pvt

PVT Analysis

Rv is known as the Oil-Gas-Ratio (OGR) or Condensate-Gas-Ratio (CGR). Substituting (9.3) and (9.4) into (9.2) and re-arranging gives:

	L		1		ost Rs gst
	(9.5)		Bo
			1		gst Rv ost
	V
			Bg

Given the (Bo, Rs), (Bg, Rv) as a function of pressure and the ( ost, gst) along with the liquid and vapour viscosities, we have all the information we need to calculate the reservoir static fluid properties – the BO tables.

In this general case, the stock tank densities are calculated as the average of the densities produced from flashing the reservoir liquid and reservoir vapour through the production system:

(9.6)	st F		st	(1 F ) st		F	VoL
	o	o	oL	o	oV	o	VoL VoV
(9.7)	st F st			(1 F ) st		F	VgV
	g	g	gV	g	gL	g	VgV VgL

The traditional BO formulation ignores the OGR, Rv. It is assumed that reservoir free gas [vapour] does not yield any liquids when brought to surface. That is, it is the same gas as the surface gas [STG] and that the properties of the STO and STG do not vary with time. This is thought to be a reasonable approximation for crude oils with an initial Solution GOR of 750 scf/STB or less. If the Solution GOR exceeds 1000 scf/STB, the STO gravity will vary with time and the fraction of STO produced from the reservoir vapour increases from zero to something approaching 90%. Hence, the need for the generalized BO table construction method.

The majority of BO reservoir simulators assume:

	(9.8)	oLst	oVst	ost	constant
		gVst	gLst	gst	constant

This may be a poor set of assumptions for a volatile oil, hence some authors have developed techniques where the BO pressure-dependent properties are adjusted to account for the non-variability of the stock tank densities, i.e. Coats.

Within a PVT program, we use one of the three depletion experiments:

CCE

CVD

DLE

to define the reservoir performance. Clearly, for a crude oil we use the DLE. For a gas condensate, because we assume that dropped-out liquid in the reservoir remains immobile due to relative permeability effects, the CVD would seem appropriate. The CCE doesn’t

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really simulate any reservoir process, however, it may valid for generating BO tables [as a function of temperature] to describe flow in a well or pipeline being modeled using the steady-state approximation, i.e. constant composition.

9.2 Compositional Modeling

The EoS models generated using the techniques outlined in sections 5-8 will typically have between 10 and 25 components. Can these models be exported directly into a compositional simulator? 10-components – yes. 25-components – no!

There are two main numerical procedures in a compositional simulator. Firstly, there is a mass conservation equation for each component, including water, which describes how that component moves around the field. Depending on the model formulation, an additional equation called the Volume Balance is then used to describe the variation in the pressure field. The second main computational effort is the Flash.

In section 6.1, we outlined the SS method for solving the Flash. It was pointed out that reservoir simulators don’t generally use this technique. Instead they employ Newtonbased techniques because they usually have a good initial estimate, i.e. the last time steps solution, and consequently a Newton will find the new solution most quickly. However, in order to use a Newton, we have to store the derivatives of the equal chemical potential condition with respect to the component mole fractions, which is a matrix of order N2. Then we must invert this N N matrix for each time step on each grid cell, where N is the number of components. Numerical analysis tells us that computationally this is an N3 operation.

For small and medium sized compositional problems in which the number of active grid cells is less than 50,000, the Flash will be the dominant computational effort. Anywhere between 50 and 80% of the total CPU will be spent in the Flash. Therefore, if we can reduce the number of components, we reduce the memory requirement in proportion to N2 and the CPU time in proportion to N3.

9.2.1 Grouping

The technique used to reduce the number of components is called grouping or pseudoization. Essentially, it consists of identifying components whose behaviour is so similar that by adding them, the predictions of the reduced EoS model are almost the same as the extended EoS in which the components are considered individually.

