Sweep and Displacement Efficiency

Dr. Siroos Azizmohammadi

Summer Course 2016

Department of Petroleum Engineering Chair of Reservoir Engineering

•

Introduction

•

Overall Recovery Efficiency

•

Displacement Efficiency

•

Sweep Efficiency

•

Areal Sweep

–

Flood Pattern

–

Estimation of Areal Sweep Efficiency

•

Vertical Sweep

–

Vertical Permeability Variations

–

Estimation of Vertical Sweep Efficiency

•

Gravity Segregation

Primary

oil recovery describes the production

of hydrocarbons under the natural driving

mechanisms present in the reservoir without

supplementary help from injected fluids such

as gas or water.

Secondary

(improved) oil recovery refers to the

additional recovery that results from the

conventional methods of water injection and

immiscible gas injection. Water flooding is

perhaps the most common method of

secondary recovery.

Tertiary

(enhanced) oil recovery is that

additional recovery over and above what could

be recovered by primary and secondary

recovery methods.

The overall recovery efficiency is defined as:

𝐸𝐸 = 𝐸𝐸

_𝐷𝐷

× 𝐸𝐸

_𝑆𝑆

‒

𝐸𝐸

_𝐷𝐷

is the displacement efficiency (microscopic)

‒

𝐸𝐸

_𝑆𝑆

is the sweep efficiency (macroscopic)

The

displacement efficiency

(microscopic displacement) is related to the

displacement of oil at the pore scale. In other words, 𝐸𝐸

_𝐷𝐷

is the fraction of

movable oil that has been displaced from the swept zone at any given time

or pore volume injected.

The

sweep efficiency

(macroscopic displacement) is the fraction of the

reservoir that is swept by the Displacing fluid. In other words, 𝐸𝐸

_𝑆𝑆

is the

The displacement efficiency is the ratio of the displaced oil to the contacted oil by Displacing

fluid (Lake, 1989).

𝐸𝐸𝐷𝐷 = 𝑉𝑉_𝑝𝑝 𝑆𝑆oi 𝐵𝐵_oi− 𝑆𝑆𝐵𝐵o_o 𝑉𝑉_{𝑝𝑝 𝐵𝐵}𝑆𝑆oi oi = 𝑆𝑆_oi 𝐵𝐵_oi− 𝑆𝑆𝐵𝐵o_o 𝑆𝑆_oi 𝐵𝐵_oi

For constant oil formation volume factor during the flood life:

𝐸𝐸_𝐷𝐷 =𝑆𝑆oi_𝑆𝑆− 𝑆𝑆o oi = 1 − 𝑆𝑆_wi− 𝑆𝑆_gi − 1 − 𝑆𝑆_w 1 − 𝑆𝑆_wi− 𝑆𝑆_gi or 𝐸𝐸𝐷𝐷 = 𝑆𝑆_w− 𝑆𝑆_wi − 𝑆𝑆_gi 1 − 𝑆𝑆_wi − 𝑆𝑆_gi

If no initial gas is present at the start of the flood:

𝐸𝐸𝐷𝐷 =𝑆𝑆_{1 − 𝑆𝑆}w − 𝑆𝑆wi wi

Displacement efficiency (microscopic displacement) is a function of:

time

,

fluid viscosities

,

relative permeabilities

and

capillary pressure

.

Displacement Efficiency Volume of oil at start of flood Volume of oil at start of flood Remaining oil volume Amount of oil displaced

Amount of oil contacted by Displacing fluid

The sweep efficiency is the ratio of the produced oil to the displaced oil (Lake, 1989).

