Fractured Reservoirs Part 3

7/28/2019 Fractured Reservoirs Part 3

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92502 Rueil-Malmaison

France

Phone: +33 1 47 08 80 00

Fax: +33 1 47 08 41 85

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Naturally Fractured Reservoirs

Part 3 Upscaling of fracture properties


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Objective: to reduce the complexity of the actual fracture

system (at fracture scale) to a few relevant equivalentparameters (at larger scale, i.e. the reservoir simulation

cell scale)

Requirement: to dispose of a detailed description of thefracture system

The equivalent parameters are the input data for the

reservoir simulation model

Upscaling of fracture properties


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Determination offf, Kftensorand (a,b,c) ors ?

Warren & Root

representation of a reservoir cell

Reservoir grid

Geological model

Geological model of a reservoir cel l

Upscaling at a reservoir cell scale


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How to upscale fracture properties ?

Introduction: fracture dynamic properties

Different types of upscaling for different problems

Dual porosity upscaling

Example


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The concept of permeability for fractures is scale dependant:

Conductivity & Permeability

ee

At the fracture scale (kf), the

intrinsic permeability can be

very high.

Main

flow

direction

The fracture network

permeability is a function of kfand connectivity

B

A

The permeability of the

fracture network at grid cell

dimension (Kf) is a function of

the fracture network

permeability and the grid cell

size.


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The concept of permeability for fractures is scale dependant.

For this reason, 3 parameters are used:

Conductivity : Cd = kfx e

kf: intrinsic fracture permeability

e : fracture aperture

Fracture permeability at grid cell scale : KfX, KfY & KfZ

Transmissibility:

Conductivity & Permeability

LSTf

B

A XZT ABf

X

Y


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The fracture aperture is the mean open thickness of an individual

fracture.

It is usually very small (between 0.1 and 0.5 mm) but can locally reach

several centimeters (enlarged fractures)

The aperture can be estimated through direct observation (Image log,

thin section, core, outcrop ) or using the Poiseuille law:

Cf, the conductivity can be derived

from the calibration of a DFN model

Cf = fracture conductivity in mD.m

e = fracture aperture in mm

Fracture aperture and porosity

ee

6

3

1098012.

eCf


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At fracture scale, the porosity is defined as :

It varies between 0 for sealed fractures and 100% for open fractures

A reservoir scale (or grid cell scale), the fracture porosity is defined as:

It is often very small (below 1%)

Example: 2 orthogonal sets of open fractures (f=100%)

Fracture density : 1 frac every meter

Fracture aperture : 0.5 mm

Estimate the network porosity (f)

Fracture aperture and porosity

f racture

vo idf V

V

rock

voidf V

V


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Parameters such as compressibility, relative permeability and capillary

pressure are difficult to measure. They are often estimated:

Fracture Compressibility Often considered to be 10 times higher than the matrix compressibility

Capillary pressure (Pc)

Often neglected (Pc=0). L.H. Reiss estimated that for an aperture higher than10 m capillarity plays little or no role in the fracture network.

Relative permeability (Kr) Like Pc, it is often neglected. Cross type relative permeability curves are

used.

Other parameters

00

1

1NORMALISED WATER SATURATION

Kr


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1Part 3 Upscaling of fracture properties





Example

F t d l & l d t


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Explicit modelling: no upscaling

e.g. : PLT or well test simulation in a DFN

Equivalent single-porosity model:

micro-fractures: Km anisotropy matrix and fractures: Km, fm, pseudo-Kr(Ref. SPE 68165)

Dual-porosity 1K or 2K :

Diffuse fractures: Kf

(tensor), ff

, s (ora, or block size) Sub-seismic faults and fracture swarms

Fracture models & upscaled parameters

Mi f t t i i t


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10 cm

No dual porosity system

Upscaling: matrix anisotropy

Micro fractures: matrix anisotropy

Si l di d K


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Sw

Krw

Sw

Krw

Sw

Krw

Fracture Matrix

Matrix + Fracture

Single grid block

X

Y

Single medium: pseudo Kr curves



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Context: dual-porosity model

Reservoir cell scale very much larger than fracture scale

At reservoir cell scale, fracture medium behaves homogeneously (as acontinuum): high fracture network connectivity

high fracture network conductivity

In this case, upscaling = homogenization

Equivalent parameters:

ff,- a fracture permeability tensor Kf,

- an equivalent block size (or a shape factor s or an exchangefactora)




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Upscaling of large scale fractures (Swarm / Faults) Upscaling of diffuse fractures

- Permeability upscaling

- Block size upscaling

Example

Upscaling of large scale fractures


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Upscaling of large scale fractures



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Upscaling of large scale fractures (Swarm / Faults) Upscaling of diffuse fractures

