JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 104, NO. B5, PAGES 10,467-10,481, MAY 10, 1999

Evolution of strength and hydraulic connectivity during dehydration' Results from a microcrack model

Guy D. H. Simpson • Institute for Mineralogy and Petrography, Swiss Federal Institute of Technology, Zurich, Switzerland

Abstract. A two-dimensional cellular automaton model based on the elastic me- chanics of tensile (mode I) microcracks was developed to investigate the evolution of rock strength and hydraulic connectivity during progressive dehydration. Fluid produced by dehydration is assumed to be accommodated by microcracks, which propagate through a simulated rock matrix owing to elevated fluid pressures. Crack propagation affects rheology by stress relaxation and interaction, and it affects hy- drology by permitting fluid flow between neighboring cracks. Numerical simulations with undrained boundary conditions show that reactions releasing small quantities of fluid (< 0.25 wt %) in a rock matrix with zero initial hydraulic connectivity induce large strength reductions (approximately 80-90 %). Strength reduction occurs abruptly at the onset of dehydration and continues until approximately 10 % reaction, when a low-strength plateau is reached. Subsequent reaction causes almost no further effect on rock strength until the percolation threshold is attained, at which point the strength drops to zero. Results with drained boundary conditions yield similar strength reductions before hydraulic connectivity of the crack network is achieved. Thereafter, fluid drainage allows partial strength recovery. The results indicate that the dominant theological response induced by dehydration is caused by the generation of fluid overpressures and is unrelated to the establishment of hydraulic connectivity coinciding with the percolation threshold. Although rocks characterized by zero initial hydraulic connectivity retain additional strength rela- tive to rocks with initial hydraulic connectivity, the magnitude of this additional strength is small (< 20 % original strength).

1. Introduction

Experimental studies have demonstrated that meta- morphic dehydration reactions can have a profound ef- fect on the mechanical properties of a rock. In undrained experiments, dehydration induces an abrupt order-of- magnitude loss of strength in response to the generation of high pore fluid pressures [Heard and Rubey, 1966; Murrell and Ismail, 1976; Olgaard et al., 1995; Raleigh and Paterson, 1965]. Drained dehydration experiments display more complex behavior whereby initial weaken- ing is followed by time-dependent strength recovery [Ko et al., 1995, 1997; Olgaard et al., 1995]. The samples upon which most dehydration experiments have been performed are near-monomineralic hydrous aggregates of gypsum, serpentine, and chlorite, which release be-

1Now at Institut Fran•ais du P6trole and Ecole Normale Sup6rieure, Paxis.

Copyright 1999 by the American Geophysical Union.

Paper number 1999JB00052. 0148-0227/99/1999JB-00052509.00

tween 10 and 13 wt % fluid during dehydration. This study focuses on determining the rheological response induced by dehydration reactions that are representa- tive of metamorphosed sediments and granites, which are the volumetrically dominant rock types of the con- tinental crust. Such rock types release a total of ap- proximately 4 wt % fluid during metamorphism and typically not more than 0.25 wt % fluid during any one reaction [Fyfe et al., 1978].

In general, the mechanical influence of fluids on de- formation is due to the dependency of strength on the effective stress, Pe -- Pt- c•pf, where Pt is the total stress, pf is the fluid pressure, and c• is the poroelastic parameter that relates stress and pore pressure. Exper- imental investigations have demonstrated that if rocks are permeable to a pore fluid that is prevented from draining, the strength decreases progressively to the standard crushing strength (strength at atmospheric pressure) as the ratio of the pore pressure to the con- fining pressure increases from 0 to I [e.g., Handin et al., 1963]. This behavior is consistent with the clas- sic effective stress concept originally developed for un- consolidated soil where the value of c• = 1 [Terzaghi,

10,467

10,468 SIMPSON: DEHYDRATION-INDUCED DEFORMATION

1923]. For low-permeability rocks in which the inter- nal pore fluid pressure is prevented from equilibrating, the strength reduction for any given pore pressure is diminished relative to predictions based on the clas- sic effective stress concept [Handin et al., 1963; Heard, 1960]. Such deviations are consistent with the scarce amount of laboratory work done on the relationship between deformation and effective stress which shows that low-permeability rocks generally have (• values of less than unity [see Fart, 1959; Christensen and Wang, 1985; Warpinski and Teufel, 1992].

One of the principal aims of this study is to de- termine whether low-permeability rocks, which release small amounts of fluid during dehydration reactions, can maintain a significant proportion of their strength. The only known data pertaining directly to this prob- lem were collected by Murrell and Ismail [1976], who experimentally investigated the strength of a dehydrat- ing granodiorite containing 2.7 wt % of structurally bound water and an additional 1.4 wt % of free wa- ter. These authors demonstrated that even before the

onset of dehydration, the small quantity of free water in the sample caused a strength decrease of 70-80 % rela- tive to samples containing no free water. The strength dropped to even lower values (approximately I - 2 % of the initial strength) during subsequent dehydration and partial melting.

In the limit that the initial permeability is negligi- ble, dehydration reactions with positive volume changes generate fluid pressures equivalent to the rock pressure after infinitesimal reaction progress. In this case, the mechanism by which reaction-induced dilational strain is accommodated depends on the rate at which fluid overpressures are produced relative to the rate at which deformation can dissipate the fluid overpressure. If heating rates are slow or if reactions take place at equi- librium conditions such that fluid pressure fluctuations may regulate the reaction rates, reaction-generated fluid overpressures may be dissipated by viscous creep [see Connolly, 1997]. If heating rates are relatively rapid or if reactions occur at conditions far from equilibrium, then the timescale at which fluid overpressures are cre- ated compared to the rate of viscous strain is such that hydro-microfracturing can occur [Nishiyama, 1989; Connolly et al., 1997]. This paper investigates the end- member case where reaction-generated fluid pressures, in rocks of negligible initial hydraulic connectivity, are assumed to be accommodated by microcracking. For a given amount of fluid released in typical reactions, principles of linear elasticity are used to calculate the fluid pressure and determine the stability of microcracks that propagate throughout a rock matrix. Crack prop- agation affects rheology by stress relaxation and inter- action, and it affects hydrology by permitting fluid flow between neighboring cracks. The approach taken here follows from similar (fluid absent) microcrack models utilizing linear elastic fracture mechanics [e.g., Spetzlcr

