a black-oil model for primary and secondary oil-recovery ... · abstract: the black-oil model is a...

A Black-Oil Model for Primary and Secondary Oil-Recovery in Stratified Petroleum Reservoirs

A. Dollari*, Ch. Chatzichristos and A. G. YiotisEnvironmental Research Laboratory, National Center for Scientific Research "Demokritos", 15341 Athens, Greece*Corresponding author: [email protected]

Abstract: The black-oil model is a multiphasefluid flow formulation extensively used in thepetroleum recovery industry for the prediction ofoil/gas recovery rates and the evolution ofpetroleum reservoir conditions. In thiscontribution, we propose a numerically stableformulation of the black-oil model withnegligible capillary pressure. The formulation isbased on the oil phase pressure and total fluidvelocity and solved numerically using the PDEinterface of COMSOL Multiphysics in thegeneral and coefficient mode for time-dependentanalysis. We apply the proposed model in aseries of typical primary and secondary oilrecovery processes ranging from the naturalpressure-driven fluid expansion and solution-gasdrive during the early-life of the reservoir tomore elaborate pressure maintenance strategiesrelying on waterflooding and gas injection in atypical homogeneous petroleum reservoir.

Keywords: black-oil model, porous media,petroleum reservoir, solution-gas drive,secondary oil-recovery

1. IntroductionBlack-oil simulators are commonly used in

petroleum reservoir engineering for theprediction of oil production dynamics, especiallyduring the earlier stages of oil-field exploitation,while also serving to guide pressure maintenancestrategies in the longer term. The mainassumption of such an approach (as opposed tocompositional flow models) is the classificationof all reservoir species into two main categorieswith respect to their physical state at surfaceconditions; 1. A mixture of heavy H/Ccomponents containing all species that exist in aliquid form at the surface (denoted as “o”), and2. A mixture of volatile H/C components thatcontain all species that exist as a gaseous phaseat surface conditions (denoted as “g”).Furthermore, the aqueous phase (w), which isabundant in petroleum reservoirs, is consideredas the third flowing phase [1, 2].

The apparent densities of the above fluidphases are expressed as functions of appropriateexperimentally-determined fluid formationvolume factors and solution ratios to implementthe effects of fluid compressibility, phase changeand species dissolution across the volatile andheavy H/C mixtures at elevated reservoirpressures (Pressure-Volume-Temperature, PVTtables). The resulting multiphase flow problem isthen treated using a phenomenologicalgeneralization of Darcy’s law, where the relativepermeability of each fluid through the porousstructure is expressed as a state function of localsaturation.

2. Theory/ Governing EquationsThe black-oil model relies on the coupled

solution of mass conservation equations for thewater (w), oil (o) and gas (g) phases (ormixtures) subject to appropriate boundary andinitial conditions, while also accounting for masstransfer of dissolved species across the (o) and(g) phases. The basic partial differentialequations for the black-oil model in a porousmedium expressed as a conservation of the massof the volatile species, the heavy species andwater reads as follows [3, 4];

where the volumetric phase velocity is given by

uα=−k rαα

μα

K (∇ pα−ρα g) ; α = g, o, w (4)

Here ϕ and K denote the porosity andintrinsic permeability tensor of the porousmedium, S α , μα , ρα , pα , uα , Bα , krαα and qα theα -phase saturation, viscosity, density, pressure,

volumetric velocity, formation volume factor,

ϕ ∂∂ t (

S g

Bg

+Rso So

Bo)+∇⋅(1Bg

ug+Rso

Bo

uo)= q g , (1)

ϕ ∂∂ t (

S o

Bo)+∇⋅(

1Bo

uo)= qo , (2)

ϕ ∂∂ t (

S w

Bw)+∇⋅(

1Bw

uw)= qw (3)

Excerpt from the Proceedings of the 2018 COMSOL Conference in Lausanne

relative permeability, and external source term,respectively. Rso is the gas solubility in the oilphase and g is the acceleration of gravity.

The system is closed by the localconstraints. The three phases jointly fill up thevoid space:S g+ S o+ Sw=1 (5)

while the phase pressures p g, po, pw are relatedby gas-oil and and oil-water capillary pressures,respectively:pcgo = pg− po , pcow= p o−pw (6)

It is also assumed that the porous media is fullysaturated but the phases are separated at thepore-scale, the porous matrix and fluids areslightly compressible, diffusion is neglected forall phases and the whole system is in localthermodynamic equilibrium [2, 5, 6].

