University:
Boston University
Course:
ENG ME 303 | Fluid Mechanics
Academic year:

2021
Views:

232

Pages:

28
Author:

Devyn Parsons

Math, Num and Comp Models Two–Phase Immiscible Flow Table of contents 1 Math, Num and Comp Models Processes to be modeled Reservoir Properties One–Phase Flow Two–Phase Immiscible Flow Black–Oil Compositional Oil Solution Schemes 2 References References Math, Num and Comp Models References Two–Phase Immiscible Flow The following assumptions are usually adopted: We consider two–phase flow where the fluids are immiscible (o, w). There is no mass transfer between the phases. One phase (w) wets porous medium more than the other (o). The two fluids jointly fill the voids: Sw + So = 1. The pressure in the wetting fluid is less than that in the non-wetting fluid. The pressure difference is given by the capillary pressure: pc = po − pw . And pc = pc (Sw ). There is no diffusion ~τ = 0. Z Extensive properties: Mα (t) = φρα Sα d~x, α = w, o. B(t) Intensive properties: ψ = φρα Sα , α = w, o. Math, Num and Comp Models References Two–Phase Immiscible Flow Balance equations: ∂(φρα Sα ) + ∇ · (φρα Sα~vα ) = qα for α = w, o. ∂t Darcy’s Law: ~uα = − krα k(∇pα − ρα G∇D) where ~uα = φSα~vα µα Recall that: Sw + So = 1 and pc = po − pw . We have six equations for six unknowns: ρα , ~uα and Sα , for α = w, o. Math, Num and Comp Models References Two–Phase Immiscible Flow Alternative differential equations: Formulation in phase pressures Assume: Sw = p−1 c (po − pw ). We use pw and po as the main unknowns: ρw ∂(φρw p−1 c ) ∇· krw k(∇pw − ρw G∇D) = − qw µw ∂t ∇· ρo kro k(∇po − ρo G∇D) µo = ∂(φρo (1 − p−1 c )) − qo ∂t This system is commonly employed in the simultaneos solution (SS) scheme. Math, Num and Comp Models References Two–Phase Immiscible Flow Formulation in phase pressure and saturation We use po and Sw as the main variables: ρw dpc ∂(φρw Sw ) ∇· krw k ∇po − ∇Sw − ρw G∇D = − qw µw dSw ∂t 1 ∇· ρw ρw dpc + krw k ∇po − ∇Sw − ρw G∇D µw dSw 1 ρo ∇· kro k (∇po − ρo G∇D) = ρo µo Sw ∂(φρw ) 1 − Sw ∂(φρo ) qw qw + − − ρw ∂t ρo ∂t ρw ρw Saturation Sw is explicitly evaluated using the first equation. The second equation can be solved for po implicitly. This is the Implicit Pressure Explicit Saturation (IMPES) scheme. Math, Num and Comp Models References Two–Phase Immiscible Flow Example: Pressure-Saturation Formulation I Phases: α = water(w) and oil(o). Phase mobility functions: λα = krα /µα P Total mobility: λ = λα P Fractional flow functions: fα = λα /λ, fα = 1 P Total velocity: ~u = ~uα po ≡ p, Sw ≡ S, pcow ≡ pc (S) The capillary pressure gradient and permeability are expressed as:  ∇pc = dpc ∇S dS No gravity: G = 0. and k11 k≡ 0 0 0 k22 0   0 k 0 = 0 k33 0 0 k 0  0 0  k Math, Num and Comp Models References Two–Phase Immiscible Flow Example: Pressure-Saturation Formulation II Incompressible rock and fluids: ρα = cte and φ = cte. Pressure equation (elliptic) dpc ∇ · −kλ∇p + kλw ∇S = qw + qo dS | {z } flux function Saturation equation (parabolic, hiperbolic for pc = 0) dpc ∂S φ + ∇ · kλw ∇S − kλw ∇p = qw ∂t dS | {z } flux function Math, Num and Comp Models References Two–Phase Immiscible Flow FVM: Pressure eq. 1D I i+ 12 Z ∇ · (−kλ∇p + kFw ∇S) dx = 0 i− 12 Fw = λw dpc dS ! dp dp dS dS − kλ − kλ + kFw − kFw dx i+ 1 dx i− 1 dx i+ 1 dx i− 1 = 0, ai pi − ai+1 pi+1 − ai−1 pi−1 − a∗i Si + a∗i+1 Si+1 + a∗i−1 Si−1 = 0. 