M S Mohan Kumar Mini Mathew & Shibani jha Dept. of Civil Engineering Indian Institute of Science

Transcription

1 Multiphase Flow in Porous /Fracture Media M S Mohan Kumar Mini Mathew & Shibani jha Dept. of Civil Engineering Indian Institute of Science Bangalore, India

2 2 Multiphase fluids ----Fluids which are immiscible and is slightly soluble. Represent as individual phase in the subsurface, flow behavior is described as multiphase problem. Common trait of these substance are NAPLs 2 NAPLs LNAPLs DNAPLs 2 The phase does not fill the pore space completely. 2 Multiphase flow--- Simultaneous flow of more than two fluids, does not take place as a piston like process. 2 Flow experiments show that Darcy s law can be extended for multiphase flows 2 Darcy s velocity for each phase in a porous medium is described d by the generalized Darcy s law.

3 Source of NAPLs 2 Subsurface leakage of hydrocarbon fuels and other immiscible organic liquids due to leaky storage tanks or pipelines. 2 Coal tar from illuminating gas production, wastes from steel industry and wood treating operation 2 Organic substances used din industries i as Mineral lfuels ( ex : Petrol, fuel oil etc), Solvents Detergents ( ex : Chlorinated hydrocarbons) Goal 2 Estimate the potential danger of NAPL infiltration & to plan eventual remediation techniques. 2 To reduce the field investigation effort and cost.

4 2 Vertical migration in the vadose zone predominantly by gravity 2 Some lateral spreading due to capillar forces and media properties 2 Migration occurs when enough pressure is available to over come the displacement pressure 2 In saturated zone the movement is by displacement of water 2 In saturated medium phase system 2 In unsaturated medium phase system or phase system with static air pressure General migration pattern and process of NAPLs (after Helmig)

5 Processes of Multiphase Multicomponent System Water Water Air Organic compound Air Air Water vapour Organic compound Air NAPL Water NAPL

6 Multiphase Multicomponent Flow in the Subsurface - Governing Equations (φ α ρ α) L(φ α,ρα, v α) = + div(φ αρα v α) ρα qα ] dg= 0 t G k (φ S ρ X ) k α α α L(vX α, α ) = + divρ α α α α pmα α α α = t G k k k k { ρ v X φs D (ρ gradx ) } r ] dg 0 φs α D k pmα vi vj = δij αlv+ (α L α T) δijϕ v i j + 3/4 S 10/3 α D N T (u ρ S ) P L(uα, Pα, T) = s s α α α pm α = t t ρ α= 1 G [ { } { α α α α (1 φ)ρ c + ϕ + divv ρ (u + ) + div λ gradt } r ] 0 α

7 Interior Conditions Interior Conditions 1 S N 1 1 α S α N = = 1 1 α X α N k = = ) (, β α β α β α S k k P P C =P = ) ( α α α S k k r r

8 Present study consists of 2 Modelling and analysis of NAPLs migration in saturated porous medium----two phase NAPL-Water system 2 Study the influence of air phase on the infiltration of water in unsaturated porous medium ---- two phase Air-Water system. 2 Modelling and analysis of NAPL migration in unsaturated porous medium. Three phase Air-NAPL-Water system Two phase NAPL-Water system with constant air pressure 2 Modelling and analysis of NAPL migration in combined saturated-unsaturated porous medium 2 Parallel computation of multiphase flow in saturated porous medium

9 NAPL Infiltration Two phase region Air - Water Three phase region Water- NAPL-Air Two phase region Air - Water Interface region Interface region Interface region Single phase region of water Two phase region Water-NAPL Interface region Single phase region of water NAPL infiltration in the subsurface

10 Two Phase NAPL-Water Simulation in Saturated Porous Media 2 Develop a robust model and numerical method for the simulation of NAPL-Water in saturated porous media 2 Models of all the combinations of pressures and saturations of the phases have been developed 2 Comparative study is made between 6 models with simultaneous and modified sequential methods. 2 Comparative study between conventional simultaneous, modified sequential, and adaptive solution fully implicit modified sequential methods using pressure and saturation of wetting fluid are made 2 Effect of different types of approximations of nodal coefficient

11 2 Effect of different types of linearization methods in numerical modelling 2 Effect of different types of iterative methods in numerical modelling 2 Influence of capillarity and heterogeneity in ht heterogeneous media 2 Effect of Peclet, Courant numbers, and convergence criteria in two phase systems 2 Effect of different types of constitutive relations in the Effect of different types of constitutive relations in the numerical simulation

