SONJA L. BRENNER & A G US T GUDMUNDSSON N G U -BULL 439, 200 2 - PA G E 7 1

Permeability development during hydrofracture propa­gation in layered reservoirs

SONJAL. BRENNER & AGUST GUDMUNDSSON

Brenner, S.L. & Gudmundsson, A. 2002: Permeabi lity development du ring hydrofracture propagation in heteroge­

neous reservo irs.Norges geologiske undersokelse Bulletin 439, 71-77.

Hydrofractures are generated by internal fluid overpressure and together with shear fractures, contribute signifi­

cant ly to the perme ability of wate r,o il or mag ma reservoirs.We use bou ndary-element models to exp lo re th e effectsof abru pt changes in layer stiffness on the propagation, arrest and aperture variation of hydrofractu res.The results

of numerical mo dels and field obse rvations indicate that changes in Young's moduli can contribute to the arrest of

hydrofractures.When th e flu id overpressure is the only load ing, hydrofract uresare more likely to propagate t hrough

st iff layers (if they are not st ress barr iers) than throug h soft layers. If most hydrofractures beco me arrested becauseof mechanica l layering in a hete roge neous flui d reservo ir, the associated fract ure system wi ll be poorly intercon­

nected and thus of low permeability. Aperture variation is of great importance in bed rock hydrogeology, particularly

because channe ll ing of th e fluid flow along th e w idest part s of a fractu re may occur.Whe n the fluid overpressure ofthe hydrof racture is th e only loading, th e hydrofrac ture ape rt ure tend s to be greater in soft layers than in sti ff layers.

These results suggest that aperture variations may encourage preferential flow (flow channelling) in layered reser­

voirs.

Sonjo Brenner &Agust Gudmu ndsson. Geological Institute, University of Bergen, Allegoten 41, N-5007 Bergen, Nor woy

(Sonj o.Brenner@geol.uib.no; Agust.Gudmundsson@geol.uib.no).

IntroductionHydrofractures (fluid-driven fractures) are partly or wholly

generated by inte rnal flu id overp ressure (net pressure or dri­

ving pressure). They are commonly extension fractures.

Exam ples inc lude dykes, mi neral-fill ed veins and man-made

hydraulic fractures as well as many joints. Hydrofractures,

together with shear fractures, contribute significantly to th e

permeabi lity of heterogeneous fluid reservoir s, w hether th e

fluid is water, oi l or magma. Although heterogeneit ies occur

at various scales in reservoirs, for the propagation of out­

crop -scale hyd rofractu res,as are discussed here, perhaps th e

most important heterogeneity in reservo irs is mechanical

layering (Economides & Nolte 2000, Gudmundsson &

Brenner 2001).

The linking up of discontinuities during hydrofracture

propagation is likely to be on e of th e main mechan isms for

generating and maintaining perm eabil ity in heterogeneous

reservoirs . Another mechanism for generating permeability

in reservo irs, the format ion of shear fractu res (faults)

through th e linking up of small fractures, has been studied

extensively in recent years (e.g., Cox & Scholz 1988,

Cartwright et al. 1995, Acocella et al. 2000, Mansfie ld &

Cartwright 2001).

To explore th e cond iti ons for propagat ion of natural

hydrofractures is important for groundwater exploration, as

well as for the use of geotherma l energy and petroleum. In

petroleum engineering, th e aim is th at th e hydraulic fractu re

prop agates only along th e target layer (t he reservoi r) to

increase its permeabili ty.Thus, t he hydraulic fracture should

be confined to th e target layer and be arrested at t he con-

tacts wit h th e layers above and belo w. Natural, arrested

hydrofractures that are confined to single layers wit h non ­

fract ured layers in between, however, cont ribute signifi ­

cantly less to th e overall permeabilit y of a reservoir tha n do

fractures tha t propagate through many layers. This fo llows

because onl y interconnected fractu re systems reach the

percolat ion th reshold (Stauffer & Aharony 1994).

Once an open fracture has form ed, for example a joint, its

permeabi lity depends much on its aperture. In part icular,

because channel ling of th e flu id flo w along th e w idest part s

of a fracture may occur (Tsang & Neretnieks 1998), aper ture

variat ion is of great importance in bedrock hydrogeology.

The apertu re depends, amo ng ot her parameters, on the

mechanical pro perties of the host rocks. Thus, in a layered

reservo ir, th e mechanical differences between th e layers are

likely to affect the size of the aperture, even for a fracture

wi th constant fluid overp ressure.

