The Role of the Intermediate Principal Stress in Petroleum

GeomechanicsRobert W. Zimmerman

Department of Earth Science and Engineering

Imperial College

SPE Geomechanics MeetingGeological Society, London

27th October 2015

Principal stresses – basic theory

Regardless of the constitutive behaviour of the rock (i.e., elastic, plastic, viscoelastic, etc.), and regardless of the boundary conditions or applied loads, at any point in a reservoir, there will always be three principal normal stresses, having the following properties:

1. They act in three mutually perpendicular directions

2. These three directions form a co-ordinate system in which all six shear stresses are zero

3. They can be labelled as 1 ≥ 2 ≥ 3, where compressive stresses are positive

Principal stresses in situ

In the reservoir, the magnitudes and the orientations of these principal stresses are determined by the tectonic stress regime, and the depth z below the surface

Typically, one principal stress acts vertically, and is equal to gz, where is the mean density of the overlying rock

The other two stresses act in two orthogonal directions lying in a horizontal plane, but their magnitudes and precise directions are not easily predicted or estimated

So, the three principal stresses are v and H ≥ h, but the ordering of the stresses cannot be determined a priori

Stresses around a borehole

For a vertical wellbore (i = 0), the stresses are given by:

For the general case of a deviated borehole, the expressions for the stresses are similar to those shown above, but much more complicated

Moreover, when a borehole is drilled, the principal stresses are altered in both magnitude and direction

Mohr’s circle for the stress on an arbitrary plane

€

=(1 +2 )

2+

(1 −2 )2

co2θ

€

τ =−(1 −2 )

2in2θ

The normal stress () and shear stress (τ) that act on a plane whose normal vector is rotated anti-clockwise by angle θ from the 1 direction are given by:

These stresses can be visualised on a Mohr’s circle diagram in the {,τ} plane, as on the right. (Note that the angle of rotation on Mohr’s diagram is twice the physical angle of rotation).

Coulomb’s failure criterion

Coulomb (1773) postulated that, as the stresses increase, failure would first occur on the plane that satisfies the condition

τ = So + tan

This criterion appears on Mohr’s diagram as a straight line with slope equal to tan

This criterion will always first be satisfied on a plane that lies in the direction of σ2

Moreover, this criterion implies that σ2 will have no effect on failure (see 3D case at right)

Mohr’s extension of Coulomb’s criterion

Mohr’s concept (1900): τ= f ()

The function f may or may not be linear, but the key assumption is that failure can be described by a curve in the (,τ) plane

A linear form of Mohr’s criterion is equivalent to the Coulomb criterion, hence it is often referred to as the “Mohr-Coulomb” criterion

Mohr’s assumption implies that the fracture plane always strikes in the σ2-direction; this was verified experimentally by Mogi (1971)

Failure under true-triaxial stress conditions

However, we now know, through the experiments of Mogi (see right), Haimson, Handin, Hoskins, et al., that the intermediate principal stress does have an influence on rock failure

Failure should therefore be described by some function of all three principal stresses

But for simplicity, and due to sparseness of data, we would like to use a failure criterion that is mathematically two-dimensional

Two issues therefore arise:

● what form of the equation to use?

● which variables to use?

Drucker-Prager Failure Criterion

€

τ oct = k + mσ oct

€

τoct = 13

(σ 1 −σ 2 )2 + (σ 1 −σ 3)2 + (σ 2 −σ 3)2

3321 ++

=oct

• Originally proposed for soils in 1952

• Often used for rocks, but never properly validated

where

Note that if 2 = 3, then Drucker-Prager reduces to

€

1 −3 =a+b1 +2 +3

3

⎛⎝⎜

⎞⎠⎟

Mogi-Coulomb Failure CriterionAccording to DP, 2–3 is the driving force for failure, and 1+2+3 is the “resisting force”

However, since the failure plane is parallel to 2, Mogi argued that the true resistive force is m,2 = (1+3)/2, not oct = (1+2+3)/3

