environmentally enhanced crack propagation in a chemically
TRANSCRIPT
Hu, M. M. & Hueckel, T. (2012). Geotechnique 62, No. 00, 1–9 [http://dx.doi.org/10.1680/geot.13.P.020]
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PROOFS
Environmentally enhanced crack propagation in a chemically degradingisotropic shale
M. M. HU! and T. HUECKEL!
Crack propagation is studied in a geomaterial subject to weakening by the presence of water, whichdissolves a mineral component. Such weakening is common when tensile microcracks develop,constituting sites of enhanced mineral dissolution. A previous concept is adopted of reactive chemo-plasticity with the yield limit depending on the mineral mass dissolved, causing chemical softening.The dissolution is described by a rate equation and is a function of variable internal specific surfacearea, which in turn is assumed to be a function of dilative plastic deformation. The crack vicinity inplane strain is subject to a constant subcritical all-round uniform radial tensile traction. The behaviourof the material is rigid-plastic with chemical softening. The extended Johnson approximation isadopted, meaning that all fields involved are axisymmetric around the crack tip, with a small,unstressed cavity around it. Initial dissolution proportional to the initial porosity activates the plasticyielding. The total dissolved mass diffuses out from the process zone and the exiting mineral massflux can be correlated with the displacement of the crack tip. A simplified semi-analytical solution forthis model is presented.
KEYWORDS: ????????
INTRODUCTIONThis paper focuses on understanding the mechanisms ofcrack propagation caused by chemically induced weakeningof intact rock material, with a view to application to shaleand industrial argillites. There are multiple geochemicalchanges that induce mechanical response in earthen materi-als; these include deformational or permeability reaction ofclays to change in ionic concentration, or dielectric constantof the invading pore fluid (Hueckel, 1992, 1997, 2002; Gajoet al., 2002, Loret et al., 2002; Wahid et al., 2011); changein mechanical properties of rocks and sands as a result oftheir inundation with water (Hueckel et al., 2001, 2005; Hu& Hueckel, 2007a, 2007b); and slope instability in responseto infiltration of acid rain water into cracks between claymineral layers (Zhao et al., 2011). The specific mechanismsof these chemo-mechanical interactions are various andinclude osmotic effects, diffuse double layer changes, ionreplacement phenomena, dissolution and precipitation ofminerals, and transformation of clay minerals (Hueckel &Pellegrini, 2002; Zhan et al., 2011).
This study addresses a possible mechanism in which rocks orcohesive soils undergo a chemical reaction of dissolutionresulting in the removal of the mineral mass and hence weaken-ing of soil/rock mass at and around the chemical reaction site.The focus is on a particular situation at a crack tip, at whichdegradation is induced in the form of an advanced irreversibledeformation aside and ahead of the crack, as a swarm of micro-fissures in mode I (Lin & Labuz, 2011). The chemicallyinduced propagation of a stressed crack under a constant mech-anical load is investigated. An extension of the Cam-Claymodel into a chemical domain is used (Hueckel, 1992, 1997,2002), in particular a concept of chemical softening, which iscoupled with irreversible deformation (Hu & Hueckel, 2007a).A simplified semi-analytical solution of an extended cavity
expansion concept is applied to model the crack vicinity; thiscan be easily coupled with axisymmetric solution for thechemical fields around the crack tip process zone.
In the past several decades, there has been a growingunderstanding that a number of problems in relation to rockmechanics and rock engineering in general could be, or needto be, tackled by addressing fracture mechanics. Some of theexamples stem from areas of energy engineering and slopestability. The technique of hydraulic fracturing has beenincreasingly used all over the world owing to its advantagesin efficiently creating fractures in the rock formation, whichdramatically enhances rock or sediment permeability andhence oil/gas production (Howard & Fast, 1970; Carter etal., 2000; Dershowitz et al., 2001; Rahman, 2008; Jackson etal., 2011,) as well as heat extraction from geothermal fields(Portier & Vuataz, 2010; Zimmermann et al., 2010). Morerecently, attention has shifted towards fracture propagationenhancement by acidising the rock (i.e. injecting chemicallyactive fluids), especially for low-permeability carbonate re-servoirs (He et al. (2010) or to enhance a geothermal system(Portier & Vuataz, 2010; Zimmermann et al., 2010).
Meanwhile, a heated debate on acidising treatment inhydraulic fracturing performance arose, owing to the potentialharm caused by disposal and transport of the toxic wastewater,which may include formaldehyde, boric acid, hydrochloricacid and isopropanol (Coffman, 2009). Poorly controlledhydraulic fracturing, often due to inadequate understanding offracture mechanisms involving acid–rock interaction, may notonly pose threat to the environment, but also result in un-necessary costs and low productivity (Rahman, 2008).
