation and free-face pressure solution. While time scale and contri-bution of different mechanisms constituting theprocess may be different in each of the phenomenamentioned, one common feature of these phenomenais removal of mass from the contact area. The spe-cific mechanisms of this removal are subject of amore or less intense debate in the respective commu-nities and subject of both theoretical and experimen-tal research. The removal is often linked to subse-quent processes such as formation of gel in pores,precipitation of the mass on exposed free surfaceswith possible changes in mechanical properties ofmaterial at a macro-scale.

A proper understanding and identification of vari-ables responsible for these interface processes is cru-cial for exploration and production in petroleum en-gineering, geotechnics in natural structured materials,and interpretation of laboratory testing. In this pa-per, we postulate asperity indentation as a dominantintergranular mechanism that involves a coupled dila-tant damage and dissolution. Experiments show mas-sive micro-cracking and micro-granulation nearstressed contact (Tada et al., 1987).

2 MODEL

We model damage enhanced dissolution of mineralsat an elasto-plastic contact between soil grains at amicro scale. To start with we assume a rigid-plasticgrain, with chemically sensitive yielding induced byan indentation of an infinitely rigid asperity. We willmake use of Johnson’s (Johnson, 1985) approxima-tion, widely accepted in contact mechanics, extend-ing it to include a chemo-mechanical coupling. Ourpurpose is to examine chemo-mechanical couplingsnecessary to see meaningful values of the principalvariables involved. For simplicity we adopt planestrain state.

Figure 1. Indentation model

Following Johnson's approximation, the contactsurface of the indenter is assumed to be encased in ahemi-spherical 'core' of radius a, within which thestress state is represented by purely hydrostatic

Chemo-mechanical coupling and damage enhanced dissolution atintergranular contact

T. Hueckel and L.B. HuDuke University, Department of Civil & Environmental Engineering, Durham, NC, USA

ABSTRACT: Many processes in geomechanics, such as structuration and aging of natural soils, compactionand pressure solution of oil bearing sediments involve mineral dissolution at grain contacts. Dissolution andsubsequent precipitation lead to a redistribution of mass within the pore space, affecting soil porosity andstiffness. We simulate dissolution at a micro scale using chemo-plasticity for penetration of asperity. We as-sume that dilatancy resulting from material damage generates new free surface around the asperity, in turn ac-celerating dissolution and material weakening.

Numerical Models in Geomechanics-NUMOG IX-Pande & Pietruszczak (eds)© 2004 Taylor & Francis Group, London, ISBN 90 5809 636 X

stress. Outside this core it is assumed that stress,strain and displacement are axisymmetric and thesame as in an internally loaded rigid-plastic tube witha, b as the inner and outer radius, respectively. Thisapproximation, with no stress applied at the externalboundary corresponds to a central part of a grain in-dented by a much stiffer mineral and away fromother contact points.

The postulated chemo-mechanical mechanism in-cludes: a field of mechanical damage induced duringdilatant plastic strain around the indenter; generationof new free surface by the micro-cracking associatedwith the plastic dilatancy; an enhancement of disso-lution of silica from the newly generated surfaces;and in turn chemo-mechanical weakening of quartz asa result the silica mass removal, followed by transferof the dissolved silica mass into the pore solution.

3 DEFORMATION ENHANCEDDISSOLUTION DURING INDENTATION

Several scenarios of contact dissolution may be envi-sioned. We shall focus on the one involving twostages: a purely mechanically induced indenter pene-tration (increase of the indenter penetration inducingyielding up an unstable phase, as described byHueckel & Mroz, 1971); followed by a phase at aconstant indentation pressure with a progress occur-ring due to chemical softening and strain hardening.The equilibrium equation and kinematic relationshipsfor the plane strain axisymmetric problem are asfollows

0=−

+rdr

d rr θσσσ,

r

u

dr

udr

−=−= θεε , (1)

where r ,θ are outward radial and circumferentialcoordinates, respectively, u is the radial displace-ment. The mechanical boundary conditions are: at

ar = , pr =σ and at br = , 0=rσ . The yield sur-face is as simple as possible, expressed via singleprincipal stress components, Fig. 2.