12 Solution Methodology used h Conventional simultaneous method h Modified sequential method h Adaptive solution fully implicit modified sequential method Simultaneous method Primary variables of the phases(pw & Sw) are solved together [ ] m [ ] m +1 A X = [ R] m

13 Sequential method Two step implicit technique Solve for pressure ( Saturation at previous iteration) [ m m + 1/2 A ] [ P ] = [ ] m W R1 1 = Solve for saturation ( Pressure at current level) [ ] m+ 1/2 [ ] m+ 1 [ ] m+ 1/ 2 A S = R 2 W 2

14 Adaptive solution fully implicit sequential method Identification of active and inactive nodes in each iteration for each primary variable. Difference in pressures between two consecutive iteration is greater than permissible level in any node or any of its 4 surrounding nodes. If the relative permeability of NAPL and water is greater than 0 and less than 1 in any node and any of its neighboring g4 nodes Difference in saturation between two consecutive iteration is greater than the permissible level in any node or any of its 4 surrounding nodes..

15 For active nodes Two step modified sequential method Solve for pressure ( Saturation at previous iteration) [ ] m [ ] m + 1/2 P [ R ] m A1 W = 1 Solve for saturation ( Pressure at current level) [ m+ 1/2 m+ 1 m+ 1/ 2 A ] [ S ] = [ R ] 2 W 2 For inactive nodes Pressure and saturation from the previous iteration

16 Verification of models Analytical solution based on McWhorter and Sunada(1990) Porosity of the medium [m 2 ] 0.35 Permeability 5E-11 Pore size distribution index 2 Displacement pressure [Pa] 2000 Residual wetting phase saturation 0.05 Wetting phase viscosity[pa-sec] 1E-03 Nonwetting phase viscosity[pa-sec] 0.5E-04 P -1/2 0t)=S 0 W (0,t) - P W (L,t) = A t - C t <= t 0 S W 0,t) W t>0

17 Verification of two dimensional models Laboratory experimental results(helmig, 1998) sand 1 sand 2 Intrinsic permeability [m 2 ] : 6.64E E-12 Porosity : Wetting Phase residual saturation : Displacement pressure : Pore size distribution index : 2.7 2

18 NAPL saturation distribution at different times T = 1000sec Experimental result T = 3000sec Present Model results T = 5000sec

19 Comparison of different types of models Between models Between numerical methods

20 Nonwetting fluid distribution profile after 2400 sec Density of NAPL : 1460kg/m 3 Viscosity of NAPL : pa-sec Experimental results Present model

21 Sand 1 Sand 2 Sand 3 Sand 4 Φ k 5.04E E E E S wr λ Pd

22 Sensitivity study for SIM, MSEM and ASMSEM

23 Nonwetting fluid distribution Host permeability : 7.08E-12m 2 Density of TCE : 1460kg/m 3 Host displacement pressure : Pa Porosity : 0.34 Residual wetting phase saturation : Pore size distribution index : 2.48 Lens displacement pressure : Pa

24 Host permeability : 9.28E-12 m 2 Density of TCE : 1400kg/m 3 Lens permeabliity : 7.53E-12 m 2 Viscosity of TCE : 1E-03 Pa-sec Host displacement pressure : 1667Pa Residual saturation : 0.12 Lens displacement pressure : 2353Pa Porosity : 0.38 Pore size distribution index : 2

25 Sand 1 Sand 2 k 6.64E E-12 Φ S Wr Pd λ ρ NW 1460kg/m 3 μ NW 0.9e-03pa-s

26 Effect of different types of approximations at the cell faces a Arithmetic mean or Harmonic mean b Harmonic mean with upstream weighing of relative permeabilities c Harmonic mean with upstream weighing of relative permeability and capillarity diffusivities d. Fully upwind method

27 Effect of different types of linearizations in the two phase system Classical Newton Raphson method Modified Newton Raphson method Picard s method

28 Effect of classical Newton Raphson method

29 Influence of capillarity and heterogeneity in two phase simulation J( s P C P P + * e ) = Pe = σ φ k 1/2 [ ] J ( S ) L ( W Pc S* - + P d - P d + 0 Wetting fluid saturation 1

