Numerical simulatio of two-phase Darcy-Forchheimer flow durig CO 2 ijectio ito deep salie aquifers Adi Zhag Feb. 4, 2013
Darcy flow VS o-darcy flow Darcy flow A liear relatioship betwee volumetric flow rate (Darcy velocity) ad pressure (or potetial) gradiet Domiat at low flow rates Φ = µ v k Φ is the flow potetial; μ is the viscosity; v is the Darcy velocity; k is the itrisic permeability No-Darcy flow Ay deviatios from the liear relatio may be defied as o-darcy flow Iterested i the oliear relatioship that accouts for the extra frictio or iertial effects at high pressure gradiets/ high velocity
No-Darcy flow equatios Forchheimer equatio µ βρ Φ = v+ v v k β is the o-darcy flow coefficiet, or Forchheimer coefficiet (Forchheimer, 1901) Baree ad Coway equatio Φ = k k d mr k d µ v (1 kmr ) µτ kmr + µτ + ρ v is absolute Darcy permeability; is the miimum permeability ratio at high flow rate; τ is the the characteristic legth (Baree ad Coway, 2004 ad 2007)
Darcy-Forchheimer flow Darcy-Forchheimer flow is defied as the flow icorporatig the trasitio betwee Darcy ad Forchheimer flows
Trasitio criteria The Reyolds umber (Type-I) applied maily i the cases where the represetative particle diameter is available Re ρdv = d is the diameter of particles µ The Forchheimer umber (Type-II) used maily i umerical models f ρkβv µ = cosistet defiitio physical meaig of the variables
Research objectives Develop a geeralized Darcy-Forchheimer model Propose a method to determie the critical Forchheimer umber for sigle ad multiphase flows Use the model ad method to aalyze the Darcyforchheimer flow i the ear well-bore area durig CO 2 ijectio ito DSA
Math model for two-phase flow ( θρ S ) t = ( ρ v ) + Q ; = w, θ : porosity; S : saturatio; Q : source ad sik. dp µ v = + βρvv ; = w, dx k k v r kr k 1 dp = - ( ( )) ; = w, µ 1+ f dx kk f r = βρ v ; = w, µ ( θρs) kr k 1 = ( ρ ( p)) + Q ; = w, t µ 1+ f
Costitutive equatios eeded S w + S = 1 Pc = P Pw P c : capillary pressure; P D : etry pressure; r S w : irreducible saturatio for water; S w : irreducible saturatio for o-wettig phase; λ : pore size distributio idex. Brooks-Corey equatios : P S = PS (1/ λ ) c D eff r = Sw Sw 1 S S eff r r w w k = ( ) r r S eff (2+ 3 λ)/ λ 2 (2+ 3 λ)/ λ k = (1 Seff ) (1 ( Seff ) )
The Forchheimer umber β = 0.005 0.005 β = = 0.005 ( k ) ( θ ) 0.5 5.5 0.5 0.5 ( kkr ) ( θ(1 Sw) ) ( kkr ) ( θs) 5.5 5.5 ( Geertsma, 1974) β = 2.923*10 ( k )( θ ) 6 τ τ Is tortuosity ( Liu et al., 1995) β β 6 2.923*10 τ = ; = w, ( Ahmadi et al., 2010) ( kkr )( θ ) Cβτ = ; = w, ( kkr )( θ S ) Evas ad Evas (1988) : a small mobile liquid saturatio, such as that occurrig i a gas well that also produces water, may icrease the o-darcy flow coefficiet by early a order of magitude over that of the dry case.
