Thermal Fracturing of Geothermal Wells and the Effects of BoreholeOrientation

Kjetil M. D. Halsa, Inga Berreb

aChristian Michelsen Research, P.O. Box 6031, NO-5892 Bergen, Norway.bDepartment of Mathematics, University of Bergen, P.O. Box 7800, NO-5020 Bergen, Norway.

Abstract

An enhanced geothermal system (EGS) expands the potential of geothermal energy by enabling the exploita-tion of regions that lack conventional hydrothermal resources. The EGS subsurface system is created byengineering enhanced flow paths between injection and production wells. Hydraulic stimulation of existingfracture networks has been successfully achieved for unconventional geothermal resources. More recentlyproposed concepts increase the use of drilled wellbores in hard rock to connect the injection and productionwells. The present work investigates the long-term thermal effects of deviated geothermal wellbores and stud-ies how the cooling of the borehole wall results in thermally induced tensile fractures. The results show thatinduced fractures are created by a combination of in situ and thermal stresses, and that the extent to whichthermally induced tensile wall fractures are created largely depends on how the wellbores are oriented withrespect to the pre-existing stresses of the reservoir. If the system is not optimized with respect to in situstresses, the risk of wellbore instability becomes severe within less than a year of production. In contrast,if the orientation of the wellbores is optimized, thermally induced instabilities can be completely excludedas potential risks for the operational lifetime of the system. Furthermore, our results show that the thermalfailure process strongly depends on the temperature of the injected water but is only weakly affected by theinjection rate.

Keywords: Enhanced geothermal system, thermal fracturing, numerical modeling

1. Introduction

Enhanced geothermal systems (EGSs) haveemerged as a promising new type of geothermalpower technology. An EGS uses hydraulic fracturingto create a fracture network in low-permeability rocklayers to enhance the connectivity between typicallydeviated injection and production wells (Abe etal., 1999). Still in the development stage, EGStechnology has been criticized because of its po-tential to cause the initiation of larger seismicevents (Majer et al., 2007). Recently, two alternativeEGS concepts have been presented (Holmberg etal., 2012; Zhang et al., 2012) that increase the use

Email addresses: kjetil.hals@cmr.no (Kjetil M. D.Hals ), inga.berre@math.uib.no (Inga Berre )

of deviated wellbores to ensure connection betweenthe production and injection wells (Fig. 1). Oneapproach avoids the use of hydraulic stimulation,thereby reducing the risk of induced seismicity, byinterconnecting the production and injection wellsvia drilled wellbores (Holmberg et al., 2012). Theother approach uses drilled boreholes to improve theconnection between the wellbores and the existingor stimulated fracture networks (Zhang et al., 2012).The present study refers to these new concepts aswellbore-based EGSs (WEGSs). The operationallifetimes of both EGSs and WEGSs largely dependon long-term effects, such as mineral scaling, cor-rosion, and fracturing resulting from thermal stress,which could alter system permeability. The presentarticle focuses on how long-term thermal effects can

Preprint submitted to Geothermics December 13, 2012

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result in borehole instabilities due to fracturing.

2000 m

z

θ

5000 m

T= 160 °C

T= 4 °C

Tinj

zbz= 5000

z= 0

Figure 1: Illustration of a WEGS concept based on connectedwellbores. The injection and production wells are intercon-nected by drilled wellbores (heat exchangers) that are tilted byan angle of θ with respect to the z axis. The length of eachheat exchanger is 2000 m. zb represents the z-direction in theborehole coordinate system, while z is the vertical direction inthe geographic coordinate system. The maximum depth of thewells is 5000 m below the surface. The temperature at 5000m is assumed to be 160 C, while the water enters the heat ex-changers with a temperature of Tinj.

