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1. 1
Chapter 4
Efficiency of a
Displacement Process

2. EOR-Chapter 4 2
Efficiency of a Displacement Process

Introduction

Microscopic Displacement of Fluid in a
Reservoir

Macroscopic Displacement of Fluids in a
Reservoir

3. EOR-Chapter 2 3
Efficiency of a Displacement Process
Trapped Oil
M
E
= = (Microscopic Efficiency)(Microscopic Efficiency) × × (Volumetric Efficiency)(Volumetric Efficiency)
ProductionProduction
InjectionInjection
E
ME
VE

4. EOR-Chapter 2 4
Overall Displacement Efficiency
DV
EEE *
Where;
E =overall hydrocarbon displacement efficiency ,the
volume of hydrocarbon displaced divided by the volume of
hydrocarbon in place at the start of the process measured at
the same conditions of pressure and temperature
DE
V
E=Macroscopic (Volumetric) displacement efficiency
=Microscopic (Volumetric) hydrocarbon displacement efficiency.

5. EOR-Chapter 2 5
Microscopic & Macroscopic sweep efficiencies
agentEORbycontactedoilofvolumereservoir
agentEORbymobilizedoilofvolumereservoir
E
D

placeinoriginallyoilofvolumereservoir
agentdisplacingbycontactedoilofvolumereservoir
E
v


6. EOR-Chapter 2 6
Efficiency of a Displacement Process

Macroscopic Displacement
E
DV
EEE 
Where; Where;
= Overall displacement efficiency= Overall displacement efficiency
VE== Macroscopic displacement efficiencyMacroscopic displacement efficiency
= Microscopic displacement efficiency= Microscopic displacement efficiency
DE

7. EOR-Chapter 2 7

However,

L
E
LAV EEE 
== Areal Sweep efficiencyAreal Sweep efficiency
= = Lateral Sweep efficiencyLateral Sweep efficiency
AE
Efficiency of a Displacement Process

8. EOR-Chapter 2 8
Therefore, using all these definitions, the oil recovery equation is
To use this equation we must have methods to evaluate the different
efficiencies.
Estimates are available from:
Correlations
Scaled laboratory experiments
Numerical simulation

)(***
O
Poi
LADP
B
VS
EEEN
Oil Recovery Equation

9. EOR-Chapter 2 9
Oil Recovery Equation
R
E
placeinnhydrocarboofVolume
displacednhydrocarboofVolume
E
R
and is the volumetric sweep efficiency defined as
typical values of the overall recovery efficiency are:
Steam injection 30%-50%
Polymer injection 30%-55%
CO
2 injection 30%-65%
Solvent injection 35%-63%
RE
RE
R
E
R
E

10. EOR-Chapter 2 10
Action on Sweep
Efficiency at the
Macroscopic Scale
By increasing water
viscosity
Polymer
flooding
By decreasing the oil
viscosity
Steam drive
In-situ combustion
Carbon dioxide drive
Action on Displacement
Efficiency at the Pore
Scale
Miscible hydrocarbon
gas flooding
Surfactant flooding
By using a miscible
displacing fluid
By reducing the
interfacial tension
By action on the
rock wettability
Alkaline flooding
Action on Sweep & Displacement Efficiency

11. EOR-Chapter 2 11
Microscopic efficiency largely determines the success or
failure of any EOR process. For crude oil it is reflected in the
magnitude of Sor ( i.e., the residual oil saturation remaining
in the reservoir rock at the end of the process).
Microscopic Displacement of Fluids

12. EOR-Chapter 2 12
This efficiency is measured directly from a coreflood (since =1). It can also be
evaluated from the Buckley-Leveret (or fractional flow theory). For an immiscible
displacement is bounded by a residual phase saturation of the displaced phase Sor.
Miscible displacements eliminate - in principle -
or
S
oilcontactedofVolume
mobilizedoilofVolume
E
D

V
E
DE
Displacement Sweep Efficiency

13. EOR-Chapter 2 13
Example
Initial oil saturation, S
oi, is 0.60 and S
or in the swept
region for a typical water flood is 0.30
E
D = (S
oi – S
or) / S
oi
E
D= ( 0.60 – 0.30 ) / 0.60
E
D
=0.50
A typical waterflood sweep efficiency, E
v, at the
economic limit is 0.70. Therefore,
 E =E
D
E
V
= 0.50 X 0.70 = 0.35

14. EOR-Chapter 2 14
Important factors relating to microscopic
displacement behavior

Capillary Forces

Surface Tension and IFT

Solid Wettability

Capillary Pressure

Viscous Forces

15. EOR-Chapter 2 15
Important factors relating to microscopic
displacement behavior

Capillary forces have a detrimental effect,
being responsible for the trapping of oil
within the pore.

Trapping is a function of the ratio of
Viscous to Capillary forces.