Based on this idea, Mogi (1971) proposed that failure should be governed by a criterion of the form: τoct = f(m,2)

Al-Ajmi and Zimmerman used a linear form of Mogi’s criterion:

τoct = a + bm,2

which reduces to Mohr-Coulomb for traditional triaxial stress states, σ2 = σ3, or σ2 = σ1. Hence, this criterion can be thought of as a natural extension of the Coulomb criterion into the true-triaxial stress domain, hence the name “Mogi-Coulomb”

The parameters a and b can be found from the traditional Coulomb parameters:

a = (2√2/3)Socos, b = (2√2/3)sin

Test of some true-triaxial failure criteriaTo test these criteria, eight sets of true-triaxial failure data were located and analysed:

• Dunham dolomite, Solenhofen limestone & Mizuho trachyte (Mogi, 1971)

• Marble (Michelis, 1985,1987)

• Shirahama sandstone and Yuubari shale (Takahashi & Koide, 1989)

• Westerly granite (Haimson & Chang, 2000)

• KTB amphibolite (Chang & Haimson, 2000)

In general, the Mogi-Coulomb criterion was found to be the most accurate:

• Does not ignore the strengthening effect of σ2, such as is done by the Mohr-Coulomb criterion • Does not predict too high a strength, as does the Drucker-Prager criterion

Model Comparison: KTB Amphibolite

0

100

200

300

400

500

600

0 100 200 300 400 500 600 700

σm,2 (MPa)

τ max

(MP

a)

0

100

200

300

400

500

600

0 100 200 300 400 500 600 700

σoct (MPa)

τ oct

(MP

a)

0

100

200

300

400

500

600

0 100 200 300 400 500 600 700

σm,2 (MPa)

τ oct

(MP

a)

Mohr-Coulomb Drucker-Prager

Mogi-Coulomb

Solid circles: 2 = 3

Open circles: 2 > 3

Test of Drucker-Prager Criterion

Dunham Dolomite k = 72.679 MPam = 0.6737 r2 = 0.9853

0

100

200

300

400

0 200 400 600

σoct (MPa)

τ oct

(MP

a)

Westerly Granitek = 32.434 MPam = 1.002r2 = 0.9984

0

100

200

300

400

500

600

0 200 400 600

σoct (MPa)

τ oct

(MP

a)

Test of Mogi-Coulomb Criterion, I

Dunham Dolomite a = 82.554 MPab = 0.4788r2 = 0.9814

0

100

200

300

400

0 200 400 600 800

σm,2 (MPa)

τ oct

(MP

a)

Solenhofen Limestonea = 86.602 MPab = 0.4123r2 = 0.9495

0

100

200

300

0 100 200 300 400

σm,2 (MPa)

τ oct

(MP

a)

Mizuho Trachyte a = 39.868 MPab = 0.4417r2 = 0.9595

0

50

100

150

200

0 50 100 150 200 250 300 350

σm,2 (MPa)

τ oct

(MP

a)

Shirahama Sandstone a = 14.867 MPab = 0.532r2 = 0.9789

0

50

100

150

0 50 100 150 200

σm,2 (MPa)

τ oct

(MP

a)

Test of Mogi-Coulomb Criterion, II

KTB Amphibolite a = 40.099 MPab = 0.6364r2 = 0.9865

0

100

200

300

400

500

600

700

0 200 400 600 800 1000

σm,2 (MPa)

τ oct

(MP

a)

Marblea = 9.1557 MPab = 0.6373r2 = 0.9789

0

20

40

60

80

100

120

140

160

0 50 100 150 200 250

σm,2 (MPa)

τ oct

(MP

a)

Yuubari Shalea = 24.069 MPab = 0.4276r2 = 0.9431

0

20

40

60

80

100

120

0 50 100 150 200

σm,2 (MPa)

τ oct

(MP

a)

Westerly Granite a = 30.186 MPab = 0.7116r2 = 0.9939

0

100

200

300

400

500

600

0 200 400 600 800

σm,2 (MPa)