3. Numerical ModelThe black-oil model is typically solved

numerically with finite volumes or finitedifferences approaches [1]. The discretizationunder the finite element framework utilized byCOMSOL requires that the non-linearity andcoupling among the equations are weakened,while preserving the main physical propertiesand constraints. Thus a more appropriate,numerically stable, reformulation of thegoverning equations is developed, which isbased on the oil phase pressure and total fluidvelocity as proposed by Chen [2].

3.1 Phase Formulation based on Oil Pressureand Total Velocity

For a better demonstration of the adoptedformulation, we first introduce the phasemobility functions λα = krαα /μα for α = g, o, w ,

and the total mobility λ =∑ λα . We also definethe fractional flow functions f α = λα /λ , so that

∑ f α=1 .Oil being a continuous phase implies that

po is well behaved, so we use the oil phasepressure as the pressure variable:p = po (7)

We now define the total velocity:u =∑ uα (8)

Then we perform the correspondingnotation substitution and we use Eqs. (7) and (8),carry out the differentiation indicated in Eqs. (1)– (3) and apply Eqs. (4) – (6) to obtain thedifferential equations:

Pressure equation

∇⋅u = ∑β=g,o,w

Bβ (qβ− Sϕ β∂

∂ t (1Bβ)−uβ⋅∇ (1Bβ

)) −Bg(Rso qo+

Sϕ o

Bo

∂Rso

∂t+

1Bo

uo⋅∇ R so) ,

(9)

u=−K λ (∇ p−G λ+∑β

f β ∇ pcβo)

Saturation equations

ϕ∂Sα

∂ t+∇⋅uα

=Bα(q α− Sϕ α∂

∂t (1Bα)−uα⋅∇ (1Bα

)) , (10)

uα=f α u+ K f α∑β

λβ (∇ ( pcβo− pcαo)−(ρ β−ρα) g )

for α = o, w , whereGλ = g∑

β

f β ρβ (11)

3.2 Numerical Implementation in COMSOLThe numerical model of Eqs. (9)-(10) is

implemented in COMSOL 5.3 using the PDEInterface in the general and coefficient form fortime dependent analysis [7]. The General FormPDE (g) is used for Eq. 9 with the total phasepressure, p, as the dependent variable. Forcompleteness we provide below the exact user-defined expressions for the built-in variablesused (maintaining COMSOL’s namingconvention):

ea=0, d a=0, Γ= [uox

uoy ] ,f = −Bg(−qg+ Sϕ g

∂∂ t (

1B g )+ugx

∂∂ x (

1Bg )+ugy

∂∂ y (

1Bg ))

−Bo(−qo + Sϕ o∂∂t (

1Bo )+uox

∂∂ x (

1Bo )+uoy

∂∂ y (

1Bo ))

−Bw(−qw + Sϕ w∂∂ t (

1Bw )+uwx

∂∂ x (

1B w )+uwy

∂∂ y (

1Bw ))

−Bg(−qo Rso+Sϕ o

Bo

∂ Rso

∂ t+

1Bo

(uox

∂ Rso

∂ x+uoy

∂ Rso

∂ y ))The Coefficient Form PDE (c) is used for

Eq. 10 with S o and S w as the dependentvariables (i.e. two equations for oil and watersaturation, respectively).

The exact user-defined expressions for theoil phase saturation read;


ea=0 , da = ,ϕ c=ε, a=0,

β =[00 ] , α= [00 ] , γ =[uox

uoy] ,

f = −B o( Sϕ o∂

∂t (1Bo

)+uox∂

∂ x (1Bo

)+uoy∂

∂ y (1Bo

))while for the water saturation we write;ea=0 , da = ,ϕ c=ε, a=0,

β =[00 ] , α= [00 ] , γ =[uwx

uwy] ,

f = −Bw( Sϕ w∂∂t (

1Bw

)+uwx∂∂ x (

1Bw

)+uwy∂∂ y (

1Bw

))Given that capillary forces among the three

fluid components are neglected, the total andindividual phase velocities can be rewritten asfollows;u = −K λ (∇ p−g ( f g ρ g+f o ρo +f w ρw ) ) ,

ug = f g u−K f g ( λo ( ρo−ρg )+λw (ρw−ρg )) g ,

uo = f ou−K f o ( λg ( ρg−ρo )+λw ( ρw−ρo ) ) g ,

uw = f w u−K fw ( λg (ρg−ρw ) +λo ( ρo−ρw )) g

(12)

Finally, the phase densities at reservoirconditions are expressed as follows;

ρo=ρOs +Rso ρGs

Bo

, ρg=ρGs

Bg

, ρw=ρWs

Bw

(13)

where the underscript "s" designates properties atstandard (surface) conditions.