Stress is induced in a reservoir that is exposedto compression or tension. The sources of inter-nal stress can be forces that are created due to achange of the reservoirs temperature or, in the caseof a porous medium, a change in fluid pressure.Large pre-existing (in situ) stresses are also typi-cally present in the reservoir due to tectonic forcesand the overburden pressure (Zoback, 2007). Tensileor compressive rock failure is observed if the stressexceeds the rock strength (Zoback, 2007). Model-ing this fracturing is complicated, because the pro-cess depends on several coupled mechanisms that acton different length and time scales (Hansen, 2005;Hals and Berre, 2012). Fracturing has been stud-ied extensively over the last few decades with re-gard to the stabilities of deviated boreholes (Perkins

and Gonzalez, 1984; Peska and Zoback, 1995). Inthe petroleum industry, one of the major operationalconcerns related to drilled wellbores has been theprevention of borehole wall failure (Cooper, 1994).This type of failure is initiated by pressure and tem-perature gradients that are created around wellboresduring drilling (Perkins and Gonzalez, 1984). Sev-eral previously published studies have investigatedthe stability of inclined boreholes. In particular,pressure-induced failures have been the focus of oil-well studies. In addition to numerical and experi-mental studies (Daneshy, 1973; Yew and Li, 1988;Baumgartner et al., 1989; Rawlings et al., 1993), theproblem of borehole wall failure has been studied an-alytically via linear elastic models (Bradley, 1979;Aadnoy, 1987). Peska and Zoback (1995) presenteda thorough investigation of borehole stabilities basedon arbitrary borehole orientations in a wide variety ofin situ stress states. They employed analytic expres-sions that describe the principal effective stresses atthe wellbore wall of a cylindrical well while consid-ering fluid pressure variations (Hiramatsu and Oka,1968; Fairhurst, 1968). Similar analytic expressionshave been used in studies of thermally induced fail-ures (Myklestad, 1942; Perkins and Gonzalez, 1984;Zoback, 2007). The effects of thermal fracturinghave also been of concern in carbon capture andstorage (CCS) technology. Recent studies have in-dicated that thermally induced stresses significantlyinfluence the safety of the seal and the fracturingof the caprock (Luo and Bryant, 2010; Goodarziet al., 2010, 2012). In the context of geothermalreservoirs, several studies have focused on thermallyinduced stresses, mainly concentrating on heat ex-traction enhancements due to thermal cracks (Mur-phy, 1979; Demuth and Harlow, 1980). Thus far,few works have attempted thorough stability analy-ses of geothermal wells that investigate the interplaybetween in situ stresses, borehole orientations, andlong-term thermal effects.

The present paper combines numerical and ana-lytical methods in the first theoretical study of thelong-term thermal effects of deviated geothermalwellbores in regions where convective fluxes be-tween geothermal reservoirs and wellbores are neg-ligible. Temperature evolution in the geothermalwell and rock reservoir are simulated using a numer-

2

ical COMSOL Multiphysics model (Littmarck andSaeidi, 1986), while stresses at the wellbore wall arecalculated from analytic expressions valid for cylin-drical wells. We show that the degree of thermaltensile fracturing is extremely sensitive to how thedrilled wellbores are oriented with respect to in situstress states. In the case of a large horizontal stress,the wellbores should be oriented along the maximumhorizontal stress direction. This orientation nearlyeliminates the risks for thermally induced boreholeinstabilities, even after several years of injecting coldwater. In contrast, when the wellbores are orientedalong the minimum stress direction, wellbore insta-bilities are observed within less than one year. Inaddition, we show that the degree of thermal fractur-ing strongly depends on the temperature of the in-jected water but is only weakly affected by the injec-tion rate. The strong fluid-temperature dependencyof the thermal failure process allows one to controlthe degree of fracturing for a given in situ stress state.Our results provide important insights for optimizingWEGSs as well as oil and geothermal wells, in gen-eral. We expect that the results can be applied to CCStechnology to enhance the safety of geological stor-age sites.

This paper is organized as follows. In Section 2,we present the theory and governing equations thatmodel the geothermal system. Section 3 provides adescription of the model system under consideration,while Section 4 presents our findings. We concludeand summarize our results in Section 5.

