EXPLORATION GEOPHYSICS

THE EXPLORATION TASK

PLAN EXPLORATION

APPROACH FOR A MATURE TREND

GATHER DATA FOR A MATURE TREND

DEVELOP PLAY

PROSPECT FRAMEWORK

INITIAL DATA GATHANAL AND

PROJECT PLANING FOR A

FRONTIER TREND

NEW DATA GATHERING FOR A FRONTIER TREND

MAKE PLAY/PROSPECT ASSESSMENT

COMMUNICATE ASSESSMENT TO MANAGEMENT

PREPARE PRELOCATION REPORT

DRILLING

EXPLORATION GEOPHYSICS

ElasticityElasticity

SourceSource

Petroleum related rock mechanicsPetroleum related rock mechanics

Elsevier, 1992Elsevier, 1992

ElasticityElasticityElasticityElasticity

Definition: The ability to resist and Definition: The ability to resist and recover from deformations recover from deformations produced by forces.produced by forces.

It is the foundation for all aspects of It is the foundation for all aspects of Rock MechanicsRock Mechanics

The simplest type of response is The simplest type of response is one where there is a linear relation one where there is a linear relation between the external forces and the between the external forces and the corresponding deformations.corresponding deformations.

StressStress

defines a force field on a materialdefines a force field on a material

Stress = Force / Area (pounds/sq. in. or Stress = Force / Area (pounds/sq. in. or psi)psi)

F / AF / AFF

Area: AArea: A

StressStressStressStress

In Rock Mechanics the sign convention In Rock Mechanics the sign convention states that the compressive stresses states that the compressive stresses are positive. are positive. 

Consider the cross section area at b, Consider the cross section area at b, the force acting through this cross the force acting through this cross section area is F (neglecting weight of section area is F (neglecting weight of the column) and cross sectional area is the column) and cross sectional area is A’. A’ is smaller than A, therefore stress A’. A’ is smaller than A, therefore stress ’ = F/A’ acting at b is greater than ’ = F/A’ acting at b is greater than acting at aacting at a

StressStress

W

F

F

F

a

b

c

A

A’

A’’

AreaAreaLoadLoad

StressStressStressStress

stress depends on the position within stress depends on the position within the stressed sample.the stressed sample.

Consider the force acting through Consider the force acting through cross section area A’’. It is not normal cross section area A’’. It is not normal to the cross section. We can to the cross section. We can decompose the force into one decompose the force into one component Fcomponent F

nn normal to the cross normal to the cross

section, and one component Fsection, and one component Fpp that is that is

parallel to the section. parallel to the section.

StressStressStressStress

Fp

Fn

F

Decomposition of forcesDecomposition of forces

StressStressStressStress

The quantity The quantity = F = Fnn /A’’ is called the /A’’ is called the

normal stress, while the quantitynormal stress, while the quantity    = F= F

pp / A’’ is called the shear stress. / A’’ is called the shear stress.

Therefore, there are two types of Therefore, there are two types of stresses which may act through a stresses which may act through a surface, and the magnitude of each surface, and the magnitude of each depend on the orientation of the depend on the orientation of the surface.surface.

General 3D State of General 3D State of Stress Stress in a Reservoirin a Reservoir

x, y, z Normal stresses

xy, yz, zx Shear stresses

x

y

z

yx

yz

zy

zx

zxxy

StressStressStressStress

= = xx xyxy xz xz

yxyx yy yzyz

zx zx zyzy zz

  

Stress tensorStress tensor

Principal StressesPrincipal Stresses

Normal stresses on planes where shear stresses are zero

v

H

h

Principal StressesPrincipal Stresses

In case of a reservoir,

= v Vertical stress,

= h Minimum horizontal stresses

= H Maximum horizontal stresses

v

H

h

Types of StressesTypes of Stresses

Tectonic Stresses: Due to relative Tectonic Stresses: Due to relative displacement of lithospheric platesdisplacement of lithospheric plates

Based on the theory of earth’s tectonic platesBased on the theory of earth’s tectonic plates Spreading ridge: plates move away from Spreading ridge: plates move away from

each othereach other Subduction zone: plates move toward each Subduction zone: plates move toward each

other and one plate subducts under the other and one plate subducts under the otherother

Transform fault: Plates slide past each otherTransform fault: Plates slide past each other

Types of StressesTypes of Stresses

Gravitational Stresses: Due to the Gravitational Stresses: Due to the weight of the superincumbent rock weight of the superincumbent rock massmass

Thermal Stresses: Due to temperature Thermal Stresses: Due to temperature variationvariation

Induced, residual, regional, local, far-field, Induced, residual, regional, local, far-field, near-field, paleo ...near-field, paleo ...

