Numerical simulation of two-phase Darcy-Forchheimer flow during CO 2 injection into deep saline aquifers. Andi Zhang Feb. 4, 2013
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1 Numerical simulatio of two-phase Darcy-Forchheimer flow durig CO 2 ijectio ito deep salie aquifers Adi Zhag Feb. 4, 2013
2 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
3 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)
4 Darcy-Forchheimer flow Darcy-Forchheimer flow is defied as the flow icorporatig the trasitio betwee Darcy ad Forchheimer flows
5 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
6 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
7 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
8 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 ) )
9 The Forchheimer umber β = β = = ( k ) ( θ ) ( kkr ) ( θ(1 Sw) ) ( kkr ) ( θs) ( Geertsma, 1974) β = 2.923*10 ( k )( θ ) 6 τ τ Is tortuosity ( Liu et al., 1995) β β *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.
10 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
11 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
12 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
13 A example plot for 0.95 water saturatio Frictio factor Forchheimer (data poit) Darcy (data poit) Forchheimer (regressio) Darcy (regressio) y = X R 2 = y = 1.138X 2 R = /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)
14 Critical Forchheimer umber for H 2 O ad CO 2 at differet saturatio values Critical Forchheimer umber for water ((f w ) c ) Water CO Critical Forchheimer umber for CO 2 ((f ) c ) Water saturatio (S w )
15 Numerical model Primary variables: P w ad S Fully-implicit scheme Discretizatio method: CVFD Cotrol volume fiite differece
16 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
17 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
18 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.
19 Darcy-Forchheimer flow f ( f) ( ) 0, if f < Darcy flow c = f, if f > f Forchheimer flow c
20 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
21 Compariso of saturatio profiles Sw distributio Aalytical: 3000 sec Aalytical: 7000 sec Aalytical: sec Numerical: 3000 sec Numerical: 7000 sec Numerical: sec Water saturatio Distace(m)
22 Applicatio problem BC type I for P w BC type I for S Row I 3.5 BC type I for P w BC type I for S I Iterface 16.5 BC type I for P w BC type I for S z 1 Row 2 1 Row 1 1 I Iterface Iterface x No flow boudary Ijectig CO 2
23 Properties of soil ad fluids Properties Values Commet Soil itrisic permeability porosity Soil 3e-9 m 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
24 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 (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
25 CO 2 saturatio ad pressure profiles Darcy-Forchheimer flow
26 f x ad f z profiles with time Darcy-Forchheimer flow
27 The evolutio of importat variables i the first row S history x P history S 0.3 Node 16 Node 17 Node t(s) P Node Node 17 Node t(s) fx fx history vx 3.5 x vx history Iterface Iterface 17.5 Iterface t(s) Darcy-Forchheimer flow 1 Iterface Iterface 17.5 Iterface t(s)
28 CO 2 saturatio ad pressure profiles Darcy flow
29 Compariso of the evolutio of importat variables S history 8.02 x P history 106 D:N16 D:N F:N16 F:N S 0.3 D:N16 D:N17 F:N16 F:N17 P t(s) t(s) 1.8 x Pc history 104 D:N16 D:N F:N16 F:N D:I16.5 D:I17.5 F:I16.5 F:I17.5 vx history Pc 1.4 vx t(s) t(s)
30 CO 2 saturatio cotour for Darcy- Forchheimer flow S distributio z(m) 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)
31 CO 2 saturatio cotour for Darcy flow 5 S distributio z(m) 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)
32 Overlappig them together 5 S distributio z(m) 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
33 Match S with Forchheimer flow regime 5 S ad fz sec 5 S ad fz sec 4 4 z(m) 3 z(m) x(m) x(m) 5 S ad fz sec 5 S ad fz sec 4 4 z(m) 3 z(m) x(m) 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
34 Compariso of CO 2 pressure betwee Forchheimer-Darcy ad Darcy flow P differece (Forchheimer-Darcy) P history for ode(16,1) t(s) P ratio (Forchheimer/Darcy) P history for ode(16,1) t(s)
35 Higher displacemet efficiecy I the Forchheimer regime for Darcy-Firchheimer flow, the total CO 2 saturatio is for the ie odes at 9100 sec. For Darcy flow, the total CO 2 saturatio is for the same ie odes at 9100 sec. The displacemet is 34.76% higher for Forchheimer flow tha Darcy flow at 9100 sec.
36 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.
37 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
38 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.
39 Thak you!
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