2. Governing Equations

The following sections, present the equations thatgovern the temperature evolution of the reservoir andthe heat transfer process between the hot rock andcold water that circulates through the wellbores. Inaddition, the analytic expressions for the stresses thatact on the wall of deviated, cylindrical wellbores inthe presence of in situ stresses, fluid pressure, andthermal stresses are presented.

2.1. Temperature Equations

The temperature equations for the system consid-ered herein are derived from the conservation of en-ergy, which is described by two coupled processes:

the energy change of the fluid and the energy changeof the rock. These two processes are coupled bythe heat transfer that occurs at the borehole wall.This heat transfer is a complicated process that de-pends on several different parameters, including thevelocity and the temperature of the fluid, the min-eral composition of the rock, and the wellbore prop-erties. The system is characterized by the followingtwo temperature equations that describe the tempera-ture, T f (r, t), of the fluid inside the wellbore, Ωw, andthe temperature, Tr(r, t), of the rock in the region, Ωr,outside the wellbore (Bejan and Kraus, 2003):

ρ f cp, f

(∂T f

∂t+ v · ∇T f

)= ∇ ·

(κ f∇T f

), x ∈ Ωw, (1)

ρrcp,r∂Tr

∂t= ∇ · (κr∇Tr) , x ∈ Ωr. (2)

Here, ρ f (ρr), cp, f (cp,r), and κ f (κr) are the density,isobaric specific heat capacity, and thermal conduc-tivity of the fluid (rock), respectively, while v is thefluid velocity. These two equations are coupled bythe heat transfer between the fluid and rock. To low-est order in the temperature difference (T f − Tr), theheat flux across the borehole wall is given by (Bejanand Kraus, 2003):

q = h(T∞, f − Ts)n, (3)

where n is the surface normal of the wall pointinginto the rock reservoir, h is the heat transfer coeffi-cient, Ts is the temperature of the borehole wall, andT∞, f is the bulk temperature of the fluid in the well-bore. The heat transfer coefficient can be expressedin terms of the Nusselt number, Nu, of the fluid bythe following relation (Bejan and Kraus, 2003):

Nu =hDh

κ f. (4)

Here, Dh is the hydraulic diameter, which is equal tothe borehole diameter, D, for a cylindrical wellbore.Several empirical models for the Nusselt number ex-ist. In the present study, we calculate Nu from theDittus-Boelter equation (see Section 3 for further de-tails).

3

2.2. Effective Stresses and Failure Criterion

In real rock reservoirs, in situ stresses are present.The instability of a borehole largely depends on itsorientation with respect to in situ principal stress di-rections. In the following analysis, we present an-alytic expressions for the effective stresses on theborehole wall for arbitrary borehole orientations andprincipal in situ stress directions. Further details canbe found in Peska and Zoback (1995) and Zoback(2007), and the references therein.

Let S i j denote the in situ stress tensor representedin the borehole coordinate system, where the zb axispoints along the borehole axis (Fig. 1), the xb axis isin the plane perpendicular to the borehole axis (ori-ented towards to the bottom side of the borehole),and the yb axis is in the same plane, but is perpendic-ular to xb (see Peska and Zoback (1995) for an illus-tration). The effective stress tensor in the boreholecoordinate system is then, σi j = S i j − bPpδi j, wherePp is the pore pressure, b is the Biot-Willis parame-ter, and δi j is the Kronecker delta. A positive stressimplies that the system is under compression. Theeffective stresses at the borehole wall of a cylindricalwellbore are:

σzbzb = σ33 − 2ν (σ11 − σ22) cos(2θ) −4νσ12 sin(2θ),

σθθ = σ11 + σ22 − 2 (σ11 − σ22) cos(2θ) −4σ12 sin(2θ) − ∆p − σ∆T

θθ,

τθzb= 2

(σ23 cos(θ) − σ13 sin(θ)

),

σrr = ∆p + σ∆Trr . (5)

