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As an example of the technique in action, consider the following case where an oil and a gas condensate are simultaneously characterized. The mole percentage and mole weight of the C7+ plus fraction were given as:

Property	Oil	Condensate
z[7+] (mol%)	36.54	1.54
Mw[7+]	198.7	141.0

The resulting split into five pseudo-components in which the fifth pseudo-component was allocated the mole weight of M = 500.0 yielded the following:

	Composition/[mol%]
Pseudo-Comp	Oil	Condensate	Mole Weight
C7+(1)	3.4999	0.3365	98.55
C7+(2)	12.7400	0.9273	135.84
C7+(3)	13.2405	0.2646	206.65
C7+(4)	5.9994	0.0115	319.83
C7+(5)	1.0600	0.0001	500.00
	C7+ Distribution	Parameters
	1.562	1.901
	90.0	90.0
	69.590	26.959
0	32.435	32.435
[for (7.9)]	0.5863	1.2252
[for (7.14)]	0.5846	1.2218

The values of were chose to match other properties: we will discuss general regression procedures in Chapter 8.

7.2 Inspection Properties Estimation

We have already seen the trends in specific gravity and normal boiling point temperature shown by the hydrocarbons: see Table/Chart 2.1.

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Back in the 1930, Watson found that within a particular gas or oil mixture, the various constituents appeared to honour the relationship:

(7.15)	K			T 1 3
		w		b

where Tb is the normal boiling point temperature in degrees Rankine. Values of the Watson K-factor, Kw, vary between 8.5 and 13.5:

Type	Lower	Upper
Paraffinic	12.5	13.5
Napthanic	11.0	12.5
Aromatic	8.5	11.0

Table 7.1: Typical Values of Watson Kw for different fluid types.

There is some overlap in these values and a mixture of paraffinic and aromatic components will produce something that looks napthanic.

Nevertheless, within a particular fluid sample, there is remarkable consistency between the value of the Kw for the plus fraction and the values of the constituent parts. Special studies on two North Sea fluids showed for a gas condensate that Kw = 11.99 0.01 and for a volatile oil Kw = 11.90 0.01. Thus, the following scheme is suggested for determining the specific gravities and boiling points of the pseudo-components derived from the splitting procedure discussed in the previous section.

Given the plus fraction mole weight and specific gravity, (MN+, N+), we calculate the Watson factor from the following correlation due to Whitson [see Søreide]:

(7.16)

Kw 4.5579M 0.15178 0.84573

Then using this value of Kw, we calculate the pseudo-component specific gravities from a re-arrangement of (7.16):

(7.17)

i

6.0108M i0.17947 Kw1.18241

The boiling points are calculated from a re-arrangement of (7.15):

(7.18)

Tbi i Kw 3

Other more complex relationships between these properties have been proposed which have proved more or less accurate: again, see Søreide for details.

The Watson factor has an important role to play in quality checking reservoir samples. Whitson showed that the Watson factor of samples taken from the same field only varied by 0.01 units. So having defined some mean value and Whitson suggests a minimum of three samples be used for this purpose, any new sample whose Kw differs from the

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established mean is probably in error. plus fraction mole-weight, adjusting sample failing this quality check.

Given the errors associated with measuring the this parameter might be appropriate for a new

7.3 Critical Property Estimation

In table 3.1, we saw how the physical properties of the first few members of the alkane series obeyed the same sort of trends we saw for the specific gravities and boiling point temperatures. Many authors have therefore suggested correlations of the form:

(7.19)

i

( i ,Tbi )

includes the set of critical pressure, critical temperature, critical volume and acentric factor. The popular correlations are due to Kesler and Lee, Cavett, Riazi and Daubert, Edmister, Twu and Søreide: see Søreide for details.

7.3.1 Normal Boiling Point Temperature

The correlation due to Riazi and Daubert firstly calculates the component normal boiling point temperatures from the values for mole weight and specific gravity:

(7.20) Tbi 6.7786M i0.40167 i 1.58262 exp 3.7741 10 3 M i 2.9840 i 4.2529 10 3 M i i

7.3.2 Critical Temperature, Critical Pressure

Given the normal point points, an equation of the form (7.19) was generated:

(7.21)	a exp bT	c dT T e f
	b	b b

where refers to the properties Tc or pc and the six constants in (7.21) are shown in the table below. Values for the Tb must be supplied in degrees Rankine. The resulting values of Tc and pc will be in degrees Rankine and psia, respectively.

	Tc	pc
a	10.6443	6.162E+06
b	-5.1747E-04	-4.725E-03
c	-0.54444	-4.8014
d	3.5995E-04	3.194E-03
e	0.81067	-0.4844
f	0.53691	4.0846

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7.3.3 Critical Volume

Critical Volume may be estimated from another correlation due to Riazi and Daubert of the form:

(7.22)	V	ci	7.0434 10	7 T 2.3829 1.683
				bi	i

Again, Tb’s must be specified in degrees Rankine: the resulting Vc’s will be in ft3/lbmole.

