Black-Oil

The black-oil model is a set of model assumptions widely used in the oil industry [83, 19, 46, 45]. It presumes three potentially present fluids oil, gas, and water—in the following indicated byh·i_o,h·i_g, andh·i_w—as well as three pseudo-components to which we also refer

Figure 3.3: Definition of the gas solubility factorRSin the black-oil model: A given volume of saturated oilVreservoir,ois brought from reservoir pressurepreservoirto the relatively low atmospheric pressurep_atm. Then, the gas solubility factor is defined as the ratio between volume of gas which emerges at the surface and the original volume of the reservoir oil. Typically, the reservoir pressure exceeds200 barwhilepatmis roughly1 bar.

to as oil, gas, and water, and which will be indicated byh·i^O,h·i^G, andh·i^W. In the terms of the thermodynamic framework of this chapter, the black-oil model presumesM =N = 3.

The first assumption of the black-oil model is that the water, and gas phases are immiscible, and that they are solely constituted by the components with the same names. In contrast, the oil phase is assumed to be a mixture of the pseudo-components gas and oil with the composition given by thegas solubility factorRS.

This factor is defined as the volume of the gas phase that emerges from a given amount of gas-saturated oil phase if the of oil is brought from the reservoir to the surface,i.e., that the pressure of the material is reduced from reservoir pressure to atmospheric pressure, as illustrated in Figure 3.3. We thus defineRSas

RS:= Vatm,g

Vreservoir,o

= S_atm,gV_total,atm Vreservoir,o

.

Given the gas solubility factor at a certain pressure, the composition of gas saturated oil at reservoir pressure can thus be determined in terms of mass fractions using

X_sat,o^G = ρ_g(p_atm)R_S(p_o)

ρsat,o(po) and X_sat,o^O = 1−X_sat,o^G

if we assume that the amount of the gas in oil is negligible at atmospheric pressure.

We also need to specify the mass densities of all phases at a given pressure. While we assume the water phase to be incompressible, the density of the gas saturated oil phaseρ_oand the one of the gas phaseρgis determined by theoil and the gas formation volume factorsBoandBg.

Model Constraints 45

These two factors express the ratio between the mass density at reservoir pressure and the density at atmospheric pressure,i.e.,

B_α(p_α) = ρsat,α(pα)

ρ_sat,α(p_atm), α∈ {o, g}, where the atmospheric densitiesρ_sat,α(p_atm)are given constants.

Fugacity Coefficients

In order to fit the set of black-oil parameters into our thermodynamic framework, we need to define fugacity coefficientsΦ^κ_αfor all componentsκand all fluid phasesαas well as the mass densities of all fluid phases. In the following, we will outline how we can achieve this goal.

The black-oil parameters assume the gas and the water phases to be immiscible,i.e., the only constituent of the gas phase is the gas component, and that of the water phase is the water component. For this reason, we can use the same approach to the fugacity coefficients as for the immiscible model constraints,i.e.,

Φ^κ_α=

1 ifκ=κ_α,

∞ else forα∈ {w, g},κ∈ {O, W, G}, and withκ_w=W,κ_g =G.

Since the oil phase is a mixture of the gas and the oil components, we first need to ensure that the oil phase does not contain any water by setting the fugacity coefficient for the water component in the oil phase to infinity. For the fugacity coefficient of the oil component in the oil phase, we can use any finite positive value; usually we use

Φ^O_o = p^O_vap p_o

wherep^O_vapdenotes a typicalvapororsaturation pressureof the oil component.

The most elaborate fugacity coefficient is the one for the gas component in the oil phase.

This is due to the facts that the gas component is the only component which is assumed to be potentially present in two fluid phases,i.e., the gas and the oil phase, and that we must choose the fugacity coefficient such that it is consistent with the black-oil parameters. We can achieve this by first calculating the mass fractionsX_sat,o^O andX_sat,o^G for gas saturated oil as outlined in the previous section. Next, we need to convert these mass fractions into mole fractions by solving the system of equations

X_sat,o^G = M^G M_ox^G_sat,o,

M_o=x^O_sat,oM^O+x^G_sat,oM^G, and x^G_sat,o+x^O_sat,o= 1

in regard tox^G_sat,o. This yields

x^G_sat,o= M^OX_sat,o^G

M^G+X_sat,o^G (M^O−M^G) and x^O_sat,o= 1−x^G_sat,o.