This paper addresses possible chemo-mechanical mechan-isms of rock/shale weakening resulting in fracture propaga-tion enhancement, to arrive at an effective predictive tool tocontrol the efficiency of hydraulic fracture and the fate ofpossible contaminants.
CONCEPT OF CHEMO-PLASTICITYSolid–fluid interaction in geomaterials has been studied
for many years, focusing on the effect of pore pressure on
Article Number: 13p020
Manuscript received 5 March 2012; revised manuscript accepted 17October 2012.Discussion on this paper is welcomed by the editor.! Duke University, Durham, North Carolina, USA.
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PROOFS
rock and soil matrix. The concept of reactive chemo-plasticity is based on the notion that a mineral massremoved from or added to a solid can affect both themedium compliance and strength (Hueckel, 2002; Hu &Hueckel, 2007a, 2007b). As one of the primary sources ofmass removal, chemical dissolution has been attracting muchattention. A key mechanism that links the chemical processand mechanical degradation has been known for some timeto be rock microcracking. Bathurst (1958) and Ostapenko(1968, 1975) noted that microcracking leads to the genera-tion of a new solid–fluid interface, where enhanced mineraldissolution takes place. Tada et al. (1987) found that microfractures and plastic deformation play an important role inthe process of pressure solution.The concept of chemical dissolution coupled with plasti-
city was addressed by Hu & Hueckel (2007a, 2007b),linking the irreversible dilatancy and the related increase offree specific surface area to the amount of mineral dissolu-tion per unit volume. This coupling mechanism, usuallyreferred to as ‘reactive chemo-plasticity’, acts at a compara-tively long time scale, as distinguished from ‘non-reactivechemo-plasticity’, that is when the mechanical properties ofa solid skeleton can be affected instantaneously by an alteredconcentration (of selected species) in the pore fluid.From a microscopic point of view, reactive chemo-
plasticity starts when a mineral mass dissolves into water atavailable dissolution sites on the solid–fluid interface. It iscommonly assumed that the dissolution reaction rate andhence mass removal rate is proportional to the specific surfacearea of the interface. It is hence postulated that the irreversibledamage-induced dilatancy is associated with the generation ofnew microcrack walls. The increased wall surface area adds tothe pre-existing specific surface area of the pores, henceadvancing the overall mineral mass production of the medium.Because mass removal affects the strength of the medium, thecentral part of the description of the chemo-mechanical cou-pling is how to quantify the amount of weakening of thematerial owing to dissolution. Therefore the coupling betweenthe irreversible strain and the chemically induced mass re-moval is a two-way coupling (see also Hueckel & Hu (2009)).To quantify the above coupling mechanisms, a representa-
tive elementary volume (REV) of the drained porous med-ium is considered, represented by a single geometrical point,to which both mechanical and chemical properties areattributed. The solid phase material is assumed as rigid-plastic. This assumption is a matter of convenience, becauseit enables a semi-analytical solution to be obtained. On theother hand the elastic stress component is much smaller thanthe irreversible one in the area of interest near the crack tip.The partial stress carried by the fluid phase is considerednegligible. The theory of plasticity states that the yield locusof a material is affected by stress ! ij and, when a sufficientload is applied, that induces initial yielding, also owing tovariation of a set of internal variables. In reactive chemo-plasticity (Hueckel, 2002), the internal chemical variablesdenoted as "n, (n " 1, 2, . . .), are the accumulated dissolvedmasses of the n th species of mineral per unit volume of theporous medium. The mechanical, strain-hardening variable ischosen as irreversible deviatoric strain intensity (#irrq ):The rate of the deviatoric strain intensity is defined as an
invariant of deviation strain rate _eirrij
_#irrq " 2
3_eirrij _e
irrij
! "1=2
(1)
where
_eirrij " _#irrij # 1
3_#irrkk$ij
With the assumption of rigid-plasticity, there are no elasticstrain rates and no strain at all for a stress state below theyield limit, that is when f (! ij) , 0: With the stress at theyield locus, the strain rate is non-zero and irreversible.Hence
f " f (! ij, pc), and_f " _f ( _! ij, _pc) (2)
_#ij " 0 when f , 0, or f " 0, _f , 0 (3)
_#ij " _#irrij 6" 0 when f " 0 and _f " 0 (4)
where the superimposed dot over a symbol denotes the timerate. Tensile stress and strain are considered as negative. pcdenotes a geometric size characteristics of the yield locus,and it plays the role of the hardening function of irreversibledeviatoric strain (#irrq ) and of the accumulated relative dis-solved mass of each mineral species ("1, . . ., "n), hencepc " pc(#
irrq , "1, . . ., "n): During the process of dissolution,
the material is chemically softened to the condition wherethere is dilatant microcracking; on the other hand plasticdeviatoric strain induces a hardening effect. This indicatesthat the two factors, plastic deviatoric strain and chemicalsoftening, affect the yield locus of the material with oppositecontributions. At an imposed constant stress at yielding, theplastic deviatoric strain hardening compensates for the dis-solution-induced strength loss.