Determie the critical f Type I: based o Reyolds umber for sigle phase (Chilto et al., 1931) The poit whe the liear relatioship begis to deviate
Determie the critical f Type II: based o the Forchheimer umber for sigle phase (Gree et al., 1951) The poit whe the liear relatioship begis to deviate
Itersectio of two regressio lies dp 1 Darcy : = 2 dxβρv f dp 1 o Darcy : = +1 2 dxβρv f Darcy : dp = dxβ ρ 1 2 v f dp 1 o Darcy : = +1 dxβ ρ 2 v f β is ot defied i the Darcy formula!!! The frictio factor is defied as dp dxβρv 2
A example plot for 0.95 water saturatio Frictio factor 18 16 14 12 10 8 6 4 2 Forchheimer (data poit) Darcy (data poit) Forchheimer (regressio) Darcy (regressio) y = 0.8953X + 1.1712 R 2 = 0.9969 y = 1.138X 2 R = 0.9969 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1/fw The two lies itercept where 1/fw=4.825, so ( ) f is 1/4.825=07 w c Data from (Sobieski ad Trykozko, 2012)
Critical Forchheimer umber for H 2 O ad CO 2 at differet saturatio values Critical Forchheimer umber for water ((f w ) c ) 0.6 0.52 4 0.36 8 Water CO 2 30 26 22 18 14 10 6 Critical Forchheimer umber for CO 2 ((f ) c ) 0.5 0.6 0.7 0.8 0.9 1 2 Water saturatio (S w )
Numerical model Primary variables: P w ad S Fully-implicit scheme Discretizatio method: CVFD Cotrol volume fiite differece
Cotrol volume fiite differece For 1D case, the two-phase mass coservatio equatios ca be discretized ito t+ 1 t+ 1 t+ 1 t+ 1 t+ 1 t Q w θs + a( p 1 ) ( 1) wi+ pwi b pwi pwi = θs t ρw ( ( wi+ 1 wi ) ( wi wi 1) ) θ S c p p d p p t+ 1 t+ 1 t+ 1 t+ 1 t+ 1 t pc t+ 1 t+ 1 pc t+ 1 t+ 1 c( Si+ 1 Si ) d( Si Si 1) t Q = θ S + t S 1 S 1 i+ ρ i 2 2 t kk 1 t kk 1 a = = ( x) 1 + f ( x) 1+ f w w r r b 2 2 µ w w 1 µ w w 1 i+ i 2 2 t kk 1 t kk 1 c= = ( x) 1 + f ( x) 1+ f r r d 2 2 µ 1 µ 1 i+ i 2 2 t+ 1 t+ 1 t+ 1 t+ 1 t+ 1 t+ 1 x = S1, Si, Sm, pw 1 pwi, p wm AX = B T t
Numerical algorithm Iitializatio t < t_max YES Update coefficiets Solve X (X-X_o)/X > tol YES X_o = X Update coefficiets NO NO Stop Output X_o = X Update Variables mb check For each iteratio of each time step, X ad other related variables i the last time step are used to update all the coefficiets icludig a, b, c, d, dpc/ds, fw, f ad all the elemets i the right had side B; The right had side term B is based o the variables i the last time step ad do t eed to be updated except for the first iterative step; Solve X Output t = t+dt
Mass balace check I C = = m t t 1 iθ[ i i ] V S S i= 1 m t t t i li, i= 1 l Γ t ( Q + q ) m t 0 Viθ [ Si Si ] i= 1 t m j j tj Qi + qli, j= 1 i= 1 l Γ ( ) ; = w, ; = w, Γ is the boudaries of the domai m is the umber of the odes; t is the umber of time steps; Q is discharge for pumpig or ijectig wells; q is the flow rate through the boudaries; For Q ad q, they are set to be positive if eterig the domai while egative if leavig the domai.