For gas saturated oil, we can now define the fugacity coefficient of the gas component in the oil phase using the definition of the fugacity coefficients (3.13), and inserting it into the condition for chemical equilibrium (3.12). Assuming that the capillary pressure is negligible, we get

Φ^G_o = Φ^G_g x^G_sat,o .

Mass Density of the Oil Phase

The presented black-oil parameters directly provide densitiesρsat,αfor all three saturated phases, but—somewhat surprisingly—we must take quite complex measures to defineρo, the mass density of potentially undersaturated oil. The reason for this is that—in contrast to the water and gas phases—the oil phase may dissolve less gas than the maximum physically possible amount but the black-oil parameters only specify oil phase densities for gas saturated oil. To defineρ_o, we take a closer look at the total derivative of the mass density of the saturated oil phase with respect to pressure. Applying the chain rule, we get

dρsat,o

dp_o = d

dp_oρo(po, X_sat,o^O (po), X_sat,o^G (po)) = ∂ρo

∂X_o^G

∂X_sat,o^G

∂p_o + ∂ρo

∂X_o^O

∂X_sat,o^O

∂p_o +∂ρo

∂p_o , whereX_sat,o^G (p_o)andX_sat,o^O (p_o)are the mass fractions of the gas and oil components of gas saturated oil at pressurep_oas specified by the black-oil parameters. Taking advantage of the fact that the mass fractions of all components sum up to one for oil, we get

dρsat,o

dp_o = ∂X_sat,o^G

∂p_o

∂ρo

∂X_o^G − ∂ρo

∂X_o^O

+∂ρo

∂p_o . (3.34)

Further, we now suppose that the compressibility of the oil phase is a given constant,i.e.,

∂ρ_o

∂po

:= const

holds, and we already assumed the mass fraction of gas in saturated oil at atmospheric pressure to be negligible. Under these preconditions, we may use the density of the oil at the surface together with the compressibility to define the partial derivative of the mass density of the oil phase regarding the mass fraction of the oil component,i.e.,

∂ρ_o

∂X_o^O :=ρsat,o(patm)

1 +∂ρ_o

∂po

(po−patm)

.

Chapter Synopsis 47

We now observe that Equation 3.34 determines the partial derivative of the gas mass fraction in saturated oil using the black-oil parameters. Resolving, we get

∂ρ_o

∂X_o^G = ∂X_sat,o^G

∂po

!−1

dρ_sat,o dpo

−∂ρ_o

∂po

+ ∂ρ_o

∂X_o^O .

Assuming that these derivatives stay constant for undersaturated oil, we can finally define the generic mass density of oil as:

ρ_o=ρ_sat,o(p_o) + X_o^G−X_sat,o^G ∂ρo

∂X_o^G + X_o^O−X_sat,o^O ∂ρo

∂X_o^O . Black-Oil Model Constraints

Now that we defined all relevant quantities using only the black-oil parameters as input, we need to specify the appropriate model constraints in order to use the abstract thermodynamic framework presented in this chapter. First, we observe that we can use the immiscibility model constraints for the water and for the gas phases,i.e.,

X

κ

x^κ_α =x^κ_α^α = 1

forα ∈ {w, g}. For the oil phase, we assume that it is always potentially present, which leads to the constraint

X

κ

x^κ_o =x^O_o +x^G_o = 1.

Like for the immiscibility model assumptions, we cannot use some of the presented defi-nitions directly in software implementations since some of the fugacity coefficients exhibit infinite values. Thus, we may either approximate the black-oil model by using very large fugacity coefficients, or we can implement it by directly incorporating them into the software.

Im Dokument Theory and numerical applications of compositional multi-phase flow in porous media (Seite 73-77)