The chemical softening parameters ("1, . . ., "n), represent-ing the accumulated mass removal of each particular mineralspecies, can be calculated from the rates of reactions inwhich the given species is involved. In reality, multiplereactions may affect the strength of any geomaterial, theprocesses of which might be simultaneous or sequential,with varied rates of progression or within different timescales. However, there is currently no widely accepted ap-proach to identify and quantify contributions of individualmineral components to the overall degradation of materialstrength or other mechanical properties. Meanwhile, it iscommon for one type of chemical reaction to be predomi-nant, as, for instance, in the case of carbonate reservoirssubjected to an acid environment or calcite exposed to water(Ciantia & Hueckel, 2013). In what follows, a single reac-tion effect with a single rate will be considered only, and thechemical softening parameter becomes the reaction progressvariable ", and hence, pc " pc(#
irrq , "): The mechanisms of
chemically induced mass removal will be detailed in thefollowing section.
The irreversible strain rate can be obtained through theassociated flow rule
_#irrij " _ @ f
@! ij(5)
where the multiplier _ , can be expressed as a function ofrate of stress and chemical reaction process
_ "@ f =@! ij
# $_! ij $ @ f =@"% & _"
# @ f
@#irrq
2
3
@ f
@skl
@ f
@skl
% &1=2 ;
_ " 0 if f , 0 or f " 0 and _f , 0
(6)
is determined by way of the extended Prager’s consistencylaw, _f (! ij, #irrq , ") " 0: The yield locus gradient @ f =@! ij
determines the mode of strain rate, whereas the magnitudeof strain rate is controlled by multiplier _ , which impliesthat a purely chemical process can promote the developmentof irreversible strain when stress is fixed (see Hu & Hueckel(2007a, 2007b) for details).
2 HU AND HUECKEL
PROOFS
MECHANISMS OF MINERAL DISSOLUTION IN ACARBONATE GEOMATERIAL COUPLED WITHVOLUMETRIC STRAIN
As one of the two major reservoir lithotypes, carbonates(the other is sandstones) have been intensely studied in thepetroleum industry (Dusseault, 2011). The most commonchemical process in carbonate reservoirs is the dissolution ofcalcite and dolomite (Andre et al., 2007). Because thedissolution of dolomite, described as
CaMg(CO3)2 $ 2H$ $ Ca2$ $Mg2$ $ 2HCO3#
is found to be geochemically similar to calcite, calcitedissolution is discussed here as a representative for carbonateminerals in general. When subjected to low-concentrationacid in an aqueous environment, calcite undergoes thepredominant reaction described as
CaCO3(s)$ H$(aq) $ Ca2$(aq)$ HCO3#(aq) (7)
This is usually the first step of the calcite–water interaction;the product HCO3
# formed in aqueous solution is unstableand inclined to convert to CO2(g) with the presence ofexcessive H$: However, since it is the most significantreaction involved in solid mass removal, the present authorsbelieve it is acceptable to omit the subsequent reactions andconsider reaction (7) as the dominant and hence only reac-tion. Based on the mass conservation of calcium, the molenumber of calcite removed from the solid phase (CaCO3) isequal to the change in number of calcium ions in thesolution, and the reaction rate is expressed as
daCa2$
dt" ~AAªCa2$ (k$aCaCO3
aH$ # k#aCa2$aHCO3# ) (8)
where ~AA is a non-dimensional quantity denoting the totalfluid–solid interface surface area per unit volume; ai areactivities and ªi are activity coefficients of the ith species;k$ and k# are rate constants of forward and backwardreactions, respectively.
The rate of calcite removal from the solid phase ofcarbonate material can be represented by the rate of changeof activity of Ca2$ in pore water, written as
_" " sdaCa2$
dt(9)
where s denotes a ratio between the number of moles ofwater and that of calcite in the initial state. To maintain thelinearity of the proposed model, the precipitation term inequation (8) is omitted.