𝐸𝐸

_𝑆𝑆

=

𝑁𝑁

𝑝𝑝

𝑉𝑉

_𝑝𝑝

_𝐵𝐵

𝑆𝑆

oi oi

− 𝑆𝑆

o

𝐵𝐵

_o

For constant oil formation volume factor during the flood life:

𝐸𝐸

_𝑆𝑆

=

𝑁𝑁

𝑝𝑝

𝐵𝐵

o

⁄

𝑉𝑉

𝑝𝑝

𝑆𝑆

_w

− 𝑆𝑆

_wi

− 𝑆𝑆

_gi

If no initial gas is present at the start of the flood:

𝐸𝐸

_𝑆𝑆

=

𝑁𝑁

𝑝𝑝

𝐵𝐵

o

⁄

𝑉𝑉

𝑝𝑝

𝑆𝑆

w

− 𝑆𝑆

wi

Sweep Efficiency

Volume of oil

at start of flood Remainingoil volume Amount of oil produced Amount of oil produced

𝑁𝑁𝑝𝑝 at the present time is known. The current water cut is known, 𝑓𝑓_w. Construct a fractional flow curve.

Draw tangent line to fractional flow curve at the current water cut, 𝑓𝑓w.

Extrapolate tangent line to the 𝑓𝑓_w = 1 (100% water cut) and obtain saturation of water at current water cut . Calculate current efficiency.

1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 𝑓𝑓w 𝑆𝑆_w 1 0.9 0.8 0.7 0.6 0.5 0.5 0.6 0.7 0.8 0.9 𝑓𝑓_w 𝑆𝑆w

𝑆𝑆

_w 𝑓𝑓w = 0.84

Sweep efficiency defined as:

𝐸𝐸

_𝑆𝑆

= 𝐸𝐸

_𝐴𝐴

× 𝐸𝐸

_𝑉𝑉

Sweep efficiency contains:

Areal

Sweep Efficiency and

Vertical

Sweep Efficiency

Areal and vertical sweep are dependent to each other.

𝑆𝑆oi Injector Producer 𝐸𝐸_𝐴𝐴 𝐸𝐸_𝑉𝑉 𝑆𝑆or

Areal sweep efficiency: controlled by four main

factors:

•

Flood pattern (injection and production wells arrangement)

•

Mobility ratio

•

Permeability heterogeneity

•

Relative importance of gravity and viscous force

Flood pattern: objective is to select the proper pattern

that will provide the injection fluid with the maximum

possible contact with oil.

Pattern types:

•

Irregular pattern

•

Peripheral pattern

•

Regular pattern

•

Crestal and basal pattern

Unswept area

Swept area

Injector

Direct Line Drive Staggered Line Drive

7-Spot 5-Spot

Peripheral Pattern

Water injection pattern in Ghawar field

The efficiency is about 70% for 𝑀𝑀 = 1 at breakthrough and becomes a lot smaller for displacement processes at 𝑀𝑀 > 1 (Most experimental works were done on 5-spot pattern)

Fassihi (1986)for 0 < 𝑀𝑀 ≤ 10 1 − 𝐸𝐸_𝐴𝐴

𝐸𝐸_𝐴𝐴 = 𝑎𝑎1ln 𝑀𝑀 + 𝑎𝑎2 + 𝑎𝑎3 𝑓𝑓w+ 𝑎𝑎4ln 𝑀𝑀 + 𝑎𝑎5 + 𝑎𝑎6 𝐸𝐸_𝐴𝐴 = areal sweep efficiency

𝑀𝑀 = mobility ratio

𝑓𝑓_w= fractional flow function

Coefficient 5-spot Direct line Staggered line 𝑎𝑎1 -0.2062 -0.3014 -0.2077 𝑎𝑎2 -0.0712 -0.1568 -0.1059 𝑎𝑎3 -0.511 -0.9402 -0.3526 𝑎𝑎4 0.3048 0.3714 0.2608 𝑎𝑎5 0.123 -0.0865 0.2444 𝑎𝑎6 0.4394 0.8805 0.3158

𝑍𝑍∗ 𝑥𝑥 = estimate of the regionalized variable at location 𝑥𝑥

𝑍𝑍 𝑥𝑥_𝑖𝑖 = measured value of the regionalized variable at position 𝑥𝑥_𝑖𝑖 𝑤𝑤_𝑖𝑖 = weight factor

