Jean-Laurent Mallet. Geomodeling

438

CHAPTER

8.

ELEMENTS

OF

STRUCTURAL

GEOLOGY

Rigid-Region-Translation

constraint

Sometimes, geologists

may

identify

regions

of the

model

that

have been merely

translated,

by an

unknown vector, without

any

deformation through geolog-

ical

time.

For

each such region

R C 0, the

rigid translation assumption

induces

the

following

constraints

on the field of

restoration vectors:

constant

or,

equivalently

It is

relevant

to

note

that

each

of

these constraints can,

in

turn,

be

transformed

into

the

following

normalized

DSI

constraint

c =

c(y,

a,

R]

defined

by

where

A"(0)

and

b

c

are

such

that

As

one can

easily check

that

\\A

C

\\

= 1,

this implies

that

the DSI

constraint

so

defined

is

actually normalized.

According

to

equation (4.51),

it can be

concluded

that

the

terms

^(a)

and

r^(a|r),

occurring

in the DSI

equation

(4.46)

at

node

a

e

R and

associated

with

the

constraint

c =

c(i>,

ce,

R), are

such

that

Observed-Strain

constraint

There

are

situations

[7]

where geologists

can

evaluate

an

approximated value

£

X

i

X:

/

(a) of a

component

of the

strain

tensor

at a

given node

a.

Using

the

notations introduced

in

section (8.6.3)

and

(8.6.5),

implies

that

the field of

restoration vectors should honor

the

following

constraint

for all the

3-cells

C =

C(a]

surrounding

a and not cut by a

fault:

If

we

note

#

X

i

X

j

(r\a,C)

the

function

of r

defined

as

follows

8.6.

UNFOLDING

A

STACK

OF

LAYERS

then

the

constraint (8.119)

can be

rewritten

as

For

example,

in the

case where

x

z

=

x and

x

j

=

y,

which

is

similar

to the

case

of the

Minimum-Deformation

constraint introduced

in

section (8.6.5),

the

above constraint generates pairs

of

terms

{7«(

c

)(/3)>r^/x(/3|r)}

and

{7w

c

)(/?)

?

r^/

c

)(/?

r)}

occurring

in the

local

form

of the

DSI

equation:

Summing

these terms

for all

3-cells

surrounding

the

node

a and not cut

by

a

fault

generates

the

following

composite terms

and

8.6.7

Deformation

analysis

Let us now

assume

that

the

deformed region

is

associated with

the

parametric

coordinate system

defined

by

equation (8.78). According

to

equation

(8.57),

the

components

of the

strain tensor

at the

center

of a

3-cell

are

The

numerical approximations (8.81), (8.82)

and

(8.83) allow each component

Z-x.i-x.3

(oi)

of the

strain

tensor

£(ot)

to be

computed numerically

at

each node

a of the

3-grid

as

follows:

Note

that

the

parameterization

denned

by

equation (8.78) implies

that

the

metric

tensor

of the

deformed region

'R,

is

constant

and

equal

to the

unit

tensor:

Consequently,

from

equation

(8.32),

it can be

concluded

that

According

to

equations (8.35),

(8.39)

and

(8.44),

the

principal strain direc-

tions

(W(i),

W(

2

),

W(

3

)},

their associated elongations

{e^),

e(

2

),e(

3

)},

and

the

dilation

coefficient

6 can

thus

be

determined easily.

439

440

CHAPTER

8.

ELEMENTS

OF

STRUCTURAL

GEOLOGY

Comment

It is

generally admitted

that

some deformations, especially extensional, induce

fractures

that

in

turn

tend

to

change

the

permeability

in the

reservoirs. From

a

practical point

of

view, this observation

can be

used

in two

different

ways:

• It can

reasonably

be

conjectured

that

the

invariants

of the

strain

tensor

(see

page

400)

are

correlated

22

with

the

permeability.

This

suggests

that these

invariants

could

be

used

as

"secondary"

functions

for

estimating

the

perme-

ability

with

the

Colocated

CoKriging

method

presented

in

section

(9.7.4).

• The

fractures

induced

by the

deformations

can be

observed

on

well

logs

but

are

invisible

in the

seismic

images.