The residual oil saturation decreases as
the ratio (Viscous force/ Capillary force)
increases.

16. EOR-Chapter 2 16
Capillary Forces: Surface Tension and IFT

Whenever immiscible phases coexist in a porous as in essentially all
processes of interest, surface energy related to the fluid interfaces
influences the saturations, distributions and displacement of the phases.
Sand Grain
Oil
Connate Water
Close up of oil water between grains of rock

17. EOR-Chapter 2 17
Capillary Forces: Surface Tension and IFT
The surface force, which is a tensile force, is
quantified in terms of surface tension

L
Liquid
Air or Vapor

The force per unit length required to create additional
surface area is the surface tension, usually expressed in
dynes/cm.

18. EOR-Chapter 2 18
Capillary Forces: Surface Tension and IFT

The term “surface tension” usually is reserved for the
specific case in which the surface is between a liquid and
its vapor or air. If the surface is between two different
liquids, or between liquid and solid, the term “interfacial
tension” is used.

The surface tension of water in contact with its vapor at
room temperature is about 73 dynes/cm.

IFT’s between water and pure hydrocarbons are about 30
to 50 dynes/cm at room temperature.

19. EOR-Chapter 2 19
Capillary Forces: Surface Tension and IFT

One of the simplest ways to measure the surface tension of liquid is to use a
capillary tube.

At the static condition the force owing to surface tension will be balanced by the
force of gravity acting on the column of fluid.



cos2
)( grh
ow


20. EOR-Chapter 2 20
Capillary Forces- Solid Wettability

Fluid distribution in porous media are affected not only by
the forces at fluid/fluid interfaces, but also by force of
fluid/solid interfaces.

Wettability is the tendency of one fluid to spread on or
adhere to a solid surface in the presence of a second fluid.

When two immiscible phases are placed in contact with a
solid surface, one phase is usually attracted to the solid
more strongly than the other phases. The more strongly
attracted phase is called the wetting phase.

21. EOR-Chapter 2 21
Capillary Forces- Solid Wettability

Rock wettability affects the nature of fluid saturations and
the general relative permeability characteristics of a
fluid/rock system.

The following figure shows residual oil saturations in a
strongly water-wet and a strongly oil-wet rock.
Water-wet System Oil-wet System

22. EOR-Chapter 2 22
Capillary Forces- Solid Wettability

Wettability can be quantitatively treated by examining
the interfacial forces that exist when two immiscible
fluid phases are in contact with a solid.
 cos
owwsos
os

ow

Water
ws

23. EOR-Chapter 2 23
Wettability
Where , , = IFT’s between
water and oil, oil and solid, and water
and solid respectively, dynes/cm.
 , contact angle, measured through
the water
os

ws
ow


 cos
owwsos

24. EOR-Chapter 2 24
Capillary Forces- Capillary Pressure
A pressure difference exists
across the interface. This
pressure, called Capillary
pressure can be illustrated by
fluid rise in capillary tube.
The figure shows rise in a glass
capillary. The fluid above the
water is an oil, and because the
water preferentially wets the
glass of the capillary, there is a
capillary rise.

25. EOR-Chapter 2 25
Capillary Pressure Equation

The difference pressure between oil water
at the oil/water interface
r
Por
rp
pghpp
ow
c
c
ow
cowwo




cos2
cos2
)(




26. EOR-Chapter 2 26
Capillary Forces- Capillary Pressure

Capillary pressure is related to

the fluid/ fluid IFT

Relative permeability of fluids (through )

Size of capillary (through r)

The phase with the lower pressure will always be the phase that
preferentially wets the capillary.

Pc varies inversely as a function of the capillary radius and
increases as the affinity of the wetting phase for the rock
surface increases.


27. EOR-Chapter 2 27
Viscous Force

Viscose forces in a porous medium are reflected in the
magnetude of the pressure drop that occurs as a result of
fluid flow through porous medium.

One of the simplest approximations used to calculate the
viscous force is to consider a porous medium as a bundle
of parallel capillary tubes.

With this assumption, the pressure drop for laminar flow
through a single tube is given by Poiseuille’s law.