In the Appendix, we provide the completeset of values for phase densities at standardconditions, the PVT phase properties as well asthe two-phase relative permeability curves forthe gas-oil and the oil-water system. In thefollowing simulations water and the solid phaseare assumed practically incompressible.

4. Numerical Simulations / ResultsWe consider an inclined homogeneous, but

anisotropic, petroleum reservoir, with a typicalporosity of φ=0.2, that exhibits a series ofstructural traps as shown in Figure 1. Thestructure is overlaid by an impermeable caprockthat imposes the trapping of petroleum and thesubsequent formation of the reservoir in thegeological time-scale.

We assume that the reservoir ischaracterized by an anisotropic permeabilitytensor where the longitudinal component followsthe direction of the lower reservoir boundary,while the transverse component is always normal

to that. Given that a fixed Cartesian coordinatesystem is used to express the gradients ofpressure that control flow in the x and y-directions, the permeability tensor needs toaccount for the direction of the longitudinalpermeability (that coincides with the bottomboundary of the reservoir) with respect to the x-axis. This effect is parametrized by the angle θbetween the two vectors as shown also in Figure1. In this domain the angle obeys the expressionθ=atan(−0.297 x+1.125)[rαad ] and this effect

generates a permeability tensor with no-zero off-diagonal terms in the x-y Cartesian system, asshown below;

K=[k maxcos

2θ+k T sin

2θ kmax cosθ sinθ - k T cosθ sinθ

kmax sinθcosθ -kT sinθ cosθ kmax sin2θ+k T cos2θ ]=[

k11 k 12

k 21 k 22]Darcy’s equation in the x,y-directions is thenexpressed as follows for the α -phases;

uα=(uαx , uαy) ,uαx=−λα (k 11(∂ p /∂ x)+k 12((∂ p /∂ y)+ρα g y))

uαy=−λα (k 21(∂ p /∂ x)+k 22((∂ p/∂ y)+ ρα g y ))

Initially the two phases are distributedequally in the pore space with an initial averageoil phase saturation of 90% PV, as shown in thenext 2-D surface plot.

Figure 1. Schematic of the down-scaled petroleumreservoir used in our numerical simulations exhibitinga series of structural traps on the top boundary belowan impermeable caprock (not shown here). Thelongitudinal permeability component follows thedirection of the bottom boundary of the reservoir.

The initial reservoir pressure is takenassuming hydrostatic conditions at a depth wherethe oil phase is under-saturated (above thebubble point in the phase envelope). The fluidsare then allowed to escape from the formation


from a production well located along the rightboundary.

In the following section, we employ thepreviously described black-oil model to solve aseries of typical oil production scenarios,including oil production due to the natural initialpressure of the formation and the pressuredepletion that allows for solution-gas drive, andsecondary oil recovery scenarios, includingwaterflooding.

Initial conditions

No flux conditions are applied in the side wallsand no external source terms are considered.

4.1 Primary Oil-recovery: Solution-gas DriveMechanism

The flow of oil and gas phases throughporous media during a primary recovery processdriven by the liberation of the dissolved gascomponent is simulated. A smooth pressuredrop-down is applied for a time period of 12000sat the right boundary.

Table 1: Primary recovery simulation data

Property Unit ValueMaximum permeability (kmax)

m2 10-12

Transverse permeability(kT)

m2 10-13

Numerical diffusion (ε)) m2/s 10-7

Boundary conditions

The oil phase saturation evolution and the fluidproductions at surface conditions are presented.

Figure 2. Oil phase saturation during primary oil-recovery for t = 8000s (left) and t = 12000s (right).

Figure 3. Fluid production rates at standard conditionsduring primary oil-recovery.