These

subseismic

fractures

are

generally

modeled

as

elliptical

pieces

of

planes simulated

at

random

in the

domain

being

studied.

Let us

assume

that

elongations

(e(i),

6(

2

),

6(3)}

associated

with

the

principal

strain

directions

(W(i),

W(2),

W($)}

are

sorted

as

follows:

To

improve

the

orientation

and the

shape

of

these

ellipses,

their

two

main

axes

may be

chosen

as

proportional

to the two first

principal

strain

directions

{W(i),

W(2)}

modulated

by

their

associated

elongations

(e(i),e(

2

)}.

8.6.8

A

test

example

figure

(8.30)

shows

an

example

of a 3D

balanced restoration obtained with

the

method presented above.

In

this example,

• the

initial

geological

model

at

current

geological

time

t = 1 is the one

presented

in

the

left-hand

side

of figure

(8.17),

• the

unfolding

of the top

horizon

was

performed

with

the

isometric

parame-

terization

method

described

in

section

(8.5)

and

shown

in the

upper

half

of

figure

(8.17),

and

• the

Mass-Preservation

and

Minimum-Deformation

constraints

were

used

with-

out

taking

into

account

the

decompaction.

This

example clearly shows

that

the

proposed method yields consistent results,

leading

us to

conclude

that

the

current model

is

probably geometrically cor-

rect except, possibly,

at one

location pinpointed

in figure

(8.30).

As

mentioned

above,

to

allow

an

objective

testing

of the

efficiency

of the

Mass-Preservation

constraint, compaction/decompaction

was not

taken into account.

It is

inter-

esting

to

note

that

both

the

area

of the

horizons

and the

volume

of the

layers

were

preserved

after

restoration with

a

relative variation

of

10~

2

.

Comment

The

geometric model used

in

this example

is a

synthetic model called

the

"EAEG-model"

23

which

was

realized

by

IFF

using gOcad software

[6].

In

22

Note

that such

a

correlation

is

positive

if

there

is no

cementation

of the

micro

fractures

induced

by the

deformation,

otherwise

it

will

be

negative.

23

EAEG

is the

acronym

for

"European Association

of

Exploration

Geophysicists,"

while

IFF

is the

acronym

for

"Institut

Francais

du

Petrole."

8.7.

CONCLUSIONS

Figure

8.30

Solid

view

showing

an

example

of

a 3D

balanced

restoration

where

the top

horizon

is

unfolded

with

the

isometric

parameterization

method.

The

initial

model

corresponds

to the

left-hand

side

fault

block

of the

example

presented

in

figure

(8.17).

practice,

for

building this model,

a

series

of

balanced

2D

parallel vertical

cross sections were

first

generated using

a

forward

modeling technique. Using

gOcad, these synthetic cross sections were next transformed

into

fault

and

horizon

triangulated

surfaces,

which

in

turn

were used

to

build

the

model

represented

in the

left-hand side

of

figure

(8.17).

8.7

Conclusions

Contrary

to

most

of the

methods presented

in the

rest

of

this book,

the

methods presented

in

this chapter

are

very

specific

to

geology

and,

more

precisely,

to the

structural geology

of

stratified media.

Because numerical structural geology

is a

wide

field of

active research,

this chapter

can be

considered

as no

more

than

a

brief

introduction limited

441

442

CHAPTER

8.

ELEMENTS

OF

STRUCTURAL GEOLOGY

to

some examples

of

important problems

associated

with sedimentary layers.

Once

again,

DSI

and the

discrete modeling approach proposed

in

this book

provide

efficient

tools

that

can be

very

useful

even

in

complex cases.

Chapter

9

Stochastic

Modeling

Important economic decisions

are

often

made

based

on

models

of the

subsur-

face.

However,

due to the

sparse

and

unprecise data, these models cannot

be

considered

fully

reliable,

and the

impact

of

uncertainties should

be

inte-

grated

into

the

decision-making process.

As

shown

in

this chapter, stochastic

methods provide

a

useful

framework

for

assessing

the

uncertainties attached

to

these models.

