Title: ock Mechanics Study of the Ghawar Khuff Reservoirs
1EXPLORATION GEOPHYSICS
2THE EXPLORATION TASK
INITIAL DATA GATHANAL AND PROJECT PLANING FOR A
FRONTIER TREND
PLAN EXPLORATION APPROACH FOR A MATURE TREND
DEVELOP PLAY PROSPECT FRAMEWORK
NEW DATA GATHERING FOR A FRONTIER TREND
GATHER DATA FOR A MATURE TREND
MAKE PLAY/PROSPECT ASSESSMENT
COMMUNICATE ASSESSMENT TO MANAGEMENT
PREPARE PRELOCATION REPORT
DRILLING
3EXPLORATION GEOPHYSICS
4Elasticity
Source Petroleum related rock
mechanics Elsevier, 1992
5Elasticity
- Definition The ability to resist and recover
from deformations produced by forces. - It is the foundation for all aspects of Rock
Mechanics - The simplest type of response is one where there
is a linear relation between the external forces
and the corresponding deformations.
6Stress
- defines a force field on a material
- Stress Force / Area (pounds/sq. in. or psi)
- s F / A
F
Area A
7Stress
- In Rock Mechanics the sign convention states that
the compressive stresses are positive. - Consider the cross section area at b, the force
acting through this cross section area is F
(neglecting weight of the column) and cross
sectional area is A. A is smaller than A,
therefore stress ? F/A acting at b is greater
than ? acting at a
8Stress
Area
Load
9Stress
- stress depends on the position within the
stressed sample. - Consider the force acting through cross section
area A. It is not normal to the cross section.
We can decompose the force into one component Fn
normal to the cross section, and one component Fp
that is parallel to the section.
10Stress
Decomposition of forces
11Stress
- The quantity ? Fn /A is called the normal
stress, while the quantity - Â ? Fp / A is called the shear stress.
- Therefore, there are two types of stresses which
may act through a surface, and the magnitude of
each depend on the orientation of the surface.
12General 3D State of Stress in a Reservoir
sx , sy , sz Normal stresses ?xy , ?yz , ?zx
Shear stresses
sx
sz
sy
13Stress
- ? ?x ?xy ?xz
- ?yx ?y ?yz
- ? zx ?zy ?z
- Â
- Stress tensor
14Principal Stresses
sv
Normal stresses on planes where shear stresses
are zero s1 gt s2 gt s3
sh
sH
15Principal Stresses
sv
In case of a reservoir, s1 sv Vertical
stress, s2 sh Minimum horizontal stresses
s3 sH Maximum horizontal stresses
sh
sH
16Types of Stresses
- Tectonic Stresses Due to relative displacement
of lithospheric plates - Based on the theory of earths tectonic plates
- Spreading ridge plates move away from each other
- Subduction zone plates move toward each other
and one plate subducts under the other - Transform fault Plates slide past each other
17Types of Stresses
- Gravitational Stresses Due to the weight of the
superincumbent rock mass - Thermal Stresses Due to temperature variation
- Induced, residual, regional, local, far-field,
near-field, paleo ...
18Impact of In-situ Stress
- Important input during planning stage
- Fractures with larger apertures are oriented
along the maximum horizontal stress
19Natural fractures
20Strain
(x, y, z)
(x, y, z)
Shifted Position
Initial Position
21Strain
- 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.
22Strain
- 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)
23Strain
24Strain
- Elongation corresponding to point O and the
direction OP is defined as - ? (L L)/L Â
- 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)
25Strain
26Strain
- ? (1/2)tan? Â
- 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
27Strain
- The shear strain corresponding to x-direction can
be written as - Â ?xy (?u/?y ?v/?x)/2
- Strain tensor
- Principal strains
28Elastic Moduli
F
X
D
L
D
L
Y
Schematic showing deformation under load
29Elastic 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 E?x
30Elastic Moduli
- This equation is known as Hookes law
- The coefficient E is called Youngs modulus.
- Youngs modulus belongs to a group of
coefficients called elastic moduli. - It is a measure of the stiffness of the sample,
i.e., the samples resistance against being
compressed by a uniaxial stress.
31Elastic 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 gt D, thus ?y and ?z become
negative. - The ratio defined as ? -?y/?x is another
elastic parameter known as Poissons ratio. It is
a measure of lateral expansion relative to
longitudinal contraction.
32Elastic 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 Â
- K is a measure of samples resistance against
hydrostatic compression. The inverse of K, i.e.,
1/K is known as compressibility
33Elastic 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 - Â ?x (? 2G) ?x ??y ??z
- ?y ??x (? 2G)?y ??z
- ?z ??x ??y (? 2G)?z
- ?xy 2G?xy ?xz 2G?xz ?yz 2G?yz
34Elastic Moduli
- Expressing strains as function of stresses
- E?x ?x - ?(?y ?z)
- E?y ?y - ?(?x ?z)
- E?z ?z - ?(?x ?y)
- G?xy (1/2)?xy
- G?xz (1/2)?xz
- G?yz (1/2)?yz
35Elastic Moduli
- In the definition of Youngs modulus and
Poissons ratio, the stress is uniaxial, i.e., ?z
?y ?xy ?xz ?yz 0. Therefore - Â E ?x/?x G (3? 2G)/ (? G) 2 Â
- ? -?y/?x ?/(2(? G)) Â 3
- Therefore from equations (1, 2, and 3), knowing
any two of the moduli E, ?, ?, G and K, we can
find other remaining moduli
36Elastic 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.