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IMMISCIBLE DISPLACEMENT IN TWO DIMENSIONS- AREAL



If permeability varies,
If gravity segregation occurs,
If capillary forces are large compared to viscous forces.
If the reservoir homogeneous and isotropic, the flood front may be vertical:
displacement.
2-D
Otherwise, the displacement is 3D.
PREDICTION OF DISPLACEMENT PERFORMANCE:
Earlier methods: quick estimate.
Today, more computer skills.
All models have one common factor: 2D fluid flow equations:
where
  ko  o Po    ko  o Po  
   o So 

 
x  o x  y  o y  t
  k w  w Pw    k w  w Pw  

 
   w S w 
x   w x  y   w y  t
Tayfun Babadagli, PhD, PEng
Short Course - EOR
kw  kw ( S w )
and
ko  ko ( S w )
Po  Pw  Pc ( S w )
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1
DISPLACEMENT OF 5-SPOT PATTERN
INJECTOR
PRODUCER
ASSUMPTIONS and SIMPLIFICATIONS:
Large reservoirs: injector / producer  1.
Injection rate = production rate
Five spot in a reservoir can be simplified by
examining the behavior of a single five-spot.
2
Tayfun Babadagli, PhD, PEng
Short Course - EOR
File-5
CORRELATIONS DEVELOPED FROM SCALED LAB MODELS: The Craig-Geffen-Morse
(CGM) Model:
Experimental data for a variety of oil and aqueous systems were correlated
empirically.
E A  E Abt  0.633 log
Wi
Wibt
E A  E Abt  0.274 ln
Wi
Wibt
EAbt : areal sweep efficiency at BT
 : fraction of area swept to an average water saturation
EA was correlated to mobility ratio (
S wf
MS)
Mobility ratios can be calculated from relative permeability data when the water permeability was
evaluated at the average water saturation behind the front as determined from the frontal advance
theory.
E Abt  0.546 
0.0317 0.3022
 M  0.0051M S
MS
e S
Tayfun Babadagli, PhD, PEng
Short Course - EOR
3
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Prediction of displacement performance:
Oil displaced by water in a five-spot pattern, no initial gas.
At BT, oil recovery is given by:


N pbt  E Abt S wf  S wi V p
S wf
is the average displacing-phase saturation at BT in a linear flood as computed
from frontal advance solution.
Production after BT:


N pbt  E Abt S w5  S wi V p
S w5 : average water saturation in a region swept by the injected fluid.
The key to this model is the assumption made to evaluate this parameter ( S w5 ). A new variable
is defined as the volume of water injected divided by the volume of the 5-spot contacted by the
*
injected water:
Qi
Tayfun Babadagli, PhD, PEng
Short Course - EOR
File-5
At BT:
*
Qibt

For 5-spot
After BT:
pattern: Qi 
dQi* 
Wi
Vp
Wibt
 S wf  S wi
E Abt
Thus,
Qibt  E Abt ( S fw  S wi )
dWi
E AV p
Integrating this equation from BT (Wi/Wibt=1.0) to (Wi/Wibt) :
Qi*
 1  E Abt
*
Qibt
Wi / Wibt

1
W 
d  i 
 Wibt   1  a e  a1 Ei( a )  Ei( a )
1
2
1
EA
Tayfun Babadagli, PhD, PEng
Short Course - EOR
5
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where
a1 = 3.65EAbt
a2 =a1+ln(Wi/Wibt) and
Wibt ≤ Wi ≤Wi100
Wi100 : value of Wi required to sweep the pattern
completely.

Ei( x )  0.577  ln( x )   x n /( nn! )
n 1
It is possible to compute
EAbt and Wi..
Tayfun Babadagli, PhD, PEng
Qi*
*
Qibt
for any
Short Course - EOR
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6
PRACTICAL ESTIMATION OF WATERFLOOD
PERFORMANCE
Material Balance: From core data and estimates of sweep efficiency.
Divide reservoir into two part: (1) swept volume, and (2) unswept volume.
The volume of oil displaced:
Ev
N d  ( So )V p
Bo
Ev : fraction of reservoir volume that is swept by the injected water when the
economic limit is reached.
So : change in average oil saturation within swept volume.
N d : oil displaced from the volume swept by the waterflood.
Assume piston like displacement: Oil saturation in the swept region is the
average residual saturation determined from flood tests. Thus, S o  So1  Sor ,
where So1is the volumetric average oil saturation in the reservoir at the beginning
of waterflood:
7
Tayfun Babadagli, PhD, PEng
Short Course - EOR
File-5
Oil remaining in the reservoir when the unswept pore space is
resaturated to the initial water saturation:
Nr 
Ev SorV p
Bo