9.1

Simulation versus interpolation

One

common problem

in the

geosciences

is

that

the set of

information avail-

able

is not

enough

to

assess

the

local complexity

of the

entities

1

to be

modeled.

In

areas

far

away

from

the

data

points,

any

interpolation method honoring

these

data

would produce

one

single smooth solution

that

does

not

reproduce

possible high-frequency variations

of the

actual phenomenon; this

is a

conse-

quence

of the

Shannon theorem [15], which

can be

approximately formulated

as

follows:

If

a

function

has

neighboring extrema, located

at a

relative

dis-

tance

d

from

each

other,

then

an

interpolation method

can

repro-

duce

these extrema only

if the

data spacing

is

less than

or

equal

to

d/2.

This implies

that,

in

regions away

from

the

data

points,

the

corresponding

"most probable" solution

can

have only

a

"smooth behavior," which

is

typ-

1

In

practice, these "entities" represent

the

geometry

and the

physical properties

of the

geological

object.

443

444

CHAPTER

9.

STOCHASTIC MODELING

Figure

9.1 For

each

equiprobable

solution

«S(u,

a>),

the

functional

"R,i

generates

a

response

rj(u;),

and the

frequency

of a

given

value

TI

for

this

response

is

equal

to

fi(n}.

ically

very

different

from

the

actual behavior

of the

entities

to be

modeled.

As

a

consequence,

it

could

be

hazardous

to

make economic decisions based

on

a

single

and

artificially smooth solution, even

if

this solution

is the

"most

probable" one.

In

practice, economic decisions

are

based

on the

values

of a

series

of

nonlinear functionals

{T^i,??^,...},

which transform

any

particular

solution

S

into

a

series

of

real numbers

{ri,

r2,...}:

For

example,

in

decisions related

to the

management

of an oil

reservoir,

the

following

functionals, where

S

represents

one

particular reservoir model,

may

be

considered (see

figure

(9.1)):

• The

time

r\

—

Tl\

(S) of the first

arrival

of

water

(breakthrough)

at a

well

location.

• The

total

volume

of oil r2 =

72.2

(<$)

produced

after

one

year

of

production,

These values

{ri,r2,...}

can be

evaluated using

a flow

simulator

fed

with

parameters (porosities, permeabilities) included

in the

model

<S.

To

avoid

catastrophic decisions, Journel [119] suggests proceeding

as

follows:

1.

Generate

a

series

of

equiprobable

elementary

statistical

events:

2

See

section

(9.2.1).

9.2.

PROBABILITIES

IN A

NUTSHELL

445

2.

Generate

an

associated

series

of

equiprobable

solutions,

all

honoring

the

data:

In

the

above

solutions

{«S(u,

u;)},

(u

e

D} is

assumed

to be a set of

points

attached

to the

object

to be

modeled

(see

figure

(9.1)).

3.

Compute

the

associated

series

of

responses

ri(u)

=

72-i(«S(-,u;)):

4.

For a

given

functional

7£i,

in

general

the

responses

are not

distributed

uni-

formly,

and the

series

{rj(u;i),

^(0*2),...}

can be

used

to

build

an

estimate

of

the

frequency

fi(ri)

of a

given

response

r*.

The

frequency

distributions

{/i(ri),

fafa), • • •}

thus obtained

can be

used

to

assess

the

risks associated with

any

decision:

the

distribution

fifa)

represents

a

measure

of the

uncertainty

functional

7t^.

This

decision-making

strategy

is

relevant only

if it is

possible

to

produce

a

series

of

equiprobable solutions

<S(u,

u^-)

called

simulations

honoring

the

two

following

fundamental constraints:

•

Each

simulation

S(u,u}j)

honors

all the

data.

•

Each

simulation

S(u,u>j)

has a

local

behavior

deemed

compatible

with

an a

priori

knowledge

of the

"style"

of

local

variations

for the

entity

to be

modeled.

In

this approach, each solution

<S(u,

u/j)

appears

as a

realization

of a

random

model

<S(u,a;)

also called

a

"stochastic

simulator"

3

or,

more simply,

a

"sim-

ulator."

Despite

the

many possible stochastic methods

[215,

58, 94, 45]

that

can be

used

for

building such random models, they

are of

practical interest

only

if

they

can

produce

a

large number

of

equiprobable solutions

at low CPU

and

memory cost.