τ oct

(MP

a)

Wellbore Stability AnalysisFirst, the stresses around the wellbore wall are calculated using linear elasticity (vertical wells: Kirsch, 1898; deviated wells: Hiramatsu & Oka, 1968)

Next, one must determine which stresses (rr, zz or θθ) correspond to 1, 2 and 3

These stresses are substituted into the failure criterion

Basic idea is that if the mud pressure in the borehole, Pw, is too high, the well will fracture, and if Pw is too low, the borehole will fail in shear and collapse

The aim of the analysis is to predict the appropriate “mud weight window” that will allow the well to be safely drilled

Wellbore Stability AnalysisWe used the Mogi-Coulomb model, because:

1.Its implementation does not require true-triaxial data; the two coefficients a and b can be obtained direct from the traditional Coulomb “cohesion” and “internal friction” parameters, based on traditional “2 = 3” data.

2.Its mathematical “linearity” allows closed-form expressions to be developed for the permissible mud weight, at least for the cases of vertical or horizontal wells.

Other true-triaxial failure criteria, such as the modified Lade criterion, may also have acceptable accuracy; the present comparison is not meant to be definitive, but merely to show the advantages of using a true-triaxial criterion.

Wanaea Oilfield, NW Continental Shelf of Australia

11.5

12.0

12.5

13.0

13.5

0 10 20 30 40 50 60 70 80 90

Borehole inclination (degrees)

Mud

den

sity

(Ib/

gal)

a=0a=30a=60a=90Actual used

Mogi-Coulomb model

Wanaea 3: vertical boreholeShale formationSo = 435 psi

= 31

n = 0.25 depth = 7028 ftσv = σH = 0.92 psi/ft

σh = 0.72 psi/ft

Po = 0.45 psi/ft

The actual mud density used was 11.85 lb/gal, with significant breakouts in a vertical borehole; this would have been predicted by our model

Pagerungan Island Gas Field, Indonesia

Data from Ramos et al. (SPE 47286, 1998) Well PGA-2, drilled successfully, at a 25 deviation from vertical Shale, So = 1800 psi, = 35, n = 0.3 Field stress system at a depth of about 6000 ft:

σv = 1.0 psi/ft

σh = 0.87 psi/ft

σH = 1.22 psi/ft

Po = 0.45 psi/ft

• Mud density used was 10.5 lb/gal

• Mohr-Coulomb predicted that density should exceed 11.35 for stability

• Mogi-Coulomb predicted that the density needed only to exceed 9.2 lb/gal

• Clear indication of superiority of Mogi-Coulomb over Mohr-Coulomb

Cyrus Reservoir, UK Continental Shelf

Data from McLean and Addis (BP, SPE 20405, 1990)

Rock (sandstone) properties:So = 860 psi = 43.8o

n = 0.3

Reservoir conditions:depth = 8530 ftσv = 1.0 psi/ft

σh = σH = 0.75 psi/ft

Po = 0.45 psi/ft

No borehole failures occurred, and again, our model is consistent with the results

8

9

10

11

12

0 10 20 30 40 50 60 70 80 90

Borehole inclination (degrees)

Mud

den

sity

(Ib/

gal)

Mohr-Coulomb

Mogi-Coulomb

Actual used

ABK Field, Offshore Abu-Dhabi

Horizontal wells drilled in the Hamalah-Gulailah oil reservoirs Field stress system at a depth of about 9705 ft:

σv = 1.0 psi/ft

σh = 1.08 psi/ft

σH = 1.52 psi/ft

Po = 0.45 psi/ft

UCS = 798 psi = 50.2n = 0.3

Two stuck pipes occurred; use of either model would have avoided this

8

9

10

11

12

13

14

15

0 10 20 30 40 50 60 70 80 90

Drilling Direction (degrees)

Mud

den

sity

(Ib/

gal)

Mohr-CoulombMogi-CoulombActual used

Comparison with Modified Lade Criterion

Example from Ewy(SPE 56862, 1998)