𝑛𝑛 = number of nearby data points 𝑍𝑍∗ _{𝑥𝑥 = �} 𝑖𝑖=1 𝑛𝑛 𝑤𝑤_𝑖𝑖𝑍𝑍 𝑥𝑥_𝑖𝑖 � 𝑖𝑖=1 𝑛𝑛 𝑤𝑤_𝑖𝑖 = 1

Well No. Permeability, 𝑘𝑘 [mD]

1 73

2 110

3 200

4 140

Inverse Distance Method Inverse Distance Squared Method Well

No. permeability𝑘𝑘, [mD] Distance, 𝑑𝑑[ft] 𝑖𝑖 1 𝑑𝑑⁄ 𝑖𝑖 𝑤𝑤𝑖𝑖= �_𝑑𝑑1 𝑖𝑖 �_𝑖𝑖=1 𝑛𝑛 1 𝑑𝑑𝑖𝑖 𝑤𝑤𝑖𝑖𝑘𝑘 𝑥𝑥𝑖𝑖 1 𝑑𝑑⁄ 𝑖𝑖 2 _𝑤𝑤 𝑖𝑖= _𝑑𝑑1 � 𝑖𝑖 2 � 𝑖𝑖=1 𝑛𝑛 1 𝑑𝑑𝑖𝑖 2 𝑤𝑤𝑖𝑖𝑘𝑘 𝑥𝑥𝑖𝑖 1 73 170 0.00588 0.3482 25.4198 0.0000346 0.4419 32.2574 2 110 200 0.00500 0.2960 32.5582 0.0000250 0.3193 35.1186 3 200 410 0.00244 0.1444 28.8765 0.0000059 0.0760 15.1938 4 140 280 0.00357 0.2114 29.5984 0.0000128 0.1629 22.8043 sum 0.01689 1.0000 116.45 0.0000783 1.0000 105.37

Inverse Distance method 𝑤𝑤𝑖𝑖 = �_𝑑𝑑1

𝑖𝑖 �_𝑖𝑖=1 𝑛𝑛

1 𝑑𝑑_𝑖𝑖

Inverse Distance Squared method 𝑤𝑤𝑖𝑖 = _𝑑𝑑1 � 𝑖𝑖 2 � 𝑖𝑖=1 𝑛𝑛 1 𝑑𝑑_𝑖𝑖 2

Vertical sweep efficiency: controlled by four main factors:

‒ Vertical permeability variations within the reservoir ‒ Mobility ratio

‒ Gravity segregation (density differences between flowing fluids) ‒ Capillary force

• A hydrocarbon formation is rarely homogeneous in a vertical

direction.

• Layers composed of various minerals and different

petrophysical properties.

• The injected fluid will seek the paths of least resistance and

will move through the reservoir as an irregular front.

• The injected fluid will travel more rapidly in the more

permeable zones and less rapidly in the tighter zones.

• This variation leads to a reduction in vertical efficiency,

because of uneven flow in the different layers.

• The most widely used descriptors are:

‒ Dykstra-Parsons permeability variation coefficient , 𝑉𝑉

‒ Lorenz coefficient, 𝐿𝐿 𝑘𝑘1 𝜙𝜙1 ℎ1 𝑘𝑘₂ 𝜙𝜙₂ ℎ₂ 𝑘𝑘₃ 𝜙𝜙₃ ℎ₃ 𝑘𝑘₄ 𝜙𝜙₄ ℎ₄ 𝑘𝑘₅ 𝜙𝜙₅ ℎ₅ 𝑘𝑘6 𝜙𝜙6 ℎ6 𝑘𝑘₇ 𝜙𝜙₇ ℎ₇