Impact of In-situ Impact of In-situ Stress Stress

Important input during planning stage Fractures with larger apertures are

oriented along the maximum horizontal stress

Natural fracturesNatural fractures

StrainStrain

(x, y, z)(x’, y’, z’)

Initial Position Shifted Position

StrainStrain

x’ = x – u y’ = y – v z’ = z – w If the displacements u, v, and w are

constants, i.e, they are the same for every particle within the sample, then the displacement is simply a translation of a rigid body.

StrainStrain

Another simple form of displacement is the rotation of a rigid body.

If the relative positions within the sample are changed, so that the new positions cannot be obtained by a rigid translation and/or rotation of the sample, the sample is said to be strained. (figure 8)

StrainStrain

L

O

P

L’

O’

P’

Initial Position Shifted Position

StrainStrain

Elongation corresponding to point O and the direction OP is defined as

= (L – L’)/L   sign convention is that the elongation is

positive for a contraction. The other type of strain that may occur

can be expressed by the change of the angle between two initially orthogonal directions. (Figure 9)

StrainStrain

PO

Q

Initial Position

P

Q

O

Shifted Position

StrainStrain

= (1/2)tan  

is called the shear strain corresponding to point O and the direction OP. We deal with infinitesimal strains.

The elongation (strain) in the x-direction at x can be written as

x = u/x

StrainStrain

The shear strain corresponding to x-direction can be written as

 xy = (u/y + v/x)/2

Strain tensor Principal strains

Elastic ModuliElastic Moduli

DLL’ D’

F

Y

X

Schematic showing deformation under loadSchematic showing deformation under load

Elastic ModuliElastic Moduli

When force F is applied on its end surfaces, the length of the sample is reduced to L’.

The applied stress is then x = F/A,

The corresponding elongation is = (L – L’)/L

The linear relation between x and x, can

be written as x = Ex

Elastic ModuliElastic Moduli

This equation is known as Hooke’s law The coefficient E is called Young’s

modulus. Young’s modulus belongs to a group of

coefficients called elastic moduli. It is a measure of the stiffness of the

sample, i.e., the sample’s resistance against being compressed by a uniaxial stress.

Elastic ModuliElastic Moduli

Another consequence of the applied stress x (Figure 10) is an increase in the

width D of the sample. The lateral elongation is y = z = (D – D’)/D. In general

D’ > D, thus y and z become negative.

The ratio defined as = -y/x is another

elastic parameter known as Poisson’s ratio. It is a measure of lateral expansion relative to longitudinal contraction.

Elastic ModuliElastic Moduli Bulk modulus K is defined as the ratio of

hydrostatic stress p relative to the

volumetric strain v. For a hydrostatic

stress state we have p = 1 = 2 = 3 while

xy = xz = yz = 0. Therefore

K = p/v = + 2G/3 1  

K is a measure of sample’s resistance against hydrostatic compression. The inverse of K, i.e., 1/K is known as compressibility

Elastic ModuliElastic Moduli Isotropic materials are materials whose

response is independent of the orientation of the applied stress. For isotropic materials the general relations between stresses and strains may be written as:

 x = ( + 2G) x + y + z

y = x + ( + 2G)y + z

z = x + y + ( + 2G)z

xy = 2Gxy xz = 2Gxz yz = 2Gyz

Elastic ModuliElastic Moduli Expressing strains as function of stresses Ex = x - (y + z)

Ey = y - (x + z)

Ez = z - (x + y)

Gxy = (1/2)xy

Gxz = (1/2)xz

Gyz = (1/2)yz

Elastic ModuliElastic Moduli In the definition of Young’s modulus and

Poisson’s ratio, the stress is uniaxial, i.e., z = y = xy = xz = yz = 0. Therefore

 E = x/x = G (3 + 2G)/ ( + G) 2  

= -y/x = /(2( + G))   3 Therefore from equations (1, 2, and 3),

knowing any two of the moduli E, , , G and K, we can find other remaining moduli

Elastic ModuliElastic Moduli For rocks, is typically 0.15 – 0.25. For

weak, porous rocks may approach zero or even become negative. For fluids, the rigidity G vanishes, which according to equation (3) implies ½. Also for unconsolidated sand, is close to ½.