et al., 1982; Brodsky et al., 1983; Costin, 1985; Rudnicki and Chau, 1996] that have proved successful in quan- titatively demonstrating that the behavior of brittle rocks deformed under compression can be interpreted in terms of tensile crack growth. Crack interaction and fluid flow are treated here using a nearest-neighbor, cel- lular automaton in a manner similar to that of Hender-

son et al. [1994] and Wilson et al. [1996]. No treatment is given here to thermally activated, time-dependent subcritical crack propagation [see $petzler et al., 1982; Brodsky et al., 1983; Costin, 1983, 1985] or to reaction kinetics [see Nishiyama, 1989; Cormoily, 1997; Hacker, 1997].

2. Governing Equations

2.1. Stress-Induced Propagation of Microcracks

The deformation of brittle materials under compres- sion is primarily controlled by the nucleation, growth, and interaction of microcracks [Kranz, 1983; Paterson, 1978]. Direct observation [Tapponnier and Brace, 1976; Wong, 1982] has shown that microcracks nucleate from preexisting flaws, such as grain boundaries and points where there are mismatches in elastic properties, which are thought to act as localized tensile stress concentra- tors. As the applied deviatoric stresses are increased, microcracks propagate predominantly as mode I ten- sile fractures in a direction parallel to the maximum compressive stress, eventually resulting in macroscopic failure [Kranz, 1983; Wawersik and Brace, 1971].

Elastic fracture mechanics postulates that a perfectly sharp crack with traction-free walls, embedded in a linear-elastic solid, propagates once the stress inten- sity factor (Kt) exceeds a critical value (Kic) at the crack tip [Lawn and Wilshaw, 1975]. Assuming that the tensile stress concentrations associated with microc-

racks are proportional to the applied deviatoric stresses, Costin [1985] formulated an approximate expression for the stress intensity factor of a penny-shaped mode I crack, which includes stress interaction with neighbor- ing cracks, which is modified here as

2 d Ki -- - • (an - pf- I[]•]Di) , (1)

•r a

I[•/] -- (1 - (2a/s)2) -« if •/-- 0 ø (i.e., colinear), I[/•]---- 1 if/•- 90 ø (i.e., orthogonal),

where a is the crack radius, d is a length scale deter- mined by the size of the tensile stress concentrations, s is the crack separation distance, rrn is the normal stress (compressive stress is negative throughout this paper), pf is the fluid pressure (py < 0), Di is the deviatoric stress (Di = rri - am), erin is the mean stress, • is the angle between neighboring cracks, and I[•] is a func- tion accounting for stress interaction (see Table I for a

SIMPSON: DEHYDRATION-INDUCED DEFORMATION 10,469

Table 1. Symbols and Typical Parameter Values

Symbol X Units Meaning Value

critical stress intensity factor stress intensity factor for mode I cracks diameter of penny-shaped crack initial crack diameter

crack diameter at which fluid redistribution occurs

crack spacing volume of individual model cell

length scale of tensile region deviatoric stress (= cri -fire) mean stress

fluid pressure Young's Modulus fluid bulk modulus Poisson's ratio

fluid mass in crack

total weight•percent of fluid produced in rock fluid density rock density

1.5

2 x 10 -4 0.4 x 10 -3

10 -3 10-9

6 x 10 -4

3.8 x 102

25 x 10 3 - 105 1 x103

0.15

< 0.25

1 x 10 3 2.6 x 10 3

summary of the notation). The mean stress in the equa- tion originally used by Costin is substituted here for the normal stress, following the suggestion by Rudnicki and Chau [1996] that the mean stress form leads to negative pressure sensitivity. The term (d/a) in (1) [after Costin, 1985] accounts for a linear decrease in the local tensile stress as cracks extend away from stress concentrators. Two crack configurations are considered here: colinear (• = 0 ø) and orthogonal (• = 90ø). The interaction between colinear arrays of cracks, which are character- ized by stress amplification, is treated with the term 1 -(23/s)2) -1/2 [after •ostin, 1985], which is an ana- lytical approximation to the pseudo-traction numerical solution obtained by Horii and Nemat-Nasser [1983]. This term causes cracks to grow at progressively lower stresses as the separation distance between cracks de- creases. This study does not consider the case when the crack length 23 equals the crack spacing s, where the interaction term for colinear cracks becomes undefined.

Stress interaction between mode I cracks oriented or-

thogonally was shown by Kachanov [1994] to be small, even when the spacing between cracks was one-tenth the crack length. On this basis, no stress interaction is assumed to occur between cracks oriented orthogonally. Colinear interaction is the most severe case and there-

fore represents an upper bound on the effect of crack interaction.

Neglecting for the moment the fluid pressure, the first term on the right-hand side of (1) is due to the com- presslye normal traction on the crack, which acts to inhibit crack growth. The second term in (1) is due to the deviatoric component of the applied stresses, which may stabilize or alestabilize cracks, depending on the crack orientation with respect to the applied stresses.

If the applied deviatoric stress is increased, cracks ori- ented perpendicular to tensile components of deviatoric stress become more unstable, while the reverse is true for cracks oriented perpendicular to compressive com- ponents of deviatoric stress. This causes cracks to pref- erentially grow parallel to the maximum compressive stress. Cracks are stationary when Kx <_ Ktc, propa- gate when Kx > K•c, and continue to grow until the condition K• _< K•c is satisfied.

crack propagation - 12

' colinear array

stress 8 __s_ __

6 '•a

KI 4 orthogonal array 2/•(rca)l/2D ß 23

0 0.2 0.4 0.6 0.8 1

2aYs

Figure 1. The effect of stress relaxation and inter- action on the stress intensity factor of a penny-shaped mode I crack in a local tensile region subject to over- all compression (computed from equation (1) using an/D = 1, d = 6 x 10 -4 m, and p/= 0). Cracks propa- gating in colinear arrays (/3 = 0 ø) are initially stable but become progressively less stable as the normalized sep- aration distance (2a/s) between cracks decreases. Or- thogonal crack arrays (/3 = 90 ø) are characterized by stable crack growth even at small separation distances.