30 Effect of without and with heterogeneity and capillary pressure effect

31 Identification of robust iterative method for the two phase simulation Explicit and alternating direction implicit methods are unstable (Peaceman, 1967) Incomplete Cholesky conjugate gradient method(iccg) Hesteness and Stiefel s conjugate gradient method(cghs) Diagonal scaling conjugate gradient(dscg) Krylov subsurface solver BiCGSTAB Strongly Implicit Procedure(SIP)

32 Influence of Peclet, Courant numbers and convergence criteria

33 Effect of different Constitutive Relations used Brooks and Corey Van-Genuchten

34 Effect of NAPL migration in Random Heterogeneous Media

35 Two Phase Air Water Flow Simulation in Unsaturated Porous Media 2 To study the effect of air flow for infiltration and distribution of water in unsaturated zone 2 Development of Numerical models of one dimensional and two dimensional using SIM, MSEM, and ASMSEM 2 Validation of one phase, quasi two phase and two phase models 2 Air water simulation in one dimensional homogeneous porous medium 2 Air water simulation in one dimensional heterogeneous porous media 2 Air water simulation in two dimensional heterogeneous porous media One phase model - Richard s equation Quasi two phase model - Two phase air water system with constant air pressure Two phase air water model - Air and water moves simultaneously

36 Verification of quasi two phase model using one phase model One phase model - Richard s equation Quasi two phase model - Arithmetic approximation - Harmonic approximation with upstream nodal relative permeabilities Flux = 3.29m/day Initial pressure = m

37

38 Validation of two phase Air-Water model with experimental results(touma and Vaucline, 1986) Soil parameters are 93.5cm Saturated water content = Residual moisture content = van Genuchten parameters α = 0.044cm -1 β = 2.2 Gas constant of air = Molecular weight = Impermeable boundary

39 Flux greater than hydraulic conductivity - 20cm/h Bounded column Two phase model One phase model

40 Open column Two phase model One phase model

41 Flux less than saturated hydraulic conductivity - 8cm/hr Two phase model One phase model

42 Soil Properties Residual moisture content : Constant head van-genuchten parameters : α= 0.044cm 044cm -1, β = Saturated moisture content : Porosity : 0.37 Intrinsic permeability :0.0145cm 2 P W =0.906 inch 3.07 ft

43 Two phase Air-Water in heterogeneous porous media Initial pressure = -100cm Flux at the inflow end = 9.5E-05cm/sec 150cm lens 20 cm 40cm Impermeable layer

44 Sand and sandy loam Open column One phase model Two phase model

45 Bounded column One phase model Two phase model

46 Sand and Silt Open column One phase model Two phase model

47 Bounded column One phase model Two phase model

48 Sand and Clay

49

50 Two dimensional two phase air water simulation in heterogeneous media Initial pressure = -150cm Flux = 1.13x cm/sec 10cm sand 50cm Low permeability soil 60cm 15cm 20cm 100 cm Test One phase Two phase Two phase Two phase problem model MSEM SIM ASMSEM

51 Sand and Sandy loam

52 Modelling and Simulation of NAPL Migration in Saturated - unsaturated us u zone oe 2 Development of three phase model of water, NAPL, and air 2 Quasi ithree phase model - Air - NAPL - Wt Water with static air pressure 2 Fully three phase model - Air- NAPL - Water with air pressure 2 Validation of the present models 2 Effect of capillary pressure and heterogeneity in the numerical modelling of three phase systems 2 Comparative study between simultaneous, sequential and adaptive modified sequential methods for the saturated unsaturated systems 2 Effect of air pressure on the numerical modelling of three phase systems

53 Model Validation Oil infiltration 3 cm 15 cm 20 cm 30 cm Watertable Infiltration : 30ml of NAPL, 20,5,5----1hr interval -----case a : 30ml ml for 3hrs case b

54 NAPL profile after 3hrs

55

56 Air-NAPL-Water simulation in variably saturated media Infiltartion rate- 5.07E-05m/sec

57

58 Comparative study between simultaneous, modified sequential and adaptive modified sequential methods. SIM MSEM ASMSEM Quasi three phase Full three phase Quasi three phase Full three phase