In this paper, we fi rst summarise the basic concepts of

perm eability of fractured rocks and hydrofractu re propa ga­

tion.Secondly, we use bou ndary-eleme nt model s to explo re

the conditions for hydrofracture propagation and their aper­

ture variation, focusing on the effects of chang es in mechan­

ical properties of th e host rock. We th en compare these

results with field observations and discuss the implications

for permeabi lity generation in fluid-filled, layered reservoirs.

Permeability and fluid flow inreservoirsIn sediments, t here is a positive correlation between th e per­

meabilit y and the (pri mary) po rosity. In solid rocks this corre-

N GU - BULL 439 , 2002 - PAGE 72

lation does normally not hold.One reason for this is t hat dia­

genet ic processes such as compaction and cementat ion

reduce the effective porosity, i.e., t he interconnected pore

space that is available for fluid flo w. Another reason, w hich

applies also to igneous and metamorph ic rocks, is th at most

of the f luid flow in solid rocks occur s through fractu res that

fo rm a secondary porosity. In the case of an impermeable

host rock,all f luids would flow through interconnected open

fractu res.In contrast to fluid flow in porous media , fluid flo w

in fractured media is still not well understood (Singhal &

Gupt a 1999).

The volumetric flo w rate (Iaminar f low) through an iso­

lated fracture wi t h smooth, parallel, fractu re walls is propor­

t iona l to the cub e of the apert ure of the fracture (the cub ic

law). From th is it follows that small changes in a fractu re

aperture may lead to great changes in its fluid transport.

Also, if the fracture has rough walls, or t he fractu re aper ture

varies much along the t race of the fracture, channelling of

the fluid flo w along the wi dest parts of the fractu re may be

important (Tsang & Neretnieks 1998).

The cubic law for single fractures can be exte nded to

fractu re sets (Bear 1993). For example, th e volumetri c f low

rate through a set of parallel fractures in near-surface condi­

tions , and away from large fault zones, can be calculated

based on the fracture opening and the dista nce between

the fractures (Singhal & Gupta 1999).

The permeability of a fractured reservoir depends on the

connectivity of its fractu re systems . If a fluid can flow

through the w hole reservoir, using an interconnected sys­

tem of op en fractures, its perco lation threshold is reached

(Stauffer & Aharony 1994).The current stress field also con­

trols flu id flo w in fractured reservoirs (Faybishenko et al.

2000, Gudmundsson 2000). Fractures are sensiti ve to

change s in the stress fie ld and deform mu ch more easily

than circul ar pores. In addition, t he stress field contributes to

the flu id overpressure. Overpressured fluids probably

deve lop many interconnected fracture systems in reservoirs

through the propagat ion of hydrofractures; but arrested

hydrofractu res crossing only one or a few layers cannot con­

t ribute much to the overall permeability of a reservoir.

HydrofracturesThe growth of a hydrofracture depends prima rily on the

mechanical propert ies of th e host rock and th e fluid over­

pressure of the fracture. Fluid overpressure is defi ned as th e

total fluid pressure minus the stress normal to the fracture.

For extens ion fractures, modelled as mode I (opening mode)

cracks, as is appropriate for most hydrofractu res

(Gudmundsson et al.2001), this is th e fluid pressure in excess

of th e minim um pri ncipa l com pressive st ress (maximum

principal tens ile stress).

To solve problems in rock mechanics, at least two elast ic

modul i must be determined. The modu li most commonly

used are Young's modulus and Poisson's ratio (Hudson &

Harrison 1997). Poisson's rat io is a measure of the absolute

SO NJA L. BREN N ER & AGU ST G UDM UNDSSON

rat io of st rain in perpen dicular direct ions; 0.25 is com mon

for many solid rocks (Jaeger & Cook 1979, Jumikis 1979).

Youn g's modulus is a measure of the st iffness of a linear elas­

t ic material, which is approxima tely th e behaviour of many

rocks up to 1-3 % st rain at low temperature and pressure

(Pate rson 1978, Farmer 1983). For these, st ress varies linearly

wi t h strain (Hooke's law) and the rat io of st ress and stra in is

the rock'sYoung's modulus.The st iffnesses of common rock

types range from very soft, E< 1 GPa, for example mudstone,

to very stiff, E > 100 GPa, for some crystall ine rocks (Bell

2000). Because hydrofracture propagat ion is norm ally slow

compared wi th the velocity of seismic waves, stat ic Young's

modu li (rather than dynamic) are appropriate and used in

the models below. In th ese models we use the laboratory

values of the Young's moduli, which may be 1.5-5 t imes

greater than the in situ values of th e same rock types (Heuze

1980). The highest Young's modul i thus yield somewha t

high er stresses tha n would occur, for the same load ing con ­

dit ions, in nature.The impli cations of the numerical models

for th e general fracture geometries and stresseswou ld, how ­

ever, not be much different if lower Young's modul i were

used.