Scalar variable ~aa is introduced to represent a normalisedamount of the increased surface area per unit volume of themedium, which is linked to the dimensionless total reactivesurface area per unit volume of fluid (denoted by ~AA) by wayof
~aa " ~AAnrwr0
where r0 " 1 kg=m3, rw is the density of water and ndenotes porosity taken at its initial value.
The newly generated specific surface area can be assumedas proportional to the irreversible dilatant volumetric strain,as suggested by a model of a two-dimensional hexagonalcrystal assembly (Hu & Hueckel, 2007b). Then we have~aa " 8=(
'''3
p$)j#irrv j ' 1m for #irrv , 0, where $ denotes an
average microcrack opening at the apex. Hence
~AA " f (#irrv ) " %j#irrv j$ %c, #irrv , 0 (10)
% " 0, #irrv . 0 (11)
where % is a proportionality constant, %c denoting thedimensionless specific surface area of the pre-existing voids
per unit volume. When the volumetric strain is compressive,it is assumed that there is no microcracking and that thespecific surface area remains constant.
NUMERICAL SIMULATION OF CRACK PROPAGATIONIN THE PRESENCE OF COUPLED DAMAGE–DISSOLUTION PROCESSES
There are at least two circumstances of industrial interestinvolving the chemical enhancement of crack propagation.One typical scenario features a remote constant radial tensilestress, which goes to zero at the inner (artificial) boundaryof the process zone, r " a. At the same time, yielding whichinduces irreversible deviatoric strain is localised near thecrack tip, also producing a proportional amount of irreversi-ble dilatancy (meaning microcracking). The latter inducesthe chemical mass removal enhanced by its rate dependence.Owing to the chemical action of the fluid, the materialweakens in that zone. The circumferential stress is alsotensile everywhere. Because of boundary constraints thestress state does not evolve significantly, hence the wholeprocess is dominated by compensation of the chemical soft-ening by the strain hardening (Hueckel, 2009).
One other scenario describes a single crack being affectedby the pressure of the fluid acting on the walls of the crack,and at the same time by a chemical infiltrating from thecrack and weakening the crack surroundings. No other loadis considered. The action of the pressure transmitted to therock causes dilatant damage near the crack tip and henceengages the chemo-mechanical coupling in that location.The stress state near the crack tip is different from theprevious one in that the stress near the crack tip is compres-sive orthogonally to the crack edges. This problem is dealtwith elsewhere.
From the field point of view, in the former scenario thechemical effect plays a primary role, whereas the stress fieldplays a secondary role. In what follows this case becomesthe focus. It corresponds to a situation of a pre-existingcrack inundated with water or an aggressive liquid (e.g. acidrain), while a tensile stress source is remote. A possible fieldanalogue could be a reactivation of a landslide with a pre-existing slip zone (see, for example, Zhao et al. (2011)). Inthe latter case both factors, chemical and mechanical, areequally important. This case is likely to occur during fluidpressure (static) pulse assisted by the chemically aggressivefluid component.
Classically, a mechanically stimulated crack extends whena sufficiently high level of stress intensity is attained in thematerial at the crack tip. The microcracking can be repre-sented macroscopically through the inelastic strain near thecrack tip and its yielding is affected by the strain softening.In contrast, in the considered case, the external tractions arekept constant, while the process is activated and driven byway of the chemical field.