Here, θ is the angle (around the borehole) measuredcounterclockwise from the xb axis, r is the radial di-rection away from the center of the cylindrical well,∆p is the difference between the borehole fluid pres-sure and pore pressure in the rock, and σ∆T

rr and σ∆Tθθ

are the thermally induced stresses. In the presentstudy, we are only concerned with the value of σ∆T

θθ,

which can be reduced to the following expression inthe steady state (Zoback, 2007):

σ∆Tθθ

= αtE∆T/(1 − ν). (6)

Here, ν is the Poisson’s ratio, αt is the linear coeffi-cient of thermal expansion, E is the Young’s modu-lus, and ∆T (t) ≡ Tbw,i − Tbw(t) is the difference be-tween the initial temperature, Tbw,i, of the boreholewall and the time-dependent temperature, Tbw(t), ofthe heated or cooled borehole wall. Although theabove equations assume that ∆p and ∆T are constantalong the borehole axis, the expressions constitutegood approximations for more general situations ifthe spatial variations of ∆T and ∆p along the bore-hole axis are weak compared to the spatial variationsalong the radial direction.

From Eq. (5), we see that increasing the boreholefluid pressure or cooling the rock reduces the tangen-tial stress σθθ. For sufficiently large values of ∆p and∆T , the system may evolve from a compressive stateinto a tensile state. Because the tensile strength ofrock is relatively weak, a common tensile failure cri-terion (for a fixed θ direction) is when the minimaleffective stress in the plane tangential to the boreholebecomes negative (Zoback, 2007). The minimal ef-fective stress is given by:

σt min =12

(σzbzb + σθθ −

√(σzbzb − σθθ)2 + 4τ2

θzb

).

(7)Therefore, the present analysis assumes the creationof tensile wall fractures when σt min < 0.

3. Model Description

This section provides a description of the modelsystem used herein to study of thermally induced in-stabilities in geothermal wells.

3.1. COMSOL Multiphysics ModelA COMSOL Multiphysics (version 4.3) model is

used to simulate the temperature evolution of a devi-ated wellbore cooled by injected fluid. In the case ofmultiple wellbores, we assume that the separationsbetween the boreholes are sufficiently large to avoidthermal interaction, such that each borehole can bemodeled separately. We only simulate the temper-ature evolution along a 2000m-long wellbore seg-ment. In the WEGS concept illustrated in Fig. 1,such a wellbore segment is provided by one of the in-clined boreholes that connects the injection and pro-duction wells. The COMSOL model of the 2000m-long borehole segment is illustrated in Fig 2. The

4

system is modeled in the borehole coordinate system,in which the location of the well segment is alwaysbetween 3000 m ≤ zb ≤ 5000 m, irrespective of thetilting angle, θ. The system is assumed to be rota-tional symmetric around the zb axis, and the radius is50 m. Rotational symmetry is also a good approx-imation in the case of a tilted wellbore, θ > 0, be-cause the difference between the temperature at thecenter of the system and at the outer surface (for afixed zb value) is ∼ 50 sin(θ)dT/dz. For the temper-ature gradient, dT/dz, considered here, this situationyields a negligible temperature difference (less than1.5 K). The diameter of the wellbore is 0.1 m (Holm-berg et al., 2012; Zhang et al., 2012). The outlet ofthe wellbore segment is always located z = 5000 mbelow the surface (in the geographic coordinate sys-tem), while the depth of the inlet depends on the ori-entation of the borehole and is related to the tiltingangle by: z = 3000 + 2000(1 − cos θ) m. The tem-perature evolution of the fluid and rock matrix aredescribed by Eq. (1) and Eq. (2), respectively. Theinjected fluid is water. The fluid properties, includingthe density, thermal conductivity, and heat capacity,are given by the built-in COMSOL functions for wa-ter. The fluid velocity, v, is assumed to be constantthroughout the entire wellbore and is calculated fromthe injection rate and fluid density at a temperature of60 C. The rock matrix has the following properties,which are typical for Fennoscandia (Slagstad et al.,2009): the heat capacity is cp,r = 850 J/kgK, the den-sity is ρr = 2600 kg/m3, and the thermal conductivityis κr = 2.8 W/mK. The initial temperature, Ti(zb), ofthe rock reservoir in Fig. 2 depends on how the well-bore is tilted with respect to the vertical z axis and isgiven by:

Ti(zb) = T0 +dTdzb

zb, (8)

T0 = Ts + 5000(1 − cos(θ))dTdz,

dTdzb

=dTdz

cos(θ).

