Fractured Reservoirs Part 3 6v191q

Naturally Fractured Reservoirs

232, Avenue Napoléon Bonaparte P.O. BOX 213 92502 Rueil-Malmaison Phone: +33 1 47 08 80 00 Fax: +33 1 47 08 41 85 [email protected]

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Part 3 – Upscaling of fracture properties

Upscaling of fracture properties

 Objective: to reduce the complexity of the actual fracture system (at fracture scale) to a few relevant equivalent parameters (at larger scale, i.e. the reservoir simulation cell scale)  Requirement: to dispose of a detailed description of the fracture system

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 The equivalent parameters are the input data for the reservoir simulation model

Part 3 – Upscaling of fracture properties

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Upscaling at a reservoir cell scale

Reservoir grid

Geological model of a reservoir cell

Geological model

Determination of ff, Kf tensor

? Warren & Root representation of a reservoir cell

Part 3 – Upscaling of fracture properties

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and (a,b,c) or s

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How to upscale fracture properties ?

 Introduction: fracture dynamic properties  Different types of upscaling for different problems

 Dual porosity upscaling

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 Example

Part 3 – Upscaling of fracture properties

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Conductivity & Permeability

 The concept of permeability for fractures is scale dependant:

A

At the fracture scale (kf), the intrinsic permeability can be very high.

The fracture network permeability is a function of kf and connectivity

B

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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e

Main flow direction

Part 3 – Upscaling of fracture properties

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Conductivity & Permeability

 The concept of permeability for fractures is scale dependant.  For this reason, 3 parameters are used: • Conductivity : Cd = kf x e • kf : intrinsic fracture permeability • e : fracture aperture • Fracture permeability at grid cell scale : KfX, KfY & KfZ

L

ΔY

A ΔX

Part 3 – Upscaling of fracture properties

Tf 

K AB.Y.Z X

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•Transmissibility: Tf  K.S

B

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Fracture aperture and porosity  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:

e3 Cf  12 x 0.98.106

Cf, the conductivity can be derived from the calibration of a DFN model Part 3 – Upscaling of fracture properties

e

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Cf = fracture conductivity in mD.m e = fracture aperture in mm

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Fracture aperture and porosity  At fracture scale, the porosity is defined as :  f 

Vv oi d Vf r ac t ur e

• 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:

ff

V  

v oi d

Vr oc k

• It is often very small (below 1%)

Example:

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• 2 orthogonal sets of open fractures (φf=100%) • Fracture density : 1 frac every meter • Fracture aperture : 0.5 mm • Estimate the network porosity (Φf)

Part 3 – Upscaling of fracture properties

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Other parameters  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 than 10 µm 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 1 used.

0

0 1 NORMALISED WATER SATURATION

Part 3 – Upscaling of fracture properties

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Kr

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How to upscale fracture properties ?

 Introduction: fracture dynamic properties  Different types of upscaling for different problems

 Dual porosity upscaling

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 Example

Part 3 – Upscaling of fracture properties

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Fracture models & upscaled parameters

 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 (or a, or block size) @Beicip-Franlab

• Sub-seismic faults and fracture swarms

Part 3 – Upscaling of fracture properties

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Micro fractures: matrix anisotropy

No dual porosity system Upscaling: matrix anisotropy

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10 cm

Part 3 – Upscaling of fracture properties

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Single medium: pseudo Kr curves

Krw

Krw

Fracture

Sw

Matrix

Sw

Krw Y

Single grid block Matrix + Fracture Part 3 – Upscaling of fracture properties

Sw

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X

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Dual porosity upscaling



Context: dual-porosity model



Reservoir cell scale very much larger than fracture scale



At reservoir cell scale, fracture medium behaves homogeneously (as a continuum): • 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 exchange factor a) @Beicip-Franlab



Part 3 – Upscaling of fracture properties

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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

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 Example Part 3 – Upscaling of fracture properties

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Upscaling of large scale fractures

Part 3 – Upscaling of fracture properties

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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

@Beicip-Franlab

 Example Part 3 – Upscaling of fracture properties

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Fracture permeability tensor

Complete K tensor



Main horizontal flow directions



Equivalent permeabilities along those directions Kh1, Kh2



 

Horizontal permeability anisotropy

Vertical permeability

k yy k yz

k zx   k zy  k zz  Full tensor

K h1 K h2

 k h1   0  0 

Kz

Vertical/horizontal anisotropy ratio

k yx

0  0 k Z 

0 kh 2 0

Diagonal tensor

z

Kz K h1  K h 2

y

h2

h1 x Part 3 – Upscaling of fracture properties

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

 k xx   k xy k  xz

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Permeability upscaling methods for diffuse fractures