To address the solution of the boundary value problem forthe system of equations describing the coupled chemo-mechanical processes around the crack tip, the extendedJohnson approximation is adopted (Johnson, 1970), assumingthat all the fields involved are axisymmetric around thecrack tip point with a small, unstressed cavity around it, asshown in Fig. 1. This is analogous to the cylindrical cavityexpansion problem often used for pile or piezocone model-ling in geotechnical problems, see, for example, Collins &Stimpson (1994) or Bigoni & Laudiero (1989). For simpli-city, the plane strain is considered. The boundary conditionsimpose the radial stress at the external boundary as constantand uniform, and zero at the inner boundary
! rjr"a " 0; ! rjr"b " const: (12)
ENVIRONMENTALLY ENHANCED CRACK PROPAGATION IN AN ISOTROPIC SHALE 3
PROOFS
Note that the other boundary conditions that one normallywould impose on a boundary value problem, as shown inFig. 1, such as zero circumferential stress at the crack faces,are ignored by virtue of adopting the Johnson approxima-tion, which makes it possible to use axial symmetry fields.While notably the obtained solutions are not exact, thebenefit from capturing principal interrelationships betweenthe variables involved and their distribution and evolution issignificant. Note also that assuming zero radial stress at theinner boundary excludes any realistic description of thestress concentration near the crack tip. However, the currentfocus is rather on the adjacent process zone of microcrack-ing, which is away from the crack tip by a non-negligibledistance, represented in this case by the radius a (Bazant &Kazemi, 1990; Lin & Labuz, 2011).The equilibrium equation and kinematic relationships for
the plane strain axisymmetric problem are given as
d! r
dr$ ! r # !&
r" 0 (13)
_#r " # d _u
dr, _#& " # _u
r, _#z " 0 (14)
where & and r denote the circumferential and radial coordi-nates (respectively) and u, which denotes a radial displace-ment, is defined as positive when pointing outwards. Thecircumferential displacement is null owing to axisymmetry.A linearised yield surface is chosen for simplicity after
Mroz & Kwaszczynska (1971) and Hueckel & Mroz (1973),as shown in Fig. 2(a). Tensile stress or strain is considerednegative in the present system, hence, the process of yield-ing starts from the third quadrant in Figs 2(a) and 2(b). Thelocus segment ED determines the relationship between ! r
and !& at the initial yielding:
!&0 " tan'4# W
! "! r0 # tan
'4# W
! "$ tan%
% &! 01
where W, % and ! 01 are parameters shown in Fig. (2a). Thelinearised yield locus undergoes an expansion with an in-crease of the shear strain intensity (equation (1)), while itshrinks with the square of the loss of the mineral mass, ":The assumed hardening rules are
! r " ! r0(1$ Æ#irrq # ("2); (15)
!& " tan'4# W
! "! r # tan
'4# W
! "$ tan%
% &
3 ! 01(1$ Æ#irrq # ("2)
(16)
It should be noticed that equation (15) and equation (16)apply exclusively for !& , ! r , 0: Æ and ( are constantcoefficients for strain hardening and chemical weakening,respectively. Deviatoric strain intensity #irrq can be calculatedusing equation (1) integrated with time. The axial stresscomponent sz is assumed not to influence yielding in theplane strain rate conditions. It can be calculated throughequation (5) from the condition d#irrz " 0 as
! z "1
2(! r $ !&)( 3BJ
1=22
where B is a constant and J2 is the second invariant ofdeviatoric stress, and the sign ( refers to either the top orthe bottom part of the yield locus, Fig. 2(a) (see also Mroz& Kwaszczynska (1971)).
Recall that the precipitation effect has been neglected.Hence the combination of equations (8) and (9) yields
_" " s ~AAªCa2$k$aCaCO3aH$ (17)
It is further assumed that the activities and activity coeffi-
r a! r b!
Crack
r
Tension, ( )uniformly distributed
p t
Conceptualisedprocess zone
Fig. 1. Schematic diagram of the process zone around the cracktip
!"
F
K
!01
#0 !r
D
A2
AA1
!B2
B1
B d 0$r "
d$
d 0$" #
E
(a)
B2
B1
B
A2
AA1
(b)
Fig. 2. (a) Linearised yield locus for the material. (b) Sketch ofstress profiles within the process zone
4 HU AND HUECKEL
PROOFS
cients are unities for the convenience of simulation, andhence _" can be rewritten as
_" " sk$(%j#vj$ %c), #v , 0 (18)
Hence, if #v is negative
("2 " ((t
0
_"dt
! "2
" (s2k2$ %(t
0
j#vjdt $ %ct
! "2
(19)
which leads to
("2 " K2
(t
0
j#vjdt! "2
$ 2KKct
(t
0
j#vjdt! "
$ K2c t
2 (20)
where K "'''(
psk$%, Kc "
'''(
psk$%c represent the overall
chemical softening rate coefficients associated with newlygenerated micro-cracking and the initial void ratio, respec-tively.
An initial dissolution, proportional to a constant specificsurface area associated with the initial porosity, even for alow stress prior to yielding, that is, for
! ij : f " f ! ij " const:, " " Kc
(t
! ", 0
through a slow mass removal _" . 0 and correspondingshrinkage of the yield locus can eventually activate theplastic yielding at ED (Fig. 2(a)), in other words, bring thecondition f " 0 around the initially subcritical crack. Atthat point, with the continuing reaction, _" . 0, but now alsoenhanced by the microcracking as manifested by _#v . 0, thecoupled process is significantly intensified.
