Sand stress analysis around a producing wellbore with a simplified capillarity model

ARTICLE IN PRESS

International Journal of Rock Mechanics & Mining Sciences 42 (2005) 1015–1027 www.elsevier.com/locate/ijrmms

Sand stress analysis around a producing wellbore with a simplified capillarity model Gang Hana,, M.B. Dusseaultb a

Terralog Technologies USA, Inc., USA University of Waterloo, Waterloo, Canada

b

Accepted 3 May 2005 Available online 1 July 2005

Abstract It is well known in oil industry that water saturation changes during production can lead to sand instability, with possible sand mobilization and flux. For unconsolidated sand with little water-sensitive cementation, two mechanisms are identified as the main reasons for this phenomenon: strength reduction because of capillarity from water–oil menisci, and stress elevation due to changes of relative fluid permeabilities. In this paper the effect of both mechanisms on sand stability is evaluated in terms of stress changes and growth of a plastic radius defining a shear-yielded zone around an oil well after water breakthrough. With only two input requirements, mean particle radius and water saturation, a simplified micromechanical model is proposed to describe the capillary effects. Using this micromechanical model, a new method to calculate pore pressure in multiphase environments and a coupled elastoplastic geomechanical model describing stress redistributions with increasing water saturation are developed. The proposed models lay down a new way to analyze rock stresses in porous rock around a wellbore and can be used to help evaluate sand production risk for unconsolidated or weakly consolidated sands in multiphase fluid flow environments. r 2005 Elsevier Ltd. All rights reserved. Keywords: Stress; Water saturation; Capillarity; Pore pressure; Strength; Stability; Sand

1. Introduction It is well known that water breakthrough during hydrocarbons production can lead to rock instability and sand production, especially for weakly consolidated sand and chalk [1]. Many experiments have been carried out to study the effect of water saturation (or moisture content, humidity, etc.) on rock strength (e.g. uniaxial compressive strength (UCS), cohesive shear strength, etc.) and elastic properties. In summary, these experiments found that:




For all rock samples, rock strength is generally found to decrease with increased water saturation [1–8]. The strength decrease has been reported to range from

Corresponding author. Tel.: +1 626 305 8460; fax: +1 626 305 8462. E-mail address: [email protected] (G. Han).

1365-1609/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijrmms.2005.05.019








8% [9] to 98% [10], depending on rock texture, mineralogy and fluid chemistry. Most of the strength decrease occurs after only a slight increase in water saturation (or moisture content) from the dry state [6,10]. Further increases in moisture content have little effect on rock strength and elastic properties. The value of the internal friction coefficient appears to remain unaltered in many cases [3,11]. Young’s modulus decreases with increased water saturation, sharing the same trend as rock strength. The behavior of Poisson’s ratio is complicated; it may increase or decrease slightly before a general increase takes place at higher saturations [9,12], or remain constant [11].

There are a number of mechanisms that have been identified to explain water-related strength reduction

ARTICLE IN PRESS 1016

G. Han, M.B. Dusseault / International Journal of Rock Mechanics & Mining Sciences 42 (2005) 1015–1027

Nomenclature

xp , yp

A area, m2 C0 cohesive shear strength, Pa C 0_init initial cohesive shear strength, Pa c1 , c2 , c3 unknown parameters only related to time UCS uniaxial (unconfined) compressive strength, Pa E Young’s modulus, Pa f fluid cut, f w þ f o ¼ 1, dimensionless F force, N h reservoir thickness, m k absolute permeability of reservoir, m2 P fluid pressure, Pa P1 , P2 fluid pressure at inner (wellbore) and outer boundary of reservoir, Pa Q production rate, m3/s rlb radius of curvature of the liquid bridge in the vertical plane, m r radial distance from wellbore, m R particle radius, m R1 , R2 inner (wellbore) and outer boundary radius of reservoir, m Rc critical radius, m S saturation, dimensionless u rock displacement, m UCS uniaxial compressive strength, Pa V volume of the unit, m2

Greek letters

and rock failure [13], including changes of surface tension and capillary forces; increased fluid pressure gradient because of decreased relative permeability of oil with saturation, leading to higher effective stresses and fluid seepage forces; chemical reactions between water and rock solids such as quartz hydrolysis, carbonate dissolution, ferruginous deposition, clay swelling, etc. and, swelling of shale and blockage of pore throats by fragments or particles plucked out of the rock skeleton by fluid flow. In many cases, several mechanisms may function simultaneously in a destabilizing direction. Modeling all mechanisms is infeasible, but some progress can be made by considering a limited number of these factors during stress calculations. These include description of strength variations through changes in the plastic modulus [11,14] or hardening factor [15,16] when considering a plasticity solution with a hardening law, through pre-assumed strength values [17], or through alterations of the relative permeability of water and oil [18–20]. Even though model calculations indicate the importance of the capillary effect [16,17] and the relative permeability on stress and rock stability [20], most studies empirically adjust parameters to match rock

a b w j f l

the spatial coordinates of point pðx; yÞ, m

Biot’s coefficient, dimensionless failure angle, radian water volume angle, radian friction angle of reservoir rock, radian porosity, dimensionless factor accounting for non-uniform particle size effects on rock strength, dimensionless porosity correlation factor, dimensionless total stress, Pa effective stress, Pa tensile strength of rock, Pa fluid viscosity, Pa s Poisson’s ratio, dimensionless surface tension between two fluids, N/m

Z s s0 sT m n g

Subscripts and superscripts c e, p h o, w oi, wc r, y

capillary properties elastic and plastic, respectively horizontal oil, water, respectively immobile oil, connate water, respectively radial and tangential, respectively

behavior, rather than using a physics-based approach (i.e., based on micromechanical processes). The discrete element method (DEM) approach can study this phenomenon at the grain scale [21]. However, it is not yet mature enough to apply to practice and design. In this paper, based on a micromechanical model that physically describes the variations of rock capillary strength and pore pressure with water saturation, a coupled elastoplastic geomechanical model is developed to describe stress redistributions and plastic zone changes after water breakthrough in unconsolidated or weakly consolidated rock.