Hydrofractures are init iated when the flu id pressure

exceeds the min imu m pr incipal compressive stress by th e

tensile strength of the host rock. Typical in situ tensile

st rengths of rocks are in the order of 0.5-6 MPa (Haimson &

Rummel 1982, Schultz 1995, Amadei & Stepha nsson 1997).

The propagation is mad e possib le by the link ing up of dis­

continuities in the host rock ahead of the hydrofracture t ip.

Discontinu it ies are significant mechanical breaks in the rock,

norma lly wi th low or neglig ib le tensile streng ths.

Analytical models of hydrofractu res in ho mogeneous,

isot ropic rocks show th at t he theoret ical t ip ten sile st resses

are normally very high so th at a cont inu ou s and buo yant

hydrofractu re should cont inue its verti cal propagation to

the surface (Gudmundsson & Brenner 2001). For examp le, in

a mathematical crack subject to constant flui d overpressure,

the th eoret ical tensile st resses at its tips approach infi nity.

Similar ly, linearly varying overpressure d istr ibution in a

hydrofractu re gives infi nite crack-t ip tensile stresses. Infi nite

stresses, however, do not occu r in rocks because fractu re-t ip

cracking and plastic flow lowers the stresses. Nevertheless,

fo r hydrofractures in homogeneous, isotropic rocks, very

high crack-t ip tensile stresses are expected; and th ese

stresses by far exceed common tensile stren gth s of rocks

(Gudmundsson & Brenne r 2001).

Analytica l mod els also indicate that, in a homogeneous

isot ropic rock, a fracture subject to constant flui d overp res­

sure opens up into a hydrofracture wit h a smooth ly varying ,

ellipt ical aperture (Sneddon & Lowengrub 1969,

Gudmundsson 2000, Maugis 2000). A hydrofracture subjec t

to a linearly varyin g fluid overpressure has a similar shape. In

heterogeneous and anisot rop ic rocks, howeve r, a greater

aper ture variat ion is expected.

SONJA L. BREN NER s AGUS T GUDM UND SSO N N G U- BU L L 439, 2002 - PAG E 73

Surface

Fig. 2. The t ip of a hydrofract ure, located in t he mode rately sti ff layer C.approaches a very st iff layer B (100 GPa) near to the free surface.Thecontours show th e maxi mum pr incipal tensile st ress s3 in megapascal s(t runcated at 1 MPaand 10 MPain all the mod els).The tensile st resscon­cent ration around the hydrofractu re t ip is very high but occurs in arath er small area.High tensile st ressesare concent rated in t he st iff layer

B, and at the sharp fractur e tip .

Fig. 1. Basic bound ary-element configurat ion used for th e models inFigs. 2 and 3. Each model has unit d imensions, a uniform Poisson's rati oof 0.25, and is fastened in it s lower corners.The fluid overpressure in th ehydrofracture varies linearly from 10 MPa at the fractu re bottom (cen­tre) to 0 MPa at the fractu re t ip, as indi cated by hor izontal arrows.Themain layer, C. hosts the hyd rofracture, is 0.92 units thi ck, and is moder­ately st iff (40 GPa).The two, thin, top layers are both 0.04 units thick.Theupp ermost layer A has the same Young's modulus as layer C in all th emodels.The st iff ness of layer Bvaries bet ween model run s.

> 1.8 > >

Hydrofracture

A

c,B

A 0.04

B 0.04

i'-Tip0.02

C

<-<--f-l

1.0 I--> 0.9

<- 1--><- I-J

'<- ---.~ --')

<--- ----><--- ~ Hydrofracture~ -l

\

Numerical modelsNumerical models are often used to simulate physical

processes, for examp le when analyt ical solutions cannot befound or, if found, are too complex to be of pract ical use.

Analyt ical solutions are not appropriate for most realisti c

probl ems concerning layered reservo irs.This applies, in par­

t icular, to problems concerning th e arrest and aperture vari­

at ion of hydrofractur es, both of which are likely to depend,

in a complex way, on th e mechanical contrast between th e

different layers.

Most numerical programs in solid mechanics are basedon the linear elast icity theory.The mo st commonly used pro­

grams in rock mechanics and engi neering are based on thefini te element method (FEM) (Zienkiew icz 1977). where th e

pro blem do main must be divided into volumetri c elements

for which properti es are defined and solutions calculated.

There is, however, an increasing use of the bounda ry ele­

ment method (BEM) (Brebbia & Dom inguez 1989). whereonly the surfaces need to be discret ised into elemen ts fo rcalculat ion and zones with uniform propert ies are defined.