10,470 SIMPSON' DEHYDRATION-INDUCED DEFORMATION

Figure 1 shows the effects of stress relaxation and in- teraction during crack propagation under uniaxial com- pression. Initial crack growth is stable for all crack con- figurations, in that K• decreases with increasing crack length. Cracks aligned in colinear arrays become pro- gressively less stable as they propagate, and they even- tually become unstable at small separation distances.

2.2. Fluid Pressure-Propagated Micro cracks

Fluids influence the stability of cracks both chemi- cally, by reducing bond energies at the crack tips, and physically, by increasing the crack opening forces [see Atkinson, 1987; Meredith, 1990]. Only the physical ef- fect of fluids is considered here. The propagation of penny-shaped fractures by fluid pressure has been con- sidered by Ab• et al. [1976]. They solved the appropri- ate equations of linear-elastic fracture mechanics and fluid dynamics to derive an approximate expression for the fluid pressure py inside a penny-shaped fracture:

3•rEQ (2) py -- an - , 7raapf 16(1_ y2)

where a is the crack radius, an is the far-field applied normal stress (an < 0), E is the Young's modulus of the rock, y is the Poisson's ratio, Q is the mass of fluid in the fracture, and pf is the fluid density. Equation (2) was derived from static considerations and strictly applies to the case when the fluid velocity is zero. In situations when the static requirements are not fully satisfied, calculations based on (2) will overpredict the fluid pressure owing to the inability of fluid to fully penetrate the entire crack.

The fluid density is also a function of the fluid pres- sure. This relationship can be obtained from the defi- nition of the fluid compressibility/•:

an - C pf -- . (6)

Equation (6) indicates that the fluid pressure in a crack with constant fluid mass is determined by the relative compressibilities of the fluid and the enclosing elastic medium. In the limit of a highly compressible fluid such as air (Ki -• 0), the fluid pressure induced by the applied external stress approaches zero. In the opposite limit (Ki -• ½x•), (6) reduces to (2). Introduction of the fluid pressure term into (1) destabilizes cracks (increases /Q) by reducing the compressive normal stress.

The Young's modulus and the Poisson's ratio are as- sumed to undergo no modification during microcrack evolution. Introduction of effective elastic constants

that decrease during microcrack propagation [see Costin, 1983] will decrease the fluid pressure and therefore the rheological response to dehydration. However, as demonstrated in section 4.1, variations in the elastic constants have little bearing on the dominant rheologi- cal response induced by dehydration.

3. Numerical Model of Microcracking in a Dehydrating Rock 3.1. Basic Structure

A cellular automaton model was developed to predict the influence of reaction-generated fluid overpressures on the compressive strength and to investigate the pro- gressive evolution of hydraulic connectivity. The model rock is represented by a two-dimensional grid consist- ing of cells of equal volume (dza). The cell dimensions are physically equated with the grain size. Each cell is assigned a penny-shaped crack of uniform initial diam- eter, 2ai, such that 2ai • dz (Figure 2). For simplicity,

•_ I Op _ I (3) p Op• Kf '

by taking a linear approximation for the density change (i.e., p2 - p•) and integrating both sides with respect to pi to obtain

p._•a = p_•_• + 1, (4) p• K•

where K• is the bulk modulus of the fluid, p• is the fluid density at reference pressure, and p• is the fluid density at the pressure p•. Equation (2) can be rewritten as

P2 pf -- an C, (5)

Px

where

C • 3•EQ

7ra3p216(1_ y2)'

Substituting (4) into (5) and rearranging gives an equa- tion for the fluid pressure inside a penny-shaped crack filled with a nonviscous compressible fluid:

o 3 ( - o m + D 3)

o 1

(-Om+D 1)

o 2

( - o + D 2) in

Figure 2. Diagrammatic representation of the model consisting of discrete cells, each containing a penny- shaped crack oriented parallel to either the maximum (a•) or minimum (a3) principal stress axis. For compu- tational purposes the applied stresses are resolved into homogeneous (am) and deviatoric (Di) parts.

SIMPSON: DEHYDRATION-INDUCED DEFORMATION 10,471

K(i,j+l ) < KiC K(i,j+l ) > KiC 2a(i,j+l ) > 2% crack stationary crack propagation fluid redistribution

(id) (i+l,j)

•1•__--I --

(i+lO+l:

O,j+Z (i+• j+z.

2a

(a) (b) (c)

progressive dehydration

(d)

Figure 3. Schematic representation of (a) a stationary crack, (b) crack propagation, (c) fluid redistribution, and (d) spreading wave of damage among nearest neighbors . Cell(i,j+•) is the only cell producing fluid by dehydration. As the reaction progresses, the increasing fluid pressure (represented by the intensity of shade) in cell(i,j+•) eventually causes the stress intensity factor (K(i,j+•)) to exceed the critical stress intensity factor (K•c), and the crack in this cell begins to propagate (Figure 3b). However, it is not until the length of crack(i,j+•) has reached the critical length 2ac (Figure 3c) that the cell(i,j+•) can equilibrate fluid mass with the neighboring cells (i.e., cell(i,j) and cell(i,j+2)). Fluid equilibration decreases the fluid pressure (and increases the stability) of cell(i,j+•) and increases the fluid pressure of the neighboring cells. In this manner, fluid produced by dehydration in localized sites spreads throughout the rock causing a wave of damage (Figure 3d).

cracks are randomly oriented parallel to either the max- imum or minimum principal stress axis and are assumed to propagate in-plane as pure mode I cracks. As shown by Gu•guen et al. [1990], more complicated crack ge- ometries consisting of initial cracks inclined with re- spect to •, from which mode I wing cracks propa- gate [e.g., Ashby and Hallam, 1986; Kemeny and Cook, 1987], result in more realistic fracture geometries but produce calculated elastic yield surfaces that are in- distinguishable from models considering only mode I cracks [Costin, 1985].