- Permeability upscaling


Example

Fracture permeability tensor


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Complete K tensor

Main horizontal flow directions

Equivalent permeabilities along those

directions Kh1, Kh2

Horizontal permeability anisotropy

Vertical permeability Kz

Vertical/horizontal anisotropy ratio

2

1

h

h

K

K

21 hh

z

KK

K

Fracture permeability tensor

zzyzxz

zyyyxy

zxyxxx

kkk

kkk

kkk

Z

h

h

k

k

k

00

00

00

2

1

x

y

z

h1

h2

Full tensor

Diagonal tensor

Permeability upscaling methods for diffuse fractures


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Two different methods:

Numerical upscaling: small (local)-scale simulation to derive

large-scale properties

Analytical upscaling: the fracture network large scalepermeability is derived from the fracture geometrical

characteristics. Two possible techniques:

Analytical abacuses: Reissabacuses to determine equivalent

fracture permeability of orthogonal fracture sets of infinite length

Oda analytical upscaling

Permeability upscaling methods for diffuse fractures



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Dual porosity upscaling Upscaling of large scale fractures (Swarm / Faults)

Upscaling of diffuse fractures- Permeability upscaling

Numerical upscaling

Analytical upscaling


Example

Permeability upscaling workflow


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Equivalent fracture permeability tensor at the scale of areservoir simulator cell

Discrete fracturenetwork model

Steady-state flow computationin 3 directions

Local fracturepermeability

ellipsoid

Permeability upscaling workflow

Equivalent permeability tensor


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Approach: Discrete fracture model, no contribution of matrix

Procedure:

- a 3D-extended resistor network

method (incompressible

steady-state flow simulation)

- fracture network discretization with

nodes at fracture intersections

- specific boundary conditions to derive apermeability tensor

Equivalent permeability tensor

Permeability tensor and flow directions


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X

Y 1

2

Kh1

Kh2

Principal flow directions 1 and 2

Kh1 and Kh2 = eigenvalues of K tensor

zz

yyxy

yxxx

k000kk

0kk

K tensor

P 0(XY)

P

P(Z)

y

Fracture permeability vs fracture density


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0

2000

4000

60008000

10000

12000

14000

16000

18000

20000

0 5 10 15 20 25 30 35 40 45

fracture density (m/m)

firstperm

eability(mD)

Transition Linear part

Percolation

density

Zero K

1 set of diffuse fractures with a given distribution of orientation and

given lognormal distributions of fracture length and conductivity

Ref.:Correlat ions Between Natural Fracture Attr ibutes and Equivalent Dual-Porosi ty ModelParameters, S. Sarda, B. Bourbiaux and

M.C. Cacas, 10th European Improved Oil Recovery Symposium, Brighton, 18-20 Aug. 1999.

y y

Fracture permeability vs fracture length


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0

5

10

15

20

25

0 5 10 15

Fracture length (m )

Fractureperm

eability(mD)

p y g



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Numerical upscaling



Example

Analytical upscaling Introduction


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rL

dP

Rate Q

Viscosity

Poiseuille law(derived from Navier-Stokes equations in the case

of a viscous incompressible fluid flow in a pipe)

Darcy law(describes an incompressible fluid flow in a porous

medium)

Qr

LP

4

8

L

Pr

r

Q

1

8

2

2 L

PK

r

Q

12

For a given fracture (here a capillary tube) the permeability can be derived

analytically directly from the geometry of the fracture (ie aperture, shape,

length)

Abacuses for fracture upscaling (Reiss)


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2Part 3 Upscaling of fracture properties 2

Abacuses based on simple relationships between ff, kf,fracture aperture and spacing (afterReiss)

1. Fracture porosity (ff), spacing (a,b,c in 3D) and fracture aperture e: ff# e[1/a+1/b+1/c]

2. Apparent (equivalent) fracture permeability in direction (i)

Flow rate through fractures (delimiting n blocks)

Poiseuille equation

Q= (ne3(a+b)/12). (P/L) Darcy law

Q= (kf.A/). (P/L) with A#nabHence:

kf= [n(a+b)/A].(e3/12)=(1/a+1/b).(e3/12) e

b

a

ac

(i)



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2Part 3 Upscaling of fracture properties 2

Abacuses established for matrix blocks (plates, bars or cubes) of lateral

dimension a

na/A=fracture length per unit cross-section area

1/a if 1 fracture set

na/A= along flow direction

2/a if 2 fracture sets

Application:

possibility to check the consistency between dynamic information

(well test results), and geological information (spacing or block size)

2 of the 4 unknowns (kf, ff, a, e) have to be fixed to infer the 2 others.