We saw in chapter 2 that isomerism makes the identification of hydrocarbon molecules containing six or more Carbon atoms a very time consuming and hence expensive process. Butane [C4] and Pentane [C5] are routinely reported in terms of their normal-

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paraffin’s and a single isomer, usually denoted iCN24. The properties of the normal C4/C5 and their isomers are so similar they are a natural for combination.

If the mole fractions of the inorganics N2, CO2 and H2S are small and they are not being considered as [a considerable part of any] injection fluid, they can be combined with one of the light hydrocarbons. N2 is very similar to C1 and the ratio of N2:C1 often exceeds 1:50 – this is a natural group. Clearly, this is not an option when modeling Ekofisk where N2 injection is being done as a way of slowing-down the compaction of the chalk. CO2 is most similar to C2 and these components should be considered as a potential group. Again, this is not an option in CO2-injection is being done as is common in the South West states of the USA. H2S is similar to C3 so that another group is possible but not for the super-giant Bab field in Abu Dhabi in which the H2S mole fraction varies from zero percent in the southwest to over 10% in the northeast.

Beyond this, care must taken. Depending on the application, a C2 plus C3 group and a C4 to C6 group is commonly used. If the C7+ plus fraction has been split into five pseudocomponents, it maybe viable to re-group the 1st/2nd pseudos and 3rd/4th pseudos. Regardless of the grouping scheme adopted, the ultimate test is that the predictions of the pseudoized system should be broadly similar to those of the original system.

Note the current trend is to work with more detailed compositional descriptions in the reservoir simulator. The Production and Process Engineers who require this more detailed description to perform their calculations and optimizations dictate this trend.

9.2.2 Mixing Rules

The simplest and easiest method of generating physical properties for a grouped component is via Kay’s rule:

(9.9)	J	z j j	z j
		j J	j J

where j is the set of components with the group J and is the usual set of properties, critical temperature, critical pressure, etc. The group specific gravity must be calculated from:

(9.10)	J z j M j	z j M j j
	j J	j J

BIPS for binary group I-J can be calculated from:

(9.11)	kIJ zi z j kij	zi z j
	i I j J	i I j J

Coats then suggests the aI and bI are determined from:

24 Remember the Alkane-Pentane has two isomers but one of these, neo-Pentane, is rarely found in naturally occurring petroleum – see section 2.2.1.

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zi z j ai a j (1 kij )

(9.12)

i I j J

aI

)2

,T

(RT

cI

I

(

I

)

pcI

rI

and:

bI

zi bi

(9.13)

i I

RTcI

pcI

The component ai and bi may include previously determined corrections via the regression process. Coats has shown it preserves the volumetric predictions made with the original EoS: Whitson recommends the method.

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References

Abramowitz, M., and Stegun, I. A., editors, “Handbook of Mathematical Functions”,

Washington D.C., National Bureau of Standards, Applied Mathematics Series-55, (1964).

Adkins, C. J.,

“Equilibrium Thermodynamics”, 3rd Edition, Cambridge University Press, 1985.

API (American Petroleum Institute) RP44, 1st Edition, “Recommended Practice for Sampling Petroleum Reservoir Fluids”, API, Dallas, Texas, January 1966.

Beggs, H. D.,

“Production Optimization”,

OGCI Publications, Tulsa, Oklahoma, 1991.

Bradley, H. B., Editor-in-Chief, “Petroleum Engineering Handbook”,

Society of Petroleum Engineers, Richardson, Texas, 1987.

Coats, K. H.,

“Simulation of Gas Condensate Reservoir Performance”, JPT, (Oct. 1985), pp. 1870-1886.

Dake, L. P.,

“Fundamentals of Reservoir Engineering”, Elsevier, Amsterdam, 1978.

Eyton, D. G. P.,

“Practical Limitations in Obtaining PVT Data for Gas Condensate Systems”,

SPE 15765, Presented at the 5th SPE Middle East Oil Show, Bahrain, March 7-10, 1987.

Firoozabadi, A.,

“Thermodynamics of Hydrocarbon Reservoirs”, McGraw-Hill, New York, 1999.