Two different methods:  Numerical upscaling: small (local)-scale simulation to derive large-scale properties  Analytical upscaling: the fracture network large scale permeability is derived from the fracture geometrical characteristics. Two possible techniques:

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• Analytical abacuses: Reiss abacuses to determine equivalent fracture permeability of orthogonal fracture sets of infinite length • Oda analytical upscaling

Part 3 – Upscaling of fracture properties

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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 @Beicip-Franlab

 Example Part 3 – Upscaling of fracture properties

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Permeability upscaling workflow

Discrete fracture network model

Steady-state flow computation in 3 directions

Part 3 – Upscaling of fracture properties

Local fracture permeability ellipsoid

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Equivalent fracture permeability tensor at the scale of a reservoir simulator cell

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Equivalent permeability tensor

• Approach: Discrete fracture model, no contribution of matrix

• Procedure: - fracture network discretization with nodes at fracture intersections - a 3D-extended resistor network method (incompressible steady-state flow simulation)

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- specific boundary conditions to derive a permeability tensor

Part 3 – Upscaling of fracture properties

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Permeability tensor and flow directions

P

(XY) 0

(Z) P

1 Kh1 X Kh2

2

K tensor

Principal flow directions 1 and 2 Kh1 and Kh2 = eigenvalues of K tensor Part 3 – Upscaling of fracture properties

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P

 kxx ky x 0     kxy ky y 0   0 0 kzz   

Y

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Fracture permeability vs fracture density

20000 18000 16000 14000 12000 10000 8000 6000 4000 2000 0

Percolation density

0

5

10

Zero K

15

20

25

30

35

40

45

fracture density (m/m²) Linear part

Transition

Ref.:“Correlations Between Natural Fracture Attributes and Equivalent Dual-Porosity Model Parameters”, S. Sarda, B. Bourbiaux and M.C. Cacas, 10th European Improved Oil Recovery Symposium, Brighton, 18-20 Aug. 1999. Part 3 – Upscaling of fracture properties

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first permeability (mD)

1 set of diffuse fractures with a given distribution of orientation and given lognormal distributions of fracture length and conductivity

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Fracture permeability vs fracture length

Fracture perm eability (m D)

25

20

15

10

5

0 0

5

10

15

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Fracture length (m )

Part 3 – Upscaling of fracture properties

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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 @Beicip-Franlab

 Example Part 3 – Upscaling of fracture properties

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Analytical upscaling – Introduction dP Rate Q Viscosity μ

r L

Poiseuille law

Darcy law

(derived from Navier-Stokes equations in the case of a viscous incompressible fluid flow in a pipe)

(describes an incompressible fluid flow in a porous medium)

Q r 1 P  2 r 8  L 2

Q 1 P K 2 r  L

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) Part 3 – Upscaling of fracture properties

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8L P  4 Q r

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Abacuses for fracture upscaling (Reiss) Abacuses based on simple relationships between ff, kf, fracture aperture and spacing (after Reiss) 1. Fracture porosity (f f), spacing (a,b,c in 3D) and fracture aperture e:  f f # e[1/a+1/b+1/c]

2. Apparent (equivalent) fracture permeability in direction (i) Flow rate through fractures (delimiting n blocks)

(i)  Poiseuille equation Q= (ne3(a+b)/12). (P/L) a a

c

Hence: kf = [n(a+b)/A].(e3/12) =(1/a+1/b).(e3/12) Part 3 – Upscaling of fracture properties

e

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 Darcy law Q= (kf.A /). (P/L) with A#nab

b

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Abacuses for fracture upscaling (Reiss) 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

Ref.: Reiss, L.H. 1980. Reservoir engineering en milieu fissuré. Editions Technip, Paris. Part 3 – Upscaling of fracture properties

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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.

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Abacuses for fracture upscaling (Reiss) L.H. Reiss abacus (p.92)

Kf 0.1darcy

a 6m e= 200 µm

f f= 0.003% Part 3 – Upscaling of fracture properties

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f f,%

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Analytical upscaling – The ODA method ; (1/3)

 Poiseuille law for a single planar « crossing » fracture of aperture e : E’

Nord azimut

A’

D’

 n

B’

 g2

 g1  g3

C’ (j)

Poiseuille coefficient

 n1 n2 1  n 22  n 2 n3

 n1 n3    n 2 n3 .P 2   1 n3 

Fluid velocity (analogous to Q/Area)

Describes the fracture geometry Part 3 – Upscaling of fracture properties

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j

2   1 n 1 2   e u λ   n1 n 2 μ    n1 n3

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Analytical upscaling – The ODA method ; (2/3)

 From a single fracture to the fracture network, assuming: • A perfect Poiseuille Behavior (λ=1/12) • That each fracture belongs to a “crossing” network and is independant of the network (i.e. uniform pressure gradient in the cell) In that case, the global fluid velocity can be obtained by a weighted average of the velocity in each fracture proportionally to its volume