28. EOR-Chapter 2 28
Viscous Force

Capillary Number

Water floods typically operates at conditions
where Nca < 10
-6
, and Nca values on the order
of 10
-7
are probably most common.
ow
w
ca
N




29. EOR-Chapter 2 29
Displacement Sweep Efficiency is a function of

Mobility ratios

Throughput or Transmissibility

Wettability

Dip angle

Capillary number

30. EOR-Chapter 2 30
All sweep efficiencies can be increased by decreasing the
mobility ratio by either:

rwOilkor i.e. steam flooding


rowater
kor
Oil recovery would still be limited by the residual or trapped
oil saturation. Methods that target to reduce this saturation
include solvent flooding.
Increasing i.e. polymer flooding
Lowering
Displacement Sweep Efficiency

31. EOR-Chapter 2 31
Experimental evidence suggests that under most conditions the residual oil saturation
(usually a non-wetting phase) can be as large as the wetting phase saturation.
The relationship between trapping wetting or non-wetting phase and a local capillary
number indicates experimental evidence of trapping in a permeable media. This
relationship is called the capillary desaturation curve.
The local capillary number is

u
N
C
Where
 = interfacial tension between displacing and displaced fuid
u = displacing superficial velocity

=displacing fluid viscosity
Trapped Oil Saturation

32. EOR-Chapter 2 32
Typical capillary desaturation curve
Trapped Oil Saturation

33. EOR-Chapter 2 33
Note that it is required a substantial increase in the capillary number to reduce the
residual oil saturation. The capillary number can be increased by either.
Lowering interfacial tension miscible/solvent methods
Increasing viscosity of displacing fluid polymer flooding.
There are physical, technical and economic limits of how much can the displacing
fluid viscosity and velocity be increased, thus solvent methods are the natural
choice to increase the capillary number and therefore lower the residual oil
saturation
Capillary desaturation curves are also affected by wettability, and pore size
distribution.
Trapped Oil Saturation

34. EOR-Chapter 2 34
Viscous Force

Viscous forces in a porous medium can be expressed in
terms of Darcy’s law:
darciestypermeabilik
porosity
ftlengthL
cpityvis
Dftvelocityaverage
psioppressuredrp
k
L
p
,
,
,cos
/,
,
))(158.0(












35. EOR-Chapter 2 35
Calculation of pressure gradient for
viscous oil flow in a rock









BA
owBA
rr
pp
11
cos2 

36. EOR-Chapter 2 36
Example: Calculation of pressure
gradient for viscous oil flow in a rock

Calculate the pressure gradient for flow of an oil with
10 cp viscosity at an interstitial flow rate of 1 ft/D. the
rock permeability is 250 md and the porosity is 0.2.

Solution: ))(158.0(
k
L
p


ftpsi
darcies
cpDft
L
p
/264.1
250.0
2.010/0.1158.0





37. EOR-Chapter 2 37
Example: pressure required to force an oil trap
through a pore throat

Calculate the threshold pressure necessary to force an oil trap
through a pore throat that has a forward radius of 6.2 micro meter
and radius of 15 micro meter. Assume that the wetting contact angle
is zero and IFT is 25 dynes/sec.

PB-PA=2*25(1/0.00062-1/0.0015)= - 47300 dynes/cm2

-47300*1.438*10^-5= - 0.68 psi

ftpsi
ft
cm
cm
psi
Lp /2073
48.30
01.0
68.0
/ 

38. EOR-Chapter 2 38
Macroscopic Displacement of Fluids in Reservoir
Volumetric Displacement Efficiency & Material
Balance
Volumetric Displacement Efficiency Expression
Definition & Discussion of Mobility Ratio
Areal Displacement Efficiency
Correlations
Vertical Displacement Efficiency
Volumetric Displacement Efficiency

39. EOR-Chapter 2 39
Macroscopic Displacement of Fluids In a Reservoir

Introduction
Oil recovery in any displacement process depends on the volume of
reservoir contacted by the injected fluid. A quantitative measure of this
contact is the volumetric displacement (sweep) efficiency defined as the
fraction of reservoir (or project )PV that has been contacted or affected by
the injected fluid. Clearly, is a function of time in a displacement
process.
Overall displacement efficiency in a process can be viewed conceptually as
a product of the volumetric sweep, ,and the microscopic efficiency,
VE
VE
DE

40. EOR-Chapter 2 40
Volumetric Displacement Efficiency and Material Balance

Volumetric displacement ,or sweep efficiency, is often used to
estimate oil recovery by use of material-balance concepts. for
example, consider a displacement process that reduces the initial
oil saturation to a residual saturation in the region contacted by the
displacing fluid. If the process is assumed to be piston-like, the oil
displaced is given by
Vp
o
o
o
o
p EV
B
S
B
S
N )(
2
2
1
1

Where ;
= oil displaced , = oil saturation at the beginning of the
displacement process, = residual oil saturation at the end of the process in
the volume of reservoir contacted by the displacing fluid, = FVF at initial
conditions, = FVF at the end of the process, and = reservoir PV

2oB
1o
B
2oS
1oS
pN
p
V

41. EOR-Chapter 2 41
Volumetric Displacement Efficiency and Material Balance
VD
p
EE
N
N
RF 
1
1
N
Where;
=OOIP at the beginning of the displacement process. if displacement
performance data are available, above Eq. also can be used to estimate volumetric
sweep. For example, if waterflood recovery data are available, the equation can be
rearranged to solve for
)(
2
2
1
1
o
o
o
o
p
p
V
B
S
B
S
V
N
E