2. A simplified micromechanical model to describe capillary strength 2.1. Rationale Based on previous work, it has been found that [13,22,23]:




Capillary strength generally decreases with increasing water saturation and eventually disappears after

ARTICLE IN PRESS G. Han, M.B. Dusseault / International Journal of Rock Mechanics & Mining Sciences 42 (2005) 1015–1027












some specific saturation level that is affected by contact fabric that defines how particles contact each other, size difference between particles, and contact angle; capillary strength increases linearly with increasing surface tension, and its increase rate becomes slower at higher water saturation; capillary strength increases with smaller particle diameters; when particles have substantially different sizes, the larger size difference results in lower capillary strengths and slower decreases with increasing saturation; the magnitude of capillary strength and its variation with saturation is influenced by phase contact angle. A larger contact angle leads to lower strength and a faster decrease rate with increasing saturation; and, capillary strength generally decreases with particle deformation (either extensional or compressional), whereas for detached particles a slight increase occurs before continuously declining with water saturation. This is in contrast to the tangential contact model, where capillary forces continuously decrease with water saturation.

From a practical point of view, i.e., assuming that water breakthrough is relatively sudden and water saturation increases quickly to some level that makes capillary forces relatively small, a ‘‘safe’’ or conservative model should be selected to describe how significant the impact of capillary force changes on rock stability is after water breakthrough. Another consideration is the limited knowledge of system parameters in practice. For example, some parameters such as contact angle, surface tension, and grain size difference may not be available from routine petrophysical tests. Hence, a model that accounts for a maximum change of capillary strength, but with modest input data requirements, should be considered. This leads to a set of analytical assumptions that would support such a conservative approach. Several assumptions inherent in such a model can be stipulated. For example, particles in the model should be set with as small a diameter as possible (linked to the lower limit of the available grain size data); the value of surface tension should be the upper limit of available data; the contact angle should be set to zero, which maximizes capillary strength; a uniform particle size should be assumed; and, a tangential contact fabric should be assumed. As a consequence, instead of pursuing complex particle combinations and interactions, a more practical model for rock capillary strength is proposed in which particles have the same size and contact tangentially, with a contact angle between fluid and particles of zero (Fig. 1).

1017

Y

Solid (0,R) (0, yp)

χ R A1

p(xp, yp)
r

A2 (xp, 0)

Oil

r

X

Water R (0,-R) Solid

Fig. 1. A 2-grain micromechanical model for capillary strength.

2.2. Basic scheme Assume that the shape of the liquid bridge between grains is a toroid characterized by radii rlb and xp (Fig. 1). Then, capillary pressure across the liquid water bridge can be calculated by
 1 1  Pc ¼ g . (1) xp rlb The precision of the toroid approximation is within 10% of the value obtained by numerical solution of the Laplace-Young equation [24]. The pressure difference method is used here to calculate the capillary cohesive force resulting from capillary pressure since it has been demonstrated to be more reasonable than surface tension method [13] F c ¼ px2p Pc .

(2)

In particulate mechanics, based on statistical-geometrical considerations, the rock tensile strength sT can be related to the cohesive force F c of a single bond among rock particles through [25,26] sT ¼ l

1  f Fc , f 4R2

(3)

where R is the radius of the spherical rock particles, f is porosity, and l is a factor accounting for non-uniform particle size effects on total rock strength. A value of l ¼ 68 is suggested for packs of particles with a narrow size range, and l ¼ 1:914:5 for packs with wider particle size distributions [26]. The above equation is based on several assumptions:




A large number of bonds exist in the stressed crosssection, and stresses are transmitted by the liquid bridge at the contact points of the particles;

ARTICLE IN PRESS G. Han, M.B. Dusseault / International Journal of Rock Mechanics & Mining Sciences 42 (2005) 1015–1027

1018






a statistical distribution of bonds exists over the cross-section and the directions in space; the particles consist of spheres distributed statistically; and, the bond strength between individual particles can be replaced by a mean value applicable throughout the whole assembly.

Based on a Mohr–Coulomb strength criterion (Fig. 3), the UCS can be approximately expressed as [13] sUCS ¼ l

1  f sin j F c , f 1  sin j 2R2

(4)

which illustrates that, for unconsolidated sand, strength is related to porosity (f), friction angle (j), capillary force (F c ), particle size (R), and size distribution (l). 2.3. Geometrical expressions for model parameters In the above equations, xp and rlb are unknown parameters related to particle radius and water saturation through geometry. The coordinates of point p are xp ¼ R sin w; yp ¼ R  R cos w.

(5)

The radius of curvature at the free interface between the two fluids is
 1 1 . (6) rlb ¼ R cos w If water in the unit cell (the area surrounded by a dotted line in Fig. 2) is present only at the liquid bridge

Y

(0,R)


 2−


/

Po Pw Pw

(i.e., low water saturation), the water saturation is the ratio of the liquid bridge volume to that of the void Sw ¼

4½xp yp  ðA1 þ A2 Þ

, Vf

(7)

where the areas A1 and A2 , shown in Fig. 1, are given by the following equations: w 2 1 R  ðR  yp Þxp , (8) 2 2  p w 1  r2  y rlb sin w. A2 ¼ (9) 4 2 lb 2 p Substituting xp , yp , and rlb from Eqs. (5) and (6), Eqs. (8) and (9) become A1 ¼

R2 ðw  cos w sin wÞ, (10) 2 p w
1  cos w2 1 ð1  cos wÞ2 A2 ¼  R2 sin w.  R2 4 2 cos w 2 cos w (11)

A1 ¼

Thus, a relationship between the water volume angle and water saturation can be established by substituting Eqs. (10) and (11) into Eq. (7) w p
1  cos w2 V fSw w 1  Z ¼  þ tan w þ , 2 2 2 4 cos w 4R2 (12) where Z is a porosity correlation factor, Z ¼ fo =f , and fo is the unit cell porosity, f is the macroscopic porosity [13]. For each value of water saturation, the water volume angle (w) can be determined, as well as the toroid radii of curvature (rlb , xp ), the capillary pressure (Pc ), and the rock strength resulting from capillary forces. It should be noted that, besides the description of rock capillary strength in unconsolidated sand after water breakthrough, these equations can be used as the basis for any calculation of capillary force between two identical spheres contacting tangentially.

