Transcript CHAPTER 2

Chapter 4
Efficiency of a
Displacement Process
1
Efficiency of a Displacement Process



Introduction
Microscopic Displacement of Fluid in a
Reservoir
Macroscopic Displacement of Fluids in a
Reservoir
EOR-Chapter 4
2
Efficiency of a Displacement Process
Production
Trapped Oil
EM
Injection
E = E M (Microscopic Efficiency) × EV (Volumetric Efficiency)
EOR-Chapter 2
3
Overall Displacement Efficiency
E  EV * ED
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
EV =Macroscopic (Volumetric) displacement efficiency
ED =Microscopic (Volumetric) hydrocarbon displacement efficiency.
EOR-Chapter 2
4
Microscopic & Macroscopic sweep efficiencies
ED 
reservoirvolum eof oil m obilizedby EOR agent
reservoirvolum eof oil contactedby EOR agent
reservoirvolum eof oil contacted by displacingagent
Ev 
reservoirvolum eof oil originally in place
EOR-Chapter 2
5
Efficiency of a Displacement Process

Macroscopic Displacement
E  EV  ED
Where;
E
= Overall displacement efficiency
EV = Macroscopic displacement efficiency
ED = Microscopic displacement efficiency
EOR-Chapter 2
6
Efficiency of a Displacement Process

However,
EV  EA  EL
EA
= Areal Sweep efficiency
EL
= Lateral Sweep efficiency
EOR-Chapter 2
7
Oil Recovery Equation
Therefore, using all these definitions, the oil recovery equation is
NP
SoiVP
 ED * E A * EL * (
)
BO
To use this equation we must have methods to evaluate the different
efficiencies.
Estimates are available from:
Correlations
Scaled laboratory experiments
Numerical simulation
EOR-Chapter 2
8
Oil Recovery Equation
and
ER
is the volumetric sweep efficiency defined as
ER 
Volum eof hydrocarbon displaced
Volum eof hydrocarbon in place
typical values of the overall recovery efficiency
Steam injection
Polymer injection
CO2 injection
Solvent injection
ER
ER
ER
ER
are:
30%-50%
30%-55%
30%-65%
35%-63%
EOR-Chapter 2
9
Action on Sweep & Displacement Efficiency
By increasing water
viscosity
Action on Sweep
Efficiency at the
Macroscopic Scale
Steam drive
By decreasing the oil
viscosity
By using a miscible
displacing fluid
Action on Displacement
Efficiency at the Pore
Scale
Polymer
flooding
By reducing the
interfacial tension
By action on the
rock wettability
EOR-Chapter 2
In-situ combustion
Carbon dioxide drive
Miscible hydrocarbon
gas flooding
Surfactant flooding
Alkaline flooding
10
Microscopic Displacement of Fluids
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).
EOR-Chapter 2
11
Displacement Sweep Efficiency
ED
Volum e of oil m obilized

Volum e of contacted oil
This efficiency is measured directly from a coreflood (since EV =1). It can also be
evaluated from the Buckley-Leveret (or fractional flow theory). For an immiscible
displacement E D is bounded by a residual phase saturation of the displaced phase Sor.
Miscible displacements eliminate - in principle -
S or
EOR-Chapter 2
12
Example






Initial oil saturation, Soi, is 0.60 and Sor in the
swept region for a typical water flood is 0.30
ED = (Soi – Sor) / Soi
ED= ( 0.60 – 0.30 ) / 0.60
ED=0.50
A typical waterflood sweep efficiency, Ev, at the
economic limit is 0.70. Therefore,
E =EDEV = 0.50 X 0.70 = 0.35
EOR-Chapter 2
13
Important factors relating to microscopic
displacement behavior

Capillary Forces

Surface Tension and IFT

Solid Wettability
Capillary Pressure


Viscous Forces
EOR-Chapter 2
14
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.
EOR-Chapter 2
15
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.
Oil
Connate Water
Sand Grain
Close up of oil water between grains
of rock
EOR-Chapter
2
16
Capillary Forces: Surface Tension and IFT

The surface force, which is a tensile force, is
quantified in terms of surface tension
Air or Vapor
L
Liquid
The force per unit length required to create additional
surface area is the surface tension, usually expressed in
dynes/cm.

