There is also a handout document to illustrate the material shown in the slides.

1. Gas-liquid pipe flow patterns as
directed graphs

2. Abstract
Abstract
From observation
From simple
to concept
to complex
Pablo Adames, Schlumberger Gas-liquid flow patterns as directed graphs Banff, Canada, June 11, 2014 2 / 34

3. Introduction
Applications of directed graphs
Optimun stream to tear Given all possible states
Upadhye, R. and E.A. Grens. An efficient algorithm for optimum decomposition of recycle systems.
AIChE Journal, Vol. 18, No. 3, 1972, pp 533-39
Pablo Adames, Schlumberger Gas-liquid flow patterns as directed graphs Banff, Canada, June 11, 2014 3 / 34

4. Introduction
Applications of directed graphs
Dynamic distillation column Accumulation in node f
rate of accumulation in edges
Smith, C.L., Pike, R. W., and P. W. Murrill. Formulation and optimization of mathematical
models. International Textbook Company, Scranton, Pennsylvania, 1970, p. 420
Pablo Adames, Schlumberger Gas-liquid flow patterns as directed graphs Banff, Canada, June 11, 2014 4 / 34

5. Introduction
Applications of directed graphs
This paper What does it look like?
Pablo Adames, Schlumberger Gas-liquid flow patterns as directed graphs Banff, Canada, June 11, 2014 5 / 34

6. The graph structure
What is a directed graph?
Definition
Abstract representation of interconnected sets
Components
1 Node: a point representing a set
2 Edge: a link connecting two nodes
Refined definition
The collection of all edges {(i, j)} such that i = j
Pablo Adames, Schlumberger Gas-liquid flow patterns as directed graphs Banff, Canada, June 11, 2014 6 / 34

7. The graph structure A simple flow pattern
A well-mixed flow pattern
Dispersed bubble Control volume
Pablo Adames, Schlumberger Gas-liquid flow patterns as directed graphs Banff, Canada, June 11, 2014 7 / 34

8. The graph structure A simple flow pattern
From control volume to DG
Pablo Adames, Schlumberger Gas-liquid flow patterns as directed graphs Banff, Canada, June 11, 2014 8 / 34

9. The graph structure A simple flow pattern
From control volume to DG
Pablo Adames, Schlumberger Gas-liquid flow patterns as directed graphs Banff, Canada, June 11, 2014 8 / 34

10. The graph structure A simple flow pattern
Well-mixed
mass balance subgraph
Control volume Mass balance subgraph
Pablo Adames, Schlumberger Gas-liquid flow patterns as directed graphs Banff, Canada, June 11, 2014 9 / 34

11. The graph structure A simple flow pattern
Well-mixed
mass balance subgraph
Mass balance subgraph Balance equations
ai,2 − a2,j = 0
a1,2 − a2,3 = 0
˙m1−2 − ˙m2−3 = 0
ρ1 v1 A − ρ2 v2 A = 0
˙m L,1−2 − ˙m L,2−3 = 0
ρ L,1 vsL A − ρ L,2 vsL A = 0
ρ L,1 v L,1 c L,1 A − ρ L,2 v L,2 E L,2 A = 0
Pablo Adames, Schlumberger Gas-liquid flow patterns as directed graphs Banff, Canada, June 11, 2014 10 / 34

12. The graph structure A simple flow pattern
Well-mixed up flow
force balance
(a) Up flow:
control volume
(b) Force
diagram
(c) Directed graph
Pablo Adames, Schlumberger Gas-liquid flow patterns as directed graphs Banff, Canada, June 11, 2014 11 / 34

13. The graph structure A simple flow pattern
Well-mixed down flow
force balance
control volume diagram
(d) Down flow:
(f) Directed graph
(e) Force
Pablo Adames, Schlumberger Gas-liquid flow patterns as directed graphs Banff, Canada, June 11, 2014 12 / 34

