history matching and rock mechanics 20 years

8/18/2019 History Matching and Rock Mechanics 20 Years

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HISTORY MATCHING

&

ROCK MECHANICS

Øystein PettersenReservoir Mechanics 20 years anniversary 30.08.02


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Real History Matching

Original photography

Official photography after Trotskij had fallen into disgrace

Now you see him -- now you don't


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What is History Matching?

Standard procedure:

From a system of equations L(u(x), a, x) = 0 with I.C. and B.C.find the solution u(x), with known parameters a.

History matching:

From a system of equations L(u(x), a, x) = 0 with I.C. and thesolution u(x) known, determine the boundary conditions andparameter set a.

Normally some of the B.C.'s and (part of) a will be known.


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History Matching in Practice

u(x) will not be known as a continuous function of acontinuous space/time variable, but only at a few points in

space at a few times. Mathematically, the solution isunknown almost everywhere.

The achieved B.C. "solution" cannot be unique.

In addition the known (sparse) solution u i(x i) is not always

reliable.(standard uncertainty, allocation errors, coarse errors).

The observed quantity may reflect a realisation of thesolution which is not possible to model, and thereforewould be wrong to honour.

Our task is to critically utilize the provided historical data inthe best possible manner, such that the parameters wedetermine by the H.M. process are the most likely ones froma physical point of view. (Diff icult, diff icult,....)


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Heterogeneity: Sandstone Outcrop


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Idealised Heterogeneous X-Section


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After Upscaling to a Simulation Grid


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Heterogeneity Modelling (generic example Field A)

Cum. oil produced

Deterministiccase

Different heterogeneity

cases


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Field A History Matching:

Region Pressure -- Best result after > 200 runs

RFT-pressures ("certain")

Less certain pressuremeasurements

Simulated Region Pressure (bar)

275

300

250

1987 1988 1989 1990 1991 1992 1993 1994 1995 1996


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210220230240250260270275280285290295300310

Pressure (bars)

0,96

0,97

0,98

0,99

1

P o r e V o l u m e M u l t i p l i e r

c f = 3E-05

Example rock compaction curves

(ROCK constant and Irreversible ROCKTAB)


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Classification of Compaction

Reversible elastic

Irreversible elastic

Pp (Pore pressure)

Compaction byOverburden

Decline period Changing pressure trends

Pc (Conf. press.)

With hysteresis

N o r m a

l i z e d P o r e V o l u m e


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Time-dependent compaction (creep) --

1 cm reservoir compaction pr. year

Pure pressure-dependant compaction

Time- & pressure-dep. compaction

RFT-pressures ("certain")

Less certain pressuremeasurements

Simulated Region Pressure (bar)

275

300

250

1987 1988 1989 1990 1991 1992 1993 1994 1995 1996


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The Influence of Rock Mechanics

By tweaking compaction curves on a per-region basis,pore volume change can be modelled to honour history

A better approach would be to compute the changes fromthe stress-strain relations

We should expect that compaction also influences permeabil ity,

including permeability isotropy

What if the soil doesn't behave like a stable soil / rock at all?

Are Darcy's law and the flow equations still valid?


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0 100 200 300 400

p' (bars)

0

200

400

600

800

1000

1200

E (

M P a )

0

0,1

0,2

0,3

0,4

0,5

Field A -- Weak Sandstone

0 50 100 150 200 250

p' (bars)

200

400

600

800

1000

1200

E ( M P a )

0

0,1

0,2

0,3

0,4

0,5

Field B -- Weak Sandstone

0 50 100 150 200 250

p' (bars)

0

200

400

600

800

1000

1200

E ( M

P a )

0

0,1

0,2

0,3

0,4

0,5

Field C -- Unconsolidated Sand

Elastic moduli

0 50 100 150 200 250 300

p' (bars)

0

100

200

300

400

500

600

E ( M

P a )

0

0,05

0,1

0,15

0,2

Field B -- Unconsolidated Sand

E

E

E

E


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Elasticity with permanent deformation (" Irreversible")

ε

ρ

Loading

Unloading

"Ideal"

irrev.