3.2. Treatment of Dehydration Reactions

Metamorphic reactions are treated by assigning a fixed proportion of cells in the model rock, each of which releases a fixed amount of fluid (x weight percent progressively during the course of the reaction. The re- maining cells produce no fluid during the reaction. The distribution of productive and unproductive cells is ran- domly specified. The model does not account for ther- mally activated subcritical crack growth [see $petzler et al., 1982; Brodsky et al., 1983; Costin, 1983, 1985] and does not consider reaction kinetics [see Nishiyama, 1989; Connolly, 1997; Hacker, 1997]. The implication of these simplifications is discussed in section 5.3 and 5.5. Changes in the elastic properties between reac- tant and product phases are not considered. This is not expected to introduce large errors for the case of a reacting quartzo-feldspathic pelite because the prod- uct and reactant phases have similar elastic properties

[Birch, 1966]. Changes in solid volume associated with reactions are neglected.

3.3. Fluid Redistribution

Initially, the fluid produced within a particular pro- ductive cell is assumed to be accommodated solely by the propagation of the microcrack within that cell (i.e., no fluid redistribution). However, as a microcrack prop- agates toward the margins of the cell, the fluid that is driving the crack will be exchanged with neighboring cells. To account for fluid redistribution, it is assumed that once cracks have reached a critical crack diameter,

2ac (here-in-after called the critical length), the fluid mass of the cell producing fluid can equilibrate (con- serving mass) with the neighboring cells (Figure 3). The ratio of the initial crack length to the critical length is a measure of the ease with which fluid can be redis-

tributed throughout the rock by flow via existing het- erogeneities versus fracture-propagated fluid flow. In the limit that the critical length is less than the initial crack length, fluid can equilibrate without crack prop- agation (i.e., flow occurs along existing heterogeneities such as grain boundaries). In the other extreme, where the critical length is large compared with the initial crack length, all fluid flow is accommodated by fracture propagation and fluid redistribution only occurs when fractures intersect. The effect of varying the critical length is discussed in section 4.1.

The assumption that the critical length may be less than the crack spacing implies that fluid flow can take

10,472 SIMPSON: DEHYDRATION-INDUCED DEFORMATION

place before cracks intersect. This assumption is justi- fied on the basis of experimental observations made by Connolly et al. [1997], who showed that fluid pressure- induced microcracks typically do not intersect in two- dimensional sections but that hydraulic connectivity is achieved via existing grain boundaries. In simulations in which the critical length is less than the crack spac- ing, no account is taken for the spatial distribution of heterogeneities or for the connectivity between hetero- geneities, which are implicit in enabling fluid redistri- bution to take place prior to crack intersection.

3.4. Choice of Intrinsic Parameters

The solution of (1) requires specification of a number of critical parameters (i.e., size of the tensile region (d), initial crack length (2ai), crack spacing (s), and the cell dimension (dz)) which are either unknown, poorly con- strained, or highly variable from one rock to another. Varying these parameters leads to large variations in the absolute calculated strength. However, because in- terest here is focused on the magnitude of the rheologi- cal response during dehydration, the results are plotted in terms of the ratio of strength (dimensionless devia- toric stress, 2/TrD•-•/Kic) during dehydration to the initial strength before dehydration (i.e., fluid absent). This strength ratio is directly related to the proportion of strength drop measured in dehydration experiments. As demonstrated in section 4.1 and 4.2, the magnitude of this strength ratio is robust with respect to order-of- magnitude parameter variation.

Critical stress intensity factors (Klc) measured at room temperature vary between 0 and 3.5 2VfPa• for a wide range of rock types [Atkinson and Meredith, 1987]. Unless stated otherwise, all computations were carried out with a K•c of 1.5 MPax/•. The presence of fluid decreases the resistance to fracture propagation causing cracks to propagate at lower stress intensities [Atkinson and Meredith, 1987]. Although this is not ac- counted for here, incorporation of time-dependent sub- critical crack growth would increase the severity of the rheological response to dehydration. The size of the cell volume was arbitrarily assigned to be I mm 3. Because each cell is assigned a single crack at the center of the volume, the crack spacing is uniform and equal to I mm. Uniform initial crack radii of 2x10 -4 m were used for

all simulations. The spatial extent of the tensile region associated with individual cracks (d) is assumed to be 5x10 -4 m.

3.5. Calculation Procedure

The method used to calculate the strength (peak de- viatoric stress) of alehydrating rocks is summarized in the flow chart of Figure 4. For every increment of re- action extent the strength calculation is carried out in two main parts'

1. A constant homogeneous stress is applied at the boundaries of the elastic rock. At a fixed extent of re-

action the amount of fluid produced in discrete cells is used to calculate the fluid pressure in individual mi- crocracks of specified initial length using (6). The sta- bility of cracks is tested using (1); cracks with K• K•c are propagated until stability is achieved, whereas cracks with K• •_ K•c remain of fixed length. If crack radii exceed the critical length, the fluid mass is equilibrated among the neighboring cracks (conserving mass), and both the fluid pressure and crack stability are reassessed. This procedure is continued iteratively until all cracks have attained stability.

2. The fluid-bearing elastic medium is loaded with monotonically increasing deviatoric stress. For every stress increment the stability of individual cracks is re- peatedly tested by calculating the local fluid pressures and stress intensity factors. Cracks with K• are propagated until stability is achieved while cracks with K• _• Klc remain stationary. Cracks with radii exceeding the critical length are allowed to equilibrate their fluid mass with neighboring cracks (conserving mass), and both the fluid pressure and crack stability are reassessed. This procedure is repeated iteratively for every increment of stress loading until all cracks have attained stability (i.e., K• • K•c). Once stability is achieved, the boundary deviatoric stresses are incre- mentally increased. This procedure (outlined in part 2) is repeated until crack growth becomes unstable (i.e., K• increases with increasing crack length). Once crack growth becomes unstable the boundary deviatoric stress at the onset of crack instability is recorded, and the cal- culation is repeated for the next increment of reaction as described in part 1 above.