7.3.4 Acentric Factor

Acentric factors are routinely calculated from one of two correlations due to Edmister and Kesler-Lee. The Edmister correlation is:

							p	ci
			log
		3			14.7
(7.23)	i									1
			T
		7
				ci
			T				1
				bi

The Kesler-Lee correlation, which is recommended, depends on the value of reduced boiling point:

T

(7.24) Tbr Tb

c

For values of Tbr < 0.8, then:

p

ci

1

1.28862lnT

0.169347T 6

ln

5.92714 6.09648

14.7

Tbri

bri

bri

(7.25)

i

15.2518 15.6875

1

13.4721lnT

0.43577T 6

Tbri

bri

bri

For values of Tbr > 0.8, then:

(7.26)

7.904 0.1352K

0.007465K 2

8.359T

(1.408 0.01063K

)T 1

i

wi

wi

wi

bri

bri

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8.Regression

Suppose we have split our plus fraction into 3-5 pseudo-components and used the techniques in sections 7.2 and 7.3 to assign physical properties to them. Can this fluid description be used with the EoS models in section 5 to simulate the laboratory experiments discussed in section 4 - No! Typically, saturation pressure can be predicted to about 10%, densities to 5% and compositions to 10%. Why is the case?

1.Insufficient detail regarding the make-up of the plus fraction

2.Inaccurate physical properties for the plus fraction pseudo-components

3.Errors in the compositional determination and/or laboratory measurements

4.The cubic EoS is only an approximation to the real fluid behaviour

The compositional determination and laboratory experiments can be checked to some degree: a couple of these procedures are discussed in sections 4.1.4 and 4.2.6. Assuming these and other rationality21 checks have been performed and there are still discrepancies between theory and measurement, what do we do? The generally accepted procedure is to regress the EoS model. That is, change some of the parameters of the model to minimize some measure of the difference between theoretical and observed behaviour.

There are generally three parts to an optimization problem such as a minimization. They are:

1.An objective function

2.A number of degrees of freedom, i.e. parameters to vary

3.Constraints, i.e. physical limitations on the variability of the parameters We will discuss these elements in turn.

8.1 Objective Function

The objective function is the single [scalar] variable we will construct and will measure the goodness of fit between our model and the measured data. Generally, the sum of squares error is used in EoS modeling.

Suppose we have a number of measured data, denoted yiobs, i = 1, , M, for which our EoS predicts the equivalent values, yiEoS. On an item-by-item basis, we can define the following residual:

21 Do the experimental results change in a predictable way? Does the results from this sample agree with those from a similar sample, if one exists?

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	(8.1)	ri	wi	( yiObs yiEoS )
				yiEoS

The wi are weighting factors which we will discuss shortly. Summing the square of the residuals from (8.1) gives us our objective function:

		1	M
(8.2)	f (x)		ri	2
		2 i 1

The factor of 0.5 is included for subsequent algebraic convenience. The vector x indicates that a number of parameters or degrees of freedom are available for adjustment. Changing one or more of these parameters will change the yiEoS: this is the mechanism by which we seek to minimize f.

In order to minimize f, we need to ensure22:

				f	M	r
	(8.3)	g j			ri	i	0	i 1,..., N
				x j		x j
					i 1

The derivatives of the residuals with respect to the set of approximated by finite differences as:

		J			r		ri (x; x j x j ) ri (x)
	(8.4)		ij		i
					x j		x j

N M

parameters, x, can be

Each of variables is perturbed and then re-set, in turn, by xj = xj: is some small number, say 10-5. The elements of (8.4) are usually called the Jacobian elements. The set of N-equations in (8.3), called the gradient, can be solved by a variety of Newton and quasi-Newton techniques. Assuming the residuals are normally distributed, i.e. there are as many positive errors as negative errors and they have the same spread of error, we can solve (8.3) by:

		N	g j
	(8.5)				xk g j		0
		k 1	xk
	where:
				g j		M
	(8.6)	H jk				Jij Jik
				xk
						i 1

To ensure the solution to (8.3) is a minimum and not a maximum or turning point, we require the Hessian matrix, (8.6), to be positive definite23. That is the eigenvalues of the Hessian should all be positive: see any standard undergraduate text on mathematics.

22The necessary condition.

23The sufficient condition.

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8.2 Variable Choice

Prior to the general introduction of the volume translation technique, see section 5.4, volumetric properties predicted by the common EoS could require some EoS parameters to be changed by 30-40%! With the use of volume translation, the changes required should be only 10%.

