Additional Quality Controls

In this section, several quality checks that are used in various PVT experiments like CCE, CVD, and DLE or sample validations are presented.

Z-factor Quality Check

To estimate the gas Z-factor, Hall and Yarborough correlations are used. The inputs of the correlations are pseudocritical properties of the gas mixture, \(T_{pc}\) and \(p_{pc}\). To calculate them, we use Sutton correlations, Standing correlations, or Kay's mixing rule. For more information, see the pages Z-Factor and Real Gas Law.

When gas composition is available, Kay's mixing rule is usually applied¹.

$$\begin{equation}\label{eq:kays_mixing_tpc} T_{pc} = \sum^{N}_{i=1}y_i T_{ci} \end{equation}$$

$$\begin{equation}\label{eq:kays_mixing_ppc} p_{pc} = \sum^{N}_{i=1}y_i p_{ci} \end{equation}$$

where the pseudocritical properties of \(\text{C}_{7+}\) can be estimated from Matthews et al. correlations²,

$$\begin{equation}\label{eq:matthews_tpc} T_{c \text{C}_{7+}} = 608 + 364 \log \Bigr(M_{\text{C}_{7+}}-71.2 \Big)\\ \quad \quad \quad +\Bigr(2450 \log M_{\text{C}_{7+}} -3800\Bigr)\log\gamma_{\text{C}_{7+}} \end{equation}$$

$$\begin{equation}\label{eq:matthews_ppc} p_{c \text{C}_{7+}} = 1188 -431 \log \Bigr(M_{\text{C}_{7+}}-61.1 \Bigr)\\ \quad \quad \quad +\Biggr[2319 -852 \log \Bigr(M_{\text{C}_{7+}}-53.7\Bigr)\Biggr] \Bigr(\gamma_{\text{C}_{7+}}-0.8\Bigr) \end{equation}$$

When significant quantities of \(\text{CO}_2\) and \(\text{H}_2\text{S}\) nonhydrocarbon components are present, pseudocritical properties are corrected using Wichert and Aziz correlations³.

$$\begin{equation}\label{eq:wichert_tpc} T_{pc} = T^{*}_{pc} - \epsilon \end{equation}$$

$$\begin{equation}\label{eq:wichert_ppc} p_{pc} = \frac{p^{*}_{pc} (T^{*}_{pc} - \epsilon)}{T^{*}_{pc} + y_{\text{H}_2\text{S}}(1-y_{\text{H}_2\text{S}})\epsilon} \end{equation}$$

where, $$\begin{equation}\label{eq:wichert_epsilon} \epsilon = 120 \Biggr[\Bigr(y_{\text{CO}_2} + y_{\text{H}_2\text{S}}\Bigr)^{0.9}-\Bigr(y_{\text{CO}_2} + y_{\text{H}_2\text{S}}\Bigr)^{1.6}\Biggr] \\ + 15 \Bigr(y_{\text{H}_2\text{S}}^{0.5}-y_{\text{H}_2\text{S}}^4\Bigr). \end{equation}$$

If only the gas gravity and nonhydrocarbon content are known, the hydrocarbon specific gravity is first calculated from

$$\begin{equation}\label{eq:hc_gravity} \gamma_{g\text{HC}} = \frac{\gamma_g - (y_{\text{N}_2}M_{\text{N}_2} +y_{\text{CO}_2}M_{\text{CO}_2} +y_{\text{H}_2\text{S}}M_{\text{H}_2\text{S}}) /M_{\text{air}}}{1-y_{\text{N}_2}-y_{\text{CO}_2}-y_{\text{H}_2\text{S}}}. \end{equation}$$

Hydrocarbon pseudocritical properties are then calculated using Sutton correlations. These values are adjusted for nonhydrocarbon content based on Kay's mixing rule. $$\begin{equation}\label{eq:hc_tpc} T_{pc}^{*} = (1-y_{\text{N}_2}-y_{\text{CO}_2} -y_{\text{H}_2\text{S}})T_{pc\text{HC}} \\ \quad \quad \quad + y_{\text{N}_2}T_{c\text{N}_2} +y_{\text{CO}_2}T_{c\text{CO}_2}+y_{\text{H}_2\text{S}}T_{c\text{H}_2\text{S}} \end{equation}$$

$$\begin{equation}\label{eq:hc_ppc} p_{pc}^{*} = (1-y_{\text{N}_2}-y_{\text{CO}_2} -y_{\text{H}_2\text{S}})p_{pc\text{HC}} \\ \quad \quad \quad + y_{\text{N}_2}p_{c\text{N}_2} +y_{\text{CO}_2}p_{c\text{CO}_2}+y_{\text{H}_2\text{S}}p_{c\text{H}_2\text{S}} \end{equation}$$

Hall and Yarborough correlations⁴ are used to calculate the Z-factor. First, the following parameters are calculated.

$$\begin{equation}\label{eq:hall_t} t=\frac{1}{T_{pr}} \end{equation}$$

$$\begin{equation}\label{eq:hall_alpha} \alpha = 0.06125\ t \exp\Bigr[-1.2(1-t)^2 \Bigr] \end{equation}$$