42. EOR-Chapter 2 42
Volumetric Displacement Efficiency and Material Balance

Where = oil produced in the waterflood.
)(
2
2
1
1
o
o
o
o
p
p
V
B
S
B
S
V
N
E


pN

43. EOR-Chapter 2 43
Volumetric Displacement Efficiency

Volumetric Displacement Efficiency Expressed as the product of
Areal and Vertical Displacement Efficiencies
Volumetric sweep efficiency can be considered conceptually as
the product of the areal and vertical sweep efficiencies. Consider
a reservoir that has uniform porosity,thickness,and hydrocarbon
saturation, but that consists of several layers. For a displacement
process conducted in the reservoir, can be expressed as
LAV
EEE *
VE

44. EOR-Chapter 2 44
Where ;

All efficiencies are expressed as fractions. is the volumetric sweep
efficiency of the region confined by the largest areal sweep efficiency in the
system.
For a real reservoir, in which porosity,thickness,and hydrocarbon saturation vary
areally, is replaced by a pattern sweep efficiency ,
LPV EEE
A
E
A
E
areaTotal
agentdisplacingbycontactedArea
E
A
Volumetric Displacement Efficiency
lengthverticalTotal
agentdisplacingbycontactedLength
E
L



45. EOR-Chapter 2 45
Where ;

=pattern sweep (displacment)efficiency,hydrocarbon pore space enclosed
behind the injected-fluid front divided by total hydrocarbon pore space in the
pattern or reservoir a real reservoir.
In essence, is an ideal sweep efficiency that has been corrected for
variations in thickness,porosity,and saturation. In either case, overall
hydrocarbon recovery efficiency in a displacement process may be expressed as
DLPEEEE
p
E
p
E
LPV
EEE
Volumetric Displacement Efficiency

46. EOR-Chapter 2 46
This figure illustrates the concept of the vertical and
areal sweep efficiency

47. EOR-Chapter 2 47
The following figure illustrate the definition
of areal sweep efficiency
areaTotal
agentdisplacingbycontractedAreal
E
A

48. EOR-Chapter 2 48
These correlations are for piston like displacements in
homogeneous, confined patterns. When the well patterns are
unconfined, the total area can be much lager and smaller .
Areal Sweep Efficiency
The most common source of areal sweep efficiency data is from
displacements in scaled physical models. Several correlations exist
in the literature. Craig (1980) in his SPE monograph “the reservoir
engineering aspects of waterflooding” discusses several of these
methods.

A
E
Oil Recovery Equation

49. EOR-Chapter 2 49
AREAL SWEEP EFFICIENCY

When oil is produced from patterns of injectors and producers, the flow is
such that only part of the area is swept at breakthrough. the expansion of
the water bank is initially radial from the injector but eventually is
focused at the producer.
The pattern is illustrated for a direct line drive at a mobility ratio of unity.At
breakthrough a considerable area of the reservoir is unswept.

50. EOR-Chapter 2 50
Parameters Affecting

The following definitions are needed to describe the effects of
reservoir and fluid properties upon the efficiencies:

Mobility Ratio

Dimensionless Time

Viscous Fingering

Injection/Production well pattern

Reservoir permeability heterogeneity

Vertical Sweep Efficiency

Gravity Effect

Gravity/ Viscous Force Ratio
AE

51. EOR-Chapter 2 51

Mobility Definition

The mechanics of displacement of one fluid with
another are controlled by differences in the ratio
of effective permeability and viscosity


The specific discharge (flow per unit cross
sectional area) for each fluid phase depends on
This is called the fluid mobility( ):

k


k

52. EOR-Chapter 2 52
Mobility Control
d
W
W
W
k


O
O
O
k


Mobility controls the relative ease with which fluids can flow through a porous
medium.
= mobility of the displacing fluid phase
= mobility of the displaced fluid phase
dM
D
/
D

53. Mobility ratio

The mobility ratio is an extremly important parameter in
any displacement process. It affects both areal and vertical
sweep, with sweep decreasing as M increases for a given
volume of fluid injected.

M <1 then favorable displacement

M >1 then unfavorable displacement
EOR-Chapter 2 53

54. EOR-Chapter 2 54
volumeporefrence
fluidinjectedOfVolumeCumulative
t
D
Re

Dimensionless Time
This variable is used to scale-up between the laboratory and the
field . The dimensionless time is defined as the
There are various definitions for the reference pore volume
according to the application.

55. EOR-Chapter 2 55
Viscous Fingering

The mechanics of displacing one fluid with another are
relatively simple if the displaced fluid (oil) has a tendency to
flow faster than the displacing fluid (water).

Under these circumstances, there is no tendency for the
displaced fluid to be overtaken by the displacing fluid and the
fluid – fluid (oil-water) interface is stable.