Shale formation

c = 705 psi

= 20.2o

n = 0.30

depth = 8000 ft

σv = 0.95 psi/ft

σh = 0.75 psi/ft

Po = 0.45 psi/ft 9.00

10.00

11.00

12.00

13.00

14.00

15.00

0 10 20 30 40 50 60 70 80 90

Borehole inclination (degrees)

Mud

wei

ght (

Ib/g

al)

Mohr-Coulomb

Mogi-Coulomb

Modified Lade

Drucker-Prager

Failure of anisotropic rocksSome rocks of relevance to the oil and gas industry, notably shales, are anisotropic, and their physical properties, such as strength, will depend on the orientation of the rock with respect to the principal stresses

Utica shale

β

Failure on a pre-existing plane of weakness

€

1 −2 =2(So +μ2 )

(1−μcoτb)in2b

Criterion for failure on a plane whose normal vector makes an angle bwith the 1 direction:

σ1

b

σ1

σ2 σ2

σ τ

Jaeger’s “Plane of Weakness” ModelJaeger (1960) assumed that for layered, transversely isotropic rocks, failure will occur either:

• at an angle given by the Coulomb criterion, in which case 1 at failure will be given by 1 = Co + q3, and will be independent of b, or

• along one of the bedding planes, in which case 1 at failure will be given by 1 = 3 + 2(So+μ3)/(1-μcotb)sin2b,and will vary with b

The mode of failure that actually occurs will be determined by the smaller of these two values of 1, i.e., the lower of the two curves

Bossier Shale: Sample Description and PetrologyBossier shale

Elastic anisotropy Eh / Ev = 3.2Strength anisotropy 2.8 (> 2.0)Lithology Argillaceous Mudstone

Clay Minerals abundant IL + I/S, Sparse KA

Fossils moderate carbonate particles

Organic Materials stringers, lenses with amorphous kerogen

Authigenic Minerals Silica, calcite, pyrite

Total organic content (wt %) Low to Moderate, (≈1–2%)

Petrographic Comments very well laminated with burrows and bioturbation

Bossier Shale: Plane of weakness model

Angle

β σ1

Actualσ3

σ1

Predict(Actual - Predicted)2 Ratio

(Actual/Predicted)

90 6,140 0 12,732 43,460,705 0.482

60 1,811 0 6,372 20,805,340 0.28445 6,874 0 7,390 266,703 0.930

45 5,653 0 7,390 3,018,672 0.765

30 6,577 0 12,732 37,889,853 0.5170 16,466 0 12,732 13,939,221 1.293

β σ1Actual

σ3 σ1Predict

(Actual - Predicted)2 Dev in Ratio

90 21,371 1,000 15,615 33,136,907 1.36960 6,672 1,000 8,756 4,344,100 0.76245 8,021 1,000 9,996 3,898,736 0.80230 13,321 1,000 15,615 5,260,296 0.8530 18,106 1,000 15,615 6,207,406 1.160

β σ1Actual

σ3 σ1Predict

(Actual - Predicted)2 Dev in Ratio

90 28,988 3,000 21,379 57,902,153 1.356

75 19,834 3,000 18,377 2,122,188 1.07960 13,751 3,000 13,524 51,449 1.01745 15,678 3,000 15,206 223,068 1.03130 19,367 3,000 21,379 4,046,750 0.90630 17,500 3,000 21,379 15,043,954 0.81915 17,103 3,000 21,379 18,281,214 0.80015 20,956 3,000 21,379 178,636 0.9800 30,060 3,000 21,379 75,365,775 1.406

β σ1Actual

σ3 σ1Predict

(Actual - Predicted)2 Dev in Ratio

90 37,437 6,000 30,025 54,940,208 1.24760 20,330 6,000 20,676 119,763 0.983

45 22,722 6,000 23,021 89,381 0.987

30 25,984 6,000 30,025 16,328,338 0.865

30 24,026 6,000 30,025 35,986,007 0.80030 23,468 6,000 30,025 42,992,069 0.78220 30,920 6,000 30,025 801,323 1.0300 36,951 6,000 30,025 47,971,778 1.231

