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

2022
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203

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20
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Tobsuchtbvow

Math, Num and Comp Models Black–Oil 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 Black–Oil The basic hypotheses of the black-oil model are, see [11]: 1 There are three fluid phases in the reservoir: w, o, and g. References Math, Num and Comp Models Black–Oil The basic hypotheses of the black-oil model are, see [11]: 1 2 There are three fluid phases in the reservoir: w, o, and g. Solubility of hydrocarbons in the water phase is negligible. References Math, Num and Comp Models Black–Oil The basic hypotheses of the black-oil model are, see [11]: 1 2 3 There are three fluid phases in the reservoir: w, o, and g. Solubility of hydrocarbons in the water phase is negligible. The water and gas phases only contain one component. References Math, Num and Comp Models References Black–Oil The basic hypotheses of the black-oil model are, see [11]: 1 2 3 4 There are three fluid phases in the reservoir: w, o, and g. Solubility of hydrocarbons in the water phase is negligible. The water and gas phases only contain one component. The oil and water components are not allowed to vaporize into the gas phase. Math, Num and Comp Models References Black–Oil The basic hypotheses of the black-oil model are, see [11]: 1 2 3 4 5 There are three fluid phases in the reservoir: w, o, and g. Solubility of hydrocarbons in the water phase is negligible. The water and gas phases only contain one component. The oil and water components are not allowed to vaporize into the gas phase. The gas component is allowed to dissolve into the oil phase. Math, Num and Comp Models References Black–Oil The basic hypotheses of the black-oil model are, see [11]: 1 2 3 4 5 6 There are three fluid phases in the reservoir: w, o, and g. Solubility of hydrocarbons in the water phase is negligible. The water and gas phases only contain one component. The oil and water components are not allowed to vaporize into the gas phase. The gas component is allowed to dissolve into the oil phase. The oil-phase contains dissolved gas and non-volatile oil. Math, Num and Comp Models References Black–Oil The basic hypotheses of the black-oil model are, see [11]: 1 2 3 4 5 6 7 There are three fluid phases in the reservoir: w, o, and g. Solubility of hydrocarbons in the water phase is negligible. The water and gas phases only contain one component. The oil and water components are not allowed to vaporize into the gas phase. The gas component is allowed to dissolve into the oil phase. The oil-phase contains dissolved gas and non-volatile oil. Mass diffusive-processes are neglected. Math, Num and Comp Models References Black–Oil The basic hypotheses of the black-oil model are, see [11]: 1 2 3 4 5 6 7 8 There are three fluid phases in the reservoir: w, o, and g. Solubility of hydrocarbons in the water phase is negligible. The water and gas phases only contain one component. The oil and water components are not allowed to vaporize into the gas phase. The gas component is allowed to dissolve into the oil phase. The oil-phase contains dissolved gas and non-volatile oil. Mass diffusive-processes are neglected. The system is in thermal equilibrium. Math, Num and Comp Models References Black–Oil The basic hypotheses of the black-oil model are, see [11]: 1 2 3 4 5 6 7 8 9 There are three fluid phases in the reservoir: w, o, and g. Solubility of hydrocarbons in the water phase is negligible. The water and gas phases only contain one component. The oil and water components are not allowed to vaporize into the gas phase. The gas component is allowed to dissolve into the oil phase. The oil-phase contains dissolved gas and non-volatile oil. Mass diffusive-processes are neglected. The system is in thermal equilibrium. There is no chemical reactions. Math, Num and Comp Models References Black–Oil The basic hypotheses of the black-oil model are, see [11]: 1 2 3 4 5 6 7 8 9 10 There are three fluid phases in the reservoir: w, o, and g. Solubility of hydrocarbons in the water phase is negligible. The water and gas phases only contain one component. The oil and water components are not allowed to vaporize into the gas phase. The gas component is allowed to dissolve into the oil phase. The oil-phase