14. The graph structure A simple flow pattern
The force balance
Mass balance subgraph Balance equations
i
ai,2 −
j
a2,j = 0
A ∆p
pressure force
− τw S ∆l
frictional force
−A ρmix g ∆l sin θ
hydrostatic force
− A ρmix vmix ∆vmix
kinetic force
= 0
Pablo Adames, Schlumberger Gas-liquid flow patterns as directed graphs Banff, Canada, June 11, 2014 13 / 34

15. The graph structure A simple flow pattern
Well-mixed complete
directed graph
Mass balance Force balance
Combined mass
and force balances
Pablo Adames, Schlumberger Gas-liquid flow patterns as directed graphs Banff, Canada, June 11, 2014 14 / 34

16. The graph structure The slip generator
Slip generator
Slip at source and sink nodes:
vslip,1 = v G,1 − v L,1 = vsG
1−c L,1
− vsL
c L,1
(1)
vslip,2 = v G,2 − v L,2 = vsG
1−E L,2
− vsL
E L,2
(2)
Mass flows through arches, area and density fixed:
v L,1 c L,1 = v L,2 E L,2
v G,1 (1 − c L,1) = v G,2 (1 − E L,2)
Any change in mechanical equilibrium will affect the slip and holdup
Pablo Adames, Schlumberger Gas-liquid flow patterns as directed graphs Banff, Canada, June 11, 2014 15 / 34

17. More complex flow patterns
Separated flow patterns
Stratified wavy
Annular mist
Pablo Adames, Schlumberger Gas-liquid flow patterns as directed graphs Banff, Canada, June 11, 2014 16 / 34

18. More complex flow patterns Separated flow pattern graph
Separated flow pattern
Annular mist
Control volume
1
6
4
3
FG-L fric
− FG-L fric
Gas core
vL
vG
Liquid film
vG vL
Pablo Adames, Schlumberger Gas-liquid flow patterns as directed graphs Banff, Canada, June 11, 2014 17 / 34

19. More complex flow patterns Separated flow pattern graph
The separated flow
directed graph
1
vmix,2 ρ2
6
vmix,5ρ5
2
(1−
cF)v
m
ix,3ρ
in
3
cFv
m
ix,4ρ
in4
511
4
E
Fv
m
ix,4ρ
out4
3
(1−
E
F)v
m
ix,3ρ
out3
7
8 9
10
12
13 14
15
Gas core
Film
Ffriction
Fhead Facceleration
∆Fpressure
slip
node
source sink
Ffriction
Fhead Facceleration
∆Fpressureslip
node
input
split
eqilibrium
mix
FG-L fric
Pablo Adames, Schlumberger Gas-liquid flow patterns as directed graphs Banff, Canada, June 11, 2014 18 / 34

20. More complex flow patterns Separated flow pattern graph
The mass balance equations
Mass balance around splitter node 2:
vmix,2 A2ρmix,2 = vin
s4
A2 ρ4 + vin
s3
A2 ρ3
After introducing the input split parameter, cF :
v2 ρ2 = cF vin
4
ρin
4
+ (1 − cF) vin
3
ρin
3
Mixture velocities into film and gas core:
vin
3
and vin
4
are relative to the area for flow going into the
slip nodes 3 and 4.
Pablo Adames, Schlumberger Gas-liquid flow patterns as directed graphs Banff, Canada, June 11, 2014 19 / 34

21. More complex flow patterns Separated flow pattern graph
The mass balance equations
If all liquid in film and all gas in core:
cF = cL
vin
3
= vsG,3
vin
4
= vsL,4
cL vsL,4 = vsL,2
(1 − cL) vsG,3 = vsG,2
And the splitter mass balance would be:
ρ2 = cL ρL + (1 − cL) ρG
This is the definition of the input mixture density to the control
volume.
Pablo Adames, Schlumberger Gas-liquid flow patterns as directed graphs Banff, Canada, June 11, 2014 20 / 34

22. More complex flow patterns Separated flow pattern graph
The mass balance equations
Similarly for the mixer, node 5:
EF vout
4
ρout
4
+ (1 − EF) vout
3
ρout
3
= v5 ρ5
If no entrainment in the gas core or film:
EL ρL + (1 − EL) ρG = ρ5
This is the definition of the equilibrium mixture density of the control
volume.
Pablo Adames, Schlumberger Gas-liquid flow patterns as directed graphs Banff, Canada, June 11, 2014 21 / 34