β σ1Actual

σ3 σ1Predict

(Actual - Predicted)2 Dev in Ratio

90 47,097 10,000 41,553 30,735,115 1.13375 39,812 10,000 39,532 78,225 1.00760 33,197 10,000 30,212 8,910,695 1.09945 32,556 10,000 33,441 783,797 0.97430 36,198 10,000 41,553 28,676,818 0.87115 42,278 10,000 41,553 525,518 1.01710 38,263 10,000 41,553 10,824,587 0.9210 42,614 10,000 41,553 1,125,564 1.026

MSE==> 4,171 0.961

Units in psi

0 10 20 30 40 50 60 70 80 900

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

4

βmin, 57.0

0

5,000

10,000

15,000

20,000

25,000

30,000

35,000

40,000

45,000

50,000

0 10 20 30 40 50 60 70 80 90

σ 1 (

psi)

angle b

Jaeger Plane of Weakness - Bossier shale

σ3= 0

σ3= 1,000

σ3= 3,000

σ3= 6,000

σ3= 10,000

JPW σ3 = 0

JPW σ3 = 1,000

JPW σ3 = 3,000

JPW σ3 = 6,000

JPW σ3 = 10,000

βmin

o = 29.0 degSo = 3750 psiw = 24.0 degSw = 2050 psi

RMSE = 4171 psi

Vaca Muerta Shale: Sample Description and Petrology

Vaca Muerta shaleElastic anisotropy Eh / Ev = 2.0Strength anisotropy 1.6 (< 2.0)

Lithology Calcareous Mudstone

Clay Minerals IL > I/S

Fossils

nondescript shell fragments, hash, echinoderm fragments, charophyte spores, phosphatic bone fragments and calcified algal remnants

Organic Materials Discrete particles

Authigenic minerals Calcite, dolomite, quartzTotal organic content (wt %) Moderate to high, (≈2–8%)

Petrographic Comments Poorly laminated; calcite lenses and fine shell. Cement in matrix

Vaca Muerta Shale: Plane of weakness model

Angle

σ3 β σ1

Actualσ1

Predict(Actual - Predicted)2 Ratio

(Actual/Predicted)

0 60 7,874 8,519 415,109 0.924

0 0 13,730 15,829 4,406,365 0.867

σ3 β σ1Actual

σ1Predict

(Actual - Predicted)2 Ratio(Actual/Predicted)

1,000 90 19,075 18,492 340,004 1.032

1,000 60 12,335 11,087 1,558,522 1.113

1,000 0 20,120 18,492 2,650,705 1.088

σ3 β σ1Actual

σ1Predict

(Actual - Predicted)2 Ratio(Actual/Predicted)

2,500 90 21,845 22,486 411,280 0.9712,500 75 19,935 20,304 136,235 0.9822,500 50 17,010 15,802 1,459,384 1.0762,500 40 18,005 21,266 10,632,787 0.8472,500 30 20,535 22,486 3,807,615 0.913

2,500 20 21,950 22,486 287,630 0.976

2,500 15 20,105 22,486 5,670,643 0.894

σ3 β σ1Actual

σ1Predict

(Actual - Predicted)2 Ratio(Actual/Predicted)

5,000 90 27,675 29,144 2,156,965 0.9505,000 80 33,640 29,144 20,217,064 1.1545,000 60 23,915 21,358 6,537,861 1.1205,000 40 29,355 29,679 105,240 0.9895,000 10 30,630 29,144 2,209,204 1.0515,000 0 30,745 29,144 2,564,287 1.055

σ3 β σ1

Actualσ1

Predict(Actual - Predicted)2 Ratio

(Actual/Predicted)20,000 90 67,000 69,088 4,358,742 0.97020,000 60 61,000 59,876 1,263,062 1.01920,000 0 69,965 69,088 769,550 1.013

