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A New Formulation of Immiscible Compressible Two-Phase Flow in Porous Media Via the Concept of Global Pressure

Brahim Amaziane1 Mladen Jurak2

1Universit´

e de Pau, Math´ ematiques, LMA CNRS-UMR 5142, France

2Department of Mathematics, University of Zagreb, Croatia

Scaling Up and Modeling for Transport and Flow in Porous Media

Brahim Amaziane, Mladen Jurak () A New Formulation . . . Dubrovnik, October 2008 1 / 24

Outline

Two-phase immiscible, compressible flow equations Fractional flow formulation New global formulation Simplified global formulation Numerical comparison of the coefficients Conclusion

Brahim Amaziane, Mladen Jurak () A New Formulation . . . Dubrovnik, October 2008 2 / 24

Two-phase immiscible, compressible flow equations

Compressible two-phase flow

We consider two-phase isothermal, compressible, immiscible flow through heterogeneous porous medium. For example, water and gas. Assumptions:

◮ Incompressible fluid: water, ρw = const. ◮ Compressible fluid: gas, ρg = cgpg. ◮ Viscosities µw and µg are constant. ◮ No mass exchange between the phases; ◮ The temperature is constant;

Note that the assumptions on the form of mass densities are not essential. Independent variables: water saturation Sw and gas pressure pg (pw = pg − pc(Sw), Sg = 1 − Sw).

Brahim Amaziane, Mladen Jurak () A New Formulation . . . Dubrovnik, October 2008 3 / 24

Two-phase immiscible, compressible flow equations

Flow equations

Mass conservation: for α ∈ {w, g}, Φ ∂ ∂t (ραSα) + div(ραqα) = 0, The Darcy-Muscat law: for α ∈ {w, g}, qα = −Kkrα(Sα) µα (∇pα − ραg), Capillary law: pc(Sw) = pg − pw, No void space. Sw + Sg = 1.

Brahim Amaziane, Mladen Jurak () A New Formulation . . . Dubrovnik, October 2008 4 / 24

Fractional flow formulation

Fractional flow formulation

Goal:

◮ Reformulate flow equations in a form giving less tight coupling

between the two differential equations, allowing a sort of IMPES (implicit in pressure and explicit in saturation) numerical treatment. There are two approaches:

• 1. Introduce total velocity: Qt = qw + qg: leads to non-conservative

form of the equations

• 2. Introduce total flow: Qt = ρwqw + ρgqg: leads to conservative form
• f the equations.

We work with total flow.

Brahim Amaziane, Mladen Jurak () A New Formulation . . . Dubrovnik, October 2008 5 / 24

Fractional flow formulation

Fractional flow formulation: equations

Total flow: Qt = −λ(Sw, pg)K (∇pg − fw(Sw, pg)∇pc(Sw) − ¯ ρ(Sw, pg)g) , Total mass conservation: Φ ∂ ∂t (Swρw + (1 − Sw)ρg(pg)) + div (Qt) = 0, Water mass conservation: Φρw ∂Sw ∂t + div(fw(Sw, pg)Qt + Kgbg(Sw, pg)) = div(Ka(Sw, pg)∇Sw).

Brahim Amaziane, Mladen Jurak () A New Formulation . . . Dubrovnik, October 2008 6 / 24

Fractional flow formulation

Fractional flow formulation: coefficients

phase mobilities λw(Sw) = krw(Sw) µw , λg(Sw) = krg(Sw) µg , total mobility λ(Sw, pg) = ρwλw(Sw) + ρg(pg)λg(Sw), water fractional flow fw(Sw, pg) = ρwλw(Sw) λ(Sw, pg) , mean density ¯ ρ(Sw, pg) = λw(Sw)ρ2

w + λg(Sw)ρg(pg)2

λ(Sw, pg) , ”gravity” coeff. bg(Sw, pg) = ρwρg(pg)λw(Sw)λg(Sw) λ(Sw, pg) (ρw − ρg(pg)), ”diffusivity” coeff. a(Sw, pg) = −ρwρg(pg)λw(Sw)λg(Sw) λ(Sw, pg) p′

c(Sw).