EOR-Chapter 2
17
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.
EOR-Chapter 2
18
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.
rh(  w  o ) g

2 cos
EOR-Chapter 2
19
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.
EOR-Chapter 2
20
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
EOR-Chapter 2
Oil-wet System
21
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.
 ow
Water
 os
 ws
 os   ws   ow  cos
EOR-Chapter 2
22
Wettability
 os   ws   ow  cos


Where  ow ,  os ,  ws = IFT’s between
water and oil, oil and solid, and water
and solid respectively, dynes/cm.
 , contact angle, measured through the
water
EOR-Chapter 2
23
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.
EOR-Chapter 2
24
Capillary Pressure Equation

The difference pressure between oil water at
the oil/water interface
p o  p w  h(  w   o ) g  p c
rp c
 ow 
2 cos
2 ow cos
or
Pc 
r
EOR-Chapter 2
25
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.
EOR-Chapter 2
26
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.
EOR-Chapter 2
27
Viscous Force


Capillary Number
w
N ca 
 ow
Water floods typically operates at conditions
where Nca < 10-6, and Nca values on the order
of 10-7 are probably most common.
EOR-Chapter 2
28
Displacement Sweep Efficiency is a function of





Mobility ratios
Throughput or Transmissibility
Wettability
Dip angle
Capillary number
EOR-Chapter 2
29
Displacement Sweep Efficiency
All sweep efficiencies can be increased by decreasing the
mobility ratio by either:
Lowering
Oil or k rw 
Increasing
water or kro 
i.e. steam flooding
i.e. polymer flooding
Oil recovery would still be limited by the residual or trapped
oil saturation. Methods that target to reduce this saturation
include solvent flooding.
EOR-Chapter 2
30
Trapped Oil Saturation
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
Where
NC 
 =displacing fluid viscosity
u

=
interfacial tension between displacing and displaced fuid
u = displacing superficial velocity
EOR-Chapter 2
31
Trapped Oil Saturation
Typical capillary desaturation curve
EOR-Chapter 2
32
Trapped Oil Saturation
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.
EOR-Chapter 2
33
Viscous Force

Viscous forces in a porous medium can be expressed in
terms of Darcy’s law:
 L
p  (0.158)(
)
k
p  pressuredrop, psi
  average velocity, ft / D
  vis cos ity, cp
L  length, ft
  porosity
k  perm eability, darcies
EOR-Chapter 2
34
Calculation of pressure gradient for
viscous oil flow in a rock
1 1
p A  p B  2 ow cos   
 rA rB 
EOR-Chapter 2
35
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:
 L
p  (0.158)(
k
)
p
0.158 1.0 ft / D  10 cp  0.2

 1.264 psi / ft
L
0.250 darcies
EOR-Chapter 2
36
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
0.68 psi 30.48 cm
p / L  

 2073 psi / ft
0.01 cm
ft
EOR-Chapter 2
37
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
EOR-Chapter 2
38
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, EV 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, E ,and the microscopic efficiency, E
V
EOR-Chapter 2
D
39
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
So1 So 2
Np  (

)V p EV
Bo1 Bo 2
Where ;
N p = oil displaced , S o1 = oil saturation at the beginning of the
displacement process, S o 2= residual oil saturation at the end of the process in
the volume of reservoir contacted by the displacing fluid, Bo1 = FVF at initial
conditions, Bo 2 = FVF at the end of the process, and Vp = reservoir PV
EOR-Chapter 2
40
Volumetric Displacement Efficiency and Material Balance
RF 
Where;
Np
N1
 ED EV
N1
=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
EV 
Np
S o1
So 2
Vp (

)
Bo1
Bo 2
EOR-Chapter 2
41
Volumetric Displacement Efficiency and Material Balance
EV 

Where
Np
Np
S o1
So 2
Vp (

)
Bo1
Bo 2
= oil produced in the waterflood.
EOR-Chapter 2
42
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, E can be expressed as
V
EV  E A * EL
EOR-Chapter 2
43
Volumetric Displacement Efficiency
EA 
Where ;
EL 
Area contacted by displacing agent
Total area
Length contacted by displacingagent
Total vertical length
All efficiencies are expressed as fractions. E A 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, E A is replaced by a pattern sweep efficiency ,
EV  E P E L
EOR-Chapter 2
44
Volumetric Displacement Efficiency
Where ;
EV  E P E L
E p =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, E p 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
E  EP EL ED
EOR-Chapter 2
45
This figure illustrates the concept of the vertical and
areal sweep efficiency
EOR-Chapter 2
46
The following figure illustrate the definition
of areal sweep efficiency
Areal contracted by displacing agent
EA 
Total area
EOR-Chapter 2
47
Oil Recovery Equation
E A 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.
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 .
EOR-Chapter 2
48
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 EOR-Chapter
reservoir is2 unswept.
49
Parameters Affecting E A