 1 (f) U   V Vfrac

With



(j)

Volume of a given fracture (s,j)

 1 u ).dx.dy.dz V  1  n12  j N    n1 n2  n n  1 3

nb _ set



(s)

s 1

 n1 n2 1  n22  n 2 n3

nb _ frac

 j 1

j  λsj ( . V.e². N).P μ

 n1 n3    n 2 n3  1  n32 

Volume of rock Part 3 – Upscaling of fracture properties

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Global « large scale » fluid velocity in the network

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Analytical upscaling – The ODA method ; (3/3)

 From there, the upscaled permeability can be written: • In Dual media:  K eq  K    (f)

nb_j oints j  1 e² j j ( . V N).  (m) (f) V V j1 12 

• In Single media: (f)

K @Beicip-Franlab

K eq  K 

(m )

Part 3 – Upscaling of fracture properties

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Conductive network with two fracture sets

Fracture set n°1 and fracture set n°2: - different orientations - different geological history - different conductivities

Example: the network is non conductive if one of the fracture sets is non conductive If C1 >> C2, Kf tensor is

Sensitivity studies are needed to analyse the effect of fracture sets’ conductivities on equivalent permeability (role of each fracture set) Part 3 – Upscaling of fracture properties

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If C2 >> C1, Kf tensor is

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Analytical vs. Numerical upscaling time in s 10000

Kh in mD 60000 2 orthogonal sets with mean conductivity of 10000 mDm

Computationnal time for equivalent parameters

50000 1000

numerical time computation analytical time computation

40000

30000

100

20000

numerical Kh analytical Kh

10

10000

0

1

2

4

6

8

10

12

0

1

1,33333333 1,81818182

5

10

density

density in m-1

Permeability is slightly more optimistic with the analytical upscaling, especially in low density (poorly connected) networks (the method assumes total connectivity)

Analytical upscaling is much quicker than numerical upscaling

Part 3 – Upscaling of fracture properties

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0

35

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 @Beicip-Franlab

 Example Part 3 – Upscaling of fracture properties

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Block size upscaling methods for diffuse fractures

Two different methods:  Geometrical averaging: block dimensions are deduced from the 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 @Beicip-Franlab

• Slower but more accurate technique generally used in combination with a numerical upscaling

Part 3 – Upscaling of fracture properties

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Block size upscaling : geometrical averaging

 Equivalent block size: • a = mean spacing in the principal direction of flow

a

(here the yellow fractures)

• b = mean spacing in the perpendicular direction

b

(here the green fractures)

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• c = mean height of fractures

Part 3 – Upscaling of fracture properties

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Block size upscaling : image processing

Horizontal block dimensions determined in each layer on the basis of capillary imbibition with 2 assumptions: + piston-type invasion of homog. isotr. matrix blocks: R <--> A + Distance of invasion vs. time independent of shape: t <--> X

X(t)

Conclusion:

R(t) <======> A(X) Part 3 – Upscaling of fracture properties

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X(t)

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Geometrical method: implementation Actual fractured medium

Equivalent medium

b a Image processing

A(X)

Aeq(a,b,X) )

Analytical expression

Aeq i

Ai X

Xi

X

Minimization of

J(a, b)   [A(x i )  Aeq(a, b, x i )]2 i

a and b solutions Part 3 – Upscaling of fracture properties

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Xi

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How to upscale fracture properties ?

 Introduction: fracture dynamic properties  Different types of upscaling for different problems

 Dual porosity upscaling

@Beicip-Franlab

 Example

Part 3 – Upscaling of fracture properties

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Outcrop photograph

Part 3 – Upscaling of fracture properties

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Upscaling & REV: example

100 m

Z X @Beicip-Franlab

Y

Part 3 – Upscaling of fracture properties

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REV study of equivalent permeability

0,4

Ky

0,3

0,2

Kz

0,1 Kx 0,0 0

20

40

60

80

Subvolume horizonta l dimension, m

Part 3 – Upscaling of fracture properties

10 0 @Beicip-Franlab

Equivalent fracture permeability, md

0,5

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REV study of equivalent block size

10 La yer 3

6

La yers 1 to 9

4

2 La yer 7 0 0

20

40

60

80

Subvolume horizonta l dime nsion, m

Part 3 – Upscaling of fracture properties

10 0 @Beicip-Franlab

Equivalent block size a, m

8

45

Top view of layer 3

X Part 3 – Upscaling of fracture properties

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Y

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Top view of layer 7

X Part 3 – Upscaling of fracture properties

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Y

47

Full field equivalent parameters

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•Kx map •Ky map …

Part 3 – Upscaling of fracture properties

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Final dual porosity simulation model

Sw fracture Part 3 – Upscaling of fracture properties

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Sw matrix

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