Dykstra and Parsons (1950) introduced the concept of the permeability variation coefficient 𝑉𝑉, which describes the degree of heterogeneity within the reservoir and it is a statistical measure of non-uniformity of permeability data. Dykstra-Parsons procedure is introduced as follows:

1. Arrange the permeabilities in descending order from highest to lowest

2. For each sample, calculate the percentage of thickness with permeability greater than this sample

3. Plot the data from Step 2 on log-probability paper

4. Draw the best straight line through data (with less emphasis on points at the extremities, if necessary)

5. Determine the permeability at 84.1% probability (𝑘𝑘84.1) and the mean permeability at 50% probability (𝑘𝑘50)

6. Compute the permeability variation, 𝑉𝑉: 𝑉𝑉 = 𝑘𝑘50_𝑘𝑘− 𝑘𝑘84.1 50 𝑉𝑉 = 0 completely homogeneous 𝑉𝑉 = 1 completely heterogeneous 10 100 1000 0 20 40 60 80 100 Pe rme ab ilit y [mD ]

% of thickness with greater k

𝑉𝑉 = 𝑘𝑘50_𝑘𝑘− 𝑘𝑘84.1 50 =

69 − 28

Schmalz and Rahme (1950) introduced a single parameter that describes the degree of heterogeneity within a pay zone section. The term is called Lorenz coefficient.

The following steps summarize the methodology of calculating Lorenz coefficient:

1. Arrange the permeabilities in descending order from highest to lowest

2. Calculate the cumulative permeability capacity ∑ 𝑘𝑘ℎ and cumulative volume capacity ∑ 𝜙𝜙ℎ 3. Normalize both cumulative capacities such that

each cumulative capacity ranges from 0 to 1

4. Plot the normalized cumulative permeability capacity versus the normalized cumulative volume capacity on a Cartesian scale

𝐿𝐿 = _{area below the straight line}area above the straight line 𝐿𝐿 = 0 completely homogeneous 𝐿𝐿 = 1 completely heterogeneous Increasing heterogeneity Normalized ∑ 𝜙𝜙ℎ No rm aliz ed ∑ 𝑘𝑘ℎ 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Variation, 𝑉𝑉 Lor enz coef fic ient

Johnson (1956)

developed a simplified graphical approach for the Dykstra-Parsons method.

𝐸𝐸𝑉𝑉1 − 𝑆𝑆wi = 0.4 𝑀𝑀 =𝑘𝑘_𝑘𝑘rw ro 𝜇𝜇o 𝜇𝜇w 𝑉𝑉 WOR = 1 𝑀𝑀 =𝑘𝑘_𝑘𝑘rw ro 𝜇𝜇o 𝜇𝜇w 𝑉𝑉 WOR = 5 𝐸𝐸𝑉𝑉1 − 0.72𝑆𝑆wi = 0.45 𝐸𝐸𝑉𝑉1 − 0.52𝑆𝑆wi = 0.5 𝑀𝑀 =𝑘𝑘_𝑘𝑘rw ro 𝜇𝜇o 𝜇𝜇w 𝑉𝑉 WOR = 25 𝑀𝑀 =𝑘𝑘_𝑘𝑘rw ro 𝜇𝜇o 𝜇𝜇w 𝑉𝑉 WOR = 100 𝐸𝐸𝑉𝑉 1 − 0.4𝑆𝑆wi = 0.5

de Souza and Brigham (1981) 𝑌𝑌 = WOR + 0.4 18.948 − 2.499𝑉𝑉 (𝑀𝑀 + 1.137 − 0.8094𝑉𝑉)10𝑓𝑓 𝑉𝑉 𝑓𝑓 𝑉𝑉 = −0.6891 + 0.935𝑉𝑉 + 1.6453𝑉𝑉2 Fassihi (1986) 𝑌𝑌 = 𝑎𝑎₁(𝐸𝐸_𝑉𝑉)𝑎𝑎2_{(1 − 𝐸𝐸} 𝑉𝑉)𝑎𝑎3 𝑎𝑎₁ = 3.334088568 𝑎𝑎2 = 0.7737348199 𝑎𝑎₃ = −1.225859406

Fassihi, M. R., 1986, “New Correlations for Calculation of Vertical Coverage and Areal Sweep Efficiency”, SPE Res. Eng.