Numerical solutions were obtained by solving (usingMatlab#) the above-mentioned system of equations, given theinitial and boundary conditions, as specified earlier. The inputparameters are Æ " 1, ( " 1, W " '=6, tan% " 0:2, b=a " 10,Kc " 2:03 10#8 s#1, and K is set to be 5:03 10#6 s#1,unless it is a subject of a parametric study. The displacementua at the internal circumference is used to represent propaga-tion of the crack tip and its dimensionless form ua=a, as shownin Fig. 3. If the value of chemical softening coefficient, K, isincreased by one order of magnitude, the displacement uawithin 1 month (30 days) starts growing exponentially, mean-ing an accelerated crack propagation, which is an expectedresult of the damage–dissolution coupling. Fig. 4 further
demonstrates the relationship between the crack propagationas represented by ua=a and a total integrated dissolved mass
1
b# a
(b
a
(t
0
_"dtdr
Notably, within the range studied at an advanced massremoval, a decelerating crack propagation is observed for asufficiently high valued K: As seen also, for the sameaccumulated mass of dissolution, a smaller chemical coeffi-cient K leads to a deeper crack penetration, as _" is propor-tional to K and #irrv , whereas the latter is linked to ua: Theevolution of the distribution of relative mass dissolution aswell as of dissolution rate is shown in Fig. 5 and Fig. 6,respectively. Fig. 5 indicates that the evolution of theaccumulated dissolved mass distribution along the radius isconfined to the near-crack-tip area. The maximum accumu-lated relative mass removal reaches 0.67, which is very high.Note that dissolution of carbonate rock can be very fast andadvanced (see, for example, Ciantia & Hueckel (2013)). The
0
0·01
0·02
0·03
0·04
0·05
0·06
0·07
0·08
0·09
0·10
0 200 400 600 800 1000
ua
a/
t: h
K 2·5 10 s! $ % %6 1
K 5 10 s! $ % %6 1
K 1 10 s! $ % %6 1
Fig. 3. Propagation of the crack tip
0
0·005
0·010
0·015
0·020
0·025
0·030
0·035
0·040
0·045
0 0·02 0·04 0·06 0·08 0·10 0·12 0·14 0·16
ua
a/
K 1 10 s! $ % %6 1
K 2·5 10 s! $ % %6 1
K 5 10 s! $ % %6 1
1 b
ab a" # 0d d
tt r# !%
Fig. 4. Influence of total mass removal on crack propagation
0
0·1
0·2
0·3
0·4
0·5
0·6
0·7
%
r a/
t 1100 h!
t 550 h!
t 0!
1 2 3 4 5 6 7 8 9 10
Fig. 5. Evolution of the distribution of relative mass dissolution
ENVIRONMENTALLY ENHANCED CRACK PROPAGATION IN AN ISOTROPIC SHALE 5
PROOFS
confinement to the near-tip zone is even sharper in thedissolution rate distribution; as Fig. 6 demonstrates, the rateof chemical dissolution in the inner area of the process zoneundergoes a huge change, while that outside the area isalmost negligible, because it comes mainly from the pre-existing surface area of the initial porosity. The rate distribu-tion is clearly linked to that of dilatancy, along the radius,which (as shown in Fig. 7) develops in a similar fashion,indicating that the near-crack-tip area is subject to the mostsevere damage. Given the linear form of the yield locus andthe normality rule, there is no compaction in the processzone. Fig. 8 and Fig. 9 show the evolution of the radial andcircumferential stress component distribution, respectively.Because the radial stresses on both the interior and exteriorboundaries are fixed, there is little freedom for any evolu-tion. However, the distribution of !& undergoes a noticeablechange, specifically from t " 550 h to t " 1100 h, duringwhich the crack penetrates almost ten times deeper (asobserved in Fig. 3). It is interesting to notice that in thenear-crack-tip area the tensile (negative) stress experiences a
small increase before it collapses back, which indicates aninitially dominating strain-hardening followed by the pro-gressive overtaking by chemical softening. This is visible atthe stress profiles for two representative moments in theevolution delineated in Fig. 2(b), where A1, A2 stand forstress state at the interior, r " a, and B1, B2 represent thecounterpart exterior, r " b, respectively. The exterior of theprocess zone is subjected to consistent but much moremodest chemical weakening mainly linked to the backgrounddissolution, because the deformational effect is very small.
MODELLING OF DIFFUSION OF DISSOLVED CALCITECOUPLED WITH IRREVERSIBLE DAMAGE INSIDE THEPROCESS ZONE
Inside the process zone, the phenomenon of diffusionoccurs once the solute, that is the calcite, is dissolved inwater. The reference condition is set as a uniform dissolutionat the surface of pre-existing voids within a REV. Intercon-nected voids within the process zone form microchannels,providing pathways for calcite transport. At the crack tip, it
0
0·2
0·4
0·6
0·8
1·0
1·2
1·4
r a/
t 1100 h!
t 550 h!
t 0!
1 2 3 4 5 6 7 8 9 10
%:10
$%
6
.