4. Model Results: Evolution of

Strength and Hydraulic Connectivity During Dehydration

4.1. No-Flow Boundaries

The first runs presented were carried out with no fluid flow across the external boundaries. Model results for a dehydration reaction releasing 0.05 wt % fluid are pre- sented in Figure 5. For the case in which the initial crack lengths are smaller than the critical crack length for fluid redistribution (i.e., 2ai • 2at), strength de- creases abruptly at the onset of reaction and continues to decrease until a low-strength plateau is reached by approximately 20 '• reaction extent. Subsequent reac- tion causes no further strength reduction, so that the simulated rock still retains approximately 6 % of its original strength after reaction is complete. Also pre- sented in Figure 5a for comparison is the strength pro- file for a simulation in which the initial crack lengths exceed the critical crack length for fluid redistribution (i.e. 2ai • 2ac). This simulation represents an extreme situation in which fluid can be redistributed through- out within the sample without inducing microcracking,

SIMPSON: DEHYDRATION-INDUCED DEFORMATION 10,473

Reaction extent (x %)

[ calculate fluid mass at x% [ I

initial crack dimensions

and elastic parameters

initial boundary stresses

[calculate fluid pressure (equation (6))

I cacuat stress intensity factor (equation (1

if K I > KiC if K I <_ KiC

I crack propagation[

if K ! (_ Ki_ 1 (i.e. crack stable)

-•- if2a<2a c if2a:l:2a c

• [fluid redistribution[

if K I > Ki_ 1 (i.e. crack unstable)

when all cracks have

satisfied this condition

increase boundary de viatoric steedses[

compressire yield strength (maximum differential stress)

Figure 4. Summary flow chart of the procedure used to calculate the compressive yield strength at a fixed extent of reaction. The calculation is repeated for different extents of reaction to map-out a failure envelope.

but no fluid can flow out of the external boundaries. In

this scenario, strength also drops abruptly at the onset of dehydration reaching values of 20 % original strength after less than 5 % reaction. The subsequent rheological response is characterized by monotonically decreasing strength with increasing reaction extent. The simulated rock has zero strength after approximately 40 % extent of reaction.

The abrupt (70 %) strength reduction that occurs at the onset of dehydration is due to the generation of fluid pressures exceeding the rock pressure and a consequent reduction in the effective confining pressure within fluid- producing regions of the rock (Figure 5b). For the case in which 2ai > 2ac, strength reduction occurs almost linearly in response to monotonically increasing fluid

pressure as the reaction progresses. This near-linear behavior consistent with the classic concept of effective stress occurs because fluid produced by dehydration at localized reaction sites is redistributed throughout the entire rock. This response differs from the simulation in which 2ai < 2ac, which displays an initial strength drop that takes place almost independently of fluid pressure, followed by almost no change in either fluid pressure or strength for the rest of the reaction. In this case, the constancy of fluid pressure and strength with increas- ing reaction extent occurs because fluid overpressures generated by dehydration at reaction sites are accom- modated by local fluid redistribution among neighbor- ing cells and not by the entire rock. It is the lack of pervasive connectivity that enables the rock to retain

10,474 SIMPSON' DEHYDRATION-INDUCED DEFORMATION

(a)

lOO

80

60

40

20

o

2a i < 2a c

normalised crack length

0.0 0.1 0.2

(ii)

(iii)

(iv) (v)

20 40 60 80

REACTION PROGRESS (%)

lOO

(b) lOO

80 1

4o

R • % '-', • 1

approximately 25 % reaction extent •• • 0 , I • I • I • I • I • /

o 0.5 1 .o

NORMALISED FLUID PRESSURE (Pf/•m)

Figure 5. (a) Calculated strength versus reaction ex- tent and (b) strength versus fluid pressure (solid stars represent conditions after negligible reaction progress), for a dehydration reaction releasing 0.05 wt % fluid in rocks with no initial hydraulic connectivity (i.e., 2ai < 2ac, solid line) and pervasive initial hydraulic connectivity (i.e., 2ai > 2ac, dashed line). The rock strength is calculated as 2/•rDx/-•/Kzc and is shown as a percentage ratio of the strength during reaction to the initial strength before dehydration. Parameters used in the calculations were E/Kf = 25, dis = 0.5, 2ai/2ac - 0.5 (solid line) and 2ai/2ac - 2 (dashed line). The inset images in Figure 5a (calculated for 2ai < 2ac) show the progressive microcrack damage (normalized crack length - (a - ai)/(s - ai)) as a func- tion of reaction extent. At high stresses before the onset of dehydration (i.e., images i and ii), most crack growth has occurred in an orientation parallel to the (horizon- tal) maximum principal stress, whereas crack growth is more random at lower stresses (i.e., images iii-v). Note that the deformation during dehydration is both more intense and more distributed than deformation in the absence of fluid (compare images iii-v with i-ii).

strength even after the reaction has gone to completion. This behavior is qualitatively similar to departures from the classic effective stress concept observed in the exper- imental deformation of low-permeability fluid-bearing rocks [e.g., Handin et al., 1963; Heard, 1960].

