Slides gas liquid flow patterns as directed graphs
- 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