3. Calculations of pore pressures and effective stresses X

Po

(0,-R)

Fig. 2. Sketch of pore pressure calculations based on the micromechanical model.

After water breakthrough, two pressures exist: water pressure and oil pressure. Assuming steady-state fluid flow in an infinite reservoir, they can be calculated using
 Qw m w R2 Pw ðrÞ ¼ P2  ln , (13) 2pkkrw h r
 Qo mo R2 ln , (14) Po ðrÞ ¼ P2  2pkkro h r where the subscripts w and o represent water phase and oil phase, respectively, P is fluid pressure at distance ‘‘r’’ from the wellbore, P2 is far-field flowing pressure at

ARTICLE IN PRESS G. Han, M.B. Dusseault / International Journal of Rock Mechanics & Mining Sciences 42 (2005) 1015–1027

distance R2 , h is reservoir thickness, k is absolute permeability, kr is fluid relative permeability, Q is fluid flow rate, and m is fluid viscosity. The difference of the two pressures is equal to the capillary pressure Pw  Po ¼ Pc .

(15)

Following the development of effective stress theory [27,28], a relationship of the form PðrÞ ¼ Pw ðrÞS w  Po ðrÞð1  Sw Þ is conventionally used to determine pore pressure in multiphase environments [16,18–20]. However, this has not been physically confirmed and is open to question, as saturation is a concept of volume, compared to pore pressure that is an areal concept. Furthermore, it is well known that when water saturation reaches a certain level (less than 100%), capillary bridges will collapse and capillary forces will disappear. This fact has been overlooked in current theories of pore pressure calculation. From Fig. 2, the pore fluid pressure acting on the particle surface is Aw Ao þ Po ðrÞ , (16) A A where Aw and Ao are areas on which water and oil are acting, respectively, and A is the particle surface area. The ratio of Aw =A and Ao =A can be derived within the dashed frame of Fig. 2 as PðrÞ ¼ Pw ðrÞ

Aw =A ¼ 2w=p; Ao =A ¼ 1  2w=p.

where P2 is far-field flowing pressure at distance R2 , Q is production rate (assumed to be a constant ¯ w Þ ¼ QxðS w Þ=2pkh is a variable Q ¼ Qw þ Qo ), KðS related only to water saturation, and
 2w f w 2w fo xðSw Þ ¼ þ 1 . p krw =mw p kro =mo Furthermore, f w and f o are water and oil fractions in fluid production, respectively, and can be related to each other through f w ¼ 1  f o . The water fraction can be calculated through fw ¼

Qw ðAkkrw =mw ÞðdPw =drÞ ¼ . Ak½ðkrw =mw ÞðdPw =drÞ þ ðkro =mo ÞðdPo =drÞ

Q (19)

With Eq. (15), considering capillary pressure to be only related to water saturation (i.e. dPc =dr ¼ 0), the above equation becomes fw ¼

1 . 1 þ ðkro mw =krw mo Þ

specific value of pore pressure PðrÞ for each value of water saturation. It should be noted that since w varies from zero to some level (as will be shown later, see Fig. 7), the above equation only holds within a certain range of w. Beyond this range, the capillary pressure diminishes, and flow becomes monophasic again.

4. A coupled geomechanical model for multiphase flow 4.1. Coupled poroelastic solutions For elastic isotropic rock, stress equilibrium around a borehole in a coaxial cylindrical coordinate system can be expressed as qs0r s0r  s0y qP , (21) þ ¼a qr qr r where a is the Biot effective stress coefficient, defined such that s0 ¼ s þ aP, and varying from f (porosity) to 1 depending on rock lithology and consolidation. Usually, for unconsolidated or weakly consolidated sandstone, a is approximately 1. The solutions for the effective stresses can be derived as [29] Ec1 ðtÞ Ec2 ðtÞ 1  ð1 þ nÞð1  2nÞ ð1 þ nÞ r2 Z 1  2n a r  rPðr; tÞ dr, 1  n r2

s0r ¼ aPðr; tÞ þ

(17)

Finally, with constant production rate, the expression for pore pressure can be written as
 ¯ w Þ ln R2 , PðrÞ ¼ P2  KðS (18) r

(20)

Since the value of the water volume angle w is related to water saturation through Eq. (12), there will be a

1019

n Ec1 ðtÞ Pðr; tÞ þ 1n ð1 þ nÞð1  2nÞ Z Ec2 ðtÞ 1 1  2n a r þ þ rP dr. ð1 þ nÞ r2 1  n r2

ð22Þ

s0y ¼ a

ð23Þ

Assuming that water saturation is only a function of time (i.e., saturation is not linked to radius) and substituting the pore pressure from Eq. (18) into these two equations, the effective stresses for multiphase fluid flow in elastic porous media are s0r ¼

s0y ¼

0:5a Ec1 Ec2 1 Pþ  1n ð1 þ nÞð1  2nÞ ð1 þ nÞ r2 ¯ wÞ 0:5  n KðS , þa 1n 2 0:5a Ec1 Ec2 1 Pþ þ 1n ð1 þ nÞð1  2nÞ ð1 þ nÞ r2 ¯ wÞ 0:5  n KðS . a 1n 2

