Sweep and Displacement Efficiency

Dr. Siroos Azizmohammadi

Summer Course 2016 Department of Petroleum Engineering Chair of Reservoir Engineering

September 15, 2016

•

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

•

Cross-Flow in Layered Reservoir

Dr. Siroos Azizmohammadi

Summer Course 2016 | Department of Petroleum Engineering Reservoir Engineering Module: Sweep and Displacement Efficiency

September 15, 2016

2

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.

Dr. Siroos Azizmohammadi

Summer Course 2016 | Department of Petroleum Engineering Reservoir Engineering Module: Sweep and Displacement Efficiency

15 September 2016

3

The overall recovery efficiency is defined as: 𝐸𝐸 = 𝐸𝐸𝐷𝐷 × 𝐸𝐸𝑆𝑆

‒ 𝐸𝐸𝐷𝐷 is the displacement efficiency (microscopic) ‒ 𝐸𝐸𝑆𝑆 is the sweep efficiency (macroscopic)

The displacement efficiency (microscopic displacement) is related to the displacement of oil at the pore scale. In other words, 𝐸𝐸𝐷𝐷 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, 𝐸𝐸𝑆𝑆 is the overall fraction of the flood pattern that is contacted by the injected fluid.

Dr. Siroos Azizmohammadi

Summer Course 2016 | Department of Petroleum Engineering Reservoir Engineering Module: Sweep and Displacement Efficiency

15 September 2016

4

The displacement efficiency is the ratio of the displaced oil to the contacted oil by Displacing fluid (Lake, 1989). Displacement Efficiency

Amount of oil contacted by Displacing fluid

𝐸𝐸𝐷𝐷 =

𝑉𝑉𝑝𝑝

Remaining oil volume

Volume of oil at start of flood

Amount of oil displaced

Volume of oil at start of flood

𝑆𝑆oi 𝑆𝑆o 𝑆𝑆oi 𝑆𝑆o − − 𝐵𝐵oi 𝐵𝐵o 𝐵𝐵oi 𝐵𝐵o = 𝑆𝑆oi 𝑆𝑆oi 𝑉𝑉𝑝𝑝 𝐵𝐵oi 𝐵𝐵oi

For constant oil formation volume factor during the flood life: 𝐸𝐸𝐷𝐷 =

1 − 𝑆𝑆wi − 𝑆𝑆gi − 1 − 𝑆𝑆w 𝑆𝑆oi − 𝑆𝑆o = 𝑆𝑆oi 1 − 𝑆𝑆wi − 𝑆𝑆gi

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

or

𝐸𝐸𝐷𝐷 =

𝑆𝑆w − 𝑆𝑆wi − 𝑆𝑆gi 1 − 𝑆𝑆wi − 𝑆𝑆gi

𝑆𝑆w − 𝑆𝑆wi 1 − 𝑆𝑆wi

Displacement efficiency (microscopic displacement) is a function of: time, fluid viscosities, relative permeabilities and capillary pressure. Dr. Siroos Azizmohammadi

Summer Course 2016 | Department of Petroleum Engineering Reservoir Engineering Module: Sweep and Displacement Efficiency

15 September 2016

5

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

Amount of oil produced

Sweep Efficiency

Volume of oil at start of flood

Amount of oil displaced

𝐸𝐸𝑆𝑆 =

𝑉𝑉𝑝𝑝

Remaining oil volume

𝑁𝑁𝑝𝑝

𝑆𝑆oi 𝑆𝑆o − 𝐵𝐵oi 𝐵𝐵o

For constant oil formation volume factor during the flood life: 𝐸𝐸𝑆𝑆 =

𝑁𝑁𝑝𝑝 𝐵𝐵o ⁄𝑉𝑉𝑝𝑝

𝑆𝑆w − 𝑆𝑆wi − 𝑆𝑆gi

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

Dr. Siroos Azizmohammadi

𝑁𝑁𝑝𝑝 𝐵𝐵o ⁄𝑉𝑉𝑝𝑝

𝑆𝑆w − 𝑆𝑆wi

Summer Course 2016 | Department of Petroleum Engineering Reservoir Engineering Module: Sweep and Displacement Efficiency

15 September 2016

6

1 0.8

1

0.6

𝑓𝑓w

𝑆𝑆w

0.9 𝑓𝑓w = 0.84

0.4

0.8

0.2 0

𝑓𝑓w 0.2

0.4

𝑆𝑆w

0.6

0.8

0.7

1 0.6

𝑁𝑁𝑝𝑝 at the present time is known.

