9. fracture gradients

TAMU - Pemex

Well Control

Lesson 9

Fracture Gradients

2

Contents

Allowable Wellbore Pressures

Rock Mechanics Principles

Hooke’s Law, Young’s Mudulus, Poisson’s Ratio

Volumetric Strain, Bulk Modulus, Compressibility

Triaxial Tests

3

Contents – cont’d

Rock Mechanics Principles (con’t.)

Rock Properties from Sound Speed in Rocks’

Mohr’s Circle

Mohr-Coulomb Failure Criteria

4

Fracture Gradients

Read:

“Fracture gradient prediction for the new generation,” by Ben Eaton and Travis Eaton. World Oil, October, 1997.

“Estimating Shallow Below Mudline Deepwater Gulf of Mexico Fracture Gradients,” by Barker and Wood.

5

Lower Bound Wellbore Pressure

Lower bound of allowable wellbore pressure is controlled by:

Formation pore pressure

Wellbore collapse considerations

This sets the minimum “safe” mud weight.

6

Upper bound allowable wellbore pressure may be controlled by:

The pressure integrity of the exposed formations (fracture pressure)

The pressure rating of the casing

The pressure rating of the BOP

Chapter 3 deals with fracture gradient prediction and measurement

Upper Bound Wellbore Pressure

7

Fracture Gradients

May be predicted from:

Pore pressure (vs. depth)

Effective stress

Overburden stress

Formation strength

8

Rock Mechanics

How a rock reacts to an imposed stress, is important in determining Formation drillability

Perforating gun performance

Control of sand production

Effect of compaction on reservoir performance

Creating a fracture by applying a pressure to a wellbore!!!

9

Elastic Properties of Rock

10


11


The vertical stress at any point can be calculated by:

21

4

d

F

A

F aaa

1

21a L

LL

1

21

d

ddtr

The axial and transverse strains are:

12


Hooke’s Law:

= E

Young’s Modulus:

E = /e = (F/A)/(L/L)

E = (F*L)/(A*L)

13

Hooke’s LawElastic Limit

Permanent strain or plastic deformation

Failure

14

Typical Elastic Properties of Rock

15

Poisson’s Ratio

Poisson’s Ratio

= transverse strain/axial strain

= -(x/z)

Over the elastic range, for “most metals”, ~ 0.3

Over the plastic range, increases, and may reach the limiting value of 0.5

16

Volumetric Strain

i

ifV V

VV

17

Bulk Modulus and Compressibility values in rock

18

Shear Modulus (G)

G is the ratio of shear stress to shear strain

G is intrinsically related to Young’s modulus and Poisson’s ratio

G = = E/[2*(1+)]

19

Bulk Modulus (Kb)

Kb is the ratio between the average normal stress and the volumetric strain

Kb can be expressed in terms of Young’s modulus and Poisson’s ratio.

Kb = average normal stress/ volumetric strain

Kb = E/[3*(1-2) = [(x+ y+z)/3]/v

20

Bulk Compressibility (cb)

cb is the reciprocal of the bulk modulus

cb = 1/Kb

= 3*(1-2)/E

= v / [(x+ y+z)/3]

21

Metals and Rocks

Metallic alloys usually have well-defined and well-behaved predictable elastic constants.

22

Metals and Rocks

In contrast, rock is part of the disordered domain of nature. It’s response to stress depends on (e.g.):

Loading history

Lithological constituents

Cementing materials

Porosity

Inherent defects

23

Metals and Rocks

Even so, similar stress-strain behavior is observed.

Triaxial tests include confining stress

24

Rock Behavior Under Stress

From 0-A, microcracks and other defects are closed

From A-B, linear elastic behavior is observed

Beyond B, plastic behavior may occur.

25

Young’s Modulus for a Sandstone

Ei = Initial Modulus

= initial slope of

curve

Es = secant modulus = (Total Stress/Total Strain) at any point

Et = instantaneous slope at any specific stress (tangent method)

26

Transverse Strains for SS in Fig. 3.5

Young’s Modulus & Poisson’s Ratio are stress dependent.