Here, θ is the tilting angle, Ts = 4 is the surfacetemperature, and dT/dz = 0.0312 K/m is the tem-perature gradient. The water in the wellbore is ini-tially assumed to be in thermal equilibrium with therock (before the simulation starts and cold water is

zb

Ti(z

b)

Rock reservoir

Tinj

50 m0.05 m

20

00

m

Ti(3000)

Ti=160°

zb=3000

zb=5000 ∆

(κfT)=0n∙

∆

(κfT)=0n∙

Ωw

Ωr

Figure 2: Illustration of the COMSOL model. The system isrotational symmetric around the zb axis. The model consistsof a rock reservoir with a radius of 50 m. The drilled well-bore, which has a diameter of 0.1 m, is located at the centerof the reservoir. The borehole and rock reservoir are sepa-rated by a thin, thermally resistive layer (red line). The sys-tem is thermally insulated at the upper and lower surfaces (i.e.,n·∇(κ f T ) = 0, where n is the surface normal). The outer surfacetemperature is assumed to be the initial temperature, Ti(zb), ofthe reservoir. The injected water has a temperature of Tinj andmaintains a uniform velocity profile through the wellbore.

injected into the wellbore with a temperature of Tinj).The boundary conditions are shown in Fig. 2. Theupper and lower surfaces of the rock matrix are ther-mally insulated; thus, it is assumed that the thermalcurrents along these two surfaces are oriented alongthe radial direction. To model a rock system that isin contact with an infinite rock reservoir, the temper-ature along the outer surface is set equal to the initialtemperature of the reservoir.

Heat transfer between the rock and fluid is mod-eled by a thin, thermally resistive layer at the bore-hole wall, such that the heat flux across the layer isgiven by:

q = κ fT f − Tr

dn. (9)

T f and Tr are the temperatures of the fluid and rockon each side of the thermally resistive layer, respec-tively, and d is the thickness of the layer. The thick-ness is determined by equating Eq. (9) and Eq. (3),which yields d = D/Nu. The Dittus-Boelter equa-

5

tion, known to be valid for fully developed, turbulentflow, is used to calculate the Nusselt number, as fol-lows (Bejan and Kraus, 2003):

Nu = 0.023Re0.8Pr0.4. (10)

Re and Pr are the Reynolds number and Prandtl num-ber of the fluid, respectively. We assume that thesetwo numbers and the heat transfer coefficient are con-stant along the wellbore axis, and are determined bythe fluid velocity and fluid properties evaluated at atemperature of 60 C.

The main interest of the present study is the stabil-ity of the wellbore. As explained in Zoback (2007),an unstable wellbore is defined as a borehole inwhich failure produces so much failed material fromaround the wellbore that it cannot be eliminated bymud circulation. To measure the instability, we cal-culate the cooling of the rock located one boreholediameter (i.e., 10 cm) into the rock reservoir (awayfrom the borehole wall) and use this temperature de-crease to check the failure criterion in Eq. (7). If thefailure criterion is fulfilled, we expect a failure zonewith a maximum thickness of approximately 10 cmaround the wellbore. In this case, the thermal frac-turing can potentially produce failed materials withsizes on the order of the borehole diameter. There-fore, as a borehole instability criterion, we check tosee whether the failure criterion in Eq. (7) is fulfilledat a depth of one borehole diameter into the rock.

To examine the convergence of the COMSOL pro-gram, we performed a simulation with a double-resolution grid and found that the difference betweenthe two temperature solutions was less than 1%.