0.5

0.5

0.6

0.7

0.8

0.9

𝑆𝑆w The current water cut is known, 𝑓𝑓w . Construct a fractional flow curve. Draw tangent line to fractional flow curve at the current water cut, 𝑓𝑓w . Extrapolate tangent line to the 𝑓𝑓w = 1 (100% water cut) and obtain saturation of water at current water cut . Calculate current efficiency. Dr. Siroos Azizmohammadi

Summer Course 2016 | Department of Petroleum Engineering Reservoir Engineering Module: Sweep and Displacement Efficiency

15 September 2016

7

Sweep efficiency defined as:

𝐸𝐸𝑆𝑆 = 𝐸𝐸𝐴𝐴 × 𝐸𝐸𝑉𝑉 Sweep efficiency contains: Areal Sweep Efficiency and Vertical Sweep Efficiency Producer

𝐸𝐸𝐴𝐴

𝑆𝑆oi

Injector

𝑆𝑆or

𝐸𝐸𝑉𝑉

Areal and vertical sweep are dependent to each other.

Dr. Siroos Azizmohammadi

Summer Course 2016 | Department of Petroleum Engineering Reservoir Engineering Module: Sweep and Displacement Efficiency

15 September 2016

8

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

Producer

Unswept area

Flood pattern: objective is to select the proper pattern that will provide the injection fluid with the maximum possible contact with oil. Swept area

Pattern types: • Irregular pattern • Peripheral pattern • Regular pattern • Crestal and basal pattern

Dr. Siroos Azizmohammadi

Injector

Summer Course 2016 | Department of Petroleum Engineering Reservoir Engineering Module: Sweep and Displacement Efficiency

15 September 2016

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Peripheral Pattern

Direct Line Drive

Staggered Line Drive

Water injection pattern in Ghawar field

5-Spot 7-Spot

Dr. Siroos Azizmohammadi

Summer Course 2016 | Department of Petroleum Engineering Reservoir Engineering Module: Sweep and Displacement Efficiency

15 September 2016

10

(Caudle and Witte, 1959)

(Caudle and Witte, 1959)

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 − 𝐸𝐸𝐴𝐴 = 𝑎𝑎1 ln 𝑀𝑀 + 𝑎𝑎2 + 𝑎𝑎3 𝑓𝑓w + 𝑎𝑎4 ln 𝑀𝑀 + 𝑎𝑎5 + 𝑎𝑎6 𝐸𝐸𝐴𝐴

𝐸𝐸𝐴𝐴 = areal sweep efficiency 𝑀𝑀 = mobility ratio 𝑓𝑓w = fractional flow function Dr. Siroos Azizmohammadi

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

0.123

-0.0865

0.2444

𝑎𝑎6

0.4394

0.8805

0.3158

𝑎𝑎5

Summer Course 2016 | Department of Petroleum Engineering Reservoir Engineering Module: Sweep and Displacement Efficiency

15 September 2016

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𝑛𝑛

𝑛𝑛

𝑍𝑍 ∗ 𝑥𝑥 = � 𝑤𝑤𝑖𝑖 𝑍𝑍 𝑥𝑥𝑖𝑖

� 𝑤𝑤𝑖𝑖 = 1 𝑖𝑖=1

𝑖𝑖=1

𝑍𝑍 ∗ 𝑥𝑥 = estimate of the regionalized variable at location 𝑥𝑥 𝑍𝑍 𝑥𝑥𝑖𝑖 = measured value of the regionalized variable at position 𝑥𝑥𝑖𝑖 𝑤𝑤𝑖𝑖 = weight factor 𝑛𝑛 = number of nearby data points

Inverse Distance Squared method

Inverse Distance method 𝑛𝑛

1 𝑤𝑤𝑖𝑖 = 𝑑𝑑𝑖𝑖

1 1 𝑤𝑤𝑖𝑖 = �� 𝑑𝑑𝑖𝑖 𝑑𝑑𝑖𝑖 𝑖𝑖=1

Well No.

Distance, 𝑑𝑑𝑖𝑖 [ft]

73

170

0.00588

2

110

200

3

200

4

140

Dr. Siroos Azizmohammadi

2

𝑛𝑛

1 �� 𝑑𝑑𝑖𝑖

1⁄𝑑𝑑𝑖𝑖

𝑛𝑛

𝑖𝑖=1

73

2

2

𝑖𝑖=1

1 1 𝑤𝑤𝑖𝑖 = �� 𝑑𝑑𝑖𝑖 𝑑𝑑𝑖𝑖

Permeability, 𝑘𝑘 [mD]

1

Inverse Distance Method

permeability 𝑘𝑘, [mD]

1

Well No.