Fig. 6. Evolution of the distribution of dissolution rate
%0·35
%0·30
%0·25
%0·20
%0·15
%0·10
%0·05
0
1 2 3 4 5 6 7 8 9 10
$v
r a/
t ! 550 ht 0!
t 1100 h!
Fig. 7. Evolution of the distribution of irreversible volumetricstrain
%0·7
%0·6
%0·5
%0·4
%0·3
%0·2
%0·1
0
1 2 3 4 5 6 7 8 9 10
!!
r/01
r a/
t ! 550 h
t 0!
t 1100 h!
Fig. 8. Evolution of the radial stress distribution
%0·62
%0·60
%0·58
%0·56
%0·54
%0·52
%0·50
%0·48
%0·46
%0·44
%0·42
1 2 3 4 5 6 7 8 9 10
!!
"/
01
r a/
t 550 h!
t 0!
t 1100 h!
Fig. 9. Evolution of the circumferential stress distribution
6 HU AND HUECKEL
PROOFS
is assumed that within the macro-pore, water washes awaythe excess mineral, and for simplicity zero concentration isassumed, hence xCa2$ " 0 at r " a: The exterior of the pro-cess zone is an artificial boundary and is assumed to stay asit was at the initial state of the material, when dissolutionwas limited to the pre-existing micro-pores, hencexCa2$ " x0 at r " b: An initial background steady-state diffu-sion field is created with the imposed boundary conditions.
The reactive diffusion is approximated as an axisymmetricfield of concentrations (molar fraction of calcite) withuniformly distributed sources (reaction sites) described bythe following governing partial differential equation withgiven boundary conditions as well as the initial conditions.
@xCa2$
@t" D
@2xCa2$
@r2$ 1
r
@xCa2$
@r
! "$ k$(%j#v r, t% &j$ %c)
(21)
where xCa2$ is the molar fraction of calcium in the fluidphase within the process zone, and D represents the calciumdiffusion coefficient, which is assumed a constant within theprocess zone. As seen in equation (21), the source termdepends on the amount of volumetric deformation, hence itssolution is coupled to the solution of independent equations(12)–(21).
The damage-affected zone of the material induces evolu-tion of the concentration of solutes, coupled with theevolving deformation (microcracking) (see Fig. 10). Thedistribution of mineral concentration and the flux ofthe solute mass are hence affected by the solid mechanics ofthe problem. The diffusive flux at the inner boundary,representing the influence of the mechanical damage on thenet dissolved mass, is of particular interest, and it can becorrelated with the displacement of the crack tip, or crackpropagation. Figs 11 and 12 show the evolution of mass fluxJa at r " a and its dimensionless counterpart. M is a coeffi-cient representing the effect of deformation/damage asso-ciated with the dilatant volumetric strain on the masstransfer, M " s%k$a2=Dx0: M " 0 means a pure diffusionprocess without the presence of chemo-mechanical coupling.As indicated in both Fig. 11 and Fig. 12, the reactivedissolution coupled with the irreversible damage of themedium plays an important role in the diffusion processes.Finally, it is possible to set the crack propagation against theflux of the solute at the near-tip perimeter, Fig. 13. Becausethe total flux of mass removal is experimentally measurable,
it is possible to verify the simulation results against anexperiment and perform an in situ calibration.
CONCLUSIONSA damage-enhanced reactive chemo-plasticity model was
adopted to describe a simplified scenario of a chemicallyinduced propagation, of an otherwise subcritical crack in apre-stressed geomaterial. An open system was selected,meaning that the local precipitation is neglected, and chemi-cal dissolution was considered as a form of mass removal.The chemo-plasticity coupling effect was investigated withan assumption of simple hardening rules, linear form ofyield condition and rigid behaviour prior to yielding; yet anumber of meaningful conclusions were reached. The resultshave demonstrated a significant role of the compensatorymechanism between strain hardening and chemical softening.It was possible to obtain the rate of crack propagationagainst time, total accumulated relative mass removal andagainst the exiting mineral flux. Chemical softening inten-
Coupled diffusion–damage-
affected range
r b x x, const.! ! !solute 0
r a x, const. 0! ! !solute
Microchannels providingpathways for the transport
Fig. 10. Coupled diffusion–damage-affected zone around thecrack tip
0
0·5
1·0
1·5
2·0
2·5
3·0
3·5
4·0
0 20 40 60 80 100
j
&
M 5!
M 2·5!
M 0!
Fig. 11. Evolution of the dimensionless mass flux at the crack tip
0
0·5
1·0
1·5
2·0
2·5
0 200 400 600 800 1000
J: m
ol/m
per
s10
26
$%
t: h
M 5!
