Chemo-mechanical coupling and damage enhanced dissolution at intergranular contact

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1 Numeical Models in Geomechanics-NUMOG IX-Pande & Pietuszczak (eds) 4 Taylo & Fancis Goup, London, ISBN X Chemo-mechanical coupling and damage enhanced dissolution at integanula contact T. Hueckel and L.B. Hu Duke Univesity, Depatment of Civil & Envionmental Engineeing, Duham, NC, USA ABSTRACT: Many pocesses in geomechanics, such as stuctuation and aging of natual soils, compaction and pessue solution of oil beaing sediments involve mineal dissolution at gain contacts. Dissolution and subseuent pecipitation lead to a edistibution of mass within the poe space, affecting soil poosity and stiffness. We simulate dissolution at a mico scale using chemo-plasticity fo penetation of aspeity. We assume that dilatancy esulting fom mateial damage geneates new fee suface aound the aspeity, in tun acceleating dissolution and mateial weakening. 1 INTRODUCTION Seveal basic pocesses in geomechanics depend on the micoscopic level dissolution of mineal at stessed integanula contact. They include: stuctuation, soil aging in laboatoy tests and pessue solution in oil beaing sediments. The mechanisms of the emoval include: dissolution of mineal at the inteface between gains, plastic defomation and feeface pessue solution. While time scale and contibution of diffeent mechanisms constituting the pocess may be diffeent in each of the phenomena mentioned, one common featue of these phenomena is emoval of mass fom the contact aea. The specific mechanisms of this emoval ae subject of a moe o less intense debate in the espective communities and subject of both theoetical and expeimental eseach. The emoval is often linked to subseuent pocesses such as fomation of gel in poes, pecipitation of the mass on exposed fee sufaces with possible changes in mechanical popeties of mateial at a maco-scale. A pope undestanding and identification of vaiables esponsible fo these inteface pocesses is cucial fo exploation and poduction in petoleum engineeing, geotechnics in natual stuctued mateials, and intepetation of laboatoy testing. In this pape, we postulate aspeity indentation as a dominant integanula mechanism that involves a coupled dilatant damage and dissolution. Expeiments show massive mico-cacking and mico-ganulation nea stessed contact (Tada et al., 1987). MODEL We model damage enhanced dissolution of mineals at an elasto-plastic contact between soil gains at a mico scale. To stat with we assume a igid-plastic gain, with chemically sensitive yielding induced by an indentation of an infinitely igid aspeity. We will make use of Johnson s (Johnson, 1985) appoximation, widely accepted in contact mechanics, extending it to include a chemo-mechanical coupling. Ou pupose is to examine chemo-mechanical couplings necessay to see meaningful values of the pincipal vaiables involved. Fo simplicity we adopt plane stain state. Figue 1. Indentation model Following Johnson's appoximation, the contact suface of the indente is assumed to be encased in a hemi-spheical 'coe' of adius a, within which the stess state is epesented by puely hydostatic