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
EOR-Chapter 2
50
Mobility Definition

The mechanics of displacement of one fluid with
another are controlled by differences in the ratio
of effective permeability and viscosity k


The specific discharge (flow per unit cross
sectional area) for each fluid phase depends on k

This is called the fluid mobility(  ):
EOR-Chapter 2
51
Mobility Control
W 
kW
W
O 
kO
O
Mobility controls the relative ease with which fluids can flow
through a porous medium.
M  D / d
D
= mobility of the displacing fluid phase
 d = mobility of the displaced fluid phase
EOR-Chapter 2
52
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
Dimensionless Time
This variable is used to scale-up between the laboratory and the
field . The dimensionless time is defined as the
Cum ulativeVolum e Of injected fluid
tD 
Re frence pore volum e
There are various definitions for the reference pore volume
according to the application.
EOR-Chapter 2
54
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.
EOR-Chapter 2
55
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.
EOR-Chapter 2
56
Viscous Fingering
E A - 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.
EOR-Chapter 2
57
Flooding Patterns
EOR-Chapter 2
58
Flooding Patterns
EOR-Chapter 2
59
Flooding Patterns
EOR-Chapter 2
60
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.
EOR-Chapter 2
61
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.
EOR-Chapter 2
62
Correlations Based on ….
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
M
D
EOR-Chapter 2
63
Producing well
Injection well
Breakthrough
Pore Volumes Injected
Pore Volumes Injected
Breakthrough
M=0.151
M=1.0
Figure-1: Miscible displacement in a quarter of
a five-spot pattern at mobility ratios<=1.0
EOR-Chapter 2
64
BT
BT
PV
PV
0.3
0.3
0.2
0.2
0.1
0.06
0.1
M=4.58
M=2.40
• 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)
EOR-Chapter 2
65
BT
BT
0.15
0.05
M=71.5
M=17.3
• 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)
EOR-Chapter 2
66
Correlations Based on ….
Habermann presented values of EA 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.
EOR-Chapter 2
67
Areal Sweep Efficiency, EA%
Pore Volume Injected, Vi/ Vp
Figure-4: Areal sweep efficiency after breakthrough as a function of
mobility ratio and PVs injected
EOR-Chapter 2
68
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
EOR-Chapter 2
69
Figure-5:Areal sweep efficiency as a function of mobility ratio;
EOR-Chapter 2
70
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.
EOR-Chapter 2
71
Areal Sweep Efficiency at Breakthrough ,EAbt%
Water-Gas
Gas-Oil
Miscible
Mobility Ratio,M
Figure-6: Areal sweep efficiency at breakthrough as a function of mobility
ratio( immiscible fluid displacement);
EOR-Chapter 2
72
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
EOR-Chapter 2
73
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.
EOR-Chapter 2
74
Prediction Based on Piston –Like Displacement
Figure 7 through 9 show data from the experiments. In Figure 7, EA
is given as a function of M for various values of injected PVs.
The ratio Vi/Vpd is a dimensionless injection volume defined as
injected volume divided by displaceable PV, Vpd. For a waterflood,
Vpd is given by
Vpd  Ah( Soi  Sor )
Figure 8 gives EA as a function of M for different values of the
fractional flow of the displacing phase ,fD, at the producing well.
EOR-Chapter 2
75
Prediction Based on Piston –Like Displacement

Figure 9 presents the conductance ratio,  , as a function of M for various
values of EA, 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, q
p


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).
EOR-Chapter 2
76
Prediction Based on Piston –Like Displacement




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 EA 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:
EOR-Chapter 2
77
Prediction Based on Piston –Like Displacement
0.0 0 1 5 3 8
k k rd hp
i 
d
 d (lo g
 0.2 6 8 8)
rw
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; p =pressure drop, psi; 
=viscosity of displaced phase, cp;d=distance measured between
injection and production wells ,ft; and rw= wellbore radius, ft.
At any point in the flood, the flow rate is given by
q  i
EOR-Chapter 2
78
Areal Sweep Efficiency,EA
Figure-7: Areal Sweep efficiency as a function of mobility ratio
and injected volume.
EOR-Chapter 2
79
Areal Sweep Efficiency,EA
Mobility Ratio,M
Figure-8:Areal sweep efficiency as a function of mobility ratio and
fractional flow at displacing phase
EOR-Chapter 2
80
Conductance Ratio,
Mobility Ratio,M
Figure-9:Conductance ratio as a function of mobility ratio and areal sweep.
EOR-Chapter 2
81
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:
h  2 0 ft
  0 .2
S oi  0.8
S or  0.2 5
 o  1 0 cp
 w  1 cp
Bo  1.0 R B / S TB
k  50 m d
k rw  0.2 7( a t R OS )
k ro  0.9 4( a t S wi )
p  1 2 5 0 p si
rw  0.5
ft
EOR-Chapter 2
82
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
EOR-Chapter 2
83
Solution