Because exact values are calculated at th e boundaries, and

not ext rapo lated from the inside of volumet ric elements likein th e FEM, th e BEM gives more accurate solut ions for

bou ndary problems (e.q., surface stresses). To obta in infor­

mation on areas inside of the zones of the defined probl em,

for examp le th e st ress concentrat ion around a fracture ti p,

solut ions for internal points are calculated and plotted.

We use th e softwa re BEASY (1991),which is based on th e

BEM, to explore the effects of abrupt change s in layer sti ff­ness on th e propagation, arrest and aper ture variation of

hydrofractur es. Abrupt changes are common in layered

reservoirs. For exampl e, layers wi th relat ively hig h Young'smoduli like lim estone, sandsto ne, basalt and gneiss may

alternate with softer layers such as marl, shale, tu ff and

amp hiboli te.l n th e models, extreme differences of mechani­

cal properties of layers were used to emphasi se th e effects

of interest.

The fir st mod els (Figs. 1-3) show how layers with greatly

diffe rent Young's moduli influence th e st ress field aroundthe t ip of a hydrofracture as well as th e shape of th ehydrofracture it self. The models are of unit length and

height and are fastened in th e lower corners to avoid rigid

body translation and rotation. Poisson's rat io is 0.25 in all th e

layers in all the models. In each model, th e hydrofracture is

subject to a fluid overpressure with a linear variation fro m 10

MPa at th e bottom (the fracture centre) to 0 MPa at th e tip; it

is th e only loading .The th ick layer hosting the hydrofractureC and the surface layer A have th e same st iff ness,40 GPa, in

all mod els.This is a typical value for common rocks (Jumik is1979, Bell 2000). The sti ff ness of layer B changes betweenmodel runs.

In th e f irst model (Fig. 2).a hydrofracture is appr oaching

a th in layer B of a hig h st iff ness, 100 GPa.The tensile st ress

near the tip is much greater than th e tensile st rength of

commo n rocks and would thus lead to a further propaga-

N GU-BULL 4 39 , 2002 - PAG E 74 SONJA L. BRENN ER & AGUST GUD M UN D SSO N

AB

c

.8sile stressesare generated,but they are still lower than those

generated in the stiff layer Bof the fir st model (Fig.2).

The next model focuses on the influence of layers with

different st iffnesses on the aperture of a hydrofracture.The

mod el (Figs.4-5) isof unit height and wi th a Poisson's rat io of0.25. Here we used ten layers of equal thic kness (each 0.09

units ).The lowermost layer J is very stiff (100 GPa), the next

layer above I is very soft (1 GPa), whereas the th ird layer H is

moderately st iff (10 GPa). This three-layer sequence is

HydrofractureSurface

x

,:H·:: ~... . ~ : ,- ~::~ :•: ~: ~~ Ho~•

I 10

.09

•

Fig.4 . Basic boundary-element config uration used for the model in Fig.5.The flui d overpressure in the hydrofracture, indicated by horizontalarrows, is the only loadi ng and varies linearly from 10 MPa at the frac­tu re centre (located at the contact between layers Eand F) to 0 MPaatthe upper and lower fracture tips.The model has unit height (vert icaldim ension); all the 10 layers, A-J, have the same thickness (0.09 units)and a uniform Poisson's rat io of 0.25.The thin layers at the top and bot ­tom of the model are used to fasten it (as indicated by crosses) and toconfine the hydrofractu re. A succession of 3 layers - st iff (100 GPa), soft(1 GPa), and moderate (lO GPa) - is repeated to the top, so that layer Ahasa Young'smodu lus of 100GPa.

~

1' 0.09

•

x x x xX

x x x x< x x

xx

xx x

x x x x x x 0 .09x x x ;x x x

xx x x x

x

x x x x x x xx x x

x1.0 .05r

\--+~I~-+I Hyd ro-Io.09IXX~~~~~x~ ~~-x:-;

1+-+lI---<~'/l x x x x x x 0 .09 ;.I x x x xJ; x x: x x x x x x

.. x .. x x x

c

- - -

Fx x

.cx x

xx

xX -c

XX

XX

Xx

<>G xx x x x

x x x x x x

x x x xx

xX A X X X

xx

tion of the hydrofracture tip.The tip is th in and sharp and

the fracture aperture increases downwards. The largest

aperture, however, does not occur in the centre of the

hydrofracture (bott om of the mode l) because the model is

fastened at the bottom and the simulated host rock thuscannot deform freely. In the stiff layer B above, high tensile

st ress is concent rated, so that the area in which 10 MPa is

exceeded becomes very large.In the softer (40 GPa) topmostlayer A the tensile stress is much lower. Under these loading

condit ions, th e hydrofractu re wo uld normally be able to

propagate through the stiff (100 GPa) layer.