Results for a reaction producing 0.1 wt % fluid in a simulated rock with zero initial connectivity are pre- sented in Figure 6. As observed in the simulations pre- sented in Figure 5, strength drops abruptly at the onset

of dehydration and continues to decrease until a low- strength plateau is reached after approximately 10 % reaction. The strength profile in Figure 6 differs from the profile in Figure 5 in that the low-strength plateau is terminated at approximately 70 % extent reaction as the strength drops to zero. The images presented in Fig- ure 6 show that the drop to zero strength coincides with attainment of the hydraulic percolation threshold. Be- fore the percolation threshold is reached, the crack net-

SIMPSON: DEHYDRATION-INDUCED DEFORMATION 10,475

z m lOO

z 80

o

o 60

z 40

m 20

o o z

I I ' I '

i I

0 20 40 60 80

80 ¸

40 •

100

REACTION PROGRESS (%)

Figure 6. Calculated strength (2/xDx/•/KIc) (solid line) and size of the largest connected crack network (dashed line) plotted versus reaction extent for a dehydration reaction releasing 0.1 wt % fluid (E/Ki = 25, d/s = 0.5, 2ai/ac = 0.5, and wtpB = 0.1 wt % H20). Inset images show the evolution of the largest crack cluster (white area) during progressive dehydration. When the size of the largest cluster reaches approximately 90 % of the total sample volume, fluid can be redistributed pervasively through the entire sample, and the strength drops to zero.

work is made up of finite-sized clusters, so that fluid re- distribution only occurs locally around fluid-producing sites. The size of the largest crack cluster increases as the threshold is approached with increasing reaction, until approximately 90 % volume of the sample is hy- draulically connected, at which point the strength drops to zero.

The presence and exact position of the percolation threshold in reaction extent space are determined by the rate with which the crack density increases during dehy- dration and the magnitude of the critical crack length at which fluid redistribution can take place. Rapid crack growth is favored either by large total amounts of fluid released during dehydration (WtpB) or by large ratios of the Young's modulus (E) to the bulk modulus of the fluid (Kf) (Figure 7), both of which affect rheology

tions in material parameters. Results of numerical ex- periments within which the normalized initial fracture density (2a•/dz a) and the normalized size of the tensile region (d/s) were each varied by 2 orders of magnitude are presented in Figure 8. The greatest sensitivity oc- curs at high initial crack densities where the calculated strength drop is reduced to approximately 30 % of the original strength. However, all simulations investigated display large and abrupt reduction in strength coincid- ing with the onset of dehydration.

4.2. Influence of a Drained Boundary

The no-flow boundaries investigated in the previous examples are unrealistic if the fluid produced within dehydrating rocks can drain away. To determine the

through the fluid pressure term (equation (6)). The ,rheological effect of fluid drainage during dehydration, percolation threshold occurs at smaller extents of reac- a drained boundary was introduced by removing the tion for conditions favoring rapid crack growth and for fluid mass from any cell connected in a continuous net- small critical crack lengths. work to the drained boundary. This implies that cracks

The magnitude of strength reduction occurring at the longer than the critical length remain open and conduc- onset of dehydration is robust with respect to varia- rive with respect to fluid transport.

10,476 SIMPSON: DEHYDRATION-INDUCED DEFORMATION

1.0

0.8

0.6

0.4

0.2

'[ wt•B = •.05 •o [/• ' ,

wtPB=0.1% [ Kf dz 3

0 20 40 60 80 100

REACTION EXTENT (%)

Figure 7. Calculated crack density as a function of reaction extent for different total amounts of weight percent fluid released by the reaction (wtpB) and dif- ferent ratios of Young's modulus (E) to the bulk mod- ulus of the fluid (Ki) (using d/s = 0.5). The perco- lation threshold occurs when the crack density reaches the critical density (2a3c/dz3).

Model results for a drained dehydration reaction re- leasing 0.05 wt % fluid are presented in Figure 9. In gen- eral, the strengths display abrupt decreases at the onset of dehydration. Weakening persists until a connected crack network has been established with the drained

boundary. Development of connectivity to the drained boundary and a consequent reduction in fluid pressure result in partial strength recovery. For any one reac- tion the magnitude of weakening during dehydration is dependent on the ratio of the initial crack length to critical length (Figure 9b). However, only if the critical length is less than or equal to the initial crack length (i.e., no cracking) does the rock show no response to dehydration. Strength reductions on the order of half the initial strength are predicted, even for small critical lengths.

5. Discussion

5.1. Evolution of Strength during Dehydration

Numerous experimental studies have demonstrated that the dehydration of near-monomineralic hydrous aggregates such as serpentine, gypsum, and chlorite (which release between 10 and 13 wt % H20) induces a large and abrupt loss of strength. Murrell and Is- mail [1976] showed that a similar rheological response was induced by the dehydration of a granitic rock con- taining a small quantity of water (approximately 4 wt % or less). In fact, even prior to dehydration, sam- ples containing only 1.4 wt % free water were weaker by a factor of 70-80 % relative to samples containing no free water. This latter result suggests that the re- lease of large quantities of fluid during dehydration is not required to induce a large rheological response. One of the main aims of the investigation presented in this

[-

z

z 80

z

20

¸ o z

(a)

2ai 3

:0.64 . .............................................. ,

• • =0.064 2a3 '

0 20 40 • 80 1•

REACTION PROGRESS (%)

[-

z

z 80

60

z 4o

20

¸ o z

(b) !

• • s -- =005

, , r - 'q"•;'•"--•'z'z;u'z'V:':(•'- ..':"•":"•

0 20 40 60 80

•ACTION PROGRESS (%)

Figure 8. Effects of material parameter variation on the strength versus reaction extent profile for a reaction releasing 0.05 wt % fluid (E/Ky - 25, and 2ai/ac - 0.5). (a) Variation in the initial crack density, 2a•/dz 3 (d/s - 0.5). (b) Variation in the spatial extent of the tensile region normalized to the crack spacing, d/s (2a•/dz 3 - 0.064)

SIMPSON: DEHYDRATION-INDUCED DEFORMATION 10,477

(a)

lOO

80

60

4o

20

drained

boundary

' "' '•nectivity

yield

100 >•

<

z

80 ¸

Z

60 .•

40 >.