59 Effect of capillarity and heterogeneity in the numerical modelling of three phase systems LNAPL Flux = 7.1x10-4 m/sec

60 DNAPL

61 Effect of with and without air pressure LNAPL Open column

62 LNAPL Bounded column

63 Open column DNAPL

64 DNAPL Bounded column

65 Parallel computation of NAPL migartion in saturated porous media Need of Parallel Computation To reduce the computational time To simulate bigger domain multiphase flow simulations which is not possible by conventional computation General Parallel Architecture - Interaction between the processors Shared memory programming g Distributed memory with Message Passing Interface programming Data parallel programming

66 Parallel machine used for computing 32 nodes Scalable ab Parallel(SP2) a system Nodes 1-8 : IBM RS 66MHz with 512MB RAM Nodes 9-24 : IBM RS 77MHz with 256MB RAM Nodes : IBM RS 133MHz with 256MB RAM Distributed memory machine with Message Passing Interface Programming(MPI) Parallelization Methodology Divide the domain into sub-domains Identify the initial and boundary conditions of each sub-domain Compute the primary variables(p W &S W ) Communicate primary variables between sub-domains using Message Passing Interface (MPI) programming

67 Domain Decomposition methods used Overlapping region Row wise Column wise

68 1.0m Low permeability soil Source 1.5m Impermeable boundary Flux at source = 5.16E-06 m/s m m

69

70 Properties of media and fluid Properties Sand 1 Sand 2 Porosity Displacement pressure[pa] Intrinsic permeability [m 2 ] 5.04E E-12 Residual saturation pore size distribution index Density of NAPL[kg/m 3 ] 1460 Viscosity[Pa-s ] 0.90E-03

71 Nonwetting fluid saturation distribution after 1day after 3days

72 Modelling of Multiphase Flow in a Fracture Media

73 Purpose of the Study A numerical model to simulate multiphase flow within a fracture zone To study the conditions under which a DNAPL can enter a rough walled, initially water saturated fracture To study the subsequent behaviour of DNAPL within the fracture A rough walled fracture defined in terms of aperture distribution A single pair of parallel plates (homogeneous) A set of parallel plate pairs (spatially correlated) Model study for 1D variable aperture fracture Extended for 2D homogeneous and variable aperture fracture The model has been studied for various aperture e distribution with different correlation length Model study for sensitivity to fluid and fracture properties on the migration rate of DNAPL through fractures.

74 Model Conceptualization Capillary yp pressure (Bear, 1972) P c =P nw -P w Parallel plate P e =(2σcosθ)/e Circular P e =(4σcosθ)/e Entry for DNAPL P >P P c > P e H d =(2σ)/(Δρge)

75 Height of Pool versus Aperture Invaded 10 2 σ = N/m ( μm ) FRACTURE APE ERTURE INVADED ( σ = N/m σ = N/m σ = N/m σ = 005 N/m HEIGHT OF DNAPL POOL (m)

76 Mathematical Equations Mass conservation - (ρ w q wi e) x i =( (φs w ρ w )/t) e - (ρ nw q nwi e) x i = ( (φ S nw ρ nw ) t)e Darcy s law q wi = - (k ij k rw /μ w )( P w / x j +ρ w g y/ x j ) q nwi = - (k ij k rnw /μ nw ) ( P nw / x j +ρ nw g y/ x j ) Continuity equations ( / x i )[(ek ij k rw /μ w )( P w / x j + (ρ w g) ( y/ x j ))] = eφ( S w / t) ( / x i )[(ek ij k rnw /μ nw )( P nw / x j + (ρ nw g) ( y/ x j ))] = eφ( S nw / t)

77 Continuity Equations in Terms of P w and S w as Dependent Variables Saturation constraint S w +S nw = 1.0 For wetting phase ( / x)[(ekk rw /μ w )( P w / x)] + ( / y)[(ekk rw /μ w ) ( P w / y+ρ w g)] = eφ( S w / t), For Non-Aqueous phase ( / x)[(ekk rnw /μ nw )( (P w+ P c )/ x)] + ( / y)[(ekk rnw /μ nw ) ( (P ( w + P c )/ y + ρ nw g)] = - eφ( S w / t),

78 Boundary Conditions Constant mass flow rate or constant pressure-saturation Neumann boundary with zero flux Dirichlet boundary Dirichlet boundary Neumann boundary with zero flux a) Dirichlet boundary condition: u 1 = f 1 and u 2 = f 2 b) Neumann boundary condition: (k k /μ )( P / x +ρ g y/ x )=f - (k ij k rw /μ w ) ( P w / x j +ρ w g y/ x j ) = f 3 -(k ij k rnw /μ nw ) ( P nw / x j +ρ nw g y/ x j ) = f 4