56. EOR-Chapter 2 56
Viscous Fingering

If the displacing fluid has a tendency to move faster than
the displaced fluid, the fluid-fluid interface is unstable.
tongues of displacing fluid propagate at the interface.
This process is called viscous fingering.

57. EOR-Chapter 2 57
Viscous Fingering
- Decreases when the mobility ratio increases because the displacement front
becomes unstable. This phenomena, known as viscous fingering results in an
early breakthrough for the displacing fluid, or into a prolonged injection to
achieve sweep-out. The next figure illustrates this phenomena, which is
commonly observed in solvent flooding.
A
E

58. EOR-Chapter 2 58
Flooding Patterns

59. EOR-Chapter 2 59
Flooding Patterns

60. EOR-Chapter 2 60
Flooding Patterns

61. EOR-Chapter 2 61
Permeability Heterogeneity

It is often has a marked effect on areal sweep.
This effect may be quite different from
reservoir to reservoir, however, and thus it is
difficult to develop generalized correlations.

Anisotropy in permeability has great effect on
the efficiency.

62. EOR-Chapter 2 62
Effect of Mobility Ratio

The following figures show fluid fronts at different
points in a flood for different mobility ratios. These
results are based on photographs taken during
displacements of one colored liquid by second,
miscible colored liquid in a scaled model.

63. EOR-Chapter 2 63
Correlations Based on Miscible Fluids, Five-Spot Pattern.
Figure 1 shows fluid fronts at different points in a flood for
different mobility Ratios. The Viscosity Ratio varied in different
floods and, because only one phase was present, M is given by
Equation.
D
d
M



Correlations Based on ….

64. EOR-Chapter 2 64
Figure-1: Miscible displacement in a quarter of
a five-spot pattern at mobility ratios<=1.0
M=0.151 M=1.0
Breakthrough
Breakthrough
P
o
r
e

V
o
l
u
m
e
s

I
n
j
e
c
t
e
d
P
o
r
e

V
o
l
u
m
e
s

I
n
j
e
c
t
e
d
Injection well
Producing well

65. EOR-Chapter 2 65
M=2.40
PV
BT
BT
M=4.58
0.3
0.2
0.1
0.06
0.2
0.3
0.1
PV
• PRODUCING WELL PV=PORE VOLUME INJECTED
X INJECTION WELL BT=BREAKTHROUGH
Figure 2: Miscible displacement in a quarter of a five-spot pattern at mobility
ratios>1.0,viscous fingering (from Habermann)

66. EOR-Chapter 2 66
M=17.3
M=71.5
BT
BT
0.15
0.05
• PRODUCING WELL PV=PORE VOLUME INJECTED
X INJECTION WELL BT=BREAKTHROUGH
Figure-3: Miscible displacement in a quarter of a five-spot pattern at mobility
ratios>1.0,viscous fingering (from Habermann)

67. EOR-Chapter 2 67
Habermann presented values of E
A
as a function of
dimensionless PVs injected,Vi/Vp,after breakthrough, as shown
in Figure 4 Results are given for M=0.216 (favorable) to 71.5
(unfavorable).
Correlations Based on Miscible Fluids, Other Patterns
Numerous modeling studies for patterns other than a five-spot
have been reported. Craig gives a summery listing of references.
As an example of such studies, Figure 5 shows one reported result
of areal sweep as a function of mobility ratio for one-eighth of a
nine-spot pattern.
Correlations Based on ….

68. EOR-Chapter 2 68
Pore Volume Injected, Vi/ Vp
Figure-4: Areal sweep efficiency after breakthrough as a function of
mobility ratio and PVs injected
A
r
e
a
l

S
w
e
e
p

E
f
f
i
c
i
e
n
c
y
,

E
A
%

69. EOR-Chapter 2 69
Correlation Based on
Miscible Fluids
Numerous modeling studies for
patterns other than a five-spot have
been reported.
One-eight of a nine-spot pattern is
shown as an example.
This study was conducted with
miscible liquids and the X-ray
shadowgraph method

70. EOR-Chapter 2 70
Figure-5:Areal sweep efficiency as a function of mobility ratio;

71. EOR-Chapter 2 71
Correlations Based on Immiscible
Fluids, Five –Spot Pattern

Craig et al. conducted an experimental study of areal
displacement efficiency for immiscible fluids consisting
of oil, gas, and water.The study was conducted in
consolidated sandstone cores, and fronts were monitored
with the X-ray shadowgraph technique.

Figure 6 compares areal sweep efficiency at breakthrough
as a function of mobility ratio to the data of Dyes et al.,
which were obtained with miscible fluids.