20 •2 Z Z

o

20 40 60 80 100

REACTION PROGRESS (%)

(b)

z lOO

< z 80

60

z

,._1 20 <

z o

ß I ' I ' I ' I '

2ai/2a c = 1

i

I

I

I I

20 40 60 80

REACTION PROGRESS (%)

2ai/2a c = 0.8

/ 7• •'• --"•' •- .-'"• _•._.--_ _•.. 2ai/2a c = 0.67 2ai/2ac = 0.57

I ß

t ! 2ai/2a c =0.5

i I I I I I , -

lOO

Figure 9. Results for dehydration with a simulated drained boundary. (a) Evolution of strength (solid line) and the degree of connectivity to the drained boundary (dashed line) as a function of the reaction progress for a reaction releasing 0.05 wt % fluid (2ai/ac = 0.67, and E/Kf = 25). The calculated images depict cracks hydraulically connected (white area) and unconnected (black area) to the drained boundary (top surface). (b) The magnitude of weakening and strength recovery dur- ing drained dehydration depends on the ratio of the initial crack length, 2ai, to the critical length, 2ac (the crack length at which fluid can be redistributed between neighboring cells). All peak-strength profiles were cal- culated for the same reaction and elastic parameters.

paper was to further investigate the rheological effect of dehydration reactions releasing small quantities of fluid. Numerical simulations indicate that dehydration reactions releasing less than 0.25 wt % fluid induce a profound rheological effect. Dehydration simulations of a rock matrix having zero initial hydraulic connectivity and undrained boundary conditions display an abrupt loss of strength (approximately 80-90 %) coinciding with the onset of dehydration. Strength continues to decrease until a low-strength plateau is reached after which there is no further strength reduction with in- creasing reaction progress until the percolation thresh- old is attained. With the exception of the observed strength drop coinciding with the percolation threshold, which has not been observed experimentally, the model generates results in good agreement with experiments [Heard and Rubey, 1966; Murrell, 1985; Murrell and Is- mail, 1976]. Simulations of dehydrating rocks with a drained boundary display abrupt weakening at the on- set of dehydration followed by strength recovery once connectivity with the drained boundary is established. Again, these features are in good agreement with ex- perimental observations [Ko et al., 1997].

The results presented here indicate that whenever fluid flow is obstructed (by poor hydraulic connectiv- ity and/or undrained conditions) such that excess fluid pressures are generated, dehydration reactions in elas- tic rocks will induce embrittlement. The generation of excess fluid pressures creates a mechanical instability which becomes increasingly unstable with increasing differential stress. Thus, unless the differential stress is removed, microcrack growth becomes unstable and leads to macroscopic fragmentation. The results do not necessarily imply that if dehydration reactions in- duce embrittlement, the time-averaged rock strength must become low. Indeed, one can envisage a sce- nario whereby dehydration reactions induce embrittle- ment which immediately creates permeability, enabling fluid to drain away and thus for high effective stresses to become reestablished. This behavior has been ob-

served in dehydration experiments with drained bound- aries [e.g., Ko et al., 1997]. The time-averaged strength of elastic rocks during dehydration depends on the abil- ity of fluid to flow out of the dehydrating system once (and if) embrittlement has been induced.

5.2. Relationship Between Strength Evolution and the Development of Connectivity

One of the advantages of the approach taken here is that the evolution of the pore fluid pressure and the state of hydraulic connectivity can be quantified as the reaction progresses. This evolution demonstrates that the major rheological response induced by dehydration does not coincide with the hydraulic percolation thresh-

10,478 SIMPSON: DEHYDRATION-INDUCED DEFORMATION

old. In fact, whereas the major rheological response oc- curs abruptly at the onset of dehydration, almost no further weakening occurs during the period when the state of hydraulic connectivity is evolving toward the percolation threshold. This result differs from the view held by Nishiyama [1989], who proposed that the gener- ation of reaction-induced hydrofractures coincides with the percolation threshold. The investigation carried out here indicates that the scenario envisaged by Nishiyama could only be realised if rocks are subjected to exter- nally applied differential stresses low enough to ensure that they would not experience embrittlement at the onset of reaction. Given that most deforming rocks, at least in the upper crust, are probably stressed close to their yield stress [Brace and Kohlstedt, 1980; McGarr and Gay, 1978], dehydration reactions in low perme- ability elastic undrained rocks are likely to induce fail- ure at the onset of dehydration. The implication of this behavior is that for naturally deforming rocks, the percolation threshold is unlikely to be achieved by the gradual evolution of the microstructure during progres- sive dehydration in initially low-permeability rocks. On the contrary, connectivity is most likely to be created by embrittlement occurring at or close to the onset of de- hydration. In this case, the subsequent response of the rock will depend on the stress drop during failure and on the ability of fluids to drain from the dehydrating system.

5.3. Influence of Initial Connectivity on the Evolution of Strength

Given that experimental studies have demonstrated that low-permeability rocks display a diminished re- sponse of fluid pressure on rock strength [Handin et al., 1963; Heard, 1960], this study investigated whether rocks characterized by an initially low degree of hy- draulic connectivity can maintain a significant propor- tion of their strength when they undergo dehydration reactions. The ratio of the initial crack lengths to the critical crack length for local fluid redistribution was used to vary the state of initial hydraulic connectivity. This parameter is a measure of the ease with which fluid can be redistributed throughout the rock by flow without microcracking (i.e., 2ai • 2ac) versus fracture- propagated fluid flow (i.e., 2ai • 2ac). The model re- suits for undrained boundary conditions indicate that although samples with no initial hydraulic connectivity retain strength relative to samples with pervasive ini- tial connectivity, the magnitude of this residual strength is small (< 20 % of the original strength). The resid- ual strength is maintained until the largest hydrauli- cally connected crack cluster reaches the size of the en- tire dehydrating system (i.e., the hydraulic percolation threshold), at which point the sample strength drops to zero.

In rocks which have no initial hydraulic connectivity, the dominant rheological response induced by dehydra- tion coincides with the generation of fluid overpressures

and not with the creation of hydraulic connectivity (Fig- ure 6). On the basis of this result, the incorporation of time-dependent subcritical crack growth into the model is anticipated to have little influence on the major rhe- ological response induced by dehydration. The addi- tion of subcritical crack growth to the model would make it unlikely that alehydrating rocks, which contain a fluid phase pressured to, or exceeding, the rock pres- sure, could maintain poor hydraulic connectivity. In such a case, poorly connected rocks are anticipated to naturally evolve toward states of pervasive connectivity.