Brahim Amaziane, Mladen Jurak () A New Formulation . . . Dubrovnik, October 2008 7 / 24

Fractional flow formulation

Decoupling of the system

In total flow we want to eliminate saturation gradient: Qt = −λ(Sw, pg)K

• ∇pg − fw(Sw, pg)p′

c(Sw)∇Sw − ¯

ρ(Sw, pg)g

• ,

◮ Idea: introduce a new pressure-like variable that will eliminate ∇Sw

term (Chavent-Jaffr´

ee: Mathematical Models and Finite Elements for Reservoir Simulation)

◮ Find a new pressure variable p, called global pressure, and a function

ω(Sw, p) such that: ∇pg − fw(Sw, pg)p′

c(Sw)∇Sw = ω(Sw, p)∇p

(1)

Brahim Amaziane, Mladen Jurak () A New Formulation . . . Dubrovnik, October 2008 8 / 24

Fractional flow formulation

Decoupling of the system

In total flow we want to eliminate saturation gradient: Qt = −λ(Sw, pg)K

• ∇pg − fw(Sw, pg)p′

c(Sw)∇Sw − ¯

ρ(Sw, pg)g

• ,

◮ Idea: introduce a new pressure-like variable that will eliminate ∇Sw

term (Chavent-Jaffr´

ee: Mathematical Models and Finite Elements for Reservoir Simulation)

◮ Find a new pressure variable p, called global pressure, and a function

ω(Sw, p) such that: ∇pg − fw(Sw, pg)p′

c(Sw)∇Sw = ω(Sw, p)∇p

(1)

Brahim Amaziane, Mladen Jurak () A New Formulation . . . Dubrovnik, October 2008 8 / 24

New global formulation

New global formulation

We introduce unknown function π such that pg = π(Sw, p), where p is global pressure to be defined. Then (1) reads ∇pg = ω(Sw, p)∇p + fw(Sw, π(Sw, p))p′

c(Sw)∇Sw,

• r,

∂π ∂Sw (Sw, p)∇Sw + ∂π ∂p (Sw, p)∇p = ω(Sw, p)∇p + fw(Sw, π(Sw, p))p′

c(Sw)∇Sw.

Brahim Amaziane, Mladen Jurak () A New Formulation . . . Dubrovnik, October 2008 9 / 24

New global formulation

New global formulation

We introduce unknown function π such that pg = π(Sw, p), where p is global pressure to be defined. Then (1) reads ∇pg = ω(Sw, p)∇p + fw(Sw, π(Sw, p))p′

c(Sw)∇Sw,

• r,

∂π ∂Sw (Sw, p)∇Sw + ∂π ∂p (Sw, p)∇p = ω(Sw, p)∇p + fw(Sw, π(Sw, p))p′

c(Sw)∇Sw.

Brahim Amaziane, Mladen Jurak () A New Formulation . . . Dubrovnik, October 2008 9 / 24

New global formulation

New global formulation

Since p and Sw are independent variables we must have: ∂π ∂Sw (Sw, p) = fw(Sw, π(Sw, p))p′

c(Sw)

(2) ∂π ∂p (Sw, p) = ω(Sw, p). (3) Conclusion:

• 1. To calculate π(Sw, p) solve the Cauchy problem:

     dπ(S, p) dS = ρwλw(S)p′

c(S)

ρwλw(S) + cgλg(S)π(S, p), 0 < S < 1 π(1, p) = p.

• 2. Get ω(Sw, p) from (3).

Brahim Amaziane, Mladen Jurak () A New Formulation . . . Dubrovnik, October 2008 10 / 24

New global formulation

New global formulation: Remarks

• 1. It is more natural to replace saturation Sw with capillary pressure

u = pc(Sw) as an independent variable. Then:      dˆ π(u, p) du = ρw ˆ λw(u) ρw ˆ λw(u) + cg ˆ λg(u)ˆ π(u, p) , u > 0 ˆ π(0, p) = p. and π(Sw, p) = ˆ π(pc(Sw), p) [Hat denotes the change of variables.]