During the initial stages only the oilcomponent and the dissolved gas component areproduced since the petroleum reservoir is stillunder-saturated. As the average reservoirpressure declines, gas is generated causing aphase segregation due to gravitational forces.From that point and on the oil follows a decayingrate till the end of simulation study. Similarly,the gas production follows an increasing rateand then declines smoothly.

4.2 Secondary Oil-recoveryThe next problems simulate the flow

behavior during waterflooding and gas injectionprocesses in petroleum reservoirs. Althoughnormally the fluids are injected only from theupper or bottom part of the injection wellrespectively for better sweep efficiency, in thecurrent studies they are injected from the wholeside of the left boundary of the domain fordemonstration purposes. Accordingly, theproduction well is placed in the right boundary.

4.2.1 WaterfloodingThe simulation of a waterflooding process

in an anisotropic reservoir is studied. The wateris injected for 10000s.

Outletp = pout

−n⋅(− ε∇ So +uo )=−uo

−n⋅(− ε∇ Sw + uw)=−uw

wherαe pout = pin−0.2⋅pin⋅flc 2hs ( t−t1 ,t2 ) [ Pa ]

forα t 1=7000s, t2=6800 s

p ( t 0 ) = p in=105+ρw g (2860− y ) [ Pa ]

S o ( t0 )=1−Swc

S w (t0 ) =Swc =0 .1


Boundary conditionsInlet Outlet

−n⋅u=u in p= pout=pin

S o=Sor −n⋅(− ε∇ So+uo)=−uo

S w=1−S or −n⋅(−ε∇ S w+uw )=−uw

Table 2: Waterflooding simulation data

Property Unit ValueMaximum permeability (kmax)

m2 5 10-11

Transverse permeability (kT)

m2 5 10-12

Oil residual saturation (Sor)

- 0.3

Inlet velocity (uin) m/s 10-4

Numerical dispersion (ε)) m2/s 5 10-6

The evolution of the oil phase saturationduring the water-injection and the fluidproductions at surface conditions are presented.

Figure 4. Oil phase saturation during waterfloodingfor t = 1500s (left) and t = 10000s (right)

Figure 5. Fluid production rates at standard conditionsduring waterflooding.

As water floods enter the reservoir, thesaturation of the oil phase gradually decreases inthe area near the injector leading to a largerswept zone with time. During this time period

the oil is produced with constant rate, followedby an abrupt decrease which is indicative of thewater breakthrough. The two rates exhibit then areverse proportional relationship till the end ofthe simulation study, when the efficientsweeping is visually verified.

4.2.2 Gas InjectionThe simulation of a gas injection process

in an isotropic reservoir is studied. The gas isinjected for a time period of 45000 s.

Boundary conditionsInlet Outlet

−n⋅u=u in p= pout=pin

S o=Sor −n⋅(− ε∇ So+uo)=−uo

S w=Swc Sw=Swc

Table 3: Gas injection simulation data

Property Unit ValueIsotropic permeability (kiso = kmax = kT)

m2 10-12

Inlet velocity (uin) m/s 10-5

Numerical diffusion (ε)) m2/s 8 10-7

Following are the evolution of the oilphase saturation during the gas injection and theproduction rates at surface conditions.

Figure 6. Oil phase saturation during gas injection fort = 4860s (left) and 45000s (right).

The lighter gas floods travel quickly to theupper part of the reservoir due to buoyancy, thusdisplacing non-uniformly the oil phase towardsthe production well. The oil recovery exhibits aconstant rate till the gas breakthrough, whichcauses a smooth reduction till the end of thesimulation study. Gas is produced at surfaceconditions with a constant rate from thebeginning of the injection as it exists initially assolution in the oil phase in reservoir conditions.The abrupt increase indicates that the gas floodshave reached the production well, thus they are


recovered together with the oil phase until theend of the injection process.

Figure 7. Fluid production rates at standard conditionsduring gas injection.

By the end, the gas phase has occupied theupper part of the reservoir, leaving however asignificant volume of oil in the half bottom part.The sweeping efficiency is clearly smaller in thecurrent case, due to the formation of an upperpreferable gas path by the time of thebreakthrough.