( 1  Ev )SoiV p
Bo
Oil recovered by waterflooding,N pw ,
V
N pw  So1  Ev Sor  ( 1  Ev )Soi  p
Bo
Oil potentially recoverable by waterflooding, as STB:
N pw  ( N  N p )  N
Boi
Bo

 Sor



1

E

1

v
S
 oi


N : Initial oil in place, STB.
Np : Oil produced during primary operations, STB.
Tayfun Babadagli, PhD, PEng
Short Course - EOR
File-5
8
Waterflooding performance prediction requires estimation of
the residual oil sauration (core data) and volumetric sweep
efficiency (VSE, EV).
VSE can be estimated by three methods:
1)
Using
Vp
N pw  So1  Ev Sor  ( 1  Ev )Soi 
Bo
2) VSE and ASE : Form interpretations of well logs fluid-rock properties,
and simplified displacement models.
Ev  EI E A
EA
EI
: Fraction of the reservoir area within the vertical portion of the reservoir
that has been swept to residual oil saturation.
: Fraction of the reservoir cross section that has been displaced by the
injected water.
3) Using correlations based on simulation of waterflood performances
(Hirasaki et al, SPE 13415, see two figures on Page 10).
Tayfun Babadagli, PhD, PEng
Short Course - EOR
File-5
9
Engineering approaches: Scaled-Model Correlations:
CGM (for 5-spot). Kimbler et al (JPT, Feb. 1964) developed
correlations for 9-spot.
They used the following graphs and parameters:
 krw 



 w  S or
M
 kro 
 
 o  S oi
Vd : number
Qi
Wi : PV’s of
Vd 
Wi
V p ( 1  S wi  Sor )
of displaceable HC pore volumes:
water injected, dimensionless.
: Cumulative volume of water injected, bbl.
Tayfun Babadagli, PhD, PEng
Vd 
Short Course - EOR
Qi
( 1  S wi  Sor )
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10
EXERCISE #8
(G.P. Willhite, Waterflooding Ex. 6.3):
Waterflood in a 9-spot pattern. Vp =38,790 bbl
(pore volume). M is based on end points of the
relative permeability curves is 1.0. Soi =0.7 and
Sor = 0.25. Plot Qi and WOR/Recovery.
Tayfun Babadagli, PhD, PEng
Short Course - EOR
File-5
11
After (Hirasaki et al, SPE 13415)
Tayfun Babadagli, PhD, PEng
Short Course - EOR
File-5
12
NUMERICAL MODELING OF EOR APPLICATION
 Pressure
 Saturation for each grid.
By solving the continuity equations for each phase numerically.
TWO PARAMETERS :
Numerical Grids
Injection Well
Production Well
13
Saturation contour lines
Tayfun Babadagli, PhD, PEng
Short Course - EOR
File-5
CHALLENGE : Data Preparation

Permeability – porosity distribution (2 or 3
Dimensional)
 Relative permeability and capillary pressure curves
 Fluid properties (reservoir conditions)
1-D and 2-Phase Flow

 (  o u o )  q o  (S o  o )
t
 (  wu w )  q w 
(OIL)

(S w  w )
t
(WATER)
uo and uw are darcy velocities. Neglecting gravity:

S
uo
 o
t
x

S
u w
 w
t
x
Tayfun Babadagli, PhD, PEng
(OIL)
(WATER
)
Short Course - EOR
14
File-5
uin1  uin11 Sin1  Sin

x
t
(Solve for P and S)
DATA NEEDED:


k (distribution), kr and Pc (uncertainty and up-scaling involved!)
PVT (reliable lab measurement)
Other numerical models for EOR:
IMMISCIBLE FLOW (DISPLACEMENT) Darcy’s law (Black-oil)
MISCIBLE FLOW (DISPLACEMENT) Fick’s law
THERMAL INJECTION Fourier’s law
Tayfun Babadagli, PhD, PEng
Short Course - EOR
15
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