• 2. ω is strictly positive and less than 1:

ω(Sw, p) = exp

• −

pc(Sw) cgρw ˆ λw(u)ˆ λg(u) (ρw ˆ λw(u) + cg ˆ λg(u)ˆ π(u, p))2 du

• ,
• 3. p ≤ π(Sw, p) ≤ p + pc(Sw) and therefore pw ≤ p ≤ pg.

Brahim Amaziane, Mladen Jurak () A New Formulation . . . Dubrovnik, October 2008 11 / 24

New global formulation

New global formulation: Remarks

• 1. It is more natural to replace saturation Sw with capillary pressure

u = pc(Sw) as an independent variable. Then:      dˆ π(u, p) du = ρw ˆ λw(u) ρw ˆ λw(u) + cg ˆ λg(u)ˆ π(u, p) , u > 0 ˆ π(0, p) = p. and π(Sw, p) = ˆ π(pc(Sw), p) [Hat denotes the change of variables.]

• 2. ω is strictly positive and less than 1:

ω(Sw, p) = exp

• −

pc(Sw) cgρw ˆ λw(u)ˆ λg(u) (ρw ˆ λw(u) + cg ˆ λg(u)ˆ π(u, p))2 du

• ,
• 3. p ≤ π(Sw, p) ≤ p + pc(Sw) and therefore pw ≤ p ≤ pg.

Brahim Amaziane, Mladen Jurak () A New Formulation . . . Dubrovnik, October 2008 11 / 24

New global formulation

New global formulation: Remarks

• 1. It is more natural to replace saturation Sw with capillary pressure

u = pc(Sw) as an independent variable. Then:      dˆ π(u, p) du = ρw ˆ λw(u) ρw ˆ λw(u) + cg ˆ λg(u)ˆ π(u, p) , u > 0 ˆ π(0, p) = p. and π(Sw, p) = ˆ π(pc(Sw), p) [Hat denotes the change of variables.]

• 2. ω is strictly positive and less than 1:

ω(Sw, p) = exp

• −

pc(Sw) cgρw ˆ λw(u)ˆ λg(u) (ρw ˆ λw(u) + cg ˆ λg(u)ˆ π(u, p))2 du

• ,
• 3. p ≤ π(Sw, p) ≤ p + pc(Sw) and therefore pw ≤ p ≤ pg.

Brahim Amaziane, Mladen Jurak () A New Formulation . . . Dubrovnik, October 2008 11 / 24

New global formulation

New global formulation: Flow equations

Total flow: Qt = −λn(Sw, p)K(ω(Sw, p)∇p − ¯ ρn(Sw, p)g). where the superscript n stands for new. Total mass conservation: Φ ∂ ∂t (Swρw + cg(1 − Sw)π(Sw, p)) + divQt = 0. Water mass conservation: Φρw ∂Sw ∂t + div(f n

w(Sw, p)Qt + Kgbn g(Sw, p)) = div(Kan(Sw, p)∇Sw).

Brahim Amaziane, Mladen Jurak () A New Formulation . . . Dubrovnik, October 2008 12 / 24

New global formulation

New global formulation: Coefficients

New coefficients are obtained from the old ones by replacing gas pressure pg by π(Sw, p): total mobility λn(Sw, p) = λ(Sw, π(Sw, p)), water fractional flow f n

w(Sw, p) = fw(Sw, π(Sw, p)),

mean density ¯ ρn(Sw, p) = ¯ ρ(Sw, π(Sw, p)), ”gravity” coeff. bn

g(Sw, p) = bg(Sw, π(Sw, p)),

”diffusivity” coeff. an(Sw, p) = a(Sw, π(Sw, p)).

Brahim Amaziane, Mladen Jurak () A New Formulation . . . Dubrovnik, October 2008 13 / 24

Simplified global formulation

Simplified global formulation

The problem “Find a mean pressure p and a function ω(Sw, p) such that”: ∇pg − fw(Sw, pg)p′

c(Sw)∇Sw = ω(Sw, p)∇p

(4) can be solved by introducing a simplification:

◮ Gas density ρ(pg) can be replaced by ρ(p) without introducing a

significant error. Consequently, fractional flow function fw(Sw, pg) can be replaced by fw(Sw, p). Then, (4) reduces to: ∇pg = ω(Sw, p)∇p + fw(Sw, p)p′

c(Sw)∇Sw.