3.2. In Situ Stress StateConsider a reservoir under the large horizontal tec-

tonic stresses typical in the studies of Fennoscan-dia (Stephansson et al., 1989). The maximum (σH)and minimum (σh) principal horizontal stresses aregiven by (Fejerskov et al., 2000; Stephansson et al.,1989):

σH(z) = 2.8 + 0.04z MPa, (11)σh(z) = 2.2 + 0.024z MPa. (12)

The maximum horizontal stress is oriented north-west/southeast (see the red, dotted line in Fig. 3).The vertical stress is given by σh/σV = 1.1.

We assume that the water in the wellbore diffusesinto the low-permeability rock surrounding the bore-holes. Bernoulli’s principle states that the velocityof the fluid results in a pressure decrease. The ve-locity yields a relative correction to the hydrostaticpressure (on the order of v2/gz). At the depths (i.e.,greater than 3000 m) and fluid velocities (on the or-der 1 m/s) considered in this paper, the relative ve-locity correction to the fluid pressure is on the order10−4. We, therefore, neglect the difference betweenthe borehole fluid pressure and pore pressure in therock, i.e., ∆p = 0, and set the pore pressure in the ef-fective stress tensor equal to the hydrostatic pressure,i.e., Pp = ρ f gz, where g is the gravitational accelera-tion.

We use values for the Young’s modulus andPoisson’s ratio that are typical for Fennoscan-dia (Stephansson et al., 1989) and a thermal ex-pansion coefficient that corresponds to a silica con-tent of 73% (Zoback, 2007): E = 6.0 · 104 MPa,αt = 8.5 · 10−6 K−1, and ν = 0.2. For simplicity, theBiot-Willis parameter is set to one (b = 1.0).

4. Results

First, consider the minimum cooling of the bore-hole wall required to induce thermal tensile fractures.When the wellbore is oriented along one of the prin-cipal stress directions and ∆p f = ∆T = 0, the mini-mum value of the effective stress in Eq. (7) becomesσt min(θmin) = 3σ22 −σ11. A wellbore that is directedalong the σh, σH, and σV directions, results in theminimum effective stresses:

σhmin = 3σV − σH (13)

σHmin = 3σV − σh (14)

σVmin = 3σh − σH (15)

For the principal stress values and depths consideredhere, we find that σh

min < σVmin < σH

min. Thus, weexpect thermal failure to be most dominant in well-bores that are oriented along the minimum horizontalstress direction. This premise is verified by Fig. 3,which shows the minimum ∆T that induces thermalfailures for all possible wellbore orientations. Theresults are calculated from Eq. (7) for three differentdepths: 3000 m, 4000 m, and 5000 m below the sur-face. The secant method is used to find the ∆T value

6

that yields σt min = 0. We see that the magnitude of∆T depends strongly on how the wellbore is orientedwith respect to in situ stresses. As expected from theabove arguments, the minimum ∆T appears for well-bores that are directed along the minimum principalstress direction. Wellbores that are tilted along themaximum stress direction require the largest valuesof ∆T to initiate thermal failure. The maximum ∆T ,which is above 130 C for depths greater than 3000m, is more than three times the minimum ∆T . Thelarge ∆T values along the σH direction imply thatthermal failure is completely absent for wellboresoriented along this axis. In contrast, for wellboresoriented along the minimum stress direction, thermalfracturing is expected for temperature changes in therange of 40-60 C. In deep geothermal systems withreservoir temperatures greater than 140 C, a temper-ature change of this order is feasible. Thus, for themodel system described in Section 3, we expect ther-mal fracturing to be present for boreholes tilted alongthe minimum stress direction.