In th e next model , a hydrofracture approaches a soft

layer with a Young's modulus of 5 GPa (Fig. 3).The surface

layer A is again moderately stiff with a Young's modulus of

40 GPa. The results here are very different fro m th ose in the

above model (Fig. 2). Here the lower layer B takes up very lit­

t le tensile st ress and th e maximum value, around 6 MPa, isreached inside a very small area.The tensile stress concen­

trat ion around the hydrofracture tip wi thin the moderately

st iff layer C itself is greater than in the first model (Fig. 2); in

parti cular, the area in which 10 MPa is exceeded is here

much larger.The hydrofractur e is able to propagate so as to

reach the conta ct with the soft layer B. But w ithout a

favourably orientated discontinuity in the soft layer tha t

could open up and be used as a pathway for the hydrofrac­ture, it is unlikely that its propagat ion wou ld cont inue

through th e soft layer.The t ip of th e hydro fracture is more

rounded than in Figure 2. ln the stiff surface layer A,high ten-

Fig. 3. Same model as in Fig. 2, except that layer B is now soft (5 GPa).Litt le tensile stress is transferred into thi s soft layer.The area with hightensile stress concentrat ion around the hydrofracture tip inside themoderately st iff layer C is larger than in the model with the stiff layer B(Fig. 2).The tensile stress concentr ation in the top most layer A, wit h amoderate sti ffness (40 GPa), is lower, and th e fracture tip is there morerounded, than in th e previous model (Fig.2).

SONJA L. BRENN ER & A GUS T GU D MUNDSSON NGU-BULL 439 , 2002 - PAG E 75

xx

DiscussionNumerical models are valuable tools to explore complex

phy sical prob lems that are not tractabl e with analytical

methods.The inpu t parameters,such as the modelled geom­

etry and the mechanical properties, must , however, be cho ­

sen very carefu lly so as to represent the real geological situ­

ation. The test implicat ions of the numerical mo dels must

also be checked by field observat ions; for exam ple, as

regards nat ural fracture systems .

In heterogeneous and anisot ropic rocks, many, and per­

haps most, hydro fractures become arrested, at various

dep ths, at discon t inui ties or contacts between rocks with

different mechan ical prop ert ies. Hydraulic fracture experi­

men ts in petrol eum engineering indicate tha t th e vertical

pro pagation of a hydraulic fracture is commonly arrested at

contacts between layers, parti cularly when the layers have

strong mechani cal and stress contrasts (Charlez 1997, Yew

1997, Economi des & Nolte 2000).

Similarly, field observations show that in layered rocks

many hydrofractures become arrested at conta cts between

mechanically different rock layers. Figure 6 shows an

arrested joint in Precambrian gne isswith amphibolite layers,

exposed in th e city of Bergen, West Norway.The joint, which

Fig. 6. Arrested open join t in a metamorphic rock, at a road-cut inBergen, West Norway. Fractures are commonly arrested at contactsbetween layers of different mechanical properties, such as here at thecontact between amp hibolite and gneiss. View north-northeast, thehand provides a scale.

st iff (A, D, G) layers.But the apertu re in the soft layers (C, I) is

clearly larger than in the adjacent layers.

x x x x xx x x x x x

x x x x x x xx x x x x x x

x xx x x x

xx

x xX

XX

XX

X X X X

XX

XX

XX

X

XX

XX

XX

XX X x x

x xx

x x x

.:- -

x x x xx

XX X x

<A x x x x x xx x x x x

xx

xx

x x xx X X X

--. ------- -- -- - -

~~:-~---=Hydro-

fracture -x x X

X X XX X X

X .xXx xXxxX

G' x x x >x X X xX x

x x. x

F

c

-

B=::=::=::=::=::=::=::=::=::=::=:~=

Fig.5. The aperture is greatest where a high flu id overpressure occurs inthe soft layer F.Generally, the aperture decreasesfrom th e fracture cen­tre to its tips,and so does the app lied fluid overpressure.However, in thesoft layer C the apert ure is much larger than in the stiffer adjacent lay­ers,and the decrease in aperture in the soft layer I is much more abruptthan in the adjacent layers. This aperture variation would normallyencourage channelling of subsequent horizontal fluid flow .

repeated up to the surface, so th at the topmost layer A has a

Young 's modulus of 100 GPa.The hydrofracture is confined

and th us cannot propagate into the to p and bottom layers,

in whi ch th e model is fastened. The flu id overpressure is

applied along the enti re height or dip dimension of the

hydrofracture and varies linearly from 10 MPa at th e centre

to 0 MPa at the fractu re tips (Fig.4).