Darcy-Forchheimer flow f ( f) ( ) 0, if f < Darcy flow c = f, if f > f Forchheimer flow c
Validatio: Buckley-Leverett problem with iertial effect Both fluids ad the porous medium are icompressible; Capillary pressure gradiet is egligible; Gravity effect is egligible; Semi-aalytical solutio with iertial effect (Wu, 2001; Ahmadi et al., 2010) No flow boudary No flow boudary Water floodig BC type I for P w BC type I for S x No flow boudary
Compariso of saturatio profiles 0.9 0.8 0.7 Sw distributio Aalytical: 3000 sec Aalytical: 7000 sec Aalytical: 10000 sec Numerical: 3000 sec Numerical: 7000 sec Numerical: 10000 sec Water saturatio 0.6 0.5 0.3 0 0.1 0.3 0.5 0.6 0.7 0.8 0.9 1 Distace(m)
Applicatio problem BC type I for P w BC type I for S Row 15 16 I 3.5 BC type I for P w BC type I for S I 2.5 16 Iterface 16.5 BC type I for P w BC type I for S z 1 Row 2 1 Row 1 1 I 1.5 14 15 15 Iterface 17.5 16 17 Iterface 18.5 16 17 18 31 x No flow boudary Ijectig CO 2
Properties of soil ad fluids Properties Values Commet Soil itrisic permeability porosity Soil 3e-9 m 2 0.37 Pore size distributio idex 3.86 Brook-Corey Water residual saturatio S wr = 0.35 No-wettig phase (NWP) residual saturatio S r = 0.05 Fluid Water desity 994 kg/m 3 NWP desity 479 kg/m 3 Water viscosity 7.43e-4 Pa s NWP viscosity 3.95 e-5 Pa s
Modelig parameters Properties Values Commet Boudary coditio Water pressure at x=0.5 m Water pressure at x=31.5 m Water pressure at z=0.5 m Water pressure at z=15.5 m CO 2 saturatio at x=0.5 m CO 2 saturatio at x=31.5 m CO 2 saturatio at z=0.5 m CO 2 saturatio at z=15.5 m CO 2 ijectig rate @ (16,1) Iitial coditio Water saturatio NWP saturatio Water pressure Space discretizatio Domai size, Legth Domai size, Depth Domai size, Width Space step size P w = 8 M Pa, BC Type I P w = 8 M Pa, BC Type I No flow boudary P w = 8 M Pa, BC Type I Left boudary Right boudary Bottom boudary Top boudary S = 0.1, BC Type I S = 0.1, BC Type I No flow boudary S = 0.1, BC Type I 1*10-3 m 3 /s Per meter ormal to the 2D domai S w = 0.9 S = 0.1 P w = 8 M Pa L=31 m W=15 m 1 m dx =dz=1 m Saturated with water iitially Time discretizatio Simulatio time Time step size T= 9000 s dt=1 s
CO 2 saturatio ad pressure profiles Darcy-Forchheimer flow
f x ad f z profiles with time Darcy-Forchheimer flow
The evolutio of importat variables i the first row S history 0.6 8.02 x P history 106 0.5 8.018 S 0.3 Node 16 Node 17 Node 18 0.1 0 2000 4000 6000 8000 10000 t(s) P 8.016 8.014 Node 16 8.012 Node 17 Node 18 8.01 0 2000 4000 6000 8000 10000 t(s) fx 35 30 25 20 15 fx history vx 3.5 x vx history 10-3 3 2.5 2 1.5 10 Iterface 16.5 5 Iterface 17.5 Iterface 18.5 0 0 2000 4000 6000 8000 10000 t(s) Darcy-Forchheimer flow 1 Iterface 16.5 0.5 Iterface 17.5 Iterface 18.5 0 0 2000 4000 6000 8000 10000 t(s)
CO 2 saturatio ad pressure profiles Darcy flow
Compariso of the evolutio of importat variables 0.6 0.5 S history 8.02 x P history 106 D:N16 D:N17 8.018 F:N16 F:N17 8.016 S 0.3 D:N16 D:N17 F:N16 F:N17 P 8.014 8.012 0.1 0 2000 4000 6000 8000 10000 t(s) 8.01 0 2000 4000 6000 8000 10000 t(s) 1.8 x Pc history 104 D:N16 D:N17 1.6 F:N16 F:N17 0.03 0.025 0.02 D:I16.5 D:I17.5 F:I16.5 F:I17.5 vx history Pc 1.4 vx 0.015 1.2 0.01 0.005 1 0 2000 4000 6000 8000 10000 t(s) 0 0 2000 4000 6000 8000 10000 t(s)