79 Relative and Fracture Permeabilities Brooks and Corey model k rw =S (2+3λ)/λ e k rnw =(1-S e ) 2 (1-S e (2+3λ)/λ ) Flow between parallel plates F wi = - (e 3 /12μ w )( P w / x j + ρ w g y/ x j ) S e =(S w -S r )/(1-S r ), 0<=S e <=1 F nwi = - (e 3 /12μ nw ) ( P nw / x j + ρ nw g y/ x j ) S e =(P c /P d ) -λ Fracture permeability k = e 2 /12 P d= P e

80 One Dimensional Model Equivalent porous media approach k=(e 2 )/12( R 1.5 r ) (Marshily, 1986) For moderately rough fracture plane R r = 0.1 Solution domain

81 Fluid and Fracture Properties Properties Values Units Wetting phase viscosity Pa.s Nonwetting phase viscosity Pa.s Wetting phase density Kg/m 3 Nonwetting phase density Kg/m 3 Interfacial tension N/m Porosity Pore size distribution index Wetting phase residual saturation 0.1 -

82 Model Verification Pool Height (meters) present model for density = 1460 kg/m 3 Kueper s model for density = 1460 kg/m 3 present model for density = 1200 kg/m Kueper s model for density = 1200 kg/m Aperture (microns) present model for density = 1460 kg/m 3 Kueper s model for density = 1460 kg/m 3 present model for density = 1200 kg/m 3 Kueper s model for density = 1200kg/m Time (hours) DNAPL pool vs time Time (hours) Aperture vs time Kueper s Result Present Model es) Fracture Dip (degree Time (hours) Fracture dip vs time

83 Sensitivity with DNAPL Pooled Above Fracture Opening DNAPL pool height = 0.5 m DNAPL pool height = 1.0 m DNAPL pool height = 1.5 m DNAPL pool height = 2.0 m DNAPL pool = 0.50 m DNAPL pool = 0.40 m DNAPL pool = 0.30 m DNAPL pool = 0.35 m Fracture Ap perture (microns) DNAP PL Saturation Time (hours) Length of Fracture (m) Fracture aperture vs travel time DNAPL saturation at t = 15000s

84 Sensitivity with aperture of Fracture Opening Fracture Aperture = 100 μm 0.45 Fracture Aperture = 75 μm Fracture Aperture = 50 μm Fracture Aperture = 25 μm DNAPL Poo ol Height (meters) DNAP PL Saturation e = 25 μm e = 50 μm e = 75 μm e = 100 μm Time (hours) Length of Fracture (m) DNAPL Pool vs travel time DNAPL saturation at t = 15000s

85 Sensitivity With Fracture Dip Dip = 45 o Dip = 30 o Dip = 15 o Dip = 0 o 0.35 Dip = 60 o APL Saturation DN Dip = 90 o Length of Fracture (m) DNAPL saturation at t = 15000s

86 Effect of Viscosity and Density Fracture Dip = 90 degree Aperture = 75 μ m DNAPL Pool = 0.5 m Viscosity = 0.57E 03 Time = 1000 seconds Time = 2000 seconds Time = 5000 seconds Time = seconds Fracture Dip = 90 degree Aperture = 75 μ m DNAPL Pool = 0.5 m DNAPL Density = 1460 kg/m 3 Time = 1000 seconds Time = 2000 seconds Time = 5000 seconds Time = seconds DNAPL Saturation DNAPL Saturation Length of Fracture (meters) Length of Fracture (meters) Fracture Dip = 90 degree Aperture = 75 μ m DNAPL Pool = 0.5 m Viscosity = 0.9E 03 Time = 1000 seconds Time = 2000 seconds Time = 5000 seconds Time = seconds Fracture Dip = 90 degree Aperture = 75 μ m DNAPL Pool = 0.5 m DNAPL Density = 1600 kg/m 3 Time = 1000 seconds Time = 2000 seconds Time = 5000 seconds Time = seconds uration DNAPL Satu aturation DNAPL Sa Length of Fracture (meters) Length of Fracture (meters) DNAPL saturation plots for viscosity effect DNAPL saturation plots for density effect