23. More complex flow patterns The force balance equations
The force balance equations
Force balance around node 11:
a11,3 − a11,4 = 0
τI S ∆l −
1
2
fI ˆρ |vR| vR = 0
The interfacial shear force is equal to the friction force due to a rough
interface
Pablo Adames, Schlumberger Gas-liquid flow patterns as directed graphs Banff, Canada, June 11, 2014 22 / 34

24. More complex flow patterns Intermittent flow pattern graph
Intermittent flow pattern
Slug flow
Control volume
10
8
9
Gas core (SC)
Liquid film(SF)
Liquid slug (D)
Pablo Adames, Schlumberger Gas-liquid flow patterns as directed graphs Banff, Canada, June 11, 2014 23 / 34

25. More complex flow patterns Intermittent flow pattern graph
The intermittent flow
directed graph
1 172 16
5
6
4
3
14
15
10
12
13
7 1122
9
8
18
19 20
21
23
24 25
26
27
28 29
30
Film (F)
Bubble (S)
Liquid slug (D)
vmix,2 ρmix,2
(1 −
c LU
) v L,3
ρG
c
LU v
L,5 ρ
L
γ v sGS,4
ρG
(1
−
γ)vsGD,6
ρG
γ
v
sLS,4ρ
L
(1 −
γ) v
sLD,6 ρ
L
v mix,4
ρmix,4
v
mix,6 ρ
mix,6
v mix,10
ρmix,10
c
Fv
m
,7ρ
m
,7
(1
−
cF
)vm
,7
ρm
,7
(1
−
E
F)v
m
,7ρ
m
,9
EF
vm
,7
ρm
,8
v
mix,11 ρ
mix,11
β
vsGS,15
ρG
β v
sLS,13 ρ
L
(1
−
β)v
sLD,13ρ
L
(1 −
β) v sGD,15
ρG
E
LU v
LU,16 ρ
L
(1 −
E LU
) v GU,16
ρG
vmix,17 ρU,17
Pablo Adames, Schlumberger Gas-liquid flow patterns as directed graphs Banff, Canada, June 11, 2014 24 / 34

26. More complex flow patterns Intermittent flow pattern graph
Description of intermittent
directed graph
1 A long bubble section, noted as S
2 A dispersed flow section or slug, noted as D
3 A periodic slug unit, S+D
4 A timed-averaged ratio called intermittency, β = lS
lS+lD
5 An input intermittency, γ =
lin
S
lin
S +lin
D
Pablo Adames, Schlumberger Gas-liquid flow patterns as directed graphs Banff, Canada, June 11, 2014 25 / 34

27. More complex flow patterns Intermittent flow pattern graph
The mass balance equations
Input section Features
1 172 16
5
6
4
3
14
15
10
12
13
7 1122
9
8
18
19 20
21
23
24 25
26
27
28 29
30
Film (F)
Bubble (S)
Liquid slug (D)
vmix,2 ρmix,2
(1 −
c LU
) v L,3
ρG
c
LU v
L,5 ρ
L
γ v sGS,4
ρG
(1
−
γ)vsGD,6
ρG
γ
v
sLS,4ρ
L
(1 −
γ) v
sLD,6 ρ
L
v mix,4
ρmix,4
v
mix,6 ρ
mix,6
v mix,10
ρmix,10
c
Fv
m
,7ρ
m
,7
(1
−
cF
)vm
,7
ρm
,7
(1
−
E
F)v
m
,7ρ
m
,9
EF
vm
,7
ρm
,8
v
mix,11 ρ
mix,11
β
vsGS,15
ρG
β v
sLS,13 ρ
L
(1
−
β)v
sLD,13ρ
L
(1 −
β) v sGD,15
ρG
E
LU v
LU,16 ρ
L
(1 −
E LU
) v GU,16
ρG
vmix,17 ρU,17
1 Node 2 uses the input liquid
fraction CLU
2 Nodes 3 and 5 use the input
intermittency, γ
3 Node 7 is the source node for
the long bubble, S
4 Node 6 is the source node for
the well-mixed region, D
Pablo Adames, Schlumberger Gas-liquid flow patterns as directed graphs Banff, Canada, June 11, 2014 26 / 34