ð24Þ

ð25Þ

4.2. Coupled poro-elastoplastic stress solutions Weak or unconsolidated sandstones are more likely to yield and be mobilized by stresses and fluid flow, and

ARTICLE IN PRESS G. Han, M.B. Dusseault / International Journal of Rock Mechanics & Mining Sciences 42 (2005) 1015–1027

1020

this may lead, for example, to sand influx during fluid production. Many experiments have been carried out to study the relevant failure mechanisms, yet these are not fully understood. It is generally believed that sand does not start to flow until most of the weak mineral bonds between sand particles are destroyed, so that the fluid seepage force can then pluck sand grains out of the rock skeleton under low stress and carry them into the wellbore. Shear distortion is largely blamed for rock cohesion loss, whereas tensile failure is considered to be the result of fluid flux that plucks sand grains at the lowor no-cohesion stage. Therefore, whether or not sand can withstand the induced shear stresses is the first step in assessing the potential for sand production. The wellestablished Mohr–Coulomb failure criterion (Fig. 3) is used to delineate the occurrence of shear failure although the procedure is general and any yield criterion can be used: s0y ¼ 2C o tan b þ s0r tan2 b,

(26)

where C o is cohesive shear strength, and b is the failure angle which is related to the friction angle through b ¼ p=4 þ j=2. After water breakthrough, the friction angle can be assumed to be constant as shown in Fig. 4 [1,3,11]. However, there is a decrease in cohesive

strength that can be divided into two parts [22]: one results from chemical reactions, the other from changes in capillary force. If the effect of chemical reactions is neglected (e.g., for clean sands), the cohesive shear strength after water breakthrough can be approximately expressed as C o ðS w Þ ¼ C o_init þ sT ðS w Þ tan j or C o ðS w Þ ¼ C o_init þ l tan j

1  f F c ðS w Þ f 4R2

(27)

where C o_init is the initial cohesive shear strength before water breakthrough. To avoid complexity in this theoretical treatment, such that analytical expressions can be developed, a simple way to describe stress distributions inside the Coulomb zone around a borehole has been adopted. Assuming that the rock stresses inside the Coulomb zone satisfy the Mohr–Coulomb failure criterion represented by Eqs. (26) and (27), the effective stress equilibrium Eq. (21) becomes ¯ wÞ ds0r 1  tan2 b 0 2C o ðS w Þ tan b þ aKðS sr ¼ þ r r dr

(28)

yielding the solutions τ

s0r ðrÞ ¼



T

s0y ðrÞ ¼

Co ′T

UCS 3

1



Fig. 3. Various strength concepts defined in Mohr–Coulomb criterion.

SHEAR STRESS τ

1b/1sq. 1a

40 000

dried oven CaCl2

submerged in water

10 000

0 0

c3 ð1  oÞ o r o ¯ wÞ 2C o ðS w Þ tan b þ ð1  oÞaKðS , þ o

10 000

20 000

30 000

NORMAL STRESS 

40 000

50 000

60 000

ð30Þ

(31)

Following similar derivations published previously [29], three unknown constants, c1 , c2 in Eqs. (24) and (25), and the radius (Rc ) defining the boundary between elastic zone and Coulomb zone can be solved as
ð1 þ nÞð1  2nÞ 0 0:5a sh  PðR2 Þ c1 ¼ E 1n  ¯ wÞ Ec2 1 0:5  n KðS þ a , ð32Þ 1n 2 ð1 þ nÞ R22

1b/sq. 1a

Fig. 4. Experimental findings of Mohr fracture envelopes at two saturation states [3].

(29)

where o ¼ 1  tan2 b. Since at the inner boundary the radial effective stress must be zero, i.e., s0r ðR1 Þ ¼ 0, the constant c3 can be solved ¯ w ÞÞRo c3 ¼ ð2C o ðS w Þ tan b þ aKðS 1.

30 000

20 000

¯ wÞ c3 o 2C o ðS w Þ tan b þ aKðS , r þ o o

c2 ¼ 


 1 þ n 0:5a ¯ KðSw Þ þ c3 Ro R2c , c 2E 1  n

(33)

ARTICLE IN PRESS G. Han, M.B. Dusseault / International Journal of Rock Mechanics & Mining Sciences 42 (2005) 1015–1027

¯ w Þ 0:5a KðS ¯ wÞ 2C o ðSw Þ tan b þ aKðS þ o
1n 2  c3 2o 1 1 o ¼ 2 Rc þ  c3 R c o 2 2R2 
 ¯ w Þ R2c 0:5aKðS Rc þ  ln . 2 1n R 2R2 2

s0h 

ð34Þ

Eq. (34) is a nonlinear equation for Rc , in the form of 2 zhfl; ðR2o ; Ro c c ; Rc ; lnðRc ÞÞ ¼ 0, which can be solved iteratively with the aid of a mathematical software such as Matlab, Mathematica, etc. The above stress solution treats the cohesive shear strength in the Coulomb zone as being irrelevant to rock deformation, which conflicts with the fact that sand becomes softer and weaker after shear yield, leading to a reduced cohesion or even a cohesionless state, as well as a condition of increased compliance. Therefore, Eqs. (29) and (30) give the upper limit of stresses inside the plastic zone, and should be treated as conservative solutions. One common approach to compensate for this shortcoming is to add a plastic strain to the elastic strain calculated by Hooke’s Law,

Table 1 Input parameters for capillarity model R (m)

g (N/m)

y

1  104

0.036

0

Table 2 Relative permeabilities vs. saturation Sw

krw

krow

0.32 0.375 0.415 0.4555 0.495 0.535 0.575 0.615 0.655 0.694 0.734

0 0.003 0.008 0.017 0.028 0.057 0.091 0.134 0.184 0.242 0.301

1 0.653 0.436 0.311 0.214 0.14 0.089 0.049 0.019 0.001 0

1021

this type of strain being defined by plastic theory [20,30]. However, mathematical plasticity theory focuses on empirically matching nonlinear stress– strain relationships, rather than capturing the effect of specific physical mechanisms, and intensive calibration is needed before such ‘‘laws’’ can be used in practice. Many authors believe that a nonlinear theory based on rock moduli and other properties that change with loading path is more convincing and reasonable. Some developments have been made during the past few decades [31–33], but additional experiments are needed in order to determine the parameters applied in models, presenting substantial challenges for geomechanics modelers.