87 Spatially Correlated Aperture Field for One Dimensional Fracture Aperture (mm) Length of Fracture (meters) Mean aperture e 75 μm Std.dev σ Correlation length 0.2m Maximum e μm Minimum e 19.1 μm Aperture distribution along the length of fracture

88 DNAPL and Water Migration T=5000s T=10000s T=25000s T=50000s DNAPL Velocity (m/s) DNAP PL Saturation Length of Fracture (meters) DNAPL velocity T=5000s T=10000s T=25000s T=50000s Time = 5000 s 0.3 Time = s Time = s Time = s Length of Fracture (meters) s) Water Velocity (m/s DNAPL distribution ib ti at various times Length of Fracture (meters) Water velocity

89 Effect of DNAPL Pool and Fracture Dip in a Correlated Aperture Field L Saturation DNAP DNAPL Pool = 0.15m DNAPL Pool = 0.20m DNAPL Pool = 0.25m DNAPL Pool = 0.50m DNAPL Pool = 0.75m DNAPL Saturation Fracture Dip = 90 o Fracture Dip = 60 o Fracture Dip = 45 o Fracture Dip = 30 o Fracture Dip = 15 o Fracture Dip = 1 o Length of Fracture (meters) Length of Fracture (meters) DNAPL distribution ib ti at t = 50000s DNAPL distribution ib ti at t = 50000s for various pool height for various fracture dip

90 Effect of Correlation Length on DNAPL and Water Migration Fracture Aperture (mm) Nodal Discretization = 250 mm correlation length = 100 mm correlation length = 200 mm correlation length = 300 mm correlation length = 400 mm DNAPL Velocity (m/s) correlation length =200mm correlation length =300mm correlation length=400mm correlation length =100mm Length of Fracture (meters) Aperture distribution Length of Fracture (meters) DNAPL velocity at t = 50000s on DNAPL Saturatio correlation length = 200mm correlation length = 300mm Water Velocity (m m/s) correlation length = 400mm correlation length = 300mm correlation length = 200mm correlation length = 100mm 0.3 correlation length = 400mm correlation length = 100mm Length of Fracture (meters) DNAPL distribution at t = 50000s Length of Fracture (meters) Water velocity at t = 50000s

91 Range of Apertures for Various Fracture Aperture fields generated Correlation Maximum Minimum Aperture at Aperture just lengths mm aperture μm aperture μm the top of fracture μm below the top cell μm

92 Two Dimensional Model DNAPL Saturation Time = 8000 s Time = s Time = s Time = s Length of fracture along y axis (m) DNAPL saturation along vertical centre line Saturation of DNAPL 0.5 Solution domain for homogeneous fracture plane Time = 8000 s Time = s Time = s Time = s n DNAPL Saturatio Length of fracture along x axis (m) DNAPL saturation along top boundary

93 ture along y axis (m) Length of Fract 0.35 Time = 8000 s Length of Fracture along x axis (m) Length of Fracture along y axis (m) DNAPL Velocity vector plot Time = 8000 s Length of Fracture along x axis (m) Length of Fractu ure along y axis (m) 0.35 Time = s Length of Fracture along x axis (m) along y axis (m) Length of Fracture DNAPL Velocity vector Time = s Length of Fracture along x axis (m) Length of Fracture along y axis (m) 0.35 Time = s Length of Fracture along x axis (m) g y axis (m) Length of Fracture alon DNAPL Velocity vector plot Time = s Length of Fracture along x axis (m) DNAPL distribution at various times DNAPL velocity vector at various times

94 Length of Frac cture along y axis (m) 0.35 Fracture Aperture = 25 μm Length of Fracture along x axis (m) along y axis (m) Fracture Aperture = 35 μm Base case Length of Fracture Length of Fracture along x axis (m) Le ength of Fracture alon ng y axis (m) Horizontal Fracture Effect of aperture Length of Fracture along x axis (m) Effect of dip

95 DNAPL Courant Number

96 Effect of Heterogeneity Solution domain for heterogeneous fracture plane DNAPL Saturation T = 8000s T = 30000s T = 40000s turation DNAPL Sat Time = 8000 s Time = s Time = s uration DNAPL Satu Length of Fracture along y axis (m) DNAPL saturation along vertical centre line Length of Fracture along x axis (m) DNAPL saturation along top boundary