2 stess. Outside this coe it is assumed that stess, stain and displacement ae axisymmetic and the same as in an intenally loaded igid-plastic tube with a, b as the inne and oute adius, espectively. This appoximation, with no stess applied at the extenal bounday coesponds to a cental pat of a gain indented by a much stiffe mineal and away fom othe contact points. The postulated chemo-mechanical mechanism includes: a field of mechanical damage induced duing dilatant plastic stain aound the indente; geneation of new fee suface by the mico-cacking associated with the plastic dilatancy; an enhancement of dissolution of silica fom the newly geneated sufaces; and in tun chemo-mechanical weakening of uatz as a esult the silica mass emoval, followed by tansfe of the dissolved silica mass into the poe solution. 3 DEFORMATION ENHANCED DISSOLUTION DURING INDENTATION Seveal scenaios of contact dissolution may be envisioned. We shall focus on the one involving two stages: a puely mechanically induced indente penetation (incease of the indente penetation inducing yielding up an unstable phase, as descibed by Hueckel & Moz, 1971); followed by a phase at a constant indentation pessue with a pogess occuing due to chemical softening and stain hadening. The euilibium euation and kinematic elationships fo the plane stain axisymmetic poblem ae as follows d θ d du u =, ε =, εθ = (1) d whee,θ ae outwad adial and cicumfeential coodinates, espectively, u is the adial displacement. The mechanical bounday conditions ae: at = a, = p and at = b, =. The yield suface is as simple as possible, expessed via single pincipal stess components, Fig.. = ( ε, ζ ) fo 1i - tan ϕ < / < 1, θ - ctan ϕ < / < 1, < θ ( ε, ζ ), elsewhee and θ = i () The yield limit undegoes stain hadening and chemical softening, as ζ epesents the chemically induced mass loss. The flow ule is associative. We conside a linea deviatoic stain hadening and chemical softening ules, as follows (fo > ): θ = γε βζ, = γε βζ ; ; (3) whee γ and β espectively stain hadening and chemical weakening mateial constants, taken hee fo simplicity as the same fo and θ ; ε is deviatoic stain: ε = 3 εθ ε. It is essential in establishing the hadening functions (3) that the two mechanisms may compensate one anothe, in ode to simulate pocess of chemically induced stain at constant stess. Clealy, thee is also a limitation on the amount of softening and hadening, as the pincipal stesses ae not intended to change thei sign. We conside a uatz-wate system undegoing the eaction of dissolution SiO (s) H O(l) H 4SiO 4 (a) To uantify the loss of mass of silica fom the mineal, we will adopt the ate euation fo activity change in dissolution eaction afte Rimstidt and Banes (198): da ( k a a ) ζ H O k a H 4SiO4 = Aγ H 4SiO4 SiO H 4SiO4 dt (4) whee a is activity, γ is activity coefficient, k and k ae the ate constants fo the fowad and backwad eaction espectively. A is a dimensionless uantity epesenting the specific intefacial suface pe unit volume of fluid phase at which dissolution occus. The key assumption of this theoy is that, as the dilatant ievesible defomation of the mineal is linked to geneation of micofactues, such micofactues fom new inteface suface aea, at which dissolution takes place at contact with poe fluid. Hence, A = A (ε v ) whee ε v is volumetic stain. Futhemoe, the local pecipitation tem in the eaction euation is ignoed, as we conside the emote mass pecipitation in the up-scaling famewok, elsewhee. Thus, the ate of eaction in euation (4) becomes: ζ = kφε c v (5) whee φ is a popotionality constant between the specific intefacial aea and volumetic stain, while c = const. epesents the kinetic ate associated with pe-existing intefacial suface in poes, assumed constant. We will focus on the second phase of the pocess, which is dissolution at a constant indentation pessue, p, with a pogess occuing due to chemical softening and stain hadening is contolled by the dissolution time scale, while the esults fo the pecedent phase of the time independent mechanical loading fom to p is included fo completeness.

3 The esults of calculations pefomed using Matlab 6.5 ae shown in Fig. 3. The values used in this simulation ae b / a = 1, γ =. 36, tan ϕ =.. K epesents the coefficient of chemical softening dissolution associated with dilatancy elated dissolution, K = βφ k, while the chemical softening effect due to pe-existing factues and poes 1 K c = 1.e 6s is assumed as a constant. The evolution of the zones of diffeent kinematic esponse is shown in Fig. 4. Notably, the case of a one way coupling, with β =, and hence K=, epesents the situation in which thee is no effect of dissolution on hadening, but thee is still an effect of staining (dilatancy) on dissolution. Figue : Linea yield condition and stess pofiles We conside a constant inne bounday pessue p shown at point A in the stess pofile, Fig.. The solution euies existence of two zones. An inne zone foms with the stess on the citical line DA, of which D epesents the bounday at = ξ, between the inne and oute zone. Initially, unde the inceasing inne pessue, it pushes the oute zone, which is oiginally igid. The latte stats yielding with the stess state at B '' D. As the pessue is stopped at p=const, the mateial undegoes simultaneous stain hadening and simultaneous chemical softening. With time pogessing, the stess at bounday ξ gows along D' D, while physically the bounday moves inwad. Hence, the inne zone gets smalle and the oute zone of dilatancy inceases. With the choice of extemely simple yield function, the following constaints ae imposed on es. (1): fo the inne zone a ξ, ε =, and u = u = const, fo the oute zone ξ b, θ =. The two zones ae euied to be continuous, as fa as stess is concened, hence inne oute ξ =, inne oute ξ θξ = θξ. With these conditions and euations (1) though (5), we could calculate the intenal pessue-displacement elationship unde diffeent intensity of the chemical pocess, as a solution of the following system of diffeential euations: t tan ϕ βφk u dt p a b γu b ( 1 Kct) = 1 ln ξ ξ 3 ξ t p a = ( ) βφk udt γu 1 Kct ξ 3 u a u 3 p a a = In a ξ γ ξ ξ Figue 3. Penetation vs. time Figue 4. Popagation of zones As can be seen, chemical softening due to dissolution of the mateial can significantly acceleate the penetation of the indente. At lage chemical softening paamete values, the dissolution effect pevails ove the stain hadening and the penetation becomes uncontained, until small stain hypothesis ceases to hold. As a esult of dissolution-dilation coupling, dissolution acceleates the advancement of the dilatancy zone into the body of the gain. Notably, the compaction zone withdaws, and eventually, vanishes.