5. Model calculations and discussion Bearing in mind the practical limits on the availability of input data, the models presented above have deliberately been developed to require a limited number of input parameters. For example, the capillarity model only needs two parameters, namely particle radius (R) and surface tension (g), while parameter l in Eq. (3) can be selected based on the distribution of particle sizes. The values used for this analysis are listed in Table 1. The surface tension between oil and water can be set as high as 0.036 N/m (which is the value for heavy oil and water), as capillary strength is linearly related to surface tension and the peak strength is needed for calculations. As a result, only particle radius is required de facto. The relation of oil and water relative permeabilities to water saturation is usually available from petrophysical data for reservoir simulations (Table 2). If capillary pressure data have been determined at the same time, calibration of the micromechanical capillarity model is straightforward. The inputs for the geomechanical model used in this analysis are listed in Table 3. 5.1. Capillary strengths vs. water saturation Compared to the rapid decrease of capillary pressure with water saturation (Fig. 5), the rates of decrease of capillary force and strengths (i.e. UCS, tensile strength and cohesive shear strength) with saturation are much slower (Fig. 6). The maximum strength can be as high as

Table 3 Parameters used in the geomechanical (stress) model Rock mechanical properties E (Pa) 3  10

9

n 0.45

Reservoir flow properties C o (Pa)

sh (Pa) 6

28  10

j 6

0.5  10

301

fi 0.3

ki (m2) 0.3  10

Geometry parameters

P2 (Pa) 12

6

10  10

m (Pa/s) 0.01

Q (m3/s) 3

1.157  10

R2 (m)

R1 (m)

h (m)

50

0.1

10

ARTICLE IN PRESS G. Han, M.B. Dusseault / International Journal of Rock Mechanics & Mining Sciences 42 (2005) 1015–1027

1022

1

12

0.9 Water Volume Angle(Radian)

Capillary Pressure (kPa)

10 8 6 4 2

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

0

0

0

0.05

0.1

0.15 0.2 0.25 Water Saturation

0.3

0.35

0

0.05

0.1

0.15

0.2

0.25

0.3

0.336

Water Saturation

Fig. 7. Variations of water volume angle with water saturation.

Fig. 5. Calculated capillary pressure vs. water saturation.

As shown in Fig. 7, water volume angle is not linearly related to water saturation: its increase rate becomes small as saturation rises.

25

20

5.2. Pore pressure vs. water saturation

Capillary Variables

UCS (kPa) T (kPa)

15

Co (kPa) Fc (Dyne) 10

5

0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Water Saturation

A significant point pertaining to the use of the present approach is the fact that, before introducing relative permeability data, the water saturation in the micromechanical model developed above should be calibrated to experimentally determined values. Discrepancies between the model and reality results are mainly from two sources that the micromechanical model cannot address: one is connate water saturation (Swc ) and immobile oil saturation (S oi ); the other is the effect of irregular particle shape. Therefore, the calibration is carried out as

Fig. 6. Model calculations of capillary force and strengths vs. water saturation.

S 0w ¼ S wc þ S w ð1  S wc  S oi Þ=Sw0 ,

20 kPa, whereas all capillary strengths become zero around a saturation value of 0.34. However, there is a small section of the relationship at water saturation approaching zero where a short increase of strength is predicted (Fig. 6), because some volume of water is needed to build a stable liquid bridge between particles. This has been confirmed by Bianco and Halleck’s experiment, in which a stable arch is found to develop with a small increase in water saturation in a two-phase environment, whereas such an arch cannot be stable in a single-phase condition [34]. The value of the saturation is closely related to liquid–solid contact angle, the distance between particles, and the irregularity of particle surfaces [23].

where S w0 is the saturation at which capillary pressure becomes zero in the capillarity model. For example, S w ¼ 0 in the capillarity model corresponds to connate water saturation in field (S0w ¼ S wc ), while S w ¼ S w0 conforms to a fully water-saturated situation (i.e. S 0w ¼ 1  S oi ). Fig. 8 shows the calculated pressure variations with water saturation at different distances from the wellbore (i.e. r ¼ 0:2 m, 1.0, 2.55 m). Interestingly, pore pressure first decreases with saturation until some critical saturation (Sw ¼ 0:45), and the decrease in magnitude can be as high as several mPa. Subsequently, it increases continuously to a value at S w ¼ 0:734 that is higher than the initial value at S w ¼ 0:32. Correspondingly, the pressure distributions around the wellbore (Fig. 9) are first lowered by the increase of water saturation (e.g., from Sw ¼ 0:34 to 0:507), but eventually increase until leveling off (e.g., at S w ¼ 0:704). Physically,

(35)

ARTICLE IN PRESS G. Han, M.B. Dusseault / International Journal of Rock Mechanics & Mining Sciences 42 (2005) 1015–1027

Pore Pressure near Wellbore (MPa)

9 8 7 6 5 r = 0.2m 4

r = 1.0m r = 2.55m

3 2 0.3

0.35

0.4

0.45

0.5 0.55 0.6 Water Saturation

0.65

0.7

0.75

Pressure Difference between Two Methods (MPa)

10

2

1023

x 10-15

0

-2 0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

Water Saturation

Fig. 8. Variations of pore pressure with saturation at different locations around a wellbore.

