Download - Fractured Reservoirs Part 3
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Naturally Fractured Reservoirs
Part 3 Upscaling of fracture properties
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Part 3 Upscaling of fracture properties
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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Part 3 Upscaling of fracture properties
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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Part 3 Upscaling of fracture properties
How to upscale fracture properties ?
Introduction: fracture dynamic properties
Different types of upscaling for different problems
Dual porosity upscaling
Example
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Part 3 Upscaling of fracture properties
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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Part 3 Upscaling of fracture properties
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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Part 3 Upscaling of fracture properties
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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Part 3 Upscaling of fracture properties
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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Part 3 Upscaling of fracture properties
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
How to upscale fracture properties ?
Introduction: fracture dynamic properties
Different types of upscaling for different problems
Dual porosity upscaling
Example
F t d l & l d t
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Part 3 Upscaling of fracture properties
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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1Part 3 Upscaling of fracture properties
10 cm
No dual porosity system
Upscaling: matrix anisotropy
Micro fractures: matrix anisotropy
Si l di d K
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1Part 3 Upscaling of fracture properties
Sw
Krw
Sw
Krw
Sw
Krw
Fracture Matrix
Matrix + Fracture
Single grid block
X
Y
Single medium: pseudo Kr curves
Dual porosity upscaling
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1Part 3 Upscaling of fracture properties
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)
Dual porosity upscaling
How to upscale fracture properties ?
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1Part 3 Upscaling of fracture properties
How to upscale fracture properties ?
Introduction: fracture dynamic properties
Different types of upscaling for different problems
Dual porosity upscaling
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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1Part 3 Upscaling of fracture properties
Upscaling of large scale fractures
How to upscale fracture properties ?
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1Part 3 Upscaling of fracture properties
How to upscale fracture properties ?
Introduction: fracture dynamic properties
Different types of upscaling for different problems
Dual porosity upscaling
Upscaling of large scale fractures (Swarm / Faults) Upscaling of diffuse fractures
- Permeability upscaling
- Block size upscaling
Example
Fracture permeability tensor
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1Part 3 Upscaling of fracture properties
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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1Part 3 Upscaling of fracture properties
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
How to upscale fracture properties ?
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2Part 3 Upscaling of fracture properties
How to upscale fracture properties ?
Introduction: fracture dynamic properties
Different types of upscaling for different problems
Dual porosity upscaling Upscaling of large scale fractures (Swarm / Faults)
Upscaling of diffuse fractures- Permeability upscaling
Numerical upscaling
Analytical upscaling
- Block size upscaling
Example
Permeability upscaling workflow
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2Part 3 Upscaling of fracture properties
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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2Part 3 Upscaling of fracture properties
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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2Part 3 Upscaling of fracture properties
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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2Part 3 Upscaling of fracture properties
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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2Part 3 Upscaling of fracture properties
0
5
10
15
20
25
0 5 10 15
Fracture length (m )
Fractureperm
eability(mD)
p y g
How to upscale fracture properties ?
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2Part 3 Upscaling of fracture properties
Introduction: fracture dynamic properties
Different types of upscaling for different problems
Dual porosity upscaling Upscaling of large scale fractures (Swarm / Faults)
Upscaling of diffuse fractures- Permeability upscaling
Numerical upscaling
Analytical upscaling
- Block size upscaling
Example
Analytical upscaling Introduction
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2Part 3 Upscaling of fracture properties
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)
Abacuses for fracture upscaling (Reiss)
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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.
Abacuses for fracture upscaling (Reiss)
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3Part 3 Upscaling of fracture properties
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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3Part 3 Upscaling of fracture properties
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
Analytical upscaling The ODA method ; (2/3)
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3Part 3 Upscaling of fracture properties
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
Analytical upscaling The ODA method ; (3/3)
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3Part 3 Upscaling of fracture properties
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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3Part 3 Upscaling of fracture properties
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
How to upscale fracture properties ?
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3Part 3 Upscaling of fracture properties
Introduction: fracture dynamic properties
Different types of upscaling for different problems
Dual porosity upscaling Upscaling of large scale fractures (Swarm / Faults)
Upscaling of diffuse fractures- Permeability upscaling
Numerical upscaling
Analytical upscaling
- Block size upscaling
Example
Block size upscaling methods for diffuse fractures
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3Part 3 Upscaling of fracture properties
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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3Part 3 Upscaling of fracture properties
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
How to upscale fracture properties ?
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4Part 3 Upscaling of fracture properties
Introduction: fracture dynamic properties
Different types of upscaling for different problems
Dual porosity upscaling
Example
Outcrop photograph
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Upscaling & REV: example
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4Part 3 Upscaling of fracture properties
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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4Part 3 Upscaling of fracture properties
Y
X
Top view of layer 7
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Y
X
Full field equivalent parameters
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4Part 3 Upscaling of fracture properties
Kx
map
Ky map
Final dual porosity simulation model
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4Part 3 Upscaling of fracture properties
Sw matrix
Sw fracture