28. More complex flow patterns Intermittent flow pattern graph
The mass balance equations
Equilibrium section Features
1716
14
15
10
12
13
1122
9
8
19 20
21
24 25
26
28 29
30
Film (F)
Bubble (S)
Liquid slug (D)
v mix,10
ρmix,10
c
Fv
m
,7ρ
m
,7
)vm
,7
ρm
,7
(1
−
E
F)v
m
,7ρ
m
,9
EF
vm
,7
ρm
,8
v
mix,11 ρ
mix,11
β
vsGS,15
ρG
β v
sLS,13 ρ
L
(1
−
β)v
sLD,13ρ
L
(1 −
β) v sGD,15
ρG
E
LU v
LU,16 ρ
L
(1 −
E LU
) v GU,16
ρG
vmix,17 ρU,17
1 Node 11 is the sink node for
the long bubble, S
2 Node 14 is the sink node for
the well-mixed region, D
3 Nodes 12 and 14 use the
intermittency, β
4 Node 16 uses the equilibrium
liquid fraction ELU
Pablo Adames, Schlumberger Gas-liquid flow patterns as directed graphs Banff, Canada, June 11, 2014 27 / 34

29. More complex flow patterns The force balance equations
The force balance equations
1 The force balances of the S and D regions are done
independently
2 These are exact replicas of the ones done for
separated and well-mixed
3 The Tulsa Unified model, as an example, adds a force
term from 8 to 10
Pablo Adames, Schlumberger Gas-liquid flow patterns as directed graphs Banff, Canada, June 11, 2014 28 / 34

30. More complex flow patterns Similarities between separated and slug flow
Similarities between
separated and slug flow
The liquid phase is distributed according to a key
parameter:
vsL = vLEL = EF vLFELF
liquid in film region
+ (1 − EF) vLCELC
liquid in gas core region
vsL = vLEL = β vLSELS
liquid in separated region
+ (1 − β) vLDELD
liquid in dispersed region
Pablo Adames, Schlumberger Gas-liquid flow patterns as directed graphs Banff, Canada, June 11, 2014 29 / 34

31. Consequences of graph structure
Consequences of the
directed graphs
1 In separated flow there are paths for film and gas core
2 The forces controlling the degree of separation are
gravitational and inertial
3 In intermittent flow the paths are distributed in time
through regions S and D
4 Further paths exist in the separated region S of
intermittent flow
Pablo Adames, Schlumberger Gas-liquid flow patterns as directed graphs Banff, Canada, June 11, 2014 30 / 34

32. Consequences of graph structure
Consequences of the
directed graphs
5 There is a recursive nature in this representation with
well-mixed regions as the primary level
6 Model refinements can be visualized and modelled
using directed graphs for flow patterns:
1 Back mixing regions
2 Additional mass flow paths
3 Additional momentum exchange
Pablo Adames, Schlumberger Gas-liquid flow patterns as directed graphs Banff, Canada, June 11, 2014 31 / 34

33. conclusions
Conclusions
1 All major steady state flow pattern types can be
represented as directed graphs
2 The mass balance directed graph always has one
input and one output
3 This in itself proves that mass is conserved when they
are used to solve discretized pipeline models
Pablo Adames, Schlumberger Gas-liquid flow patterns as directed graphs Banff, Canada, June 11, 2014 32 / 34

34. conclusions
Conclusions
4 Directed graphs for simple flow patterns can be
reused to build more complex ones
5 Intermittency in slug flow plays an equivalent role as
film fraction in separated flow
Pablo Adames, Schlumberger Gas-liquid flow patterns as directed graphs Banff, Canada, June 11, 2014 33 / 34

35. conclusions
Thank you
Pablo Adames, Schlumberger Gas-liquid flow patterns as directed graphs Banff, Canada, June 11, 2014 34 / 34