Fig. 10. Difference of pressure variations (@ r ¼ 2:5 m) calculated by two methods.

9

2

8 7 6 Sw = 0.340

5

Sw = 0.507 Sw = 0.704

4 3 1

50

100

150

200

250

300

350

400

450

500

Dimensionless Distance from Wellbore

Pressure Difference between Two Methods (MPa)

Pore Pressure around Wellbore (MPa)

10

0

Sw = 33.97%

-2

Fig. 9. Distributions of pore pressure around a wellbore at different water saturations.

x 10-15

0

50

100 150 200 250 300 350 400 Dimensionless Distance from Wellbore (r/R1)

450

500

Fig. 11. Difference of pressure distributions (Sw ¼ 33:97) calculated by two methods.

because water is a less viscous and more mobile fluid than oil, less energy (i.e. lower pressure drawdown) is needed to drive it into the wellbore. Consequently, the increase of water relative permeability raises the pore pressure whereas that of oil relative permeability lowers it. The synthesis of both effects result in a higher pore pressure in a water-dominant fluid system than in an oil-dominant fluid system, assuming a constant flow rate. The pressure difference between this new approach based on physics at the grain scale (Eq. (18)) and the conventional method described in Section 3 is plotted in Figs. 10 and 11. The conventional method is confirmed to be precise enough to be applied in the pressure analysis.

5.3. Stresses distribution vs. water saturation Compared to the changes of pore pressure that first decrease then increase with water saturation, the stress behavior is more complicated. Fig. 12 describes the redistribution of effective stresses around a wellbore producing oil and water simultaneously. To facilitate investigation of the details, stresses in both the elastic zone near the shear yield front (Rc ) and the plastic zone are presented in Figs. 13 and 14. In the elastic zone close to the shear yield front (Fig. 13), the effective tangential stress increases to a

ARTICLE IN PRESS G. Han, M.B. Dusseault / International Journal of Rock Mechanics & Mining Sciences 42 (2005) 1015–1027

1024

3 Sw = 0.340

30

Sw = 0.704

25 20

2.5

Sw = 0.507

Stresses in Plastic Zone (MPa)

Effective Stresses near Wellbore (MPa)

35

′

15 ′r

10

2

′r

1.5

′ 1

0.5

5

0 0.3

0 5

10

15

20

25

30

0.35

0.4

Dimensionless Distance from Wellbore (=r/R1)

30

Stresses in Elastic Zone (MPa)

28 26 24 ′r

22

′

20 18 16 14 0.3

0.35

0.4

0.45

0.5 0.55 0.6 Water Saturation

0.65

0.7

0.75

Fig. 13. Variations of stresses with water saturation in elastic zone (@r ¼ 0:802 m).

peak before it declines with saturation, whereas the effective radial stress does the opposite; i.e., it first decreases to its lowest value and then increases. As such, the difference between them (i.e., shear stress  2) reaches a maximum at some specific saturation value (around S w ¼ 0:45 where connate water saturation is 0.32). Thus, at the initial stage of water breakthrough, the sand that initially behaves elastically is most likely to experience shear yield that breaks cementation among particles and propagates the yield front (defined as radius Rc ) away from the wellbore. If fluid flow forces are strong enough to carry yielded sand into the wellbore, sand production occurs. However, as shown in Fig. 12, in the elastic zone far from wellbore (i.e., r=R1 415), the effective radial stress follows the same

0.65

0.7

0.75

Fig. 14. Variations of stresses with water saturation in Coulomb zone (@r ¼ 0:131 m).

Evolution of Dimensionless Critical Radius (Rc/R1)

Fig. 12. Distributions of effective stresses around a wellbore at different water saturations.

0.45 0.5 0.55 0.6 Water Saturation

16 Co_init = 0.4 MPa

14

Co_init= 0.5 MPa 12

Co_init = 1 MPa

10 8 6 4 2 0.3

0.35

0.4

0.45 0.5 0.55 0.6 Water Saturation

0.65

0.7

0.75

Fig. 15. Variations of plastic yield front (Rc ) with water saturation.

trend as the effective tangential stress: first it increases then decreases with saturation. For the Coulomb zone (Fig. 14), both the effective tangential stress and the effective radial stress decrease with saturation, except that the former decreases more than the latter. This creates an even lower general confining stress environment around the wellbore after water breakthrough, which is much more likely to lead to fluid erosion of sand. 5.4. Propagation of the Coulomb zone with water saturation Fig. 15 shows the propagation of Rc with saturation for rocks with different initial cohesive shear strengths (C o_init ). Clearly, saturation has a large impact on the

ARTICLE IN PRESS G. Han, M.B. Dusseault / International Journal of Rock Mechanics & Mining Sciences 42 (2005) 1015–1027

Effective Stresses around Wellbore (MPa)

