Reservoir DepletionReservoir Depletion

E�ects of reservior depletionE�ects of reservior depletion

© Cambridge University Press (Fig. 10.21, pp. 380)Zoback, Reservoir Geomechanics

Estimating stress changes in depletingEstimating stress changes in depletingreserviorsreserviors

= = = + α (1 − )SHor SHmax Shmin

ν

1 − νSv Pp

ν

1 − ν

= α during productiondSHor

dPp

1 − 2ν

ν − 1

Δ = α ΔSHor

1 − 2ν

ν − 1Pp

Taking and ν = 1

4α = 1

Δ ∼ ΔSHor

2

3Pp

Comparison of theory and observationComparison of theory and observation

© Cambridge University Press (Fig. 12.2, pp. 382)Zoback, Reservoir Geomechanics

Production induced faultingProduction induced faulting

= ( + μ− (Pp − Δ )Sv Pp

( − Δ ) − (Pp − Δ )Shmin Shmin Pp

+ 1μ2‾ ‾‾‾‾‾√ )2

Simpli�cation leads to

= 1 −ΔSHmin

ΔPp

1

( + μ+ 1μ2‾ ‾‾‾‾‾√ )2

For μ = 0.6

= 0.67ΔSHmin

ΔPp

Reservoir space plotReservoir space plot

© Cambridge University Press (Fig. 12.4a, pp. 386)Zoback, Reservoir Geomechanics

GOM Field XGOM Field X

© Cambridge University Press (Fig. 12.3ab, pp. 383)Zoback, Reservoir Geomechanics

© Cambridge University Press (Fig. 12.4b, pp. 387)Zoback, Reservoir Geomechanics

Valhall �eld in North SeaValhall �eld in North Sea

© Cambridge University Press (Fig. 12.5, pp. 388)Zoback, Reservoir Geomechanics

Stress rotations with depletionStress rotations with depletion

Original Depleted

© Cambridge University Press (Fig. 12.6ab, pp. 391)Zoback, Reservoir Geomechanics

Rotation angle, Rotation angle, near the fault due to near the fault due todepletiondepletion

with

and

γ

γ = ( )1

2tan−1

Aq sin(2θ)

1 + Aq cos(2θ)

A =ΔShmin

ΔPp

q =ΔPp

−SHmax Shmin

Deformation in depleting reservoirsDeformation in depleting reservoirs

SubsidenceSubsidence

Source: -- Public DomainWikimedia Commons

Recall: Compaction curvesRecall: Compaction curves

© Cambridge University Press (Fig. 12.11, pp. 399)Zoback, Reservoir Geomechanics

Recall: End-cap modelsRecall: End-cap models

© Cambridge University Press (Fig. 4.19, pp. 119)Zoback, Reservoir Geomechanics

p = ( + + ) −1

3S1 S2 S3 Pp

= (( − + ( − + ( − ))q21

2S1 S2)2 S2 S3)2 S1 S3)2

− p + = 0M 2p2 M 2p0 q2

9P2p + (1 + ) ( + + )

9

M 2S2v S2

Hmax S2

hmin

+ (2 − ) ( + + )9

M 2SvSHmax SvShmin SHmaxShmin

+ 9 − 3(2 + )( + + ) = 0Ppp0 Pp p0 Sv SHmax Shmin

M = From Mohr-Coulomb assuming   is negligible6μ

3 − μ+ 1μ2‾ ‾‾‾‾‾√

C0

Deformation Analysis in Reservoir SpaceDeformation Analysis in Reservoir Space(DARS) model of GOM Field X(DARS) model of GOM Field X

© Cambridge University Press (Fig. 12.12, pp. 402)Zoback, Reservoir Geomechanics

Permeability change as a function ofPermeability change as a function ofporosity changeporosity change

© Cambridge University Press (Fig. 12.13a, pp. 404)Zoback, Reservoir Geomechanics

Modi�ed Kozeny-Carman relationship

For change in permeability

Including a factor for grain size reduction

κ = B(ϕ − ϕc)

3

(1 + − ϕϕc )2d 2

=κ

κi ( )ϕ − ϕc

−ϕi ϕc

3

( )1 + −ϕc ϕi

1 + − ϕϕc

2

Γ =1 − d/di

1 − ϕ/ϕi

= (1 − Γ(1 − ))κ

κi ( )ϕ − ϕc

−ϕi ϕc

3

( )1 + −ϕc ϕi

1 + − ϕϕc

2 ϕ

ϕi

Permeability loss with depletion in GOMPermeability loss with depletion in GOMField ZField Z

© Cambridge University Press (Fig. 12.14, pp. 408)Zoback, Reservoir Geomechanics

Idealized resevoir studyIdealized resevoir study

© Cambridge University Press (Fig. 12.15, pp. 410)Zoback, Reservoir Geomechanics

Viscous compaction in GOM Field ZViscous compaction in GOM Field Z

© Cambridge University Press (Fig. 12.16, pp. 411)Zoback, Reservoir Geomechanics

Geertsma (1973) dispacement solutionGeertsma (1973) dispacement solution

© Cambridge University Press (Fig. 12.17, pp. 414)

(r, 0)uz

(r, 0)ur

= − (1 − ν) Δ V1

πcm

D

( + )r2 D23/2

Pp

= + (1 − ν) Δ V1

πcm

D

( + )r2 D23/2

Pp

Zoback, Reservoir Geomechanics

Subsidence and horizontal dispacementSubsidence and horizontal dispacement

© Cambridge University Press (Fig. 12.17, pp. 414)Zoback, Reservoir Geomechanics

Leeville case studyLeeville case study

© Cambridge University Press (Fig. 12.18, pp. 416)Zoback, Reservoir Geomechanics