97 Effect of Heterogeneity on DNAPL Migration

98 Grid Convergence

99 Rough Walled Fracture Plane in Terms of Spatially Correlated Aperture Distribution Solution domain for a rough walled fracture plane Aperture in mm Aperture distribution for a rough walled fracture plane

100 Fluid Properties and Source Condition Properties Units Value Wetting phase density Kg/m DNAPL density Kg/m Wetting phase viscosity Pa.s DNAPL viscosity Pa.s Interfacial tension N/m Wetting phase pressure at source Pa 0.0 Wetting phase saturation at source - 0.5

101 Range of Apertures in the Field Generated Along the top Along the bottom 4.5 x Along the top Along the bottom 3.5 riation of Aperture (mm) Var Perm meability of Fracture (m 2 ) Length of Fracture along x axis (m) Length of Fracture along x axis (m) Aperture distribution along the top and bottom boundary Fracture permeability along the top and bottom boundary Range of apertures within a fracture Range Unit Domain Top boundary Bottom boundary Maximum μm minimum μm

102 DNAPL Migration Length of Fracture along y axis (m) Time = 5000s Length of Fracture along x axis (m) ength of Fracture along y axis (m) L Time = seconds Length of Fracture along x axis (m) DNAPL distribution at t = 5000s DNAPL distribution at t = 50000s Length of Fractu ure along y axis (m) Time = seconds Length of Fracture along x axis (m) Length of Fractu ure along y axis (m) Time = seconds Length of Fracture along x axis (m) DNAPL distribution at t = 10000s DNAPL distribution at t = s

103 Isotropic and Anisotropic Aperture Field Aperture (mm) 0.11 Fracture along y axis (mm) Length of Length of Fracture along x axis (mm) 0.02 Solution domain Aperture (mm) Correlation length (30,30)mm30) Aperture (mm) Length of Fracture along y axis (m mm) s (mm) Length of Fracture along y axis Length of Fracture along x axis (mm) Correlation length (30,20)mm Length of Fracture along x axis (mm) Correlation length (20,20)mm 0.02

104 Anisotropic Aperture Field 0.7 Length of Frac cture along y axis (mm m) Length of Fracture along x axis (mm) 0 Isotropic Aperture Field 0.7 Isotropic Aperture Field Le ength of Fracture along y axis (mm) Le ength of Fracture along y axis (mm) Length of Fracture along x axis (mm) Length of Fracture along x axis (mm) 0

105 Buoyancy flow and Capillary Trapping Local capillary trapping Gravity fingering Buoyancy flow -most dense fluid- downward, lateral and sliding movement Gravity fingering - once the residual DNAPL pools up on the top of the wedge

106 Observations Fractures provides preferential and faster pathways DNAPL enters the fracture at the points of largest aperture and continue to migrate through the larger aperture regions The certain regions of the fracture may remain void of DNAPL at all times The ability of DNAPL to enter smaller aperture regions of fracture increases as a function of fdepth of penetration Traverse time for DNAPL is inversely proportional to the fracture aperture, fracture dip and DNAPL pooled above the fracture Fracture aperture is most sensitive parameter Shallow fractures (30 o to0 o ) shows significant change in migration For aperture field with correlation length close to grid size, DNAPL migration shows homogeneous nature Anisotropic distribution of aperture field provides higher rate of DNAPL movement

107 Immiscible-miscible miscible flow in coastal reservoirs Seawater intrusion is an ideal problem to study the buoyancy flow Natural intrusion of heavier fluid with matched viscosity

108 Ghyben-Herzberg approximation for seawater intrusion Groundwater flow patterns in an idealized, homogeneous coastal aquifer. (Source: USGS) Analytical solution on the basis of Ghyben-herzberg Relationship (Bear,1972) and Single Potential Theory (strack,1976). The assumption of a sharp interface between freshwater and saltwater. Variable density flow in both time and space dimension (Frind,1982; Voss and Souza; Kolditz et al.,1998). 2/10/

109 Immiscible-miscible flow of Seawater - freshwater A freshwater confined aquifer of size 2m by 1m, Impermeable boundary conditions on the top and bottom. The left open boundary, a constant freshwater influx (q f =6.6x10-5 m 2 /s, ρ f =1000 kg/ m 3, μ f =10-3 m 2 /s) The open right hand boundary, a hydrostatic pressure distribution (ρ s =1025 kg/ m 3, μ s =10-3 m 2 /s) 50 % saltwater saturation (S S ) is assumed 25 % of the vertical dimension of the aquifer is assumed to be freshwater mouth. The interface between the 2 fluids is sharp without any smearing Seawater saturation contours at 6000 seconds