Then, the reduced-density parameter, \(y\), is obtained by solving $$\begin{equation}\label{eq:hall_y} f(y) = -\alpha p_{pr} + \frac{y+y^2+y^3-y^4}{(1-y)^3}\\ \quad \quad -(14.76t-9.76t^2+4.58t^3)y^2\\ \quad \quad +(90.7t-242.2t^2+42.4t^3)y^{2.18+2.82t} \\ \quad \quad \quad \quad = 0 \end{equation}$$

with

$$\begin{equation}\label{eq:hall_yprime} \frac{\text{d}f(y)}{\text{d}y} = \frac{1+4y+4y^2-4y^3+y^4}{(1-y)^4}\\ \quad \quad -(29.52t-19.52t^2+9.16t^3)y\\ \quad \quad +(2.18+2.82t)(90.7t-242.2t^2+42.4t^3)y^{1.18+2.82t}. \end{equation}$$

An initial value of \(y = 0.001\) can be used with a Newton-Raphson procedure to solve Equation \eqref{eq:hall_y} for \(y\)¹. Then, Z-factor is given by

$$\begin{equation}\label{eq:hall_z} Z = \frac{\alpha p_{pr}}{y} \end{equation}$$

Figure 1 shows an example of Z-factor QC for the data given in this Excel file.

Figure 1: An example of Z-factor QC.

Hoffman et al. K-value Quality Check

Hoffman et al. proposed correlations for prediction of \(K\)-values. Standing used the Hoffman et al. method to generate a low-pressure \(K\)-value equation for surface-separator calculations with \(p_{\text{sep}} < 1000 \text{ psia}\) and \(T_{\text{sep}} < 200\ ^{\circ}\text{F}\). The Standing equations are given below.

First, the parameters of the correlation are calculated.

$$\begin{equation}\label{eq:standing_a0} A_0(p) = 1.2 + (4.5 \times 10^{-4}) p + (15 \times 10^{-8}) p^2 \end{equation}$$

$$\begin{equation}\label{eq:standing_a1} A_1(p) = 0.890 - (1.7 \times 10^{-4})p - (3.5 \times 10^{-8}) p^2 \end{equation}$$

$$\begin{equation}\label{eq:standing_bi} b_i = \frac{\log(p_{ci}/p_{\text{sc}})}{1/T_{bi}-1/T_{ci}} \end{equation}$$

$$\begin{equation}\label{eq:standing_fi} F_i = b_i \Big(\frac{1}{T_{bi}}-\frac{1}{T} \Big) \end{equation}$$

For all components up to $\text{C}_6$, Table 1 is used for parameters \(b_i\) and \(T_{bi}\).

Table 1 Values of \(b_i\) and \(T_{bi}\) for use in Standing low-pressure \(K\)-value correlation¹.

Component	\(b_i\)(cycle-\(^\circ\)R)	\(T_{bi}\) (\(^\circ\)R)
N\(_2\)	470	109
CO\(_2\)	652	194
H\(_2\)S	1136	331
C\(_1\)	300	94
C\(_2\)	1145	303
C\(_3\)	1799	416
i-C\(_4\)	2037	471
n-C\(_4\)	2153	491
i-C\(_5\)	2368	542
n-C\(_5\)	2480	557
C\(_6\)	2738	610

For \(\text{C}_{7+}\) fraction, the following equations are used.

$$\begin{equation}\label{eq:standing_nC7p} n_{\text{C}_{7+}} = 7.3 + 0.00075\ T + 0.0016\ p \end{equation}$$

$$\begin{equation}\label{eq:standing_bC7p} b_{\text{C}_{7+}} = 1013 + 324\ n_{\text{C}_{7+}} - 4.25\ n_{\text{C}_{7+}}^2 \end{equation}$$

$$\begin{equation}\label{eq:standing_tbC7p} T_{b\text{C}_{7+}} = 301 + 59.85\ n_{\text{C}_{7+}} - 0.971\ n_{\text{C}_{7+}}^2 \end{equation}$$

\(K\)-values are then estimated using

$$\begin{equation}\label{eq:standing_Ki} K_i = \frac{1}{p_\text{sep}} 10^{(A_0 +A_1 F_i )}. \end{equation}$$

Figure 2 shows an example of Hoffman et al. quality check for a separator test given in this Excel file.

Figure 2: An example of Hoffman et al. QC for a separator test .

References

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1. C. H. Whitson and M. R. Brulé. Phase behavior. Volume 20. Henry L. Doherty Memorial Fund of AIME, Society of Petroleum Engineers …, 2000. ↩↩↩

2. TA Matthews, CH Roland, and DL Katz. High pressure gas measurement. Petroleum Refiner, 1942. ↩

3. Edward Wichert and Khalid Aziz. Compressibility factor of sour natural gases. The Canadian Journal of Chemical Engineering, 49$2$:267–273, 1971. ↩

4. K. R. Hall and L. Yarborough. A new equation of state for z-factor calculations. Oil Gas J, 71:82, 1973. ↩