The greatest aper t ure is reached w here th e soft layer F

coincides wi t h a high fluid overpressure near to the centre

of th e hydrofractu re at the contact between layers E and F

(Fig. 5). The aperture decreases to the fract ure t ips as the

applied flu id overpressure decreases.There is a small differ­

ence in aper ture between th e mod eratel y stiff (B, E, H) and

NGU -BU LL 439 , 20 02 - PAGE 76

may have origi nated as a hyd rofract ure, exte nds from gneiss

and ends abruptly at its contact w it h the amphibol ite .There

is norm ally a large difference in the labo rato ry stiffnesses of

gneiss and amph ibolite (Hansen et al. 1998, Myrvang 200 1).

In Norway, the st iff ness of amphibolite can vary between 30

GPa and 130 GPa, wi th the most common valu es perhaps

between 40 GPa and 110 GPa (Hansen et al. 1998). By con ­

trast, the stiffness of gneiss can vary betwe en 10 GPa and

150 GPa, w hile the most common values are pe rhaps

between 20 GPa and 80 GPa (Hansen et al. 1998). Thu s,

am phi bo lit e can be either stiffer or softer th an gneiss. In

cases w here the amphibolite was stiffer than th e gn eiss, th e

arrest of th e frac t ure at th eir con tact could be attributed to

the generally high ho rizontal compressive stre sses th at are

current ly ope rative in West Norway (Hicks et al. 2000 ,

Myrvang 2001 ),w hich would concentrate in the stiff layer. By

contrast, if t he am phi bol ite wa s softer t han th e gneiss, t he

arrest of the fracture could be attributed to its being gen er­

ated as a hydrofracture, w it h fluid ove rpressure as the only

load ing, similar to that in th e model in Fig. 3. Wh ich arrest

mechani sm ope rates depend s not only on the stiffnesses of

the rock layers, but also on th e (unknow n) t ime of fractu re

arrest because d iffere nt st ress fields opera te at different

times.

In igneous roc ks, dykes are commonly arrested at con­

tacts between lava flo ws or between lava flows and layers of

pyroclastic rocks .This is seen, for exam ple, for many dyke ti ps

in Tener ife (Canary Islands) and Iceland (Gud m undsson et al.

1999, Marinoni & Gudmundsson 2000 ). Many arrested dyke

tips have also been obs erved at bedding contacts in sedi­

ment ary rocks (Baer 1991). In sedimentary rocks, there are

also many joints and minera l veins that are strata bound

(rest ricted to one layer) (Odling et al. 1999, Gillespie et al.

2001). For example, calcite veins are abundant ins ide lime­

stone layers but become arrested at t heir con tac ts wi th

softer marllayers (Gudm undsson & Brenner 2001 ).

These field observati ons support th e results of the

numerical models (Figs. 1-3), that soft layers are effective

barriers for propagating hydrofractures. In t hese mode ls, a

hydrofracture would norma lly propagate through a stiff

layer, if it is not a st ress barr ier. A stress barrier is a layer

wh ere t he hydrofracture-norma l compressive stress is

high er th an in adjacent layers.St ress barr iers are par t icularly

common in mechanically layered rocks subject to compres­

sive stresses; th e st iff layers usually take up most of th e com­

pressive stress, become highly stressed, and arrest

hydrofractures (Gud m undsson 1990, Gudmun dsson &

Brenner 2001 ). Such barriers commonly co incide w it h

abrupt changes in Young's moduli, horizontal discontinuities

or both (Gudm undsson 1990, Gudm und sson & Brenner

2001), all of w hich can con t rib ut e to the arrest of hydrofrac­

tures (Gudm undsson & Brenner 2001 ).

At th is stage, com pari son of th e resul ts of th e nu me rical

mo dels concerning the aperture variation of hydrofractures

(Figs. 4-5) wi t h f ield obse rvati ons is, however, more d iffic ult.

SO NJA L. BRENN ER & A GUST G UD M U N DS SO N

One reason is th at the difference in aperture, particularly fo r

small-s cale hydrofractures such as mineral veins , is com­

monly to o smal l to be noticed.Very soft layers are commonly

to some extent ductile and sustain little or no tensile

stresses. Hydrofractures, such as veins , dykes or joints that

enter such layers are th us likely to tr igger failure in shear

rather th an in exte nsion. Thus, a hydrofracture that propa­

gates throug h such a soft layer is likely to follow an incli ned

shear fractu re rather th an go ver tically through the layer as

an extension fracture. Many fractures also follow exist ing

we aknesses in th e hos t rock (like cooling joints or fo liation),

in which case parts of th eir pathways may be incli ned. An

incl ined hydrofract ure is normally not perpendicular to the

ho rizon tal minimum principal com pressive stress, but rather

to th e (higher) norma l st ress, and thus becomes thinner.