( ) 0 1,< /<ctan -

0 1,< /<tan - ,

1 ⎪⎩

⎪⎨⎧

<≥

=rr

rrforir σσθσϕ

σσθσϕζεσσ

( ) elsewhere ,, and 2 i ζεσσθ = (2)

The yield limit undergoes strain hardening andchemical softening, as ζ represents the chemicallyinduced mass loss. The flow rule is associative. Weconsider a linear deviatoric strain hardening andchemical softening rules, as follows (for σr > 0):

0;0 ;01,01 02010201

≤≥≥−+=≥−+= σσβζγεσσ

βζγεσσ θ

qqr

(3)

where γ and β respectively strain hardening andchemical weakening material constants, taken herefor simplicity as the same for σr and σθ;

�

εq is devia-toric strain: rq εεε θ −= 32 . It is essential in es-tablishing the hardening functions (3) that the twomechanisms may compensate one another, in orderto simulate process of chemically induced strain atconstant stress. Clearly, there is also a limitation onthe amount of softening and hardening, as the princi-pal stresses are not intended to change their sign.

We consider a quartz-water system undergoing thereaction of dissolution

(aq)SiOHO(l)2H(s)SiO 4422 ⎯→←+

To quantify the loss of mass of silica from themineral, we will adopt the rate equation for activitychange in dissolution reaction after Rimstidt andBarnes (1980):

( ) ζγ ≡−= −+ 442244

44 2SiO SiOHOHSiOH

SiOHakaakA

dt

da(4)

where a is activity, γ is activity coefficient, +k and−k are the rate constants for the forward and back-

ward reaction respectively.A is a dimensionless quantity representing the

specific interfacial surface per unit volume of fluidphase at which dissolution occurs.

The key assumption of this theory is that, asthe dilatant irreversible deformation of the mineral islinked to generation of microfractures, such mi-crofractures form new interface surface area, atwhich dissolution takes place at contact with porefluid. Hence, A = A (

�

εv ) where

�

εv is volumetricstrain. Furthermore, the local precipitation term inthe reaction equation is ignored, as we consider theremote mass precipitation in the up-scaling frame-work, elsewhere. Thus, the rate of reaction in equa-tion (4) becomes:

ck v += +φεζ (5)

where φ is a proportionality constant between thespecific interfacial area and volumetric strain, whilec = const. represents the kinetic rate associated withpre-existing interfacial surface in pores, assumedconstant.

We will focus on the second phase of the process,which is dissolution at a constant indentation pres-sure, p, with a progress occurring due to chemicalsoftening and strain hardening is controlled by thedissolution time scale, while the results for theprecedent phase of the time independent mechanicalloading from 0 to p is included for completeness.

Figure 2: Linear yield condition and stress profiles

We consider a constant inner boundary pres-sure p shown at point A in the stress profile, Fig.2.The solution requires existence of two zones. An in-ner zone forms with the stress on the critical line

�

DA , of which D represents the boundary at

�

r = ξ ,between the inner and outer zone. Initially, under theincreasing inner pressure, it pushes the outer zone,which is originally rigid. The latter starts yieldingwith the stress state at DB '' . As the pressure isstopped at p=const, the material undergoes simulta-neous strain hardening and simultaneous chemicalsoftening. With time progressing, the stress atboundary ξ grows along DD' , while physically theboundary moves inward. Hence, the inner zone getssmaller and the outer zone of dilatancy increases.With the choice of extremely simple yield function,the following constraints are imposed on eqs. (1): forthe inner zone

�

a ≤ r ≤ ξ , 0=rε , andconstuu == 0 , for the outer zone br ≤≤ξ ,

ϕσσθ tan=− r . The two zones are required to becontinuous, as far as stress is concerned, hence

outerr

innerr ξξ σσ = , outerinner

θξθξ σσ = . With these conditionsand equations (1) through (5), we could calculate theinternal pressure-displacement relationship underdifferent intensity of the chemical process, as a solu-tion of the following system of differential equa-tions:

( )ξ

γβφ

ξξϕσ

ϕb

ru

r

dtukbtK

apt

c ln32

11tan

00 0tan1

01 ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−−⎟⎟⎠

⎞⎜⎜⎝

⎛−−=⎟⎟⎠

⎞⎜⎜⎝

⎛ ∫ ++

( )⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−−−=⎟⎟⎠

⎞⎜⎜⎝

⎛ ∫ ++

r

u

r

dtuktK

apt

c 3

21 00 0

tan1

01

γβφ

ξσ

ϕ

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

+⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎠⎞⎜

⎝⎛

+=−

++

ξξϕσ

γξ

ϕϕra

r

apu

a

ua Intan12

3tan1tan1

010

The results of calculations performed using Mat-lab 6.5 are shown in Fig. 3. The values used in thissimulation are 10/ =ab , 36.2=γ , 2.0tan =ϕ . Krepresents the coefficient of chemical softening dis-solution associated with dilatancy related dissolu-tion, += kK βφ , while the chemical softening effectdue to pre-existing fractures and pores

1s60.1 −−= eKc is assumed as a constant. The evo-lution of the zones of different kinematic response isshown in Fig. 4. Notably, the case of a one waycoupling, with

�

β=0, and hence K=0, represents thesituation in which there is no effect of dissolution onhardening, but there is still an effect of straining (di-latancy) on dissolution.