Apply the correlations in Figs 7 through 9
1. Calculate oil recovered



2. Calculate total water injected. From Fig 7, Vi/Vpd=2.5 (at EA=.94)



M=2.9, fD=20/21=.95 From Fig 8, EA=.94
Np=321000 STB
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



0.001538k k rd hp
d
 d (log  0.2688)
rw
i=63.4 B/D
From Fig. 9, γ=2.7, from
q  i =
63.4x2.7 = 171 B/D
4. Calculate initial water injection rate

i=63.4 B/D
EOR-Chapter 2
84
Calculation of EA 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
EOR-Chapter 2
85
Comparison of calculated and experimental
results, 5 spot pattern (Douglas et al.)
EOR-Chapter 2
86
Vertical Displacement Efficiency
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.
EOR-Chapter 2
87
Vertical Displacement Efficiency
EOR-Chapter 2
88
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
EOR-Chapter 2
89
Heterogeneity
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.
EOR-Chapter 2
90
Heterogeneity:Location of the water front
at different Location
EOR-Chapter 2
91
Heterogeneity:Dykstra-Persons
model
EOR-Chapter 2
92
Gravity Segregation in Horizontal Bed

Water tongue
Water

Gas umbrella
Gas
EOR-Chapter 2
93
Gravity Effect
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
1
satisfied when RL is large.
v
2
L
L k
R 
( )
H k
(2) When gravity forces are strong compared to viscous forces. This is satisfied
when the gravity number Ng is large.
Ng
kg

u
EOR-Chapter 2
94
Gravity Effect

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).
EOR-Chapter 2
95
Gravity Effect
Tonguing will occur when M < 1 as long as R Land 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
EOR-Chapter 2
96
Gravity Effect
Density of displacing fluid higher than density of displaced fluid
EOR-Chapter 2
97
Effect of Gravity Segregation and Mobility Ratio on
Vertical Displacement Efficiency
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.
EOR-Chapter 2
98
Gravity Segregation in displacement processes
Displacing
Phase
Displaced
Phase
Displaced
Phase
Gravity Override (a)  D   d
Displacing
Phase
Displacing
Phase
Displaced
Phase
Gravity Underride (b)  D   d
Figure-10: Gravity Segregation in displacement processes.
EOR-Chapter 2
99
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
EOR-Chapter 2
100
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.
EOR-Chapter 2
101
Vertical sweep efficiency at breakthrough as a function
of the ratios of viscous/gravity forces, Linear system
(from Craig et al.)
EOR-Chapter 2
102
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:
EOR-Chapter 2
103
Solution
EOR-Chapter 2
104
Mathematical Model

Spivak used a 2D and 3D numerical model to study
gravity effects during water flooding and gas
flooding
EOR-Chapter 2
105
Gravity Segregation in two-phase flow
EOR-Chapter 2
106
The correlations of Craig et al. and Spivak on
gravity segregation
The correlations of Craig et al. and Spivak indicate the
following effects of various parameters on gravity
segregation, as summarized by Spivak





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.
EOR-Chapter 2
107
Flow Regions in Miscible Displacement at Unfavorable
Mobility Ratios
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 RV / g 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 RV / g .At larger RV / g
values, in region II, a single gravity tongue still exists, but vertical sweep is
relatively insensitive to the value of the viscous/gravity ratio.
SOLVENT
Oil
(A) REGIONS I AND II
EOR-Chapter 2
108
Flow Regions in Miscible Displacement at Unfavorable
Mobility Ratios
The transition to region III occurs at a particular critical RV / g 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 RV / g
SOLVENT
SOLVENT
Oil
Oil
(C) REGION IV
(B) REGION III
EOR-Chapter 2
109
Flow Regimes in Miscible Displacement
EOR-Chapter 2
110
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
EOR-Chapter 2
111
Volumetric Displacement Efficiency
EOR-Chapter 2
112
Calculation of volumetric sweep with
Numerical Simulators
EOR-Chapter 2
113