1 Chapter 4 Efficiency of a Displacement Process.

1

Chapter 4

Efficiency of a Displacement

Process

EOR-Chapter 4 2

Efficiency of a Displacement Process

Introduction Microscopic Displacement of Fluid in a

Reservoir Macroscopic Displacement of Fluids in a

Reservoir

EOR-Chapter 2 3


Trapped Oil

ME

= = (Microscopic Efficiency)(Microscopic Efficiency) × × (Volumetric Efficiency)(Volumetric Efficiency)

ProductionProduction

InjectionInjection

E ME VE

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

DEVE =Macroscopic (Volumetric) displacement efficiency

=Microscopic (Volumetric) hydrocarbon displacement efficiency.

EOR-Chapter 2 5

Microscopic & Macroscopic sweep efficiencies

agentEORbycontactedoilofvolumereservoir

agentEORbymobilizedoilofvolumereservoirED

placeinoriginallyoilofvolumereservoir

agentdisplacingbycontactedoilofvolumereservoirEv

EOR-Chapter 2 6


Macroscopic Displacement

E

DV EEE Where; Where; = Overall displacement efficiency= Overall displacement efficiency

VE == Macroscopic displacement efficiencyMacroscopic displacement efficiency

= Microscopic displacement efficiency= Microscopic displacement efficiencyDE

EOR-Chapter 2 7

However,

LE

LAV EEE

== Areal Sweep efficiencyAreal Sweep efficiency

= = Lateral Sweep efficiencyLateral Sweep efficiency

AE


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

PoiLADP B

VSEEEN

Oil Recovery Equation

EOR-Chapter 2 9


RE

placeinnhydrocarboofVolume

displacednhydrocarboofVolumeER

and is the volumetric sweep efficiency defined as

typical values of the overall recovery efficiency are:

Steam injection 30%-50%Polymer injection 30%-55%CO2 injection 30%-65%Solvent injection 35%-63%

RE

RERE

RE

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

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

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 -

orS

oilcontactedofVolume

mobilizedoilofVolumeED

VE

DE

Displacement Sweep Efficiency

EOR-Chapter 2 13

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 14

Important factors relating to microscopic displacement behavior

Capillary Forces

Surface Tension and IFT Solid Wettability Capillary Pressure

Viscous Forces

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.

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

EOR-Chapter 2 17


The surface force, which is a tensile force, is quantified in terms of surface tension

LLiquid

Air or Vapor

The force per unit length required to create additional surface area is the surface tension, usually expressed in dynes/cm.

EOR-Chapter 2 18


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 19


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

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.

EOR-Chapter 2 21


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

EOR-Chapter 2 22


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

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 wsow

cos owwsos

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.

EOR-Chapter 2 25

Capillary Pressure Equation

The difference pressure between oil water at the oil/water interface

rPor

rp

pghpp

owc

cow

cowwo

cos2cos2

)(

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.

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.

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

wcaN

EOR-Chapter 2 29

Displacement Sweep Efficiency is a function of Mobility ratios Throughput or Transmissibility Wettability Dip angle Capillary number

EOR-Chapter 2 30

All sweep efficiencies can be increased by decreasing the mobility ratio by either:

rwOil kor 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

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 capillarynumber indicates experimental evidence of trapping in a permeable media. Thisrelationship is called the capillary desaturation curve.The local capillary number is

u

NC Where

= interfacial tension between displacing and displaced fuid

u = displacing superficial velocity

=displacing fluid viscosity

Trapped Oil Saturation

EOR-Chapter 2 32 Typical capillary desaturation curve


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 methodsIncreasing 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 34

Viscous Force

Viscous forces in a porous medium can be expressed in terms of Darcy’s law:

darciestypermeabilik

porosity

ftlengthL

cpityvis

Dftvelocityaverage

psioppressuredrpk

Lp

,

,

,cos

/,

,

))(158.0(

EOR-Chapter 2 35

Calculation of pressure gradient for viscous oil flow in a rock

BAowBA rr

pp11

cos2

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

Lp

ftpsidarcies

cpDft

L

p/264.1

250.0

2.010/0.1158.0

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

ftpsift

cm

cm

psiLp /2073

48.30

01.0

68.0/

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

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

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

Vpo

o

o

op EV

B

S

B

SN )(

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

2oB1oB

2oS1oSpN

pV

EOR-Chapter 2 41


VDp EE

N

NRF

1

1N

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

op

pV

BS

BS

V

NE

EOR-Chapter 2 42


Where = oil produced in the waterflood.