Common sense suggests that since the majority of the uncertainty connected with the fluid characterization derives from the plus fraction, we should concentrate our efforts here. Various authors have suggested different combinations of parameters connected with the plus fraction. Whitson suggests the EoS-multipliers for the A- and B-

coefficients, which we denoted A , B , should be selected: one pair for each pseudocomponent split-out of the plus fraction. We would recommend this approach.

As an alternative, the user might want to consider the parameters in the GDM. In particular, we have discussed back in section 4.2.1 that errors in the mole weight can be10%. The value of the distribution parameter, , is generally unknown. To complete the set we might consider the plus-fraction specific gravity. This approach has some appeal as we are concentrating our efforts on the real measurements, namely the plusfraction molar distribution rather than adjusting the properties of an inappropriate distribution.

Viscosities should be matched as a separate exercise once the match to phase and volumetric behaviour has been obtained. The standard model used for viscosity prediction, due to Lohrenz, Bray and Clark, is a fourth order polynomial in reduced molar density:

	* 0.0001 1 4	4
(8.7)		ak rk

k 0

Adjustment of either the component critical volumes or the coefficients of the model [a0, a1, a2, a3, and a4] are recommended. The reduced molar density is calculated from:

V mix

(8.8) r VcEoS

m

The mixture critical volume is estimated from a linear mixing rule:

	N
(8.9)	Vcmix ziVci

i 1

See Lohrenz et al. for details.

In the past, it was usually possible to achieve a physically consistent match to reliable laboratory measurements for a single sample. Multi-sample matching involving fluids from different parts of the reservoir, i.e. gas cap and oil leg, was not so easy. By using the modified Whitson GDM, multi-sample matching should not present a problem.

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8.3 Constraints

It is often possible to achieve a near-perfect match to the set available measured data only to find when the tuned EoS is applied at some combination of conditions not represented with the matching set its predictions are poor, or even worse non-physical. This is often because the parameters selected have been allowed to vary too far from their initial values. Generally, constraints should be applied and despite the temptations, they should not be relaxed from their default settings. The programmer based on the experience of many users has usually set these defaults.

The quality check performed on the CVD experiment suggests we plot the component K- values as a function of pressure. The K-values should vary in a smooth, monotonic, noncrossing fashion with the largest K-values corresponding to the most volatile components, etc. Whitson has suggested the tuned EoS should behave in a similar way.

On those rare occasions where there is a surplice of measured data, some might be held back to be used as quality check for the EoS tuned to the other measured data.

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9.Export for Simulation

The two major classes of reservoir simulation performed today are:

1.Black Oil (BO)

2.Compositional (EoS)

Young and Hemanth-Kumar showed that the BO model maybe considered as a special case of a 2-component EoS model. In section 9.1, we will see how to generate a BO model suitable for use in a reservoir simulator of that type.

The EoS models developed in the previous chapters could be used directly in simulation models like MORE and Eclipse 300. However, a 15-component system derived from a system consisting of N2, CO2, C1, C6 and 5 pseudo-components generated from C7+ split would generally be considered as computationally too expensive. A technique called pseudoization or grouping must be considered. This is discussed in section 9.2.

9.1 Black Oil Modeling

In a BO model of a hydrocarbon fluid, we represent the system by 2-components which are identified as Stock Tank Gas (STG) and Stock Tank Oil (STO). These components are generated by the production system when well stream fluid is processed through the separator train, ultimately yielding the stock tank products: see the diagram in section 4.2.4. From the modeling point of viewing, this process is the mechanism by which we turn moles of well stream into volume of STO and STG.

Let us consider the most general case where we have a two-phase reservoir system consisting of a mass mL of liquid and mV of vapour. If produced to the surface, the liquid will yield a mass of moL of STO and a mass of mgL STG whilst the vapour will yield a mass of moV of STO and a mass of mgV of STG according to:

	(9.1)	mL moL mgl
		mV moV mgV

by the law of conservation of mass. (9.1) can be written in terms of reservoir and surface volumes:

		L	V	L	stV	oL	stV	gL
(9.2)					o		g
			V		stV		stV
		V				oV			gV
				V	o		g

where ( L, V) are the reservoir liquid and vapour densities and ( ost, gst) are the STO and STG densities.

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Figure 42: Schematic of the Generalized BO Table Construction.

Bo reservoir volumes of liquid liberates 1 surface volume of STO and Rs surface volumes of [dissolved] STG:

	(9.3)	Bo	VL	Rs	VoV
			Vol		VoL

By analogy, Bg reservoir volumes of vapour, liberates1.0 surface volume of STG and Rv surface volumes of [vapourized] STO:

(9.4)	B	g		VV	R	VoV
			VgV		V	VgV

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