Hall, K. R. and Yarborough L.,

“A New Equation of State for Z-factor Calculations”, Oil and Gas J., (June 18, 1973), pp. 82-90.

Hoffman, A. E., Crump, J. S., and Hocott, C. R., “Equilibrium Constants for a Gas Condensate System”, Trans. AIME, (1960), 219, pp. 313-319.

Lohrenz, J., Bray, B. G., and Clark, C. R.,

“Calculating Viscosities of Reservoir Fluids from their Compositions”, JPT, Oct. 1964, pp. 1171-1176.

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McCain, W. D. Jr.,

“The Properties of Petroleum Fluids”, 2nd Edition, Penn Well Books, Tulsa, Oklahoma, 1990.

Pedersen, K. S., Fredenslund, A., Thomassen, P., “Properties of Oils and Natural Gases”,

Gulf Publishing Company, Houston, Texas, 1989.

Peneloux, A., Rauzy, E., and Freze, R.,

“A Consistent Correction for Redlich-Kwong-Soave Volumes”,

Fluid Phase Equilibria, 8, (1982).

Søreide, I.,

“Improved Phase Behavior Predictions of Petroleum Reservoir Fluids From a Cubic Equation of State”,

Ph.D. Thesis, Department of Petroleum Technology and Applied Geophysics, Norwegian Institute of Technology, Trondheim, April 1989.

Turner, R. G., Hubbard, M. G., and Dukler, A. E.,

“Analysis and Prediction of Minimum Flow Rate for Continuous Removal of Liquids from Gas Wells”,

JPT, November 1969.

UKOOA (United Kingdom Offshore Operators Association),

“Sampling and Analysing Gas/Condensate Reservoir Fluids”, Complied by Hearn, R. S., March 1986.

Wichert, E., and Aziz, K.

“Compressibility Factor of Sour Natural Gases”, Can. J. Chem. Eng., (1971), 49, p. 267.

Whitson, C.H.,

“Characterizing Hydrocarbon Plus Fractions”, SPEJ, (August 1983), pp. 683-694.

Whitson, C. H., and Michelsen, M. L., “The Negative Flash”,

Paper presented at the 5th International Conference on Fluid Properties and Phase Equilibria for Chemical Process Design, Banff, Alberta, (April 31 – May 5, 1989).

Whitson, C. H., and Torp, S. B.,

“Evaluating Constant Volume Depletion Data”, JPT, (March 1983), pp. 610-620, Trans. AIME, 275.

Wilson, G. M.,

“A Modified Redlich-Kwong Equation of State, Application to General Physical Data Calculations”,

Paper No. 15c, presented at AIChE 65th National Meeting, Cleveland, (May 4-7, 1969).

Young, L. C., and Hemanth-Kumar, K., “High Performance Black Oil Computations”,

Paper SPE 21215, presented at 11th SPE Symposium on Reservoir Simulation”, Anaheim, Feb. 17-20, 1991.

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Appendix A: Classical Thermodynamics

In order to calculate fluid properties, we need a thermodynamic model for fluid behaviour. However the model we require is difficult to develop. Derivations are often too mathematical, too abstract or both. Instead, we will quote some results and point the interest reader at the book by Firoozabadi.

A.1 Abstractions

The basis for the development of our thermodynamic model is sound, namely the law of conservation of energy. However, it is usually expressed as:

(A1)

dU dQ dW

U is the Internal Energy, Q is the Heat Content and W the Work Done. Of these quantities, only W is readily understandable. Imagine the piston of a frictionless cylinder is moved by a distance dx. If the cylinder’s cross-sectional area is A and its gas is at a pressure p, then:

(A2)	dW F.dx ( pA)dx pdV

F is the force moving the piston and dV is the volume change.

The change in heat content is related to the Temperature, T, by:

(A3)

dQ TdS

S is another abstract quantity called Entropy. Other quantities can be introduced such as Helmholtz Energy, A, Enthalpy, H, and Gibbs Free Energy (GFE), G. Why we need all these different abstract quantities becomes clearer when we consider what information we know in advance.