Figure 3. Penetration vs. time

Figure 4. Propagation of zones

As can be seen, chemical softening due to dissolu-tion of the material can significantly accelerate thepenetration of the indenter. At larger chemical sof-tening parameter values, the dissolution effect pre-vails over the strain hardening and the penetrationbecomes uncontained, until small strain hypothesisceases to hold. As a result of dissolution-dilationcoupling, dissolution accelerates the advancement ofthe dilatancy zone into the body of the grain. Nota-bly, the compaction zone withdraws, and eventually,vanishes.

4 TRANSPORT OF MASS AWAY FROM THEDAMAGE/DISSOLUTION ZONE

The accelerated dissolution as a result of generationof free surface area around the asperity due to thematerial damage is expected to have a considerableeffect on the mass transfer during the indentationprocess. The effect of the dissolution in the totaltransfer of mass outside of the grain is simulated us-ing the reactive transport equation as follows:

FCDt

C +∇=∂∂ 2 (6)

where C is the molar fraction concentration of thesolute 44SiOH , D is the diffusion coefficient, F isthe rate of mass production. As discussed above, Fis enhanced by the mechanical deformation duringthe process. The process is approximated as the lin-ear function of dilatant volumetric strain, ζ in Eq.(5) and by ignoring the local precipitation.

Figure 5. Flux across the external grain surface: (a) short term;(b) long term

Assuming that the transport is uniquely drivenby the concentration gradient created by the dis-

solved mass, that is imposing at the inner boundary aconstant concentration, aCC = at ar = , 0=C at

br = , eq. (5) and (6) were numerically integratedusing Matlab 6.5 to yield the flux generated at ar = .Coefficient M represents the deformation effect as-sociated with dilatant volumetric strain on the masstransfer,

aDCakM 2+= φ . The results are pre-

sented in Fig. 5a and b in terms of the actual massflux simulated for the following data: ma 510−= ,

mb 410−= , 112101 −−+ ×= sk , smD /101 213−×=

(Rimstidt & Barnes, 1980; Shimizu, 1995), still,10/ =ab , 2.0tan =ϕ , and 36.2=γ .

It should be pointed out that the deviatoric strainhardening in the constitutive functions plays an im-portant role to compensate the weakening effect in-duced by dilatancy which causes more material todissolve and transfer. The following Figure 6(a) and(b) demonstrate the mass flux simulated with thesame data used in Fig.5(a) and (b) and 5=M , butunder different deviatoric strain hardening coefficientγ .

Figure 6. Flux across the external grain surface with differentdeviatoric strain hardening coefficient: (a) short term; (b) longterm

5 CONCLUSIONS

It is seen that the deformation enhances the masstransport significantly. Clearly, the solutions pre-sented are based on very simple constitutive func-tions. Indenter penetration is dramatically enhancedby mineral dissolution, even at constant pressure.When 0=M , transport is purely diffusive, and fluxreaches steady state after a period. However, underthe increasing mechanical damage (deformation), sub-stantially more material is dissolved and transferredthrough the free surface to pore solution.

ACKNOWLEDGMENT

Portion of this work was supported by a grant fromthe National Science Foundation, Geomechanics andGeotechnical Systems Program, Division of Civil &Mechanical Systems.

REFERENCES

Hueckel, T. & Mroz, Z. 1973. Some boundary value problemsfor variable density materials. In W.K. Nowacki (ed),Problèmes de la Rhéologie: 173-191, Warsaw: PWN.

Johnson,.K.L. 1985. Contact Mechanics. Cambridge: Cam-bridge University Press

Rimstidt, J.G. & Barnes, H.L. 1980. The kinetics of silica-water reactions. Geochimica et Cosmochimica Acta 44:1683-1699

Shimizu, I. 1995. Kinetics of pressure solution creep inquartzu: theoretical considerations. Tectophysics 245:121-134

Tada, R., Maliva,R. & Siever, R. 1987. A new mechanism forpressure solution in porous quartzose sandstone. Geo-chimica et Cosmochimica Acta 51: 2295-2301