4 4 TRANSPORT OF MASS AWAY FROM THE DAMAGE/DISSOLUTION ZONE The acceleated dissolution as a esult of geneation of fee suface aea aound the aspeity due to the mateial damage is expected to have a consideable effect on the mass tansfe duing the indentation pocess. The effect of the dissolution in the total tansfe of mass outside of the gain is simulated using the eactive tanspot euation as follows: C t = D C F (6) whee C is the mola faction concentation of the solute H 4SiO4, D is the diffusion coefficient, F is the ate of mass poduction. As discussed above, F is enhanced by the mechanical defomation duing the pocess. The pocess is appoximated as the linea function of dilatant volumetic stain, ζ in E. (5) and by ignoing the local pecipitation. Assuming that the tanspot is uniuely diven by the concentation gadient ceated by the dissolved mass, that is imposing at the inne bounday a constant concentation, C = Ca at = a, C = at = b, e. (5) and (6) wee numeically integated using Matlab 6.5 to yield the flux geneated at = a. Coefficient M epesents the defomation effect associated with dilatant volumetic stain on the mass tansfe, M = φ k a DC. The esults ae pesented in Fig. 5a and b in tems of the actual mass a flux simulated fo the following data: a = 1 5 m, b = m, k = 1 1 s, D = 1 1 m / s (Rimstidt & Banes, 198; Shimizu, 1995), still, b / a = 1, tan ϕ =., and γ =. 36. It should be pointed out that the deviatoic stain hadening in the constitutive functions plays an impotant ole to compensate the weakening effect induced by dilatancy which causes moe mateial to dissolve and tansfe. The following Figue 6(a) and (b) demonstate the mass flux simulated with the same data used in Fig.5(a) and (b) and M = 5, but unde diffeent deviatoic stain hadening coefficient γ. Figue 5. Flux acoss the extenal gain suface: (a) shot tem; (b) long tem Figue 6. Flux acoss the extenal gain suface with diffeent deviatoic stain hadening coefficient: (a) shot tem; (b) long tem

5 5 CONCLUSIONS It is seen that the defomation enhances the mass tanspot significantly. Clealy, the solutions pesented ae based on vey simple constitutive functions. Indente penetation is damatically enhanced by mineal dissolution, even at constant pessue. When M =, tanspot is puely diffusive, and flux eaches steady state afte a peiod. Howeve, unde the inceasing mechanical damage (defomation), substantially moe mateial is dissolved and tansfeed though the fee suface to poe solution. ACKNOWLEDGMENT Potion of this wok was suppoted by a gant fom the National Science Foundation, Geomechanics and Geotechnical Systems Pogam, Division of Civil & Mechanical Systems. REFERENCES Hueckel, T. & Moz, Z Some bounday value poblems fo vaiable density mateials. In W.K. Nowacki (ed), Poblèmes de la Rhéologie: , Wasaw: PWN. Johnson,.K.L Contact Mechanics. Cambidge: Cambidge Univesity Pess Rimstidt, J.G. & Banes, H.L The kinetics of silicawate eactions. Geochimica et Cosmochimica Acta 44: Shimizu, I Kinetics of pessue solution ceep in uatzu: theoetical consideations. Tectophysics 45: Tada, R., Maliva,R. & Sieve, R A new mechanism fo pessue solution in poous uatzose sandstone. Geochimica et Cosmochimica Acta 51: 95-3