110

3

200

4

140

Inverse Distance Squared Method

𝑤𝑤𝑖𝑖 𝑘𝑘 𝑥𝑥𝑖𝑖

1⁄𝑑𝑑𝑖𝑖

2

0.3482

25.4198

0.0000346

0.00500

0.2960

32.5582

410

0.00244

0.1444

280

0.00357

sum

0.01689

1 𝑤𝑤𝑖𝑖 = 𝑑𝑑𝑖𝑖

2

𝑛𝑛

1 �� 𝑑𝑑𝑖𝑖 𝑖𝑖=1

2

𝑤𝑤𝑖𝑖 𝑘𝑘 𝑥𝑥𝑖𝑖

0.4419

32.2574

0.0000250

0.3193

35.1186

28.8765

0.0000059

0.0760

15.1938

0.2114

29.5984

0.0000128

0.1629

22.8043

1.0000

116.45

0.0000783

1.0000

105.37

Summer Course 2016 | Department of Petroleum Engineering Reservoir Engineering Module: Sweep and Displacement Efficiency

15 September 2016

12

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 petrophysical properties.

•

The injected fluid will seek the paths of least resistance and will move through the reservoir as an irregular front.

•

•

•

various

minerals

and

different

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 Dr. Siroos Azizmohammadi

Summer Course 2016 | Department of Petroleum Engineering Reservoir Engineering Module: Sweep and Displacement Efficiency

𝑘𝑘1

𝜙𝜙1 ℎ1

𝑘𝑘3

𝜙𝜙3 ℎ3

𝑘𝑘2 𝑘𝑘4 𝑘𝑘5 𝑘𝑘6 𝑘𝑘7 15 September 2016

𝜙𝜙2 ℎ2 𝜙𝜙4 ℎ4 𝜙𝜙5 ℎ5 𝜙𝜙6 ℎ6 𝜙𝜙7 ℎ7 13

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

1000

Permeability [mD]

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. DykstraParsons procedure is introduced as follows:

100

10

0

20

6. Compute the permeability variation, 𝑉𝑉: 𝑘𝑘50 − 𝑘𝑘84.1 𝑉𝑉 = 𝑘𝑘50 𝑉𝑉 = 0 completely homogeneous 𝑉𝑉 = 1 completely heterogeneous

Dr. Siroos Azizmohammadi

60

80

100

% of thickness with greater k

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 )

40

𝑉𝑉 =

Summer Course 2016 | Department of Petroleum Engineering Reservoir Engineering Module: Sweep and Displacement Efficiency

𝑘𝑘50 − 𝑘𝑘84.1 69 − 28 = 0.59 = 𝑘𝑘50 69

15 September 2016

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The following steps summarize the methodology of calculating Lorenz coefficient:

Normalized ∑ 𝑘𝑘ℎ

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.

Increasing heterogeneity

1. Arrange the permeabilities in descending order from highest to lowest 2. Calculate the cumulative permeability capacity ∑ 𝑘𝑘ℎ and cumulative volume capacity ∑ 𝜙𝜙𝜙

1

3. Normalize both cumulative capacities such that each cumulative capacity ranges from 0 to 1

𝐿𝐿 = 0 𝐿𝐿 = 1

𝐿𝐿 =

area above the straight line area below the straight line

completely homogeneous completely heterogeneous

Dr. Siroos Azizmohammadi

Warren and Price (1961)

0.8

Lorenz coefficient

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

Normalized ∑ 𝜙𝜙ℎ

0.6 0.4 0.2 0

0

0.2

Summer Course 2016 | Department of Petroleum Engineering Reservoir Engineering Module: Sweep and Displacement Efficiency

0.4

0.6

Variation, 𝑉𝑉

0.8

1

15 September 2016

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Dykstra-Parson method provides optimistic estimates of vertical sweep efficiency, 𝐸𝐸𝑉𝑉 , for layered systems. Green, D. W., and Willhite, G. P., 1998, “Enhanced Oil Recovery”

Dr. Siroos Azizmohammadi

Summer Course 2016 | Department of Petroleum Engineering Reservoir Engineering Module: Sweep and Displacement Efficiency

15 September 2016

16

Johnson (1956) developed a simplified graphical approach for the Dykstra-Parsons method. WOR = 5