M 2·5!
M 0!
Fig. 12. Evolution of the mineral mass flux at the crack tip
5
6
ENVIRONMENTALLY ENHANCED CRACK PROPAGATION IN AN ISOTROPIC SHALE 7
PROOFS
sity, dominating the rate of mineral dissolution, has beenproved essential to the progress of crack propagation, as wellas that of soil/rock degradation.Most importantly, it is seen that with the mass removal,
the rate of crack penetration initially substantially increases.However, at some point (around 700 h) the penetration ratestarts decelerating (for K " 53 10#6 s#1). This time corre-sponds to the moment when the quadratic nature of thechemical hardening formula in the hardening law starts tobe felt. In other words, starting from that time, there needsto be much more of the mass loss to compensate for thelinear strain hardening term (which is a linear function ofthe crack tip displacement). The findings of this paper maybe applied to several energy source harvesting areas includ-ing petroleum/natural gas and geothermal engineering (tosuch processes as acid-enhanced hydraulic fracturing orreservoir engineering), as well as in the area of slopestability, and so on, where the soil/rock mass is subjected tochemically induced degradation.
ACKNOWLEDGEMENTSThe authors acknowledge the partial support given to
Duke University by ENI Spa, Milan, Italy. The support ofUS National Science Foundation through grant no. 0700294is also gratefully acknowledged.
NOTATIONA1, A2 stress states at the interior, r " a
~AA non-dimensional quantity denoting total fluid–solid interface surface area per unit volume
a radius~aa scalar variableai activities of i th speciesB constant
B1, B2 stress states at the exterior, r " bD calcium diffusion coefficient_eirrij invariant of deviation strain ratef
@ f =@! ij yield locus gradientJaJ2 second invariant of deviatoric stressK "
'''(
psk$%, overall chemical softening rate
coefficient associated with newly generatedmicrocracking
Kc "'''(
psk$%c, overall chemical softening rate
coefficients associated with initial void ratiok$, k# rate constants of forward and backward
reactions respectivelyM coefficient representing the effect of
deformation/damage associated with dilatantvolumetric strain on the mass transfer
n porosity taken at its initial valuepc geometric size characteristics of the yield locusr radial coordinates ratio between number of moles of water and that
of calcite in the initial stateu radial displacement
xCa2$ molar fraction of calcium in the fluid phasewithin process zone
Æ constant coefficient for strain hardening( constant coefficient for chemical weakeningªi activity coefficients of i th species$ average microcrack opening at the apex
(#irrq ) irreversible deviatoric strain intensity& circumferential coordinateW_
" reaction progress variable("1, . . ., "n) accumulated relative dissolved mass of each
mineral species"n, (n " 1, 2, . . .) accumulated dissolved masses of the n th
species of mineral per unit volume of porousmedium
rw density of water n denotes porosity taken at itsinitial value
! ij stress!r radial stress component!z axial stress component!& circumferential stress component! 01
% proportionality constant%c dimensionless specific surface area of pre-
existing voids per unit volume
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1: Please provide keywords from approved list, http://www.icevirtuallibrary.com/upload/geotechniquekeywords.pdf2: Zhan et al not on refs list, please provide full ref details3: Zimmermann cited as 2010, listed as 2011. Please check year4: Zimmermann again cited as 2010, listed as 20115: Fig 10 was not cited in manuscript. Citation has been added here,OK?6: In Fig 11 vertical access is ’j’, in Fig. 12 ’J’ - text refers to Ja -should these be consistent?7: Notation list has been added, please check and amend/complete asnecessary8: Bazant - issue no.?9: Ciantia - please clarify this ref. Is it a Geotechnique journal publishedpaper? If so, please give vol/issue/pp10: Coffman - Committee of what organisation? Located where?11: Dusseault - issue no.?12: Hori et al does not seem to be cited in text - please cite or deletefrom list13: Hu and Hueckel 2007a - issue no.?14: Hueckel 1992 - issue no.?15: Hueckel 1997 - issue no.?16: Hueckel and Hu 2009 - issue no.?17: Hueckel et al 1998 does not seem to be cited in text. Please cite ordelete from list18: Johnson - issue no.?19: Lin and Labuz - issue no.?
20: Mura - this reference does not seem to be cited in text, please citeor delete from list21: Ostapenko1968 and 1975 - issue no.?22: Rahman - issue no.?23: Tada et al - issue no.?
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ENVIRONMENTALLY ENHANCED CRACK PROPAGATION IN AN ISOTROPIC SHALE 9