Local changes in the fluid overpressure because of stress

changes normal to th e fract ure, fo r example due to stress

barr iers, may also lead to aperture changes.

The apert ure (t hickness) variat ion of dykes propagating

th rough mecha nically layered rocks in Iceland was observed

by Gudmundsson (1984). His resul ts indicated tha t dyke

ape rture inside soft pyroclast ic rock layers tends to be

great er than w here th e dyke cuts through the centre of a

sti ff basa lt ic lava flow. In layer ed rock masses subject to

exte nsion, where stiff layers take up mos t of the tens ile

stress, th e aperture in the st iff layers may be greater than in

the soft layers (Gudm undsson & Brenner 2001 ). We plan to

carr y out more field studies to invest igate the effects of

chang ing elasti c properties of the hos t rock in layered reser­

voirs that are explored in the numerica l models in th is paper.

AcknowledgementsWe th ank Helge Ruistuen and Arn e My rvang for helpful comment s.This

work was suppo rted by grant s from the Norwegian Research Council,th e Europ ean Commission (cont ract EVR1-CT-1999-40002) and Statoil,

ReferencesAcocella,v.,Gudmundsson, A.& Funi ciello, R. 2000: Inte ract ion and link­

age of exten sion fractures and nor mal fault s: exam ples from the rift

zone of Iceland .Journal of Structural Geology 22,1 233-12 46.

Amad ei, B. & Stephansson, O. 1997: Rock Stress and its Measurement .Chapma n & Hall, Lond on, 490 pp.

Baer, G. 1991: Mechanism s of dike propagation in layere d rocks and inmassive, porou s sedime ntary rocks.Journal ofGeophysical Reseorch

96,11 911-11929.

Bear, J. 1993: Modelli ng flow and contam inant t ranspor t in fract ured

rocks. In: Bear, J., Tsang, C F. & DeMarsily, G. (eds): Flow andContominont Transport in Fractured Rock.Academic Press, New York,

1-37.BEASY 1991: The Boundary Element Analys is System User Guide.

Computat ional Mech anics, Boston.Bell, F.G. 2000: Engineering Propert ies ofSoils and Rocks, 4" ed. Blackwell,

Oxford ,482 pp .Brebbia , CA . & Dominguez, J. 1989: Boundary Elements: an Intr oductory

Course, 2"" ed.Computational Mechan ics, Boston, 313 pp.Cartwright,J.A.,Trudgill, B.o.& Mansfie ld, CS. 1995: Fault growth by seg­

ment linkage - an explanation for scatter in maxim um disp lace­

ment and tr ace length dat a from the Canyon land s Grabens of SE

Utah. Journa l of Structural Geology 17,1319-1326.

SONJA L. BRENNER & AGUST G U D M U N D SSO N

Charlez, P.A. 1997: Rock Mechanics, Volume 2: Petroleum Appl ications.Editi on sTechnip, Paris, 66 1 pp.

Cox, 5.J.D.& 5cho lz, C H. 1988:On th e formation and growt h of fau lts : an

experime nta l st udy.Journa l ofStructura l Geology 10,413-430.Economides, M.J. & No lte , K.G. (eds), 2000: Reservoir Stimulation, 3'd ed.

Wil ey, New York.

Farmer, I. 1983 :Engineering Behaviour ofRocks, 2"" ed. Chapma n and Hall,

London, 208 pp.

Faybishenko, B.,Wi therspoon, P.A. & Benson, 5.M. (eds) 2000 :Dyna mics of

Fluids in Fractu red Rocks. American Geophys ical Uni on, Wash ington

0 (, 400 pp.

Gillespie, P.A., Walsh, J.J., Wat te rson, J., Bonson, CG. & Manzocchi, T. 2001 :

Scaling relat ionship s of joint and vein array s fro m The Burren, Co.

Clare, Ireland.Journal ofStructural Geology 23,183-20 1.

Gudmundsson , A. 1984: A study of dykes, fissures and faults in selectedareas of Iceland. PhD The sis, Universit y of London, 268 pp.

Gudmundsson , A., 1990 : Emplacement of dikes, sills and crus tal m agma

chambers at d ivergent p late boundaries. Teetonophysics 176, 257 ­

275.