To study thermally induced wellbore instabilities,we performed a numerical COMSOL simulation ofthe reservoir’s temperature evolution. Fig. 4 showsthe minimum value of the wellbore instability crite-rion along the 2000m-long wellbore segment for allpossible orientations of the wellbore after one yearand 20 years of geothermal production. In each fig-ure, the blue lines confine two blue pockets that rep-resent the critical orientations for which thermallyinduced instabilities are present. Within less than oneyear of geothermal production, borehole instabilitiesoccur for the wellbores that are tilted by an angle ofθ > 45 along the minimum stress direction. After20 years of production, this critical region expandsto include wellbores that are tilted by an angle of ap-proximately θ > 35. Thus, for the most part, the setof critical borehole orientations is mainly fixed afterone year of production, and only minor changes inthe set are observed after 20 years of production be-cause the cooling of rock close to the borehole beginsas a rapid process and quickly equilibrates towardsa quasi-stationary process in which the temperaturechanges slowly.

We also studied of how the instability criterion de-pends on the injection rate and injection temperatureof the fluid. Fig. 5a shows the time evolution of the

x (north)

y (east)

z (down)

φ

θ

a b

c d

x

x

120

40

80

(K)

N

E

S

W

60°

30°

0°

60

110

160

(K) (K)

200

60

130N

E

S

W

0°

30°

60°60°

30°

0°S N

W

E

zb

Figure 3: (a) All possible wellbore orientations are representedas points on the unit disk. A well that intersects the lower hemi-sphere of the unit sphere at the point x, which is parameterizedby the angles θ and φ, is mapped to the point x

′

by a projectiononto the xy plane. (b) The temperature change of the rock (atthe borehole wall) required to induce thermal fractures for allpossible orientations of the wellbore at a depth of 3000 m. (c)The temperature change of the rock (at the borehole wall) re-quired to induce thermal fractures for all possible orientationsof the wellbore at a depth of 4000 m. (d) The temperaturechange of the rock (at the borehole wall) required to inducethermal fractures for all possible orientations of the wellbore ata depth of 5000 m. In (b) - (d), the red dotted line shows the di-rection of the maximal horizontal stress, while the inner (outer)black circle represents wellbores that are tilted by an angle of30 (60) with respect to the z axis.

minimum value of the wellbore instability criterionalong the 2000m-long borehole segment for injectionrates of 5, 15, and 30 L/s. The wellbore is orientedalong the direction θ = 60, φ = 75 and the in-jection temperature is 45 C. The figure shows thatthe value of the stability criterion increases weaklywith decreasing injection rate. This result is logi-cal, given that a lower injection rate implies a lowerrate of heat extraction from the reservoir. The phys-ical reason for this weak injection-rate dependencyis that the heat extraction rate is predominantly gov-erned by the heat transfer coefficient, h, which scaleswith the Reynolds number of the fluid as h ∝ Re0.8.Fig. 5b shows the time evolution of the minimumvalue of the wellbore instability criterion along the2000m-long borehole segment for injection temper-atures of 30 C, 45 C, and 60 C. The orientation

7

40

-20

0

20

(MPa) (MPa)

-20

0

20

40

ba

NN SS

EE

WW

0° 0°

30°30°

60°60°

Figure 4: (a) The figure shows the minimum value (along thewell segment) of the wellbore instability criterion for all possi-ble orientations of the wellbore after one year of operation. (b)The figure shows the minimum value (along the well segment)of the wellbore instability criterion for all possible orientationsof the wellbore after 20 years of operation. In both plots, the reddotted line shows the maximum horizontal stress direction. Ineach plot, the two pockets that are confined by the blue dottedlines represent the orientations for which we expect thermallyinduced borehole instabilities. The injection temperature of thewater is 45 C, and the injection rate is 15 L/s.

of the wellbore is the same as in Fig. 5a, and the in-jection rate is 15 L/s. We see that the value of theinstability criterion depends strongly on the injec-tion temperature of the fluid. The fluid-temperaturedependency is mainly determined by the elasticityparameters of the reservoir. For the model systemstudied in the present paper, a temperature changeof 15 K yields a thermal stress change on the orderof αtE15/(1 − ν) = 9.6 MPa. This estimate is ingood agreement with the injection-temperature de-pendency shown in Fig. 5b. As a consequence of thislarge thermal stress-change, the set of critical bore-hole orientations is strongly affected by the injectiontemperature of the fluid. Adding or subtracting athermal stress of ∼ 9 MPa in Fig. 4 provides an esti-mate for the case where the injection temperature is60 C or 30 C. Even when the injection temperatureis as high as 60 C, a significant number of boreholeorientations become unstable.