27

Example 3.1

Using Fig. 3.5, determine Young’s Modulus and Poisson’s ratio at an axial stress of 10,000 psi and a confining stress of 1,450 psi.

From Fig 3.5, the given stress conditions are within the elastic range of the material (e.g. linear stress-strain behavior)

28

SolutionEt = d/d = (15,000-5,000) /(0.00538-0.00266)

Et = 3.7*106 psi

= -x/z

= -(-0.00044/0.00404)

= 0.109

29

Rock Properties

Rocks tend to be more ductile with increasing confining stress and increasing temperature

Sandstones often remain elastic until they fail in brittle fashion.

Shales and rock salt are fairly ductile and will exhibit substantial deformation before failure

30

Rock Properties

Poisson’s ratio for some plastic formations may attain a value approaching the limit of 0.5

Rocks tend to be anisotropic, so stress-strain behavior depends on direction of the applied load.

31

1. An alternate form of Eq. 3.6 gives the dynamic Poisson’s ratio:

2. Use Eq. 3.7 to determine the dynamic Young’s Modulus:

)240.01(*407,7*38.2*0268.0 2 E

psiE 610*34.4

)1(0268.0 2 sb vE

32

Fracturing is a static or quasistatic process so elastic properties based on sonic measurements may not be valid.

33

We can orient a cubic element under any stress state such that the shear stresses along the six orthogonal planes vanish. The resultant normal stresses are the three “principal stresses”

3 = minimum principal stress2 normal to the page is the intermediate principal stress and is considered to be inconsequential to the failure analysis

Along an arbitrary plane , a shear stress will exist.

34

231

max

2sin2

31a

13 a

2

max

2cos22

3131a

a

35

tanff c

c = cohesion

= angle of internal friction

37

231

max

39

Note that the failure plane approaches 45o with increasing confining stress

40

Hydraulic Fracturing

Hydraulic fracturing while drilling results in one form of lost circulation (loss of whole mud into the formation).

Lost circulation can also occur into:

vugs or solution channels

natural fractures

coarse-grained porosity

41

For a fracture to form and propagate:

The wellbore pressure

must be high enough to overcome the tensile strength of the rock.

must be high enough to overcome stress concentration at the hole wall

must exceed the minimum in situ rock stress before the fracture can propagate to any substantial extent.

42

In Situ Rock Stresses

The simplest model assumes the subsurface stress field is governed solely by the rock’s linear elastic response to the overburden load.

When loaded, the block would strain in the x and y transverse directions according to Hooke’s Law.

43


horizontal signifies Hsubscript the where

isotropic, is material the If

Hyx

zxyy

zyxx

EEE

andEEE

E

zyxx

44


EE

zH

1

z

1H

ThusEEE

zHHH

Constraining the block on all sides prevents lateral strain.

Setting H = 0,

Eliminating E and rearranging yields the fundamental relationship

45


The above stressed block is analgous to a buried rock element if the material assumptions remain valid.

Using the book’s nomenclature for overburden stress and substituting Terzaghi’s effective stress equation leads to:

46


ppobH

pobpH

pp

pp

1

1

desired. if term pressure pore the to

applied be mayconstant cityporoelasti The

z

1Hpe ps

,TerzaghiFrom

(with s = 1)

47

Fig. 3.13

Rock properties assumed constant

with depth

48

ob is the max. principal stress

Fig. 3.14

Failure (fracture) occurs perpendicular to the least principal stress

49

H > ob can be created by

• Tectonic forces

• Post-depositional erosion

• Glacial action or melting of glacier

H might be locked in while ob reduces

Fracture Pressure

Fig. 3.15

50

Lower ob

Is figure drawn correctly? Or should rock sample come from right side fault?

Effect of tectonic movements on stresses

Fig. 3.16

51

Effect of topography on obFig. 3.17

52

Overburden stress is not significantly changed by abnormal pressure

Small Tolerance

Under abnormal pore pressure, the difference between pore pressure and the least horizontal stress (fracture pressure) get very small.

53

Subnormal pressures have little effect on overburden stress …

… But, result in a decrease in fracture pressure

54

Stress concentrations around a borehole in a uniform stress field

Tension

Additional compression

9. fracture gradients

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