Wet gas over-reading correction for ultrasonic flow meters

Abstract

Oil and gas operators rely on accurate flow rate measurements to optimize production and generate more from their reservoirs, particularly in wet gas fields. A cost-effective solution for these flow measurements is the use of single-phase measurement technologies with an over-reading correction that corrects the gas flow rate for the presence of the liquid phase. Traditional flow measurement technologies in wet gas fields are Venturi meters and orifice plate meters that involve differential pressure measurements. Over the years, a higher installed base of ultrasonic flow meters is observed in wet gas fields. Ultrasonic flow meters have advantages over conventional wet gas technologies; however, an over-reading correction method for this measurement technology has not yet been derived. The current work is a first attempt to devise a correction method based on a large data set of ultrasonic measurements in horizontal configuration at conditions comparable to field applications. The correction method is a physical model for the gas void fraction and is based on the dominant dimensionless numbers in wet gas flows that originate from the fundamental equations of multiphase flow dynamics. This approach leads to the definition of the over-reading correction in different flow regimes in terms of these dimensionless numbers and is supported by an extensive set of measurement data and evidence from visual observation of the flow patterns. The correction method is capable of correcting the ultrasonic over-reading with a resulting uncertainty of about 4% for a 95% confidence interval for a range of conditions relevant to the oil and gas industry.

Graphical abstract

Appendix: Equations for stratified gas–liquid flows

The derivation of the stratified gas–liquid flow equations is elaborated here and follows largely the derivation in the seminal paper of Taitel and Dukler (1976) with modifications proposed by Barnea and Taitel (1993) and Tae-Hwan et al. (2015). It considers a steady-state two-phase gas–liquid flow in a horizontal circular conduit, as depicted in Fig. 6, in the axial direction denoted by x. The dimensional momentum balance equations for gas and liquid in a pipe with unit cross-sectional area are:

$$\begin{aligned} -\frac{\mathrm{d}p}{\mathrm{d} x}-\tau _{\mathrm{gw}}\frac{S_{\mathrm{g}}}{\alpha _{\mathrm{g}}}-\tau _{i}\frac{S_i}{\alpha _{\mathrm{g}}}&= 0,\nonumber \\ -\frac{\mathrm{d}p}{\mathrm{d} x}-\tau _{\mathrm{lw}}\frac{S_{\mathrm{l}}}{\alpha _{\mathrm{l}}}+\tau _{i}\frac{S_i}{\alpha _{\mathrm{l}}}&= 0. \end{aligned}$$

(24)

Combining both equations leads to the shear stress balance:

$$\begin{aligned} \tau _{\mathrm{gw}}\frac{S_{\mathrm{g}}}{\alpha _{\mathrm{g}}}- \tau _{\mathrm{lw}}\frac{S_{\mathrm{l}}}{\alpha _{\mathrm{l}}}+\tau _i S_i\left( \frac{1}{\alpha _{\mathrm{g}}}+\frac{1}{\alpha _{\mathrm{l}}}\right) =0. \end{aligned}$$

(25)

The shear stresses are evaluated as

$$\begin{aligned} \tau _{k\mathrm{w}}=f_k\frac{\rho _k u_k^2}{2},\quad \quad \text{for}\quad k=\mathrm{g},\mathrm{l};\qquad \tau _i=f_i\frac{\rho _{\mathrm{g}} (u_{\mathrm{g}}-u_{\mathrm{l}})^2}{2}, \end{aligned}$$

(26)

where $u_k$ is the velocity of phase k. The k-phase friction factors for turbulent flow is given by

$$\begin{aligned} f_k=\frac{0.046}{\widehat{{\textit{Re}}}_k^{\frac{1}{4}}}. \end{aligned}$$

(27)

The Reynolds number, $\widehat{{\textit{Re}}_k}$, for the k-phase is calculated based on the actual velocity and the hydraulic diameter of phase k:

$$\begin{aligned} \widehat{{\textit{Re}}}_k=\frac{u_k D_{\mathrm{H},k}}{\nu _k}={\textit{Re}}_k\frac{D_{\mathrm{H},k}}{D}\alpha _k \end{aligned}$$

(28)

with $\nu _k$ the kinematic viscosity of phase k. The hydraulic diameter is a commonly used term when handling flow in non-circular geometries and is used to represent the equivalent diameter of the gas and liquid phases inside the circular cross section of the pipe. For the liquid, an open channel flow is assumed and for the gas phase a closed conduit, see Fig. 6, leading to

$$\begin{aligned} D_{\mathrm{h},\mathrm{l}}=\frac{4\alpha _{\mathrm{l}}}{S_{\mathrm{l}}}, \quad D_{\mathrm{h},\mathrm{g}}=\frac{4\alpha _{\mathrm{g}}}{S_{\mathrm{g}}+S_i}. \end{aligned}$$

(29)

The interface friction factor can be modeled as $f_i=cf_{\mathrm{g}}$ with $c\ge 1$ based on the detailed analysis of Tae-Hwan et al. (2015) for different flow regimes. The stratified flow regimes encountered during the experimental campaign were mainly wavy with concave shape and, therefore, a good estimate is $c=4$, see Tae-Hwan et al. (2015).

The wetted perimeters $S_{\mathrm{l}}$ and $S_i$ for a pipe of unit area can be written in terms of $S_{\mathrm{g}}$ as

$$\begin{aligned} S_{\mathrm{l}}=2\sqrt{\pi }-S_{\mathrm{g}}, \quad S_i=\frac{2}{\sqrt{\pi }}\sin \left( \frac{S_{\mathrm{g}}\sqrt{\pi }}{2}\right) . \end{aligned}$$

(30)

Using Eqs. (26)–(30) in the shear balance equation (25), leads to

$$\begin{aligned}&(1-\alpha _{\mathrm{g}})^3S_{\mathrm{g}}\nonumber \\&\quad -\left( \frac{{\textit{Re}}_{\mathrm{l}}}{{\textit{Re}}_{\mathrm{g}}}\right) ^{-\frac{1}{4}}\left[ \frac{S_{\mathrm{g}}+\frac{2}{\sqrt{\pi }}\sin \left( \frac{S_{\mathrm{g}}\sqrt{\pi }}{2}\right) }{2\sqrt{\pi }-S_{\mathrm{g}}}\right] ^{-\frac{1}{4}}\nonumber \\&\qquad \times X_{\mathrm{LM}}^2\alpha _{\mathrm{g}}^3\left( 2\sqrt{\pi }-S_{\mathrm{g}}\right) \nonumber \\&\quad +\frac{2c}{\sqrt{\pi }}\left[ S_{\mathrm{g}}\sqrt{\pi }-\sin \left( S_{\mathrm{g}}\sqrt{\pi }\right) \right] ^2\sin \left( \frac{S_{\mathrm{g}}\sqrt{\pi }}{2}\right) =0, \end{aligned}$$

(31)

which contains the unknowns $\alpha _{\mathrm{g}}$ and $S_{\mathrm{g}}$. The geometrical relation between these variables can be constructed:

$$\begin{aligned} \alpha _{\mathrm{g}}-\frac{1}{2\pi }\left[ S_{\mathrm{g}}\sqrt{\pi }-\sin \left( S_{\mathrm{g}}\sqrt{\pi }\right) \right] =0, \end{aligned}$$

(32)

which closes the system of equations for stratified flow.