CO 2 saturatio cotour for Darcy- Forchheimer flow S distributio 5 4.5 4 3.5 z(m) 3 2.5 2 1.5 1 11 12 13 14 15 16 17 18 19 20 21 x(m) Time 1000 sec (solid lie), 3000 sec (dot lie), 5000 sec (dash lie), 7000 sec (bold solid lie), 9000 sec (dash dot lie)
CO 2 saturatio cotour for Darcy flow 5 S distributio 4.5 4 3.5 z(m) 3 2.5 2 1.5 1 11 12 13 14 15 16 17 18 19 20 21 x(m) Time 1000 sec (solid lie), 3000 sec (dot lie), 5000 sec (dash lie), 7000 sec (bold solid lie), 9000 sec (dash dot lie)
Overlappig them together 5 S distributio 4.5 4 3.5 z(m) 3 2.5 2 1.5 1 11 12 13 14 15 16 17 18 19 20 21 x(m) Time 1000 sec (solid lie), 3000 sec (dot lie), 5000 sec (dash lie), 7000 sec (bold solid lie), 9000 sec (dash dot lie); Darcy i red; Darcy-Forchheimer i black
Match S with Forchheimer flow regime 5 S ad fz distributio @1000 sec 5 S ad fz distributio @3000 sec 4 4 z(m) 3 z(m) 3 2 2 1 12 14 16 18 20 x(m) 1 12 14 16 18 20 x(m) 5 S ad fz distributio @5000 sec 5 S ad fz distributio @7000 sec 4 4 z(m) 3 z(m) 3 2 2 1 12 14 16 18 20 x(m) 12 14 16 18 20 x(m) The white cotour lies are for saturatio cotour lies while the rectagles demostrate whether a ode is of Forchheimer (grey rectagle) or Darcy (white rectagle) flow 1
Compariso of CO 2 pressure betwee Forchheimer-Darcy ad Darcy flow P differece (Forchheimer-Darcy) 5000 4000 3000 2000 1000 P history for ode(16,1) 0 0 5000 10000 t(s) P ratio (Forchheimer/Darcy) 1.0006 1.0005 1.0004 1.0003 1.0002 1.0001 P history for ode(16,1) 1 0 5000 10000 t(s)
Higher displacemet efficiecy I the Forchheimer regime for Darcy-Firchheimer flow, the total CO 2 saturatio is 4.5916 for the ie odes at 9100 sec. For Darcy flow, the total CO 2 saturatio is 3.4072 for the same ie odes at 9100 sec. The displacemet is 34.76% higher for Forchheimer flow tha Darcy flow at 9100 sec.
Implicatios Importat to icorporate Forchheimer effect ito the umerical simulatio of multiphase flow Crucial to determie the critical Forchheimer umber ad to decide the extet to which Forchheimer effect ca ifluece the trasport of CO 2 i deep salie aquifers. The higher displacemet efficiecy by CO 2 is good ews for CO 2 sequestratio ito deep salie aquifers. The higher ijectio pressure required i Forchheimer flow is bad ews for CO 2 sequestratio.
Summary & coclusios Darcy flow is a special case of a geeralized Darcy- Forchheimer flow; Sice both the Forchheimer coefficiet ad umber are fuctios of saturatio, there is a critical Forchheimer umber for trasitio for a specific saturatio for each phase i multiphase flow; The good agreemet betwee the umerical solutio ad the semi-aalytical solutio validates the umerical tool developed i this study
Summary & coclusios The Forchheimer flow ca improve the displacemet efficiecy ad ca icrease the storage capacity for the same ijectio rate ad volume of site. The higher ijectio pressure required i Forchheimer flow is bad ews for CO 2 sequestratio because the pressure will cotiue to icrease ad might eve exceed the litho-static stress ad the risk for fracturig the porous media would icrease.
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