Brahim Amaziane, Mladen Jurak () A New Formulation . . . Dubrovnik, October 2008 14 / 24

Simplified global formulation

Simplified global formulation

Now we can write solution: pg = p + Sw

1

fw(s, p)p′

c(s) ds,

(5) ω(Sw, p) = 1 + ∂ ∂p Sw

1

fw(s, p)p′

c(s) ds.

(6) Note that (5) is nonlinear equation w.r.t. p to be solved.

Brahim Amaziane, Mladen Jurak () A New Formulation . . . Dubrovnik, October 2008 15 / 24

Simplified global formulation

Simplified global formulation: equations

The system written in unknowns p and Sw. Total flow: Qt = −λ(Sw, p)K(ω(Sw, p)∇p − ¯ ρ(Sw, p)g), Total mass conservation: Φ ∂ ∂t (Swρw + cg(1 − Sw)p) + divQt = 0, Water mass conservation: Φρw ∂Sw ∂t + div(fw(Sw, p)Qt + Kgbg(Sw, p)) = div(Ka(Sw, p)∇Sw). Note that the equations are the same, only the coefficients are different.

Brahim Amaziane, Mladen Jurak () A New Formulation . . . Dubrovnik, October 2008 16 / 24

Simplified global formulation

Simplified global formulation: Applicability

◮ Global pressure is well defined for all pg ≥ 0 and Sw ∈ (0, 1] such that

pw = pg − pc(Sw) ≥ 0;

◮ pw ≤ p ≤ pg; ◮ ω(Sw, p) > 0 holds only if certain additional condition is fullfield. For

example: p ≥ pmin > 0 and ∀Sw ∈ (0, 1], 1

Sw

cgρwλw(s)λg(s) (ρwλw(s) + cgλg(s)pmin)2 |p′

c(s)| ds < 1.

Brahim Amaziane, Mladen Jurak () A New Formulation . . . Dubrovnik, October 2008 17 / 24

Numerical comparison of the coefficients

Numerical comparison

The difference in the coefficients of new and simplified models depends on the difference between π(Sw, p) and p. 0 ≤ π(Sw, p) − p ≤ pc(Sw) ρw ˆ λw(u) ρw ˆ λw(u) + cgpˆ λg(u) du Usually permeability functions depend on dimensionless variable v = u/Pr, where Pr is some pressure constant (entry pressure in Brooks-Corey or van Genuchten models). Then we have: 0 ≤ π(Sw, p) − p ≤ Pr +∞ ρw ˆ λw(v) ρw ˆ λw(v) + cgpˆ λg(v) dv

Brahim Amaziane, Mladen Jurak () A New Formulation . . . Dubrovnik, October 2008 18 / 24

Numerical comparison of the coefficients

Numerical comparison

The difference in the coefficients of new and simplified models depends on the difference between π(Sw, p) and p. 0 ≤ π(Sw, p) − p ≤ pc(Sw) ρw ˆ λw(u) ρw ˆ λw(u) + cgpˆ λg(u) du Usually permeability functions depend on dimensionless variable v = u/Pr, where Pr is some pressure constant (entry pressure in Brooks-Corey or van Genuchten models). Then we have: 0 ≤ π(Sw, p) − p ≤ Pr +∞ ρw ˆ λw(v) ρw ˆ λw(v) + cgpˆ λg(v) dv

Brahim Amaziane, Mladen Jurak () A New Formulation . . . Dubrovnik, October 2008 18 / 24

Numerical comparison of the coefficients

Numerical comparison

As an example we show the difference in coefficients: λ(Sw, p) and λ(Sw, π(Sw, p)), fw(Sw, p) and fw(Sw, π(Sw, p)), ¯ ρ(Sw, p) and ¯ ρ(Sw, π(Sw, p)), bg(Sw, p) and bg(Sw, π(Sw, p)), a(Sw, p) and a(Sw, π(Sw, p)), ωsimp(Sw, p) and ωnew(Sw, p), for van Genuchten functions with parameters (Couplex test case):

n = 1.54, Pr = 2 MPa, µw = 7.98 · 10−3 Pas, µg = 9 · 10−6 Pas, ρw = 103 kg/m3 and cg = 0.808.