110 Miscible and immiscible models y dimension in (m) Immiscible model for linear retention laws x dimension in (m) Mixing accelerates the front movement Immiscible-Miscible model for linear retention laws 110

111 Miscible and immiscible models Immiscible model for linear retention ti laws Mixing accelerates the front movement Immiscible-Miscible model for linear retention laws 111

112 With buoyancy Effect of freshwater flux Without buoyancy Effect of buoyancy Buoyancy in a multiphase model acts as a dispersive mechanism Freshwater flux reduced to half

113 Effect of relative permeability on the seawater-freshwater transition zone Brooks - Corey Linear relation Quadratic relation Cubic relation shows nearest resemblance to Brooks - Corey Cubic relation

114 Effect of relative permeability on the seawater-freshwater transition zone

115 3-Phase Immiscible Flow in Coastal Reservoir (Buoyancy Flow and Capillary Trapping) Seawater intrusion and oil migration in a coastal aquifer This study is done to demonstrate buoyancy flow and capillary trapping in immiscible flow

116 Conceptual Model The pore space is completely filled by all the three existing phases S w + Sn + Ss =1 P P = n P n P s P w P s P cnw = w P = P = P P cns cnw cns csw P csw Interface conditions Capillary equilibrium This leads to capillary trapping 1. Possibility of interface curvature: DNAPL most nonwetting among three 2. Possibility of capillary trapping (fingering) w - Freshwater (most wetting phase) n - DNAPL (most non-wetting phase) s- Seawater (intermediate phase)

117 P P cns cnw = P = P d d ( Ss + Sn Srn) ( 1.0 Srn Srw Sw + Sn Srw S (1.0 S rn S ) ( rn) ( rw ) Linear relation for capillary pressure (conditions for the interfaces) Non-linear relation (Brooks-Corey) for relative permeability of phases (conditions for the interfaces) k rs = k ( ) 4 rw = S ew ( S ) 3 es (2.0 Ses ) ( S ) 3 (1.0 S S ) k = + rn en es ew S ew = ( Sw Srw ) ( 1.0 S S ) S es = rw S rn S ( 1.0 S S ) S en = s rw S n rn S ( 1.0 S S ) rw rn rn

118 3-Phase buoyancy flow and capillary trapping y dimension in (m) Saltwater wedge moved seaward 2. Some local blobs (trapping) and fingering developed in three-phase region x dimension in (m) y dire ection in (m) Initial condition of saline aquifer x direction in (m)

119 Buoyancy flow and Capillary Trapping Local capillary trapping Gravity fingering Buoyancy flow -most dense fluid- downward, lateral and sliding movement Gravity fingering - once the residual DNAPL pools up on the top of the wedge

120 Study of miscible and non-isothermal flow models Fracture plane developed due to external stresses Idealized fracture plane P e =(2σcosθ)/e k = 2 e 12 δ n 1 = k n 2 2 [ σ cos β + σ β ] 1 3 sin Brady and Brown, 1993 Normal deformation components of a joint

121 Coupling Deformation with Fluid Pressures Assumption δ n 1 = 1 3 sin β k n [ σ β + σ P ] 2 2 cos Modified normal df deformation If P >P If P w >P n w If P w <P n δ 1 [ σ β + σ 2 2 cos β ] n = 1 β 3 sin β P n kn Coupling Deformation with Immiscible flow Model Aperture of fracture, e t e t = e 0 ± δ 121 n

122 Influence of deformation on phase distribution Deformation of the medium enhances the diffusion process Phase distribution is more diffusive

123 Influence of deformation on energy transfer Deformation of the medium enhances the diffusion process

124 Influence of deformation on mass transfer Deformation of the medium enhances the diffusion process Mass transfer diffuses very fast In the storage system this kind of diffusive process secures better storage

125 Immiscible flow with miscible, non-isothermal and medium deformation This test problem shows that immiscible flow system should be coupled with miscible and non-isothermal flow which can represent the complete multiphase system Also the medium deformation through hydromechanical coupling should be considered if the multiphase flow system has to beconsidered d in any geological environment, shallow or deep

126 Thank You