)(2

2

1

1

o

o

o

op

pV

BS

BS

V

NE

pN

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

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 AE

AE

areaTotal

agentdisplacingbycontactedAreaEA


lengthverticalTotal

agentdisplacingbycontactedLengthEL

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

DLP EEEE

pE

pE

LPV EEE


EOR-Chapter 2 46

This figure illustrates the concept of the vertical and areal sweep efficiency

EOR-Chapter 2 47

The following figure illustrate the definition of areal sweep efficiency

areaTotal

agentdisplacingbycontractedArealEA

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 EfficiencyThe 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.

AE


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.

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

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

EOR-Chapter 2 52

Mobility Control

d

W

WW

k

O

OO

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

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

EOR-Chapter 2 54

volumeporefrence

fluidinjectedOfVolumeCumulativetD 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.

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.

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.

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.

AE

EOR-Chapter 2 58

Flooding Patterns

EOR-Chapter 2 59

Flooding Patterns

EOR-Chapter 2 60

Flooding Patterns

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.

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.

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

dM

Correlations Based on ….

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

BreakthroughBreakthrough

Po

re V

olu

mes

Inje

cted

Po

re V

olu

mes

Inje

cted

Injection well

Producing well

EOR-Chapter 2 65

M=2.40

PV

BT BT

M=4.58

0.3

0.2

0.10.06

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)

EOR-Chapter 2 66

M=17.3 M=71.5

BTBT

0.150.05

• 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)

EOR-Chapter 2 67

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.

Correlations Based on ….

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

Are

al S

wee

p E

ffic

ien

cy,

EA

%

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

EOR-Chapter 2 70

Figure-5:Areal sweep efficiency as a function of mobility ratio;

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.

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

Are

al S

wee

p E

ffici

ency

at B

reak

thro

ugh

,EA

bt%

Mobility Ratio,M

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

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.

EOR-Chapter 2 75

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.

)( oroipd SSAhV

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

Prediction Based on Piston –Like Displacement

EOR-Chapter 2 76

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,

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


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 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 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

wd

rd

r

dphkk

i

iq

At any point in the flood, the flow rate is given by

p


EOR-Chapter 2 79

Are

al S

wee

p E

ffic

ien

cy,E

A

Figure-7: Areal Sweep efficiency as a function of mobility ratio and injected volume.

EOR-Chapter 2 80

Mobility Ratio,M

Are

al S

wee

p E

ffic

ien

cy,E

A

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

EOR-Chapter 2 81

Mobility Ratio,M

Co

nd

uct

ance

Rat

io,

Figure-9:Conductance ratio as a function of mobility ratio and areal sweep.

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

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

EOR-Chapter 2 84

Solution Apply the correlations in Figs 7 through 9 1. Calculate oil recovered

M=2.9, fD=20/21=.95 From Fig 8, EA=.94 Np=321000 STB

2. Calculate total water injected. From Fig 7, Vi/Vpd=2.5 (at EA=.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

wd

rd

r

dphkk

i

EOR-Chapter 2 85

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 86

Comparison of calculated and experimental results, 5 spot pattern (Douglas et al.)

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

EOR-Chapter 2 88


EOR-Chapter 2 89


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 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

EOR-Chapter 2 91

Heterogeneity:Location of the water front at different Location

EOR-Chapter 2 92

Heterogeneity:Dykstra-Persons model

EOR-Chapter 2 93

Gravity Segregation in Horizontal Bed

Water tongue

Water

Gas umbrella

Gas

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

LR v

L

(2) When gravity forces are strong compared to viscous forces. This is satisfied when the gravity number Ng is large.

u

gkN g

LR

Gravity Effect

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 fluidThe displacing fluid will tend to flow to the top of the reservoir and bypass the fluid in the lower region (tongue over).

Gravity Effect

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

EOR-Chapter 2 97

Density of displacing fluid higher than density of displaced fluid

Gravity Effect

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

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

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

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.

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.)

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:

EOR-Chapter 2 104

Solution

EOR-Chapter 2 105

Mathematical Model Spivak used a 2D and 3D numerical model to study

gravity effects during water flooding and gas flooding

EOR-Chapter 2 106

Gravity Segregation in two-phase flow

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

EOR-Chapter 2 108

Flow Regions in Miscible Displacement at Unfavorable Mobility Ratios

gVR /

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.

gVR /

EOR-Chapter 2 109

Flow Regions in Miscible Displacement at Unfavorable Mobility Ratios

(B) REGION III (C) REGION IV

SOLVENTSOLVENT

OilOil

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 /

EOR-Chapter 2 110

Flow Regimes in Miscible Displacement

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

EOR-Chapter 2 112


EOR-Chapter 2 113

Calculation of volumetric sweep with Numerical Simulators

1 Chapter 4 Efficiency of a Displacement Process.

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Transcript of 1 Chapter 4 Efficiency of a Displacement Process.