9.2

Probabilities

in a

nutshell

This

section

briefly

introduces

the

minimum notions

of

probability required

for

an

understanding

of the

models presented

in the

next sections.

For an

extensive

presentation

of the

probability theory,

the

reader

is

referred

to

[78,

101,

139].

For

simplicity's sake, mathematical details such

as

"measura-

bility"

or

"integrability"

of

functions

or

"completeness"

of

<j-algebras

will

not

be

addressed.

In

other words,

it

will

be

assumed

that

all the

necessary

sub-

tle

mathematical conditions needed

for

ensuring

the

validity

of the

concepts

presented

in

this chapter

are

implicitly honored

if not

explicitly mentioned.

9.2.1

Notion

of

Probabilized Space

In

probability theory,

a

Probabilized

Space

consists

of any

triplet

(15,

A,

P}

where:

3

These

"stochastic" simulators should

not be

confused

with

"flow"

simulators used

in

the oil and gas

industry.

1

446

CHAPTER

9.

STOCHASTIC MODELING

•

O

is the

set

4

of all the

elementary

statistical

events

uj

corresponding,

for ex-

ample,

to

experiments

that

may be

performed

in the

context

of the

problem

to be

considered.

• A is a

a-algebra

on

IS,

that

is a

family

of

subsets

of

13

stable

for the

union

and

the

intersection:

Any

subset

A

belonging

to A is

called

an

event.

•

IP

is

a,

probability

denned

on A

that

is an

application

from

A to

[0,1]

such

that

• The two

following

notations

are

traditionally

used

for the

probability

of an

event

A

£

A:

For

any

pair

(A, B) of

parts

of A

such

that

IP(B]

^

0, the

probability

of A

"conditioned"

by B is

noted

as

JP(A\B}

and

defined

as follows:

This

conditional probability

JP(A\B}

represents

the

probability

that

any

event

u)

belongs

to A

when

it is

already known

that

uj

belongs

to B.

9.2.2

Notion

of a

Random

Variable

An

application

Z

from

13

to the

real parametric space

1R

is

said

to be a

Random Variable, (RV), relative

to

(15,

A,

P}

if, and

only

if, for any

interval

]a,

b]

of

IR

there

is an

event

A(a,

b]

€ A

such

that

The

notion

of

expected

value

E(Z),

also called

the

mean,

is

defined

as the

average

(if it

exists)

of the

values

of Z on

15

weighted

by the

probability

distribution

JP:

4

Traditionally,

in

books dedicated

to

probability

theory,

this

set is

noted

as

£7.

In

this

book,

it was

decided

to

note

it as

13

to

avoid

confusion

with

the set of

nodes

£1

used

in the

Discrete

Smooth Interpolation theory (see

section

(1.2)).

9.2.

PROBABILITIES

IN A

NUTSHELL

Figur e

9.2 An

example

of

a

one-dimensional

cdf

function

Fz(z)

and its

associ-

ated

pdf

function

fz

(z).

Note

that,

in the

case where

13

is a finite set of

elementary

statistical

events,

the

above

Lebesgue

integral degenerates into

a finite

sum:

The

mean value

mz

and the

variance

<r|

of Z

(when

they exist)

are

then

defined

as

By

definition,

the

square root

o~z

of

a\

is

called

a

"standard

deviation"

of

Z,

and it is

said

that

For any RV

Z,

there

is a

Cumulative Distribution Function (cdf) sometimes

called

the

"law"

of

Z,

defined

on

TR

as

From

the

definition

of

JP,

it is

easy

to

infer

the

following

properties

for any

cdf:

When

FZ(Z)

is

derivable relative

to z, its

derivative

is

a

nonnegative

function

called

the

Probability Distribution Function (pdf)

of

Z.

Figure (9.2) shows

an

example

of cdf and its

associated pdf.

It can be

shown

that,

for any

function

5

g(z),

the

following

formula

called

the

formula

of

transfer

is

valid:

5

Strictly

speaking,

any

A-measumble

function,

see

[101].

447

Jean-Laurent Mallet. Geomodeling

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