2 2 2 2 Math, Num and Comp Models References Two–Phase Immiscible Flow FVM: Pressure eq. 1D II ai+1 = a∗i+1 = k k (λ) 1 ; ai−1 = (λ) 1 ; ai = ai+1 + ai−1 . ∆x i+ 2 ∆x i− 2 k k (Fw )i+ 12 ; a∗i−1 = (Fw )i− 21 ; a∗i = a∗i+1 + a∗i−1 . ∆x ∆x Fw = λw dpc dS Math, Num and Comp Models References Two–Phase Immiscible Flow FVM: Saturation eq. 1D I i+ 12 n+1 Z Z 1 ∂S φ dtdx − ∂t i− 12 n φ n+1 Z i+ Z 2 dpc ∇ · kλw ∇p − kλw ∇Sw dxdt = dSw 0, n i− 1 2 Sin+1 − Sin n n ! ∂p ∂p − kλw ∆t ∆x − kλw ∂x i+ 1 ∂x i− 1 2 2 n n ! ∂S ∂S kFw − kFw ∆t ∂x i+ 1 ∂x i− 1 2 + = 0 2 n n Sin+1 = Sin + b∗i Sin − b∗i+1 Si+1 − b∗i−1 Si−1 − bi pni + bi+1 pni+1 + bi−1 pni−1 . Math, Num and Comp Models References Two–Phase Immiscible Flow FVM: Saturation eq. 1D II bi+1 = k∆t k∆t (λw )ni+ 1 ; bi−1 = (λw )ni− 1 ; bi = bi+1 + bi−1 ; . 2 2 φ∆x2 φ∆x2 b∗i+1 = k∆t k∆t (Fw )ni+ 1 ; b∗i−1 = (Fw )ni− 1 ; b∗i = b∗i+1 + b∗i−1 . 2 2 φ∆x2 φ∆x2 Fw = λw dpc dS Math, Num and Comp Models References Two–Phase Immiscible Flow FVM: IMPES The coefficients are not constant and depend on λ, λw and Fw : (λ)n , i± 1 2 (λw )n , i± 1 2 Fw = λw (Fw )n i± 1 2 dpc dS IMPES Algorithm 1: while (t < Tmax ) do 2: Calc. coeff. of pressure equation. 3: Solve the pressure equation implicitly. 4: Calc. coeff. of saturation equation. 5: Solve the saturation eq. explicitly. 6: t ← t + ∆t 7: end while while ( t <= Tmax) { pressure . calcCoefficients ( ) ; S o l v e r : : TDMA1D( p r e s s u r e ) ; p r e s s u r e . update ( ) ; saturation . calcCoefficients ( ) ; Solver : : solExplicit ( saturation ) ; s a t u r a t i o n . update ( ) ; t += dt ; } Math, Num and Comp Models References Two–Phase Immiscible Flow Inheritance I General Equation: Conservative form: ∂ ∂t Z Z ψd~ x+ B(t) ∇ · f~d~ x= B(t) Z qd~ x B(t) Discrete general equation: an+1 ψPn+1 = P n+1 n+1 ae ψe + an+1 ψwn+1 + an+1 ψnn+1 + an+1 ψsn+1 + an+1 ψfn+1 + an+1 ψbn+1 + qPn w n s f b Two phase immiscible and incompressible fluids: Pressure and saturation equations are derived from general equation: ∇· φ −kλ∇p + kλw ∂S +∇· ∂t dpc ∇S dS = qw + qo dpc kλw ∇S − kλw ∇p = qw dS Math, Num and Comp Models References Two–Phase Immiscible Flow Inheritance II Inheritance is a way to share and reuse code by defining collections of attributes and behaviors, bundled into classes (subclasses), based on previously created classes (superclasses). Inheritance gives rise to hierarchies: complexity is reduced, but efficiency can be spoiled. Math, Num and Comp Models References Two–Phase Immiscible Flow Object declaration Pressure TwoPhaseEquation< BLIP1 > p r e s s u r e ( p , A, b , mesh . g e t D e l t a s ( ) ) ; p r e s s u r e . setDeltaTime ( dt ) ; pressure . setPermeability ( permeability ) ; pressure . setPorosity ( porosity ); p r e s s u r e . s e t S r w ( Srw ) ; p r e s s u r e . s e t S r o ( Sro ) ; p r e s s u r e . s e t V i s c o s i t y w ( mu w ) ; p r