Our numerical simulations reveal two import con-siderations regarding long-term thermal effects onEGS wellbores. First, the system is extremely sen-sitive to how the wellbores are drilled with respectto the in situ stress state. For boreholes that are ori-ented along one of the critical directions, thermallyinduced fractures with risk of causing wellbore in-stabilities are observed after only a short time of pro-duction via the geothermal system. In other words,prior knowledge of the reservoir’s stress state is cru-

2 4 6 8 10 12 14 16 18 20

0

0.5

1

1.5

2

Time (years)

Str

ess (

MP

a)

15 L/s, Tinj

= 45 Celsius

5 L/s, Tinj

= 45 Celsius

30 L/s, Tinj

= 45 Celsius

2 4 6 8 10 12 14 16 18 20

−6

−4

−2

0

2

4

6

8

Time (years)

Str

ess (

MP

a)

15 L/s, Tinj

= 45 Celsius

15 L/s, Tinj

= 30 Celsius

15 L/s, Tinj

= 60 Celsius

2 8

82

14

14 20

20

Time (years)

Time (years)

Str

ess (

MP

a)

Str

ess (

MP

a)

0

1.0

2.0

0

6.0

-6.0

15 L/s, Tinj

= 45 °C

30 L/s, Tinj

= 45 °C

5 L/s, Tinj

= 45 °C

15 L/s, Tinj

= 45 °C

15 L/s, Tinj

= 30 °C

15 L/s, Tinj

= 60 °C

a

b

Figure 5: (a) The figure shows the time dependence of theminimum value of the wellbore instability criterion along thewellbore for injection rates of 5, 15, and 30 L/s. The injectiontemperature of the water is 45 C. (b) The figure shows the timedependence of the minimum value of the wellbore instabilitycriterion along the wellbore for injection temperatures of 30,45, and 60 C. The injection rate is 15 L/s. In both plots, thewellbore has an orientation of θ = 60, φ = 75.

cial for optimal design. Second, the extent to whichthermal fracturing appears is largely determined bythe temperature of the injected fluid. This stronginjection-temperature dependency opens a path forcontrolling the thermal failure process. Our resultsprovide important insights regarding long-term ther-mal effects that are essential for optimal engineer-ing of WEGSs as well as geothermal wells and oilwells, in general. Importantly, our results can alsobe interpreted in terms of CCS technology. Oneof the major concerns regarding CCS technology isfracturing of the caprock caused by the CO2 injec-tion (Zoback and Gorelick, 2012). Recent studieshave indicated that the thermal stresses induced by

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injecting cold CO2 play an important role in deter-mining whether the caprock will fracture (Luo andBryant, 2010; Goodarzi et al., 2010, 2012). Our re-sults imply that drilling through the caprock along anoptimal direction reduces the risk of thermal caprockfracturing close to the well.

5. Conclusions

This paper contains a theoretical study of thermalfracturing and considers the effects of borehole ori-entation in deviated geothermal wellbores. We foundthat the risk for thermally induced wellbore instabil-ities is highly sensitive to how the wellbores are ori-ented with respect to the in situ stress state of therock reservoir. Thermal failures can be completelyeliminated if the system is optimized with respectto the principal stress directions. In contrast, if thesystem is not optimized, thermally induced fractureswith risk of causing instabilities are observed afterless than one year of geothermal production. In ad-dition, we found that the degree of thermal fracturingdepends strongly on the temperature of the injectedwater, while it is only weakly dependent on the in-jection rate of fluid. This strong fluid-temperaturedependency provides a direct way to control the de-gree of thermal fracturing for a given reservoir stressstate.

6. Acknowledgments

We are grateful to K. E. Brun and K. Midttømmeat Christian Michelsen Research for stimulating dis-cussions regarding the COMSOL model and forFennoscandia stress data.

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