Notation: new = new model; simpl = simplified model.

Brahim Amaziane, Mladen Jurak () A New Formulation . . . Dubrovnik, October 2008 19 / 24

Numerical comparison of the coefficients

Coefficients at global pressure of 1 MPa

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 50 100 150 200 250 b_g(S,p), p=1 MPa simp new 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 10000 20000 30000 40000 50000 60000 70000 80000 a(S,p), p=1 MPa simp new 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 20000 40000 60000 80000 100000 120000 140000 160000 180000 200000 tot_mob(S,p), p=1 MPa simp new 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 f_w(S,p), p=1 MPa simp new 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 100 200 300 400 500 600 700 800 900 1000 rho_mean(S,p), p=1 MPa simp new 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00

• mega(S,p), p=1 MPa

simp new

Brahim Amaziane, Mladen Jurak () A New Formulation . . . Dubrovnik, October 2008 20 / 24

Numerical comparison of the coefficients

Coefficients at global pressure of 5 MPa

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 5 10 15 20 25 30 35 40 45 50 b_g(S,p), p=5 MPa simp new 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 20000 40000 60000 80000 100000 120000 140000 a(S,p), p=5 MPa simp new 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 50000 100000 150000 200000 250000 300000 350000 400000 450000 500000 550000 tot_mob(S,p), p=5 MPa simp new 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 f_w(S,p), p=5 MPa simp new 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 100 200 300 400 500 600 700 800 900 1000 rho_mean(S,p), p=5 MPa simp new 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.94 0.95 0.96 0.97 0.98 0.99 1.00

• mega(S,p), p=5 MPa

simp new

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Numerical comparison of the coefficients

Coefficients at global pressure of 10 MPa

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 5 10 15 20 25 b_g(S,p), p=10 MPa simp new 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 20000 40000 60000 80000 100000 120000 140000 160000 180000 a(S,p), p=10 MPa simp new 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0e+00 1e+05 2e+05 3e+05 4e+05 5e+05 6e+05 7e+05 8e+05 9e+05 1e+06 tot_mob(S,p), p=10 MPa simp new 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 f_w(S,p), p=10 MPa simp new 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 100 200 300 400 500 600 700 800 900 1000 rho_mean(S,p), p=10 MPa simp new 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.975 0.980 0.985 0.990 0.995 1.000

• mega(S,p), p=10 MPa

simp new

Brahim Amaziane, Mladen Jurak () A New Formulation . . . Dubrovnik, October 2008 22 / 24

Conclusion

Conclusion

◮ In regimes with large global pressures compared to typical capillary

pressure the difference in the coefficients is small.

◮ For small global pressures and relatively large capillary pressure the

difference between new and simplified coefficients becomes significant.

◮ Replacing

∂ ∂t (cg(1 − Sw)π(Sw, p)) by ∂ ∂t (cg(1 − Sw)p) can have large influence on mass conservation even in for small capillary pressure and elevated global pressure.

Brahim Amaziane, Mladen Jurak () A New Formulation . . . Dubrovnik, October 2008 23 / 24

Conclusion

Work in progress

◮ An extension to multiphase, multicomponent models is

straightforward and it is at present in the course of study.

◮ The new formulation is well adapted for the mathematical analysis of

the model. At present we study existence etc.

◮ The discretization of the model by a vertex-centered finite volume

scheme is currently studied. Reference: B. Amaziane, M. Jurak: A new formulation of immiscible compressible two-phase flow in porous media, C. R. A. S. M´ ecanique 336 (2008) 600-605.

Brahim Amaziane, Mladen Jurak () A New Formulation . . . Dubrovnik, October 2008 24 / 24