35 30

18 Evolution of Dimensionless Critical Radius (Rc/R1)

plastic yield zone: Rc rapidly increases with the increase of saturation. Furthermore, the lower the initial cohesive strength, the more significant the increase in critical radius. For example, for C o_init ¼ 0:4 MPa, the dimensionless critical radius (Rc =R1 ) increases from 5 to 16 when saturation rises from 0.32 (connate saturation) to about 0.45, while Rc =R1 changes from 4.2 to 7.3 for C o_init ¼ 0:5 MPa and from 2.7 to 3.2 for C o_init ¼ 1 MPa. When interpreting this effect in terms of effective stress distributions around a wellbore (Fig. 16), the magnitude of stress changes is highly significant if the initial strength varies and can be comparable in magnitude to the effect of pore pressure changes (Fig. 12) caused by relative permeabilities alterations. Thus, the initial rock strength greatly affects the extent of the effect of water breakthrough on sand stability. Concerning the decrease of the critical radius above S w ¼ 0:45 in Fig. 15, it should not be interpreted as a stabilizing factor. When sand fails with the propagation of the yielding front, there are associated geometrical changes that require re-definition of boundary conditions used in the geomechanical model. The solid lines in Fig. 17 give the relationship between dimensionless critical radius and saturation when capillary strength changes are taken into account. Comparing to the dashed lines that treat rock strength as a constant (i.e., no capillary strength appears and rock stability changes only result from pore pressure variations by virtue of relative fluid permeability changes), the capillary effect that varies the rock strength through changing water–oil menisci is far less significant than the effect of relative permeabilities, unless the initial rock strength is relatively low (e.g., C o_init ¼ 0:4 MPa). Considering that the magnitude of capillary strength (on the order of kPa) is much lower

0 4 6 8 10 12 14 16 18 Dimensionless Distance from Wellbore (=r/R1)

20

Fig. 16. Effect of cohesive shear strength on stress distributions around a wellbore (Sw ¼ 0:34).

Co_init = 0.45 MPa

8 6

Co_init = 0.5 MPa

4

0.35

0.4

0.45

0.5 0.55 0.6 Water Saturation

0.65

0.7

0.75

In this work, based on a simplified micromechanical model that needs only two input parameters (particle radius and water saturation) to physically describe capillary strength behavior with water saturation, a coupled poro-elastoplastic model has been developed that can evaluate different mechanisms affecting sand stability after water breakthrough into an oil-producing wellbore. According to model calculations, it is found that:

Co_init= 1 MPa

1 2

10

6. Conclusions

Co_init = 0.5 MPa

5

12

than rock cohesive or shear strength (on the order of MPa), effect of capillary strength on rock stability can only be significant after rock loses most of its initial strength due to rock yield or failure. For example, comparing the dotted lines with the solid lines in Fig. 17, the capillary strength can lower the plastic radius (Rc ) more significantly in the case that cohesive strength is 0.4 MPa than 0.5 MPa.




Co_init = 0.4 MPa 10

Without Capillary 14

Fig. 17. Effect of capillary strength on plastic yield front.

20 15

With Capillary

16

2 0.3




25

1025




Compared to the rapid decrease of capillary pressure with water saturation, the decline of capillary force and strength (i.e., UCS, tensile strength and cohesive shear strength) are much less acute. The capillary strengths can be in the magnitude of kPa. Because of changes in oil and water relative permeabilities, pore pressure firstly decreases with saturation until some critical point, and the magnitude of this decrease can be as high as several mPa. As water saturation increases beyond this critical point, pore pressure then increases continuously to a value even higher than its initial value (where only oil was flowing). In the elastic zone close to the shear yield front, the effective tangential stress increases to a maximum

ARTICLE IN PRESS 1026







G. Han, M.B. Dusseault / International Journal of Rock Mechanics & Mining Sciences 42 (2005) 1015–1027

before declining with saturation, whereas the effective radial stress first decreases to its lowest value and then increases. Thus, at the initial stage of water breakthrough, the sand that initially behaves elastically is most likely to experience shear yield that breaks cementation among particles and moves the yield front outward from the well. In the plastic zone, both the effective tangential stress and effective radial stress decrease with saturation. This creates a lower stress condition around the wellbore after water breakthrough, which facilitates fluid erosion of the sand. Water saturation is shown to have a large impact on changes in the plastic yield zone, and this impact becomes more pronounced with increasing water saturation. The magnitude of the initial shear strength plays a very important role in stabilizing sands. When the initial strength is low, the increase of the plastic radius with saturation becomes significant, as well as the contribution of the capillary effects. Otherwise the effect of capillarity is trivial compared to that of relative permeabilities.

Based on the micromechanical model developed in this work, the conventional method to calculate pore pressures in multiphase environments is confirmed to be precise enough to apply in pressure analysis. The model has been developed for oil wells under conditions of two-phase (oil and water) flow, without the presence of free gas, which greatly complicates the physical processes through gas bubble pore throat blockage, the Klinkenburg effect, and other factors. Nevertheless, the approach taken should be amenable to the development of a numerical model in cases of three-phase flow with three continuous phases. Also, the simplicity of the approach should make it appealing for application to other boundary value problems encountered in unsaturated sands in geotechnical engineering. This new geomechanical model facilitates the understanding of why and how rock becomes unstable after water breakthrough into an oil well, and can be used as a foundation to evaluate sand production risk in multiphase fluid environments.

Acknowledgements G. Han thanks Schlumberger for support for his Ph.D. studies, and M.B. Dusseault thanks the Natural Sciences and Engineering Research Council of Canada for financial support, as well as many industrial contacts and colleagues. In particular, inspiration from Drs. John Cook, Steve Chang, Hongren Gu, and Ian Walton from Schlumberger, and technical communications with Prof. Mario Ioannidis from the University of Waterloo, are deeply appreciated.