e s s u r e . s e t V i s c o s i t y o ( mu o ) ; p r e s s u r e . setNeumann (LEFT WALL, i n j e c t i o n ) ; p r e s s u r e . s e t D i r i c h l e t (RIGHT WALL ) ; p r e s s u r e . s e t S a t u r a t i o n ( Sw ) ; pressure . print ( ) ; Saturation TwoPhaseEquation< BLES1 > s a t u r a t i o n ( Sw , A, b , mesh . g e t D e l t a s ( ) ) ; s a t u r a t i o n . setDeltaTime ( dt ) ; saturation . setPermeability ( permeability ) ; saturation . setPorosity ( porosity ); s a t u r a t i o n . s e t S r w ( Srw ) ; s a t u r a t i o n . s e t S r o ( Sro ) ; s a t u r a t i o n . s e t V i s c o s i t y w ( mu w ) ; s a t u r a t i o n . applyBounds ( 1 , Sw . ubound ( f i r s t D i m ) − 1 ) ; saturation . setPressure (p ) ; saturation . print ( ) ; Math, Num and Comp Models References Two–Phase Immiscible Flow Test 1: Buckley–Leverett (in collaboration with M. Diaz [7]) From Diaz et al. [7] Property Length k φ µw µo Srw Sro gpin pout Value 300 m 1.0E-15 m2 0.2 1.0E-03 Pa.s 1.0E-03 Pa.s 0 0.2 3.4722E-07 m.s−1 1E+07 Pa Zero capillary pressure Pressure eq. −∇ · kλ∇p = 0 Saturation eq. φ ∂S − ∇ · kλw ∇p = 0. ∂t Math, Num and Comp Models References Two–Phase Immiscible Flow Test 1: Buckley–Leverett I Constitutive Eqs. krw = Seω ; kro = (1 − Se )ω , ω = 1, 2 Se = S − Srw 1 − Srw − Sro Coefficients: (λ)i± 1 = 2 1 Si± 1 − Srw )ω (1 − Srw − Sro )ω µw 2 (λw )n i± 1 = 2 1 µw + (1 − Sro − Si± 1 )ω n Si± 1 − Srw 2 1 − Srw − Sro 2 µo !ω ! Math, Num and Comp Models References Two–Phase Immiscible Flow Test 1: Buckley–Leverett II Upwind scheme for Si± 1 . 2 n if ( pn i+1 >= pi ) then Si+ 1 = Si+1 2 else Si+ 1 = Si 2 end if Math, Num and Comp Models References Two–Phase Immiscible Flow Test 1: Buckley–Leverett III Adaptors: BLIP1 & BLES1 template class BLIP1 : public TwoPhaseEquation > { public: inline void calcCoefficients1D(); inline void calcCoefficients2D(); inline void calcCoefficients3D(); }; template inline void BLIP1::calcCoefficients1D () { static prec_t Sw_e, Sw_w; static prec_t mult_o = k / ( (1 - Srw - Sro) * mu_o static prec_t mult_w = k / ( (1 - Srw - Sro) * mu_w aE = 0.0; aW = 0.0; aP = 0.0; sp = 0.0; for (int i = bi; i <= ei; ++i) { if ( phi_0(i+1) >= phi_0(i) ) Sw_e = S(i+1); else Sw_e = S(i); if ( phi_0(i-1) >= phi_0(i) ) Sw_w = S(i-1); else Sw_w = S(i); aE (i) = (1 - Sro - Sw_e) * mult_o + (Sw_e - Srw) aW (i) = (1 - Sro - Sw_w) * mult_o + (Sw_w - Srw) aP (i) = aE (i) + aW (i); } applyBoundaryConditions1D(); } * dx ) ; * dx ) ; * mult_w ; * mult_w ; Math, Num and Comp Models References Two–Phase Immiscible Flow Test 1: Buckley–Leverett IV template class BLES1 : public TwoPhaseEquation > { public: inline void calcCoefficients1D(); inline void calcCoefficients2D(); inline void calcCoefficients3D(); }; template inline void BLES1::calcCoefficients1D () { static prec_t Sw_e, Sw_w; static prec_t mult = k * dt / ( porosity * dx * dx * (1 - Srw - Sro) * mu_w ); aE = 0.0; aW = 0.0; aP = 0.0; sp = 0.0; for (int i = bi; i <= ei; ++i) if ( phi_0(i+1) >= phi_0(i) ) else if ( phi_0(i-1) >= phi_0(i) ) else aE (i) = (Sw_e - Srw) * mult; aW (i) = (Sw_w - Srw) * mult; aP (i) = aE (i) + aW (i); } { Sw_e Sw_e Sw_w Sw_w = = = = phi_0(i+1); phi_0(i); phi_0(i-1); phi_0(i); Math, Num and Comp Models References Two–Phase Immiscible Flow Test 1: Buckley–Leverett V // Dirichlet right side aP(ei) += aE(ei); sp(ei) = 2 * aE(ei) * p(ei+1); aE(ei) = 0; // Neumann left side aP(bi) -= aW(bi) ; sp(bi) = aW(bi) * dx * ( 3.47e-7 * mu_w / k) ; aW(bi) = 0; } 2D rectangular mesh: 300 × 10 Time steps = 1100 days and ∆t = 60 secs. (1,584,000 steps) Cases: 1 2 3 4 ω ω ω ω =1 =1 =1 =2 and and and and µw /µo µw /µo µw /µo µw /µo = 1 (lineal01) = 2 (lineal02) = 2/3 (lineal2 3) = 2/3 (quadratic2 3) Math, Num and Comp Models References Two–Phase Immiscible Flow Improved IMPES As described in Chen et al. [8] the improved IMPES consists in to take a bigger time step for the calculation of pressure. ∆tp /∆tS 1 5 10 SPp 1.0 4.7 9.6 SPT 1.00 2.50 3.19 Math, Num and Comp Models Two–Phase Immiscible Flow Overlapping method I References Math, Num and Comp Models References Two–Phase Immiscible Flow Overlapping method II CartComm<2> cart(argc, argv, rank); Isub = cart.get_I(); Jsub = cart.get_J(); num_subdom_x = cart.getNumProc_I(); num_subdom_y = cart.getNumProc_J(); // MPI::COMM_WORLD.Create_cart() SubDomain subdom(cart); double double double double ovlp_l ovlp_r ovlp_d ovlp_u = = = = subdom.createOverlap(LEFT, subdom.createOverlap(RIGHT, subdom.createOverlap(DOWN, subdom.createOverlap(UP, while (t <= Tmax) { // IMPES loop //... subdom.infoExchange1(p); pressure.updatePhi(p); subdom.infoExchange1(Sw); saturation.updatePhi(Sw); } nc_ovlp_l, nc_ovlp_r, nc_ovlp_d, nc_ovlp_u, dx); dx); dy); dy); Math, Num and Comp Models References Two–Phase Immiscible Flow Overlapping method III Results on a Quadcore computer. Procs. 1 2 4 (1) 4 (2) 4 (3) Subdom 1 2×1 4×1 1×4 2×2 CPU timep 614.51 351.45 165.61 322.26 218.45 CPU timeS 170.09 155.32 42.79 76.07 57.48 Speed up – 1.55 3.76 1.97 2.84 Eff. – 0.75 0.94 0.49 0.71 Math, Num and Comp Models References Two–Phase Immiscible Flow Multipliers–Free DDM Non–overlapping DDM : Unified theory of multipliers–free DDM methods, Herrera and Yates [9, 10] Elliptic Operator – Subdomains = 32 × 32 = 10241 Nx × Ny 50 × 50 100 × 100 150 × 150 2,560,000 10,240,000 23,040,000 Proc. K2 P3 K P K P 16 (64) 204 389 2303 2670 – 9158 32 (32) 172 401 1274 1641 5937 5178 64 (16) 133 436 745 1234 4326 3647 128 (8) 119 – 568 – 2818 – 1 Recently results from A. Carrillo and R. Yates K ≡ KanBalam 1360 AMD Opteron 2.6 GHz, 8 GB per 4 cores, Net 10 Gbps 3 P ≡ Pohualli 100 Intel Xeon 2.33 GHz, 32 GB per 8 cores, Net 1 Gbps 2 Math, Num and Comp Models Two–Phase Immiscible Flow Multipliers–Free DDM References Math, Num and Comp Models Two–Phase Immiscible Flow Test 2: Five–spots Classical five-spots pattern. Four water injection wells: a well in each corner of the domain. One oil production well: in the center of the domain. Only one quarter of the domain is simulated. No flux on the boundaries. References

Computational Methods for Oil Recovery Two-Phase Immiscible Flow

Related Documents

Recommended Documents

Description

Related Documents

Free up your schedule!

Take 5 seconds to unlock