72. EOR-Chapter 2 72
Figure-6: Areal sweep efficiency at breakthrough as a function of mobility
ratio( immiscible fluid displacement);
Water-Gas
Miscible
Gas-Oil
A
r
e
a
l
S
w
e
e
p

E
f
f
ic
ie
n
c
y

a
t

B
r
e
a
k
t
h
r
o
u
g
h

,
E
A
b
t
%
Mobility Ratio,M

73. EOR-Chapter 2 73
Prediction of Areal Displacement
Performance on the Basis of Modeling Studies

Prediction based on Piston-Like
Displacement

Caudle & Witte correlation

Claridge correlation (viscous fingering)

Mahaffey et. Al model (dispersion )

Parallel plate glass model

Mathematical Modeling-Numerical

74. EOR-Chapter 2 74
Prediction of Areal Displacement Performance
on the Basis of Modeling Studies

Prediction Based on Piston –Like Displacement.
Caudle and Witte published results from laboratory models of a
five-spot pattern in which displacements were conducted with
miscible liquids.
The performance calculations are restricted to those floods in
which piston-like displacement is a reasonable assumption; i.e.,
the displacing phase flows only in the swept region and the
displaced phase flows in the upswept region. No production of
displaced phase occurs from the region behind the front.

75. EOR-Chapter 2 75
Figure 8 gives E
A
as a function of M for different values of the
fractional flow of the displacing phase ,f
D
, at the producing well.
)(
oroipd SSAhV 
Figure 7 through 9 show data from the experiments. In Figure 7, E
A

is given as a function of M for various values of injected PVs.
The ratio Vi/V
pd is a dimensionless injection volume defined as
injected volume divided by displaceable PV, V
pd
. For a waterflood,
V
pd
is given by
Prediction Based on Piston –Like Displacement

76. EOR-Chapter 2 76

Figure 9 presents the conductance ratio, , as a function of M for various values
of E
A, but only for values of M between 0.1 and 10. Conductance is defined as
injection rate divided by the pressure drop across the pattern,

At any mobility ratio other than M=1.0,conductance will change as the
displacement process proceeds. For a favorable mobility ratio, conductance will
decrease as the area swept, EA, increases. The opposite will occur for unfavorable
M values.

The conductance ratio, shown in Figure 9 is the conductance at any point of
progress in the flood divided by the conductance at that same point for a
displacement in which the mobility ratio is unity (referenced to the displaced
phase).

p
q

Prediction Based on Piston –Like Displacement

77. EOR-Chapter 2 77

By combining Figures 7 through 9 , performance calculations can be
performed. Areal sweep, as a function of volume injected, is available
from Figure 7.

Fractional production of either phase can be determined with Figure
8.
Rate of injection may be determined as a function of E
A from Figure
9.

To apply Figure 9, however , it is also necessary to use the
appropriate expression for initial injection rate. This is given by Craig
for a five-spot pattern using parameters for the displaced phase:
Prediction Based on Piston –Like Displacement

78. EOR-Chapter 2 78
Where i=injection rate at start of a displacement process, B/D;
k=absolute rock permeability ,md;Krd=relative permeability of
displaced phase, h=reservoir thickness ,ft; =pressure drop, psi;
=viscosity of displaced phase, cp;d=distance measured between
injection and production wells ,ft; and rw= wellbore radius, ft.
)2688.0(log
001538.0



w
d
rd
r
d
phkk
i

iq

At any point in the flood, the flow rate is given by
p
Prediction Based on Piston –Like Displacement

79. EOR-Chapter 2 79
A
r
e
a
l

S
w
e
e
p

E
f
f
i
c
i
e
n
c
y
,
E
A
Figure-7: Areal Sweep efficiency as a function of mobility ratio
and injected volume.

80. EOR-Chapter 2 80
Mobility Ratio,M
A
r
e
a
l

S
w
e
e
p

E
f
f
i
c
i
e
n
c
y
,
E
A

Figure-8:Areal sweep efficiency as a function of mobility ratio and
fractional flow at displacing phase

81. EOR-Chapter 2 81
Mobility Ratio,M
C
o
n
d
u
c
t
a
n
c
e

R
a
t
i
o
,
Figure-9:Conductance ratio as a function of mobility ratio and areal sweep.