Gudmund sson , A., 2000: Fracture di mensions, d isp lacem ents and f luid

tr ansport.Journal ofStructural Geology 23, 1221-1231.

Gudmundsson, A. & Bren ne r, 5.L. 200 1: How hydrofr actures become

arre sted. TerraNova 13,456-462.

Gudm undsson, A., Ma rinon i, L.B., Marti, J. 1999 : Inject ion and arre st of

dykes: im plicat ions for vo lcan ic haza rds.Journal of Volcanology and

Geothermal Research 88, 1-13.

Gudmund sson , A., Berg, 5.5., Lyslo, K.B.& Skur tve it, E. 200 1: Fracture net ­

works an d f luid t ransport in active faul t zones.Journ al ofStructuralGeology 23,343-353 .

Haim son, B.C & Rummel, F. 1982 : Hyd rofracturi ng st ress measuremen t s

in th e Icela nd resea rch d rilling p roject dri ll hol e at Reydarfjordu r,

Icelan d.Journ al ofGeophysical Research 87,663 1-6649.

Han sen, 5.E., Serle kk, T. & Jo hann sson.A. 1998: Mechanical propert ies ofrocks (in Norwegian).5intef,Tro ndheim.

Heuze, F.E. 1980:Scale effects in the determinat ion of rock mass st reng t h

and deformability.RockMechanics 12, 167-192.

Hick s, E.C , Bungum , H. & Lindho lm, C 2000: St ress inversion of earth­

q uake focal m ech an ism so lut ions fo r onsho re and offshore Nor w ay.

Norsk Geologisk Tidskrift 80:235-250.Hudson , JA & Harri son , J.P. 1997: Engineering Rock Mechanics: an

Int roduction to the Principles.Perg am on , Oxford, 444 pp.

Jaeger, J.C & Cook , N.GW . 1979: Fundamentals of Rock Mechanics, 3'ded.

Chapman & Hall , London, 593 pp.

Jumiki s, A.R. 1979: Rock Mechanics. Tran s Tech Pub licat ion s, Clau sth al,

356 pp.

Man sfield,C & Cartwright,J. 2001: Fault g rowth by lin kage: obse rvat io ns

and implications fr om analogue models. Journa l of StructuralGeology 23,745-763.

Mari noni, L.B.& Gudmundsson, A. 2000: Dyke s, faults and palaeostresses

in th e Ten o and An aga massifs ofTene rife Canary Islands.Journ al ofVolcano logy and Geoth ermal Research 103,83-103.

Maugi s, D., 2000 :Contact,Adhesion and RuptureofElastic Solids.5p ringe r­

Verlag, Berl in, 4 14 pp.

Myr vang , A. 200 1:Rock Mechanics (in Norwegian). NTNU,Trondheim .

Od ling, N.E.,Gil lespie , P., Bou rg ine, B., Castaing ,C , Chi les,J.P., Chr ist en sen,

N.P., Fi ll ion, E., Genter, A., Olsen, C , Th rane, L., Tric e, R., Aarseth, E.,

Walsh,J .J.& Wat te rson,J. 1999:Va riations in fracture system geome­

try and th ei r implicati on s for flu id f low in fr actured hydr ocarbon

reservo irs.Petro leum Geoscience 5, 373-384.

Paterson , M.5. 1978: Experimental Rock Deformation: The Britt le Field.Springer, Berlin, 254 pp .

5chultz, RA 1995: Lim it s on st rengt h and de fo rm at io n properties of

jointed ba salt ic rock ma sses. Rock Mechani cs and Rock Engin eering28,1 -15.

5inghal, B.B.5. & Gupta , R.P. 1999: Applied Hydrogeology of FracturedRocks. Kluwer, London ,399 pp.

5neddon, I.N. & Low en g rub, M . 1969: Crack Problem s in the ClassicalTheory ofElasticity.Wil ey, New York, 22 1 pp.

NG U -BU LL 439 , 2 0 0 2 - PAG E 77

5ta uffe r, D. & Ah arony, A. 1994 : Int roduction to Percolat ion Theory, 2'ded.

Taylor &Fran cis, London , 181 pp.

Tsang , CF.& Neretn ieks, I. 1998: Flow cha nne ling in het erogeneous frac ­

tured rocks .Reviews ofGeophysics 36, 275 -298 .

Yew, CH. 1997: Mechanics of Hydraul ic Fracturing. Gulf Pub lishi ng,

Houston Texas, 183 pp.

Zie nkiewicz, O.C 1977: The Finite Element Method, 3'd ed . McGra w -Hill ,

London , 787 pp.