5. ConclusionsIn this contribution we propose a black-oil

model using COMSOL’s PDE Interface based onthe oil phase pressure and total fluid velocityformulation developed by Chen [2]. Ourimplementation is then evaluated for anhomogeneous petroleum reservoir exhibiting aseries of structural traps for typical primary andsecondary oil recovery processes, includinginitial pressure depletion (solution gas-drivemechanism), waterflooding and gas injection.

The interpretation of the physicalphenomenon is described successfully enough bythe implemented numerical model. Thewater/gas injection which is significantlyenhanced due to buoyancy that leads toaccumulation of injected water/gas in thebottom/upper part of the reservoir thus pushingthe lighter/heavier oil towards the surface andproduction wells is visually verified via 2-Dplots. Apparent is also the phase segregation inthe reservoir by the end of the pressuredepletion, as expected.

The newly-developed numerical model canhence be used as an independent PDE-module inreservoir simulation studies but also for morecomplex problems in Enhanced Oil Recovery(EOR) processes by implementing also theappropriate PDEs that describe tracer (mass)

transfer, thermal stimulation and surfuctantprocesses.

6. References

1. Z Chen, G Huan, and Y Ma, ComputationalMethods forα Multiphase Flows in Porαous Media,Computational Science and Engineering. Societyfor Industrial and Applied Mathematics (2006)2. Z Chen, Forαmulations and NumerαicalMethods of the Black Oil Model in PorαousMedia, Society for Industrial and AppliedMathematics, 38, 489-514 (2000)3. K Aziz and A Settari, Petrαoleum ReserαvoirαSimulation. Applied Science Publishers Ltd.,London UK (1979)4. D W Peaceman, Fundamentals of numerαicalrαeserαvoirα simulation. Elsevier, New York (1977)5. J Bear, Dynamics of Fluids in Porαous Media.Dover Civil and Mechanical Engineering Series,Dover (1972)6. M A Diaz-Viera, D A Lopez-Falcon, AMoctezuma-Berthier, and A Ortiz-Tapia,COMSOL implementation of a multiphase fluidflow model in porαous media, COMSOLConference, Boston (2008)7. COMSOL Mutliphysics Reference Manual,Version 5.3”, COMSOL, Inc. (2017)


7. Appendix

Table 4: Fluid densities at standard conditions

Fluid Density at SC(kg/m 3)

Gas 0.40Oil 800

Water 1000

Table 5: Oil phase properties

Pressure(MPa)

Rso

(sm3/sm3)Bo

(vol/svol)μo

(cP)8.38 24.4 1.172 1.9709.75 34.7 1.200 1.55611.1 42.9 1.221 1.39712.5 51.3 1.242 1.28015.3 66.8 1.278 1.09518.0 82.2 1.320 0.96720.8 99.4 1.360 0.84823.5 117.7 1.402 0.76226.3 137.1 1.447 0.69129.1 137.1 1.4405 0.69431.8 137.1 1.434 0.697

Table 6: Gas phase properties

Pressure(MPa)

Bg

(vol/svol)μg

(cP)8.38 0.0783 0.01249.75 0.0395 0.012511.1 0.0261 0.012812.5 0.0194 0.013015.3 0.0126 0.013918.0 0.0092 0.014820.8 0.0072 0.016123.5 0.0059 0.017326.3 0.0050 0.0187

Table 7: Water phase properties

Pressure(MPa)

Bw

(vol/svol)μw

(cP)27.52 1.0231 0.94

Table 8: Water and gas phase relative permeabilities

Sw krw Sg krg

0.1 0.0 0 00.16 0.0005 0.05 00.22 0.004 0.09 0.0320.28 0.0135 0.18 0.0890.34 0.032 0.27 0.1640.40 0.0625 0.36 0.2530.46 0.108 0.45 0.3540.52 0.172 0.54 0.4650.58 0.256 0.63 0.5860.64 0.365 0.72 0.7160.70 0.50 0.81 0.8540.80 0.667 0.9 10.90 0.833

1 1

Table 9: Oil phase relative permeabilities

Sokro (water-oil

system)kro (gas-oil

system)0.3 0.0 0

0.36 0.032 0.0010.42 0.089 0.0080.48 0.164 0.02750.54 0.253 0.0640.60 0.354 0.1250.66 0.465 0.2160.72 0.586 0.3430.78 0.716 0.5120.84 0.854 0.7290.90 1 1


a black-oil model for primary and secondary oil-recovery ... · abstract: the black-oil model is a...

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