WOR = 1 𝑉𝑉

𝑉𝑉

𝐸𝐸𝑉𝑉 1 − 𝑆𝑆wi = 0.4

𝑀𝑀 =

𝐸𝐸𝑉𝑉 1 − 0.72𝑆𝑆wi = 0.45

𝑘𝑘rw 𝜇𝜇o 𝑘𝑘ro 𝜇𝜇w

WOR = 25

𝑉𝑉

𝐸𝐸𝑉𝑉 1 − 0.52𝑆𝑆wi = 0.5

Dr. Siroos Azizmohammadi

𝑀𝑀 =

𝑘𝑘rw 𝜇𝜇o 𝑘𝑘ro 𝜇𝜇w

WOR = 100

𝑉𝑉

𝐸𝐸𝑉𝑉 1 − 0.4𝑆𝑆wi = 0.5

𝑀𝑀 =

𝑘𝑘rw 𝜇𝜇o 𝑘𝑘ro 𝜇𝜇w

Summer Course 2016 | Department of Petroleum Engineering Reservoir Engineering Module: Sweep and Displacement Efficiency

𝑀𝑀 =

𝑘𝑘rw 𝜇𝜇o 𝑘𝑘ro 𝜇𝜇w

15 September 2016

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de Souza and Brigham (1981)

𝑌𝑌 =

WOR + 0.4 18.948 − 2.499𝑉𝑉 (𝑀𝑀 + 1.137 − 0.8094𝑉𝑉)10 𝑓𝑓 𝑉𝑉

𝑓𝑓 𝑉𝑉 = −0.6891 + 0.935𝑉𝑉 + 1.6453𝑉𝑉 2

0 < 𝑀𝑀 ≤ 10

0.3 ≤ 𝑉𝑉 ≤ 0.8

Fassihi (1986) 𝑌𝑌 = 𝑎𝑎1 (𝐸𝐸𝑉𝑉 )𝑎𝑎2 (1 − 𝐸𝐸𝑉𝑉 )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.

Dr. Siroos Azizmohammadi

Summer Course 2016 | Department of Petroleum Engineering Reservoir Engineering Module: Sweep and Displacement Efficiency

15 September 2016

18

Gravity segregation occurs when density differences between Displacing (injected) and displaced fluids are large enough to induce a significant vertical flow component – even when the principal fluid flow direction is in horizontal plane. displaced phase, d

Displacing phase, D

displaced phase, d

Displacing phase, D

If density of the Displacing fluid is less than the displaced fluid’s density, the Displacing fluid overrides the displaced fluid (gravity override). Occurs at gas injection, CO2 flooding, steam injection, in-situ combustion, and solvent flooding. If density of the Displacing fluid is greater than the displaced fluid’s density, the Displacing fluid underrides the displaced 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 Dr. Siroos Azizmohammadi

Summer Course 2016 | Department of Petroleum Engineering Reservoir Engineering Module: Sweep and Displacement Efficiency

15 September 2016

19

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).

𝑅𝑅𝑣𝑣⁄g =

2050𝑢𝑢𝜇𝜇𝑑𝑑 𝐿𝐿 ℎ 𝑘𝑘g∆𝜌𝜌

𝐹𝐹g⁄𝑣𝑣 =

0.00633 𝑘𝑘𝑣𝑣 ⁄𝑘𝑘ℎ ∆𝜌𝜌𝐴𝐴 𝑞𝑞𝜇𝜇𝑑𝑑

𝑢𝑢 = [rb/(d.ft2)] , 𝜇𝜇𝑑𝑑 = [cP] , 𝑘𝑘 = [mD] , 𝜌𝜌 =[g/cm3] , 𝐿𝐿 = [ft] , ℎ = [ft]

Green, D. W., and Willhite, G. P., 1998, “Enhanced Oil Recovery”

Dr. Siroos Azizmohammadi

Summer Course 2016 | Department of Petroleum Engineering Reservoir Engineering Module: Sweep and Displacement Efficiency

15 September 2016

20

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).

β

θ

β

θ

β

θ

The criteria for stable displacement in a dipping reservoirs is called critical velocity: g 𝜌𝜌d − 𝜌𝜌D sin 𝜃𝜃 𝑢𝑢𝑐𝑐 = 𝜇𝜇d 𝜇𝜇D − 𝑘𝑘d 𝑘𝑘D

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

Dr. Siroos Azizmohammadi

Summer Course 2016 | Department of Petroleum Engineering Reservoir Engineering Module: Sweep and Displacement Efficiency

15 September 2016

21

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 𝐸𝐸𝑉𝑉

Under unfavorable mobility ratios (𝑀𝑀 > 1) ‒ cross-flow acts to reduce 𝐸𝐸𝑉𝑉

Dr. Siroos Azizmohammadi

Summer Course 2016 | Department of Petroleum Engineering Reservoir Engineering Module: Sweep and Displacement Efficiency

15 September 2016

22