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Permeability vs stress

Unconsolidated Sand Field A

69 138 207 276 345 414 483 552 621

Mean effective stress [bars]

0

20

40

60

80

100

N o r m a l i s e d p e

r m e a b i l i t y ( %

)

1. load

1. unload

2. load

2. unload

Initial permeabil ity (100%): 4138 mD


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Permeabil ity vs stress

Unconsolidated Sand Field B

69 138 207 276 345 414 483 552 621


0

20

40

60

80

100

N o r m a l i s e d p e


)

1. load

1. unload

2. load

2. unload

Initial permeability (100%): ~1.5 D


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Permeability vs stress

Unconsolidated Sand Field C

69 138 207 276 345 414 483 552 621


0

20

40

60

80

100

N

o r m a l i s e d p e


)

1. load

1. unload

2. load

2. unload

Initial permeabil ity (100%): ~3 D


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Permeabilities from transient test analysis

Field A

270275280285290295300

Reservoir pressure (perm. press. gauge)

3 000

3 500

4 000

4 500

5 000

5 500

6 000

Permeability, mD

Production tests during 2 years


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Soil strength

0

ρ

A

B

C

Dρ

0

C0

Elastic Ductile Brittle

OA: Increasing -- slightly convex AB: Increasing -- linear

OAB: Elastic (appr. linear)

BC: Increasing -- concave

Ductile: Can endure permanent

deformation without losing

ability to withstand load.

CD: Decreasing

Brittle: Ability to withstand

loading decreases with increasing deformation.

"Brittleness" : Largest slope

angle on CD.

(In theory a continuous processthrough CD. In practice sudden

failure occurs at some point on

CD: Total loss of cohesion across

a plane.)

ρ

0: Yield point (transit ion elastic ductile)

(typically B 2/3 C)

C0: Uniaxial compressive strength

S il t th


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Soil strength:

Hysteresis and deformation

0

ρ

B

C

D

P

Q

R

S

T

U

0

OB: Perfect elastic ity (No irreversible behaviour)

BC: Irreversible changes occur

Successive loading / unloading

by different curves. E.g. PQ

gives a permanently set

deformation 0.

QR PQ, but R sti ll on BC.

CD: Unloading curve ST often

results in large permanent deformation. The following load

sequence TU meets CD on a

lower stress level than S.

Characteristic for brittle matr's,

but normally hidden by failure

near point C.


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Elasto-plasticity and failure

elastic e l a s t

o - p l

a s t i c

init ial yield surface

current yield surface

failure surface

(envelope of reachable yield surfaces)

Non-achievable

states

ρ-space

Yield criterion: Specified surface inρ

-space where failure occurs.

In elasto-plasticity the yield surface can change with t ime, as the

material hardens.

Ideal plasticity: Initial yield surface = failure surface.

Sand / soil strength (Creep)


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Sand / soil strength (Creep)

Elasticity -> Plasticity -> Failure

0200

400600

8001 000

1 2001 400

1 600

Differential s tress (bar)

0

4

8

Static strain rate *E-8 (1/s)

Pc = 138bar

Pc = 207bar

Pc = 276bar Pc = 345bar

Field A, 4343m, Pp = 1bar

1 500 2 000 2 500 3 000 3 500

Differential s tress (bar)

0

1

2

3

4



Field B, 4494m, Pp = 138bar

80 120 160 200 240

Differential stress (bar)

4

8

12

16

20



Pc = 414bar

Field C, 1919m, Pp = 1bar

0 50 100 150 200 250 300 350

Differential stress (bar)

0

2

4

6

8


Pc = 207bar Pc = 276bar Pc = 414bar

Field C, 1919m, Pp = 138bar

Fl id fl i i f t (j i t )


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Fluid flow in microfractures (joints)generated by impurities

FF

Shear stress can imply

Crushing of the smallest asperities

Opening of the joint

Macroscopic enhanced perm.

in the major stress direction.

Rotation of flow direction

Field A Vertical elastic displacement


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10000 15000 20000 25000

-0,15

-0,1

-0,05

0

DEZ01

DEZ02

DEZ03DEZ04

DEZ05

DEZ06

DEZ07

Field A, Vertical elastic displacement

vs. time at top reservoir level

Field A Vertical Elastic Displacement


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PARAMETER:

DEZ07

TYPE

REAL

-0.378989

-0.294638

-0.210288

-0.125937

-0.041586

0.042764

0.127115

Field A, Vertical Elastic Displacement

N-S section, Time 1517 days (4 yrs 2 mnths)


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Conclusion

Although Newton's laws of motion are perfectlyadequate for everyday physics,

no-one would even consider using them at the

Planck scale, or at galaxy scale

history matching and rock mechanics 20 years

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