0 < 𝑀𝑀 ≤ 10 0.3 ≤ 𝑉𝑉 ≤ 0.8

Gravity segregation occurs when density differences between

D

isplacing (injected) and

d

isplaced fluids are large enough to induce a significant vertical flow component – even when

the principal fluid flow direction is in horizontal plane.

If density of the

D

isplacing fluid is less than the

d

isplaced fluid’s density, the

D

isplacing fluid

overrides the

d

isplaced fluid (

gravity override

). Occurs at gas injection, CO

₂

flooding, steam

injection, in-situ combustion, and solvent flooding.

If density of the

D

isplacing fluid is greater than the

d

isplaced fluid’s density, the

D

isplacing fluid

underrides the

d

isplaced fluid (

gravity underride

) may occur during a water flooding.

Gravity segregation leads to early breakthrough of the injected fluid and reduced vertical sweep

efficiency.

Gravity segregation increases with (1) increasing permeability (horizontal and vertical) (2)

increasing density difference (3) increasing mobility ratio (4) decreasing rate

Displacing phase, D

displaced phase, d Displacing phase, D

Gravity segregation effect can be distinguished by a dimensionless group called

(viscous/gravity) ratio or vice versa.

Effect of gravity segregation on vertical sweep efficiency studied by

Craig et. al (1957)

and

Spivak (1974)

.

𝐹𝐹g 𝑣𝑣⁄ = 0.00633_{𝑞𝑞𝜇𝜇}𝑘𝑘𝑣𝑣⁄𝑘𝑘ℎ∆𝜌𝜌𝐴𝐴 𝑑𝑑 𝑅𝑅𝑣𝑣 g⁄ =2050𝑢𝑢𝜇𝜇_{𝑘𝑘g∆𝜌𝜌} 𝑑𝑑 𝐿𝐿_ℎ 𝑢𝑢 = [rb/(d.ft2_{)] , 𝜇𝜇} 𝑑𝑑 = [cP] , 𝑘𝑘 = [mD] , 𝜌𝜌 =[g/cm3] , 𝐿𝐿 = [ft] , ℎ = [ft]

In dipping reservoir, gravity can be used to improve displacement efficiency.

If oil is displaced by injecting a less dense fluid (more mobile solvent updip) gravity forces would tend to stabilize the displacement front. If the displacement velocity is sufficiently slow, gravity would act to prevent the formation of fingers at the solvent/oil interface. Similarly, in a water flood (downdip injection of water).

The criteria for stable displacement in a dipping reservoirs is called critical velocity: 𝑢𝑢_𝑐𝑐 = g 𝜌𝜌d_𝜇𝜇− 𝜌𝜌D sin 𝜃𝜃

d 𝑘𝑘d− 𝜇𝜇

D 𝑘𝑘D

If the displacement velocity is less than the critical velocity the interface will remain stable, otherwise the displacement will be unstable.

θ θ β θ β β

All models discussed so far assumed that cross-flow between layers does not occur.

‒

This is not realistic (except for cases with permeability barriers between layers)

The effects of cross-flow are difficult to handle mathematically

‒

Can be handled with numerical simulation

Vertical displacement efficiency in layered reservoirs with cross-flow is influenced by viscous

gravity and capillary forces.

Under favorable mobility ratios (𝑀𝑀 ≤ 1)

‒

Oil recovery with cross-flow is between the

recovery predicted for a uniform reservoir and that one predicted for a layered reservoir with no cross-flow

‒

cross-flow acts to improve 𝐸𝐸_𝑉𝑉

Under unfavorable mobility ratios (𝑀𝑀 > 1)