82. EOR-Chapter 2 82
Example: Performance Calculations Based on
Physical Modeling Results

A waterflood is conducted in a five-spot pattern in which the pattern area is 20 acres. Reservoir properties are:

ftr
psip
Satk
ROSatk
mdk
STBRBB
cp
cp
S
S
fth
w
wiro
rw
o
w
o
or
oi
5.0
1250
)(94.0
)(27.0
50
/0.1
1
10
25.0
8.0
2.0
20
















83. EOR-Chapter 2 83
Required
Use the method of Caudle and Witte to calculate:
(1) the barrels of oil recovered at the point in time
at which the producing WOR=20 ,
(2) the volume of water injected at the same point
(3) the rate of water injection at the same point in
time
(4) the initial rate of water injection at the start of
waterflood

84. EOR-Chapter 2 84
Solution
Apply the correlations in Figs 7 through 9
1. Calculate oil recovered

M=2.9, f
D
=20/21=.95 From Fig 8, E
A
=.94

Np=321000 STB

2. Calculate total water injected. From Fig 7, Vi/Vpd=2.5 (at E
A=.94)

Vpd = Vp (Soi – Sor) = 341300 bbl

Vi = Vpd x 2.5 = 853300 bbl
3. Calculate water injection rate at the same point in time. From

i=63.4 B/D

From Fig. 9, γ=2.7, from = 63.4x2.7 = 171 B/D
4. Calculate initial water injection rate

i=63.4 B/D
iq
)2688.0(log
001538.0



w
d
rd
r
d
phkk
i


85. EOR-Chapter 2 85
Calculation of E
A with Mathematical
Modeling

Models are based on Numerical
analysis methods and digital
computers

Douglas et al-2D immiscible displacement. This
method is based on the numerical solution of
the PDE’s that describe the flow of two
immiscible phases in two dimensions

Higgins and Leighton mathematical model is
based on frontal advance theory

86. EOR-Chapter 2 86
Comparison of calculated and experimental
results, 5 spot pattern (Douglas et al.)

87. EOR-Chapter 2 87
Vertical sweep ( displacement) efficiency, pore space invaded by the
injected fluid divided by the pore space enclosed in all layers behind
the location of the leading edge (leading areal location) of the front.
Areal sweep efficiency, must be combined in an appropriate manner
with vertical sweep to determine overall volumetric displacement
efficiency. It is useful, however, to examine the factors that affect
vertical sweep in the absence of areal displacement factors.
Vertical Displacement Efficiency

88. EOR-Chapter 2 88
Vertical Displacement Efficiency

89. EOR-Chapter 2 89
Vertical Displacement Efficiency

Vertical Displacement Efficiency is controlled primarily
by four factors:

Heterogeneity

Gravity effect

Gravity segregation caused by differences in density

Mobility ratio

Vertical to horizontal permeability variation

Capillary forces

90. EOR-Chapter 2 90

Observation of thre figure indicates a stratified reservoir with layers of different
permeability. The displacement of the fluid is an idealized piston-flow type. Due to
the permeability contrast the displacing fluid will break through earlier in the first
layer, while the entire cross-section will achieve sweep-out at a later time, when
layer #4 breaks through.
Heterogeneity

91. EOR-Chapter 2 91
Heterogeneity:Location of the water
front at different Location

92. EOR-Chapter 2 92
Heterogeneity:Dykstra-Persons
model

93. EOR-Chapter 2 93
Gravity Segregation in Horizontal Bed

Water tongue
Water

Gas umbrella
Gas

94. EOR-Chapter 2 94
Gravity is a factor that affects the vertical efficiency not only in heterogeneous
reservoirs but in homogenous as well.
Gravity effects will be important when: (1) vertical communication is good. This
is satisfied when is large.
2
1
)(
k
k
H
L
R
v
L


(2) When gravity forces are strong compared to viscous forces. This is satisfied
when the gravity number Ng is large.
u
gk
N
g


LR
Gravity Effect

95. EOR-Chapter 2 95
Where:
= relative mobility of displacing fluid

= density difference (displaced - displacing)
u = superficial velocity
Both numbers are dimensionless.
The following figures indicate gravity effects for two different situations
1- Density of displacing fluid lower that density of displaced fluid
The displacing fluid will tend to flow to the top of the reservoir and bypass the
fluid in the lower region (tongue over).
Gravity Effect

96. EOR-Chapter 2 96
Tonguing will occur when M < 1 as long as and Ng are large. The effect of
heterogeneity and gravity can be mitigated by a favorable mobility ratio.
Gravity tonguing does not require a dipping reservoir (although dipping can be
used as an advantage when gravity is important). Gravity tonguing is important in
steam flooding applications.
Density of displacing fluid lower that density of displaced fluid
LR
Gravity Effect

97. EOR-Chapter 2 97
Density of displacing fluid higher than density of displaced fluid
Gravity Effect

98. EOR-Chapter 2 98
Gravity segregation occurs when the injected fluid is less dense
than the displaced fluid, Figure10a.
Gravity override is observed in steam displacement, in-situ
combustion, CO2 flooding, and solvent flooding processes.

