PVTsim users often ask why we do not assign any importance to Hoffmann plots in the QC module of PVTsim Nova.
The idea of a Hoffmann plot is to validate whether a gas and a liquid separator sample are in equilibrium at separator conditions.
The below table shows a recombined reservoir fluid composition for a volatile oil and the separator gas and liquid compositions at 69 bar and 65^{o}C. The recombined reservoir fluid composition will only be representative of the reservoir fluid if the separator gas and separator liquid compositions are in thermodynamic equilibrium at the separator conditions. It would be useful to have a method to clarify if that is the case.
Component

Recombined Reservoir fluid mol %

Separator Gas Mol%

Separator Liquid Mol%

N2

1.775

2.320

0.226

CO2

2.929

3.390

1.620

H2S

11.450

10.669

13.667

C1

61.314

76.443

18.340

C2

2.707

2.873

2.235

C3

2.145

1.786

3.166

iC4

0.772

0.498

1.550

nC4

1.521

0.851

3.426

iC5

0.872

0.332

2.407

nC5

0.872

0.289

2.527

C6

1.470

0.268

4.885

C7

1.666

0.134

5.871

C8

1.783

0.083

6.529

C9

1.520

0.040

5.609

C10+

7.204

0.025

27.944

Recombined reservoir fluid and separator gas and liquid compositions at 69 bar and 65 ^{o}C. The recombined fluid has a C_{7+} molecular weight of 181.5 and a C_{7+} density of 0.824 g/cm^{3 }
The method of Hoffmann et al. was published in 1953 in a paper entitled “Equilibrium Constants for a GasCondensate System” (Petroleum Transactions, AIME).
We have dug into the theory behind Hoffmann plots and found it is based on simple and approximate correlations that are otherwise no longer used in oil industry.
The Hoffmann plot assumes that a linear relation exists for
where i is a component index and
K Equilibrium constant (ratio of component mole fractions in gas and liquid)
P Pressure
P_{c} Critical pressure
T Absolute temperature
T_{B} Boiling temperature at atmospheric pressure
T_{c} Critical temperature
A Hoffmann plot for the separator gas and liquid in the above table is shown below. A perfect Hoffmann plot would have all the blue dots on the orange line, which criterion is not completely fulfilled on the below plot
Hoffmann plot for volatile oil separated at 69 bar and 65^{o}C.
To arrive at the linearity in Equation (1), the pure component vapor pressure of component i is assumed to follow the below simplified Antoine equation (1888)
where a_{i} and b_{i} are constants specific for component i.
At the normal boiling point (T_{B}) the vapor pressure equals atmospheric pressure (1 atm), giving
This allows the Antoine equation to be rewritten to
At the critical point the vapor pressure equals P_{c} and the temperature is T_{c}. This gives the following expression for b_{i}
For an ideal liquid mixture and an ideal gas, the Kfactor of component i can be calculated from Raoult’s law (1887)
which means
An ideal mixture is one, where there is no difference between the interaction between two molecules of the same chemical species and two molecules of different chemical species.
Combining Equation (8) and Equation (5) gives
This is a special case of the Hoffmann relation where the straight line assumed in the Hoffmann method follows the equation Y=X. The righthand side is independent of pressure, and a pressure correction is needed to arrive at the final Hoffmann relation.
Poynting (~1900) introduced a pressure correction through what he called a modified saturation pressure as defined below. It was later known as the pure component fugacity
is a socalled fugacity coefficient. Replacing in Equation (8) by
assume
where Con_A and Con_B are constants independent of component index. Combined with Equation (5) that gives
which reproduces Equation (1).
Several questionable approximations have been made, which makes the Hoffmann relation highly uncertain:
 The pure component vapor pressure does not in general follow the Antoine equation
 A separator gas is not an ideal gas
 A separator oil is not an ideal mixture.
 The Poynting correction is an approximation and component dependent.
Looking at the above Hoffmann plot, one may get the impression that the assumed linear relation is close to being fulfilled, but that is not quite true. The below table shows the composition of the sampled separator gas and of the gas composition you would get with the Hoffman relation, i.e. if all the blue dots on the above Hoffman plot were moved to be on the orange line. The Hoffmann gas composition deviates significantly from the measured one.
Component

Separator Gas
Mol%

Hoffmann Gas
Mol%

%Dev

N2

2.32

1.735

25.2

CO2

3.39

3.015

11.1

H2S

10.669

10.670

0.0

C1

76.443

78.383

2.5

C2

2.873

2.642

8.0

C3

1.786

1.497

16.2

iC4

0.498

0.387

22.3

nC4

0.851

0.698

18.0

iC5

0.332

0.256

22.8

nC5

0.289

0.221

23.5

C6

0.268

0.212

20.9

Measured separator gas composition at 69 bar and 65^{o}C and the one that obeys the Hoffmann relation.
The Hoffmann method was good practice at a time when computers and cubic equations of state (EoS) had not yet made their way into oil industry. Cubic equations were known at the time Hoffmann presented his method, but phase equilibrium calculations using cubic equations were out of reach until computers in the seventies became a widely used engineering tool in oil industry.
Today, more refined EoSbased QC techniques exist. Within seconds, it is possible to perform EoSbased flash and phase equilibrium calculations, which are much more accurate than the correlations forming the basis of the Hoffmann plot.
The below plot shows phase envelopes of the separator gas and separator liquid simulated using a cubic EoS. To be in equilibrium at the separator conditions the phase envelopes must have a point of intersection at the separator temperature and pressure (65^{o}C and 69 bar), which is seen to be the case.
QC check for separator conditions. The phase envelopes of a gas and liquid in equilibrium at the separator conditions must intersect at the separator T and P.
With a cubic equation, it is straight forward to perform a PT flash calculation of the recombined reservoir fluid composition at separator conditions and compare the simulated Kfactors with those derived from the sampled separator composition. A plot of experimental Kfactors versus simulated Kfactors should give a straight line with Y=X as illustrated in the below figure.
Experimental versus simulated Kfactors at separator conditions.
The two latter plots are examples of the QC checks performed by the QC module in PVTsim Nova.
Read Previous Tech Talks