MSE ==> 1,851 1.000

Units in psi

0 10 20 30 40 50 60 70 80 900

1

2

3

4

5

6

7

8x 10

4

βmin, 58.0

0

10,000

20,000

30,000

40,000

50,000

60,000

70,000

80,000

0 10 20 30 40 50 60 70 80 90

σ 1 (p

si)

angle b

Jaeger Plane of Weakness - Vaca Muerta

σ3= 0

σ3= 1,000

σ3= 2,500

σ3= 5,000

σ3= 20,000

JPW σ3 = 0

JPW σ3 = 1,000

JPW σ3 = 2,500

JPW σ3 = 5,000

JPW σ3 = 20,000

βmin

o = 27.0 degSo = 4850.0 psiw = 26.0 degSw = 2650.0 psi

RMSE = 1851 psi

3D version of Jaeger’s “plane of weakness” theory

“PA is the line (Coulomb criterion), then each point in this line gives values of the angles θ and which lie on a curve separating regions in which slip can or cannot occur”. (Jaeger, 1962)

(Jaeger et al., 2007)

(Jaeger et al., 2007)

(Jaeger et al., 1962)

Mean Squared Error = 5,216 psiw=30.5o, Sw=4031.3 psio=39.1o, So=7309.3 psi

Test of 3D JPW model: Chichibu schist (data from Mogi)

Summary, I• Most rock failure analysis is conducted using Mohr’s assumption that failure is independent of the intermediate principal stress

• However, lab data show that rocks are stronger when 2 > 3, than when 2 = 3

• The Mogi-Coulomb (i.e., linear Mogi) criterion fits many true-triaxial data sets reasonably well, and its two parameters can be found from classical 2 = 3 laboratory data

• The difference in minimum mud pressure predicted by the Mohr-Coulomb or the Mogi-Coulomb borehole failure criteria can be significant

• The Mogi-Coulomb criterion was found to be more consistent with actual field results, for several cases taken from the SPE literature

• Mogi-Coulomb gives very similar predictions to the modified Lade criterion for wellbore stability predictions

Summary, II• For anisotropic rock such as shales, it would be expected a priori that failure would depend on the intermediate principal stress

• Jaeger’s “plane of weakness” model provides a reasonable fit to failure data on shales collected under 2 = 3 conditions

• (Although not shown in this talk, Pariseau’s five-parameter continuum model for failure of transversely isotropic materials works equally well; see Ambrose’s papers for details)

• Both of these models have shown some success in modelling Mogi’s data on the failure of schist under true-triaxial conditions

• There is a lack of data on failure of shales under true-triaxial conditions, against which to test the various models

References Relation between the Mogi and the Coulomb failure criteria, A. M. Al-Ajmi and R. W. Zimmerman, Int. J. Rock Mech., vol. 42, pp. 431-39, 2005.

Stability analysis of vertical boreholes using the Mogi-Coulomb failure criterion, A. M. Al-Ajmi and R. W. Zimmerman, Int. J. Rock Mech., vol. 43, pp. 1200-1211, 2006.

Stability analysis of deviated boreholes using the Mogi-Coulomb failure criterion, with applications to some oil and gas reservoirs, A. Al-Ajmi and R. W. Zimmerman, Proc. 2006 SPE Asia-Pacific Drilling Tech. Conf., Bangkok, 13-15 Nov. 2006, paper SPE 104035.

A new well path optimization model for increased mechanical borehole stability, A. M. Al-Ajmi and R. W. Zimmerman, J. Petrol. Sci. Eng., vol. 69, pp. 53-62, 2009.

Failure of shales under triaxial compressive stress, J. Ambrose, R. W. Zimmerman, and R. Suarez-Rivera, in Proc. 48th U.S. Rock Mech. Symp., Minneapolis, 1–4 June 2014, paper ARMA 14-131.

Failure of anisotropic shales under triaxial compression and extension, J. Ambrose and R.W. Zimmerman, in Proc. 13th Int. Cong. Rock Mech., Montréal, Canada, 9–13 May 2015, paper ISRM-15-804.