Gravity segregation also occurs when the injected fluid is more
dense than the displaced fluid, as Figure10b shows for a
waterflood.
Gravity segregation leads to early breakthrough of the injected
fluid and reduced vertical sweep efficiency.
Effect of Gravity Segregation and Mobility Ratio on
Vertical Displacement Efficiency

99. EOR-Chapter 2 99
Gravity Segregation in displacement processes
dD
Figure-10: Gravity Segregation in displacement processes.
Gravity Override (a)
Gravity Underride (b) dD
Displaced
Phase
Displacing
Phase
Displacing
Phase
Displaced
Phase
Displaced
Phase
Displacing
Phase

100. EOR-Chapter 2 100
Experimental Result

Craig et al. studied vertical sweep efficiency by conducting
a set of scaled experiments in linear systems and five-spot
models. Both consolidated & unconsolidated sands were
used.

The linear models used were from 10 to 66 in. long with
length/height ratios ranging from 4.1 to 66.

Experiments were conducted with miscible and immiscible
liquids having mobility ratios from 0.057 to 200.

Immiscible water floods were conducted at M<1.

Vertical sweep was determined at breakthrough by material
balance and visual observation of produced effluent

101. EOR-Chapter 2 101
Craig et al. Results

Results of the linear displacements are shown in the next Figure,
where EI at breakthrough is given as a function of dimensionless
group called a viscous/gravity ratio.

102. EOR-Chapter 2 102
Vertical sweep efficiency at breakthrough as a
function of the ratios of viscous/gravity forces,
Linear system (from Craig et al.)

103. EOR-Chapter 2 103
Example: Relative Importance of Gravity
Segregation in a Displacement Process

A miscible displacement process will be used to displace
oil from a linear reservoir having the following
properties:

104. EOR-Chapter 2 104
Solution

105. EOR-Chapter 2 105
Mathematical Model

Spivak used a 2D and 3D numerical model to study
gravity effects during water flooding and gas
flooding

106. EOR-Chapter 2 106
Gravity Segregation in two-phase flow

107. EOR-Chapter 2 107
The correlations of Craig et al. and Spivak on
gravity segregation

Gravity segregation increases with increasing horizontal and vertical
permeability.

Gravity segregation increases with increasing density difference
between the displacing and displaced fluids.

Gravity segregation increases with increasing mobility ratio

Gravity segregation increases with increasing rate. This effect can be
reduced by viscous fingering

Gravity segregation decreases with increasing level of viscosity for a
fixed viscosity ratio.
The correlations of Craig et al. and Spivak indicate the
following effects of various parameters on gravity
segregation, as summarized by Spivak

108. EOR-Chapter 2 108
Flow Regions in Miscible Displacement at Unfavorable
Mobility Ratios
gV
R
/
gVR
/
(A) REGIONS I AND II
Oil
SOLVENT
Flow experiments in a vertical cross section in horizontal porous media have
shown that four flow regions, are possible when the mobility ratio is
unfavorable.
Region I occurs at very low values and is characterized by a single
gravity tongue, with the displacing liquid either underriding or
overriding the displaced liquid. Vertical sweep is a strong function of
.At larger values, in region II, a single gravity tongue still exists,
but vertical sweep is relatively insensitive to the value of the
viscous/gravity ratio.
gV
R
/

109. EOR-Chapter 2 109
Flow Regions in Miscible Displacement at Unfavorable
Mobility Ratios
(B) REGION III (C) REGION IV
SOLVENT
SOLVENT
Oil
Oil
The transition to region III occurs at a particular critical value.
In region III, viscous fingers are formed along the primary gravity tongue
and appear as secondary fingers along the primary gravity tongue.
Vertical sweep is improved by the formation of the viscous fingers in this
region.
In region IV ,flow is dominated by the viscous forces and by viscous
fingering. A gravity tongue does not form because of the strong viscous
fingering. The vertical sweep in this region is relatively insensitive to

gVR
/
gVR
/

110. EOR-Chapter 2 110
Flow Regimes in Miscible Displacement

111. EOR-Chapter 2 111
Volumetric Efficiency
Methods of estimating volumetric displacement
efficiency in a 3D reservoir fall into two
classifications.
Direct application of 3D models

Physical

mathematical
Layered reservoir model.

The reservoir is divided into a number of no communicating layers.

Displacement performance is calculated in each layer with
correlations of 2D.
Performance in individual layers are summed to obtain volumetric
efficiency

112. EOR-Chapter 2 112
Volumetric Displacement Efficiency

113. EOR-Chapter 2 113
Calculation of volumetric sweep with
Numerical Simulators

114. EOR-Chapter 2 114

115. Action on Sweep
Efficiency at the
Macroscopic Scale
By increasing water
viscosity
Polymer
flooding
By decreasing the oil
viscosity
Steam drive
In-situ combustion
Carbon dioxide drive
Action on Displacement
Efficiency at the
Microscopic/Pore Scale
Miscible hydrocarbon
gas flooding
Surfactant flooding
By using a miscible
displacing fluid
By reducing the
interfacial tension
By action on the
rock wettability
Alkaline flooding
Action on Sweep & Displacement Efficiency