5.4. Behavior of Naturally Deforming Rocks Undergoing Dehydration

That dehydration reactions typical of common crustal rocks such as pelites and granites are capable of in- ducing a profound brittle rheological response is consis- tent with a number of observations made in sequences metamorphosed to grades between the greenschist and amphibolite facies. Cesare [1994] and Simpson [1998] have demonstrated a direct spatial and temporal rela- tionship between dehydration reactions and brittle de- formation, represented by the formation mineral-filled fractures (veins). Numerous other studies have docu- mented the widespread occurrence of similar synmeta- morphic veins [e.g., Fisher and Byrne, 1990; Ram- berg, 1961; Sawyer and Robin, 1986; Yardley and Bot- trell, 1992], which have sometimes been attributed to reaction-generated deformation [Walther and Orville, 1982; Yardley, 1983]. The existence of extensional hy- drofractures has been interpreted as indicating litho- static fluid pressures and low differential stresses (• ap- proximately 40 MPa) during metamorphism lEtheridge, 1983]. The widespread synmetamorphic axial-planar fractures described by Simpson [1998] were interpreted as indicating fluid pressures exceeding the rock pressure and even lower differential stresses (• approximately 20 MPa) than other types of hydrofractures. Similar foliation-parallel veins have been described from a large number of regional metamorphic belts worldwide [e.g., Ague, 1994; Gratier, 1987; Kerrich, 1986; Lucas and $t-Onge, 1995; Sawyer and Robin, 1986; Yardley, 1983] and may be interpreted as indicating that low differen- tial stresses and high fluid pressures are typical during the period when rocks are undergoing dehydration re- actions.

5.5. Limitations

This study is concerned with quantifying the time- independent rheological response of rocks should ele- vated pore pressures develop. There is abundant evi- dence that elevated pore pressures exist, at least episod- ically, at upper crustal and mid crustal levels [e.g., Etheridge et al., 1984; Fyfe et al., 1978; Vrolijk, 1987]. No consideration is given here to how excess fluid pres- sures are maintained [see Gavrilenko and Gueguen, 1993; Hanshaw and Bredehoeft, 1968; Walder and Nut, 1984; Wong et al., 1997] or how fluids generated by reactions

SIMPSON: DEHYDRATION-INDUCED DEFORMATION 10,479

escape the dehydrating system [see Connolly, 1997; Gold and Sorer, 1985; Nut and Walder, 1992]. These studies have all recognized that the maintenance and eventual dissipation of excess pore fluid pressures is strongly dependent on the magnitude of the perme- ability, which was not explicitly treated in this paper. An intrinsic property of the discrete model investigated here is that the system remains impermeable until con- nectivity is achieved at the percolation threshold, af- ter which the permeability essentially increases to in- finity. That the permeability may be described by a percolation process at low porosities has recently been demonstrated by Knackstedt and Cox [1995] and Mavko and Nut [1997]. The infinite permeability above the percolation threshold is, however, unrealistic, and it is a feature that must be taken into account when con-

sidering drainage of fluids from a dehydrating system. In this respect, the model results will overestimate the magnitude of the theological response and the rate at which the strength is perturbed by dehydration. In- deed, Wong et al. [1997] have shown in a theoreti- cal analysis that the excess fluid pressure generated by dehydration is modulated over a timescale and length scale dependent upon the hydraulic diffusivity of the rock and the spatial dimension of the dehydrating sys- tem.

The accommodation of reaction-generated fluid over- pressures by microcracking represents an end-member case for the theological response of rocks to dehydration reactions. This scenario is most likely to be realised when initial porosities are low or when reaction rates are rapid relative to the rate at which fluid dissipates, and hence this scenario is favored during dehydration of low-permeability rocks undergoing rapid heating (e.g., during contact metamorphism) or if reactions have been overstepped. Even though fluid pressures equivalent to, or in excess of, the rock pressure are an inescapable con- sequence of dehydration under undrained, low-porosity conditions, embrittlement is not inevitable. Viscous creep may be important, particularly at relatively high temperatures and slow reaction-induced strain rates and if the rocks have significant tensile strengths and large initial porosities [see Cormoily et al., 1997]. Vis- cous creep would decrease the magnitude of the theo- logical response relative to the purely elastic case.

Extension of the two-dimensional model to three spa- tial dimensions is not anticipated to affect the abrupt theological response induced at the onset of dehydration because this response was demonstrated to be caused dominantly by the generation of pore pressures exceed- ing the rock pressure. The magnitude of strength reten- tion may be affected by dimensionality but the nature of this affect awaits quantification.

6. Conclusions

This study shows that rocks with zero initial con- nectivity, which release small quantities of fluid during dehydration (( 0.25 wt %), induce large and abrupt

strength reductions (approximately 80 %) at the onset of reaction. Strength reduction coincides with the gen- eration of fluid pressures equivalent to, or exceeding, the rock pressure and is unrelated to the attainment of the percolation threshold. Although rocks character- ized by a low degree of initial hydraulic connectivity can retain additional strength relative to rocks with initial hydraulic connectivity, the magnitude of this additional strength is small and probably geologically insignificant.

Acknowledgments. This research was initiated fol- lowing discussions with Jamie Connolly and Steve Miller and has since benefited from discussions with Alan Thomp- son, Jamie Cormoily, John Ridley, Yuri Podladchikov, Steve Miller, Dave Olgaard, and Ian Main. The contents of this manuscript have been greatly improved by reviews from Teng-fong Wong, an anonymous reviewer, and an Associate Editor and by uno•cial reviews performed by Alan Thomp- son, Jamie Cormoily, and John Ridley. Financial support was provided by ETH-Forschungsproject 0-20-885-94.

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(Received June 5, 1998; revised November 16, 1998; accepted January 21, 1999.)