contains dissolved gas and non-volatile oil. Mass diffusive-processes are neglected. The system is in thermal equilibrium. There is no chemical reactions. Three components: W = water component, O = liquid hydrocarbon component, G = gaseous hydrocarbon component. Math, Num and Comp Models References Black–Oil The basic hypotheses of the black-oil model are, see [11]: 1 2 3 4 5 6 7 8 9 10 11 There are three fluid phases in the reservoir: w, o, and g. Solubility of hydrocarbons in the water phase is negligible. The water and gas phases only contain one component. The oil and water components are not allowed to vaporize into the gas phase. The gas component is allowed to dissolve into the oil phase. The oil-phase contains dissolved gas and non-volatile oil. Mass diffusive-processes are neglected. The system is in thermal equilibrium. There is no chemical reactions. Three components: W = water component, O = liquid hydrocarbon component, G = gaseous hydrocarbon component. Water phase: wetting phase, Oil phase: intermediate wetting phase, Gas phase: non-wetting phase. Math, Num and Comp Models References Black–Oil Component W O G Aqueos (w) φρw Sw (ρW w ≡ ρw ) Phases Oleic (o) Gaseous (g) φρOo Sw φρGo So φρg Sg (ρGg ≡ ρg ) (dg) (fg) dg : dissolved gas; fg : free gas. Balance equations: ∂(φρw Sw ) = −∇ · (ρw ~ uw ) + qW ∂t ∂(φρOo So ) Oil component: = −∇ · (ρOo ~ uo ) + qO ∂t Water component: Gas component: dg fg flow fg z }| { z }| { z }| { ∂(φρGo So + φρg Sg )) = −∇ · ( ρGo ~ uo + ρg ~ ug ) + qG | {z } ∂t flow dg Math, Num and Comp Models References Black–Oil Darcy’s Law: ~uw = − ~uo = − krw k(∇pw − ρw G∇D) µw kro k(∇po − ρo G∇D) µo ~ug = − (where ρo = ρOo + ρGo ) krg k(∇pg − ρg G∇D) µg Saturation constrain: Sw + So + Sg = 1. Capillary pressures: pcgw = pg − pw ; pcow = po − pw ; pcgo = pg − po . We have: 3 mass conservation eqs, 3 Darcy’s Law eqs, 1 sat. constrain, 2 capillary pressure. Total: 9 eqs. We have 9 unknowns: pw , po , pg , ~uw , ~uo , ~ug , Sw , So , and Sg . Math, Num and Comp Models References Black–Oil Pressure–Saturation Formulation for Black–Oil Model Pressure equation (oil) cT X 1 ∂p +∇·u= qα − ρ0α cα uα · ∇p , α = w, o, g, ∂t ρ α α Saturation equation (water and gas) ρα φ ∂sα ∂p + ρα ∇ · uα = −(sα ρα φ0 cp + φsα ρ0α cα ) ∂t ∂t α = w, g. −ρ0α cα uα · ∇p + qα , Math, Num and Comp Models References Black–Oil Discrete pressure equation N Integrating on ∆V and ∆t: ∆x ∆z Z Z " ∆V ∆t φ0 c p + X φsα ρ 0 cα ρα α α ! n # ∂p W + ∇ · u dt dV ∂t # F Z Z "X 1 = qα − ρ0α cα uα · ∇p dt dV ρα α w f ∆V ∆t P b B e ∆y E s S ∆V = ∆x∆y∆z Using convenient approximations and θ = 1 we get : a0P + aP pn+1 − aE pn+1 − aW pn+1 P E W −aN pn+1 − aS pn+1 − aF pn+1 − aB pn+1 =S N S F B Math, Num and Comp Models References Black–Oil Coefficients aE = − X α aN = − X α aF = − ( 0 ρα,P ρn α,P ( 0 ρα,P ρn α,P ( X ρ0 α,P ρn α,P cα,P n n uα,e + uα,w n n cα,P vα,n + vα,s cα,P n n wα,f + wα,b ) 8δxe ) ∆V ∆t ∆V ∆t 8δyn ) ∆V ∆t 8δzf  0 X ρ α,P n 0 0 n cα,P sα,P  φP ∆V aP = φP cp,P ∆V +  n α ρα,P α aW = X ( 0 ρα,P α aS = X ρn α,P ( 0 ρα,P cα,P n n uα,e + uα,w n n cα,P vα,n + vα,s ) ∆V ∆t ) ∆V ∆t 8δxw ρn 8δys α,P ( 0 ) ∆V ∆t X ρα,P n n aB = cα,P wα,f + wα,b ρn 8δzb α α,P α  aP = aE + aW + aN + aS + aF + aB n n n n n n 0 S = a0P − aP pn P + aE pE + aW pW + aN pN + aS pS + aF pF + aB pB + S ! n n X qα,P wfn − wbn ue − un v n − vsn w S0 = ∆V ∆t − + n + ∆V ∆t n ρα,P ∆x ∆y ∆z α Math, Num and Comp Models References Black–Oil Discrete saturation equation (α = w, g) N Integrating on ∆V and ∆t: Z Z φ ∆V ∆t ∂sα ρ0 + ∇ · uα + cα α uα · ∇p dt dV ∂t ρα Z Z qα ρ0 ∂p = − sα φ0 cp + sα φcα α dt dV ρα ρα ∂t ∆x ∆z n W w f P n sn+1 α,P = bP sα,P + dP bP = bP (φnP , ρnα,P , pnP , pn+1 P ) dP = dP (φnP , ρnα,P , qPn , pn , unα ) e ∆y s S Using convenient approximations we get and θ = 0: B E F ∆V ∆t b ∆V = ∆x∆y∆z Math, Num and Comp Models References Black–Oil Explicit equation for saturation (α = w, g) " sn+1 α,P =snα,P − snα,P ρ0α,P φ0P cp,P n + cα,P n φP ρα,P # pn+1 P − pnP + n qα,P ρnα,P φnP n − wn n − vn wα,f vα,n ∆t unα,e − unα,w α,s α,b + + − n φP ∆x ∆y ∆z " ρ0α,P cα,P unα,e + unα,w pnE − pnP pnW − pnP − n n ∆t − + ρα,P φP 4 δxe δxw n + vn n vα,n pN − pnP pnS − pnP α,s − + 4 δyn δys # n + wn n wα,f pF − pnP pnB − pnP α,b − 4 δzf δzb ∆t Math, Num and Comp Models Black–Oil Solution Algorithm References

Computational Methods for Oil Recovery Black Oil

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