Ref.: Reiss, L.H. 1980. Reserv oi r eng ineerin g en m ilieu fis su r. Edi tio ns Technip, Paris.



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f f,%

Kf 0.1darcy

a 6m

f f= 0.003%

L.H. Reiss abacus

(p.92)

e= 200 m

Analytical upscaling The ODA method ; (1/3)


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1g2g

3g

A

C

D

E

B

n

azimutNord

( )

nnnnn

nnnnn

nnnnn

.

1

1

1

eu

2

33231

322221

3121

2

12

Poiseuille law for a single planar crossing

fracture of aperture e :

Fluid velocity (analogous to Q/Area)

Poiseuille coefficient

Describes the fracture geometry



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From a single fracture to the fracture network, assuming:

A perfect Poiseuille Behavior (=1/12)

That each fracture belongs to a crossing network and isindependant of the network (i.e. uniform pressure gradient in the cell)

)

Vfrac

_

1

_

1

jjs(j )(f)

)(

P).NV.e..

(

V

1.dx.dy.dzu

V

1U

setnb

s

fracnb

j

s

2

33231

32

2

221

3121

2

1j

11

1

Nnnnnnnnnnn

nnnnn

With

Global large scale fluid velocity in the network

Volume of rock

Volume of a given fracture (s,j)

In that case, the global fluid velocity can be obtained by a weighted average

of the velocity in each fracture proportionally to its volume



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From there, the upscaled permeability can be

written:

In Dual media:

In Single media:

.)NV.12

e(

VV

1KK

nb_joints

1j

jj

j

(f )(m )

(f )

eq

KKK(m )(f)

eq

Conductive network with two fracture sets


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Sensitivity studies are needed to analyse the effect offracturesets

conductivities on equivalent permeability (role of each fracture set)

Fracture set n1 and fracture set n2:

- different orientations- different geological history

- different conductivities

Example: the network is non conductive if

one of the fracture sets is non conductive

IfC1 >> C2, Kf tensor is

IfC2 >> C1, Kf tensor is

Analytical vs. Numerical upscaling


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2 orthogonal sets with mean conductivity of 10000 mDm

0

10000

20000

30000

40000

50000

60000

0 2 4 6 8 10 12

density in m-1

Kh in mD

numerical Kh

analytical Kh

Computationnal time for equivalent parameters

1

10

100

1000

10000

0 1 1,33333333 1,81818182 5 10

density

time in s

numerical time computation

analytical time co mputation

Permeability is slightly more optimistic with theanalytical upscaling, especially in low density

(poorly connected) networks (the method

assumes total connectivity)

Analytical upscaling is much quicker thannumerical upscaling



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Numerical upscaling



Example

Block size upscaling methods for diffuse fractures


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Two different methods:

Geometrical averaging: block dimensions are deduced fromthe average spacing between fractures in the principal

direction of flow and perpendicularly to this direction.

Very fast method used generally in combination with Oda analytical

upscaling

Image processing: Horizontal block dimensions is

determined in each layer on the basis of capillary imbibition,

with image processing analysis of the fracture network

Slower but more accurate technique generally used in combinationwith a numerical upscaling

Block size upscaling : geometrical averaging


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Equivalent block size:

a = mean spacing in theprincipal direction of flow

(here the yellow fractures)

b = mean spacing in theperpendicular direction

(here the green fractures)

c = mean height of fractures

a

b

Block size upscaling : image processing


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Horizontal block dimensions determined in each layer on

the basis ofcapillary imbibition with 2 assumptions:

X(t)

X(t)

Conclusion: R(t) A(X)

+ Distance of invasion vs. time independent of shape: t X

+ piston-type invasion ofhomog. isotr. matrix blocks: R A

Geometrical method: implementation


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a

b

Equivalent mediumActual fractured medium

A(X)

X

Ai

Xi

Image processingAeq(a,b,X))

X

Aeqi

Xi

Analytical expression

a and b solutions

Minimization of

J(a, b) [A(x ) Aeq(a, b,x )]ii

i 2



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Example

Outcrop photograph


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Upscaling & REV: example


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XY

Z

100 m

REV study of equivalent permeability


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1008060402000,0

0,1

0,2

0,3

0,4

0,5

Subvolume horizontal dimension, m

Equivalentfracturepe

rmeability,

md

Ky

Kz

Kx

REV study of equivalent block size


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1008060402000

2

4

6

8

10

Subvolume horizontal dimension, m

Equivalentbloc

ksizea,

m

Layer 3

Layer 7

Layers 1 to 9

Top view of layer 3


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Y

X

Top view of layer 7


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Y

X

Full field equivalent parameters


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Kx

map

Ky map

Final dual porosity simulation model


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Sw matrix

Sw fracture

Documents

Fractured Reservoirs Part 3