In all cases, we will know the total of feed composition, n = [n1,…,nN]. In the reservoir context or in a laboratory experiment, we will also know the pressure and temperature (p,T). In this case, it can be shown we minimize the GFE – the Isothermal Flash:

	N
(A4)	dG SdT Vdp i dni

i 1

where i is the Chemical Potential given by:

			G
	(A5)
		i	ni
				p,T ,n
				j

where nj indicates all component moles, except ni, are held constant. Chemical Potential will be explained shortly.

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In a producing well, at a given point the pressure and enthalpy are known. Here, we minimize the negative entropy [maximize the positive entropy] – the Isenthalpic Flash – from which we calculate the local temperature. The following table shows the various types of process that can be considered.

Given	Minimize	Name
p, T, n	G	Isothermal
p, H, n	-S	Isenthalpic
p, S, n	H	Isentropic
V, T, n	A	Isochoric

Table A1: Type of Flash Process Depending on Known Quantities.

In the reservoir context, when a fluid is flashed at some point, it may or may not undergo a phase transition. Generally, a phase transition will be accompanied by

A.2 Chemical Potential

Using a mathematical technique called the Reciprocity relationship [see Firoozabadi], from (A4) we can derive:

		V
(A6)						i
					p
		ni p,T ,n j					T ,n j

The term on the left-hand side of (A6) is defined as the Partial Molar Volume:

						V
	(A7)	V i
						ni p,T ,n
							j

Combining (A6) and (A7) gives:

(A8)	d i		dp T ,n
		Vi

Expression (A8) is an extremely important result since it provides a relationship between an abstract quantity, the Chemical Potential25, and the pressure and volume [and temperature and composition].

25 Sometimes called the Partial Molar Gibbs Energy.

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d i RTd ln fi

PVT Analysis

A.2.1 Fugacity

For an ideal gas, the [partial] molar volume is:

(A9) Vi RTp

Then, substituting (A9) into (A8) gives:

(A10)	d i	RT dp RTd ln p
		p

For a real gas or fluid, the real-pressure or Fugacity replaces the pressure p. Fugacity is defined by:

(A11)

And:

	Lim	f		i
(A12)
	p				1
		0 xi		p
where xi is the mole fraction of the ith component. The ratio fi								xi p i is known as
the Fugacity Coefficient.
Now subtract RTd (ln xi p) from both sides of (A11):
(A13)	d i RTd ln xi p RTd ln fi RTd ln xi p
Or:
(A14)	RTd ln i						dp RTd ln p
						Vi

In (A14), we have substituted (A8) for the first term on the right hand side and have
dropped the term RTd			ln xi					since we assume constant composition. This equation can
now be integrated to give:
(A15)	RT ln			p					RT
		i			V			i		dp
									p
				0

If we have an analytic EoS such as the cubic EoS discussed in Section 5.2, then we can substitute the appropriate expression for the partial molar volume into (A15) and perform the integral to give us an analytic expression for fugacity coefficient. In particular, the Martin’s generalized 2-parameter EoS gives rise to the following expression:

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B

1

A

A

B

Z m

B

(A16)

ln

ln Z B

i

Z 1

2

i

i

ln

2

m

m

A

i

B

2

B

B

Z m B

1

1

A.3 Equilibrium

Let’s consider a two-phase N-component system. The two-phases will be denoted using the superscript (1) and (2). The moles of each component must satisfy the material balance condition:

(A17)

ni ni(1)

ni(2)

Since the feed composition is fixed, differentiating (A17) gives:

(A18)

dni(1) dni(2)

At constant pressure and temperature, the change in GFE for the two phases will be:

		N		N
(A19)	dG(1)	i(1) dni(1)	dG(2)	i(2) dni(2)
		i 1		i 1

At equilibrium, dG(1) dG(2) 0 , which can only be satisfied if:

(A20)

i(1) i(2)

0

for

i 1, , N

From the definition of fugacity coefficient, (A20) is equivalent to:

(A21)

ln xi(1) ln i(1)

ln xi(2)

ln i(2)

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