References [1] Skjærstein A, Tronvoll J, Santarelli FJ, Jøranson H. Effect of water breakthrough on sand production: experimental and field evidence. SPE 38806, the SPE Annual Technical Conference and Exhibition, San Antonio, TX, USA, October 1997. [2] Horn HM, Deere DW. Frictional characteristics of minerals. Ge´otechnique 1962;12:319–35. [3] Colback PSB, Wiid BL. The influence of moisture content on the compressive strength of rocks. Proceedings of 3rd Canadian symposium of rock mechanics, 1965. p. 65–83. [4] Swolfs HS. Chemical effects of pore fluids on rock properties. Underground waste management and environmental implications. Proceedings of symposium of American Association of Petroleum Geologists (AAPG) Memoir 18, December 1971. p. 224–33. [5] Dube AK, Singh B. Effect of humidity on tensile strength of sandstone. J Mines Metals Fuels 1972;20(1):8–10. [6] West G. Effect of suction on the strength of rock. Q J Eng Geol 1994;27:51–6. [7] Boretti-Onyszkiewicz W. Joints in the Flysch sandstones on the ground of strength examinations. Proceedings of 1st congress of international society of rock mechanics, Lisbon, Portugal, vol. 1, 1966. p. 153–7. [8] Gutierrez M, Øino LE, Høeg K. The effect of fluid content on the mechanical behaviour of fractures in chalk. Rock Mech Rock Eng 2000;33(2):93–117. [9] Hawkins AB, McConnell BJ. Sensitivity of sandstone strength and deformability to changes in moisture content. Q J Eng Geol 1992;25:115–30. [10] Dyke CG, Dobereiner L. Evaluating the strength and deformability of sandstones. Q J Eng Geol 1991;24:123–34. [11] Papamichos E, Brignoli M, Santarelli FJ. An experimental and theoretical study of a partially saturated collapsible rock. Mech Cohesive-Frictional Mater 1997;2:251–78. [12] Priest SD, Selvakumar S. The failure characteristics of selected British rocks. Transport and Road Research Laboratory Report, 1982. [13] Han G, Dusseault MB. Quantitative analysis of mechanisms for water-related sand production. SPE 73737, the SPE International Symposium and Exhibition on Formation Damage Control, Lafayette, LA, USA, February 2002. [14] Bolzon G, Schrefler BA, Zienkiewicz OC. Elastoplastic soil constitutive laws generalized to partially saturated sates. Ge´otechnique 1996;46(2):279–89. [15] Alonso EE, Gens A, Josa A. A constitutive model for partially saturated soils. Ge´otechnique 1990;40(3):405–30. [16] Simoni L, Salomoni V, Schrefler BA. Elastoplastic subsidence models with and without capillary effects. Comput Methods Appl Mech Eng 1999;171:491–502. [17] Vaziri HH, Barree B, Xiao Y, Palmer I, Kutas M. What is the magic of water in producing sand? SPE 77683, the SPE Annual Technical Conference and Exhibition, San Antonio, TX, USA, October 2002. [18] Zhang HW, Deeres OM, Borst R, Schrefler BA. Implicit integration of a generalized plasticity constitutive model for partially saturated soil. Eng Comput 2001;18(1/2):314–36. [19] Schrefler BA, Scotta R. A fully coupled dynamic model for twophase fluid flow in deformable porous media. Comput Meth Appl Mech Eng 2001;3223–46. [20] Wang Y, Xue S. Coupled reservoir-geomechanics model with sand erosion for sand rate and enhanced production prediction. SPE 73738, the SPE International Symposium and Exhibition on Formation Damage Control, Lafayette, LA, USA, February 2002.

ARTICLE IN PRESS G. Han, M.B. Dusseault / International Journal of Rock Mechanics & Mining Sciences 42 (2005) 1015–1027 [21] Bruno MS, Bovberg CA, Meyer RF. Some influences of saturation and fluid flow on sand production: laboratory and discrete element model investigations. SPE 36534, the SPE Annual Technical Conference and Exhibition, Denver, CO, USA, 1996. [22] Han G, Dusseault MB. Strength variations with deformation in unconsolidated rock after water breakthrough. SCA 2002-22, Proceedings of the 2002 symposium of SCA (Society of Core Analysis), Monterey, CA, USA, September 2002. [23] Han G, Dusseault MB, Cook J. Quantifying rock capillary strength behavior in unconsolidated sandstones, SPE/ISRM 78170, the SPE/ISRM Rock Mechanics Conference, Irving, TX, USA, October 2002. [24] Lian G, Thornton C, Adams MJ. Effect of liquid bridge forces on agglomerate collisions. Powders Grains 1993; 59–64. [25] Capes CE. Particle size enlargement. In: Handbook of powder technology, vol. 1. Amsterdam: Elsevier; 1980. p. 23–6. [26] Schubert H. Capillary forces—modelling and application in particulate technology. Powder Technol 1984;47:105–16. [27] Bishop AW. The principle of effective stress. Teknisk Ukeblad 1959;39:859–63. [28] Bishop AW, Blight GE. Some aspects of effective stress in saturated and partly saturated soils. Ge´otechnique 1963;13(3):177–97.

1027

[29] Han G, Dusseault MB. Description of fluid flow around a wellbore with stress-dependent porosity and permeability. J Petro Sci Eng 2003;40(1/2):1–16. [30] Bradford IDR, Cook JM. A semi-analytic elastoplastic model for wellbore stability with applications to sanding. SPE 28070, the SPE/ISRM Rock Mechanics in Petroleum Engineering Conference, Delft, the Netherlands, August 1994. [31] Vaziri HH. Analytical and numerical procedures for analysis of flow-induced cavitation in porous media. Comput Struct 1995;54(2):223–38. [32] Santarelli F, Brown ET, Maury V. Analysis of borehole stresses using pressure-dependent linear elasticity. Int J Rock Mech Min Sci Geomech Abstr 1986;23(6):445–9. [33] Nawrocki PA, Dusseault MB, Bratli RK. Addressing the effects of material non-linearities on wellbore stresses using stress- and strain-dependent elastic moduli. 1995 USRM Symposium, Lake Tahoe, Nevada, 1995. [34] Bianco LCB, Halleck PM. Mechanisms of arch instability and sand production in two-phase saturated poorly consolidated sandstones. SPE 68932, the SPE European Formation Damage Conference, Hague, the Netherlands, May 2001.