**1. Introduction**

Within the petroleum industry there is a long-standing need for operationally robust, low cost, and real-time wellhead metering systems with accuracy across a broad range of produced volumes and hydrocarbon properties. Installations currently rely predominantly on accurate, yet costly, and operationally cumbersome test separators that by design deliver time-averaged multiphase flow rate data in well tests spanning several hours, thereby losing real-time flow information [1]. The real-time alternative to multiphase test separators, multiphase flow meter technology, has considerably improved in accuracy over the last decades. However, due to the complex combination of measurement technologies within these systems, these devises are highly sensitive to hydrocarbon properties and require repeated calibration in the field [2,3]. In addition, these multiphase flow meter systems tend to have a limited dynamic range in terms of produced volumes and associated multiphase flow profiles [1–3]. This poses problems in field applications as flow regimes can change over time due to natural production transients that can occur over the scale of hours or days, and inescapably occur over the lifetime of a well or due to flow restrictions caused by pipeline fouling that builds up over time during production [4]. In the absence of simple and accurate, plug-and-play well head metering solutions, many wells are operated with insufficient metering leading to suboptimal reservoir management and uncertainty in production allocation to individual wells [1–3].

For many years, magnetic resonance-based downhole logging tools have been successfully applied to in situ Earth formation evaluation [5]. These tools apply low field,

**Citation:** Tromp, R.R.; Cerioni, L.M.C. Multiphase Flow Regime Characterization and Liquid Flow Measurement Using Low-Field Magnetic Resonance Imaging. *Molecules* **2021**, *26*, 3349. https:// doi.org/10.3390/molecules26113349

Academic Editor: Robert Brinson

Received: 30 April 2021 Accepted: 29 May 2021 Published: 2 June 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

time-domain magnetic resonance (MR) technology under challenging environmental conditions, proving the robustness of the technology. In the laboratory, the same technology can be used to obtain production fluid composition information from samples [6,7]. In the last decade, considerable effort has been spent to merge the two aspects and apply MR technologies in the process industry in pursuit of industry 4.0 compatible inline process monitoring and control [8]. Real-time wellhead metering systems are a concrete example of such inline process monitoring systems. Several MR technology-based research instruments have been developed for the petroleum industry, showing specific advantages of low-field MR technology when applied to multiphase flow measurement [1,9–12]. In this article, we show that a fully integrated multiphase flow meter [13] using low-field MR technology can act as a smart and robust measurement platform that has a large dynamic range in terms of produced volumes and associated multiphase flow profiles. This multiphase flow measurement platform can be applied as a general monitoring instrument in chemical and process control industries [8].

The complex flow profiles observed in multiphase flow emerge due to the differences in densities and viscosities of the fluid phases present in the flow [2]. For the case of two-phase, gas-liquid flows, these differences are maximum and the most challenging flow profiles occur. In this article, we focus on the horizontal pipe flow of water and air at atmospheric pressure. These fluids provide several advantages: Firstly, they are chemically safe, simplifying a flow loop design and operation; secondly, at atmospheric pressure the largest difference in gas and liquid density is obtained, leading to the most challenging flow profiles; and thirdly, water-based flows are relevant to a wide variety of processes in the chemical industry, including high water-fraction oil production in the petroleum industry.

In a static situation, there is a gravity induced separation between the two phases in a gas-liquid mixture present in a horizontal pipe: Gas is concentrated at the top of the pipe and liquid is concentrated at the bottom of the pipe, see Figure 1. When a pressure gradient is added along the length of the pipe, flow is induced. Since the phases have different densities and viscosities, the flow velocity associated with a given pressure drop per unit length is different for each phase. This so-called phase slip between phases is the primary complication in two-phase flows as it creates a dynamic pressure between the two phases. Depending on the cross-sectional area occupied by the two phases, which are commonly expressed using the dimensionless liquid holdup *h*liq, see Figure 1, such that the cross-sectional area occupied by liquid is given by

$$A\_{\rm liq} = h\_{\rm liq} A\_{\rm pip\omega} \tag{1}$$

where 0 ≤ *h*liq ≤ 1, and *A*pipe is the pipe cross-sectional area, the surface tension of the fluid interface may or may not be strong enough to keep a stable interface between the two phases at a certain phase slip. If it is not, an instability in the local liquid holdup is induced. These instabilities can take the form of small, symmetric waves on a relatively stable fluid interphase, referred to as stratified flow; can give rise to large and chaotic wave patterns reminiscent of rough seas, referred to as wavy flow; and can even lead to such large instabilities that liquid is sucked up to the top of the pipe, creating so-called liquid slugs that are pushed along by the gas at high velocities.

Which flow pattern occurs in a given situation depends on many factors of which the upstream and downstream piping configuration are of paramount importance. For a given piping configuration and given volumetric liquid and gas flow rates, the flow pattern can be roughly estimated based on the superficial gas flow velocity, *u*s,gas = Qgas/*A*pipe, and the superficial liquid flow velocity, *u*s,liq = Qliq/*A*pipe, where Q*<sup>i</sup>* is the volumetric flow rate of phase *i*. The superficial flow velocity thus represents the fictitious flow velocity of a single phase of a multiphase flow that it would have if all other phases in the multiphase flow were absent from the flow. Figure 2 shows an example flow map for two-phase, gas-liquid flow in a horizontal pipe section that uses the concept of the fictitious superficial flow velocity for parametrization [2]. The purple rectangle in Figure 2 indicates the superficial gas and liquid flow velocities that can be obtained using the multiphase flow loop used in this study. Details about this flow loop are presented in Section 4. Based on this flow map, we may expect stratified, wavy, and slug flow to be observed during multiphase flow experiments.

**Figure 1.** Schematic representation of the definition of the liquid holdup *h*liq.

**Figure 2.** Schematic representation of a two-phase, gas-liquid flow map indicating the multiphase flow patterns likely to occur for a given combination of superficial flow velocities, adapted from [2]. The purple rectangle indicates the superficial gas and liquid flow velocities that can be obtained using the multiphase flow loop used in this study.

Based on the preceding discussion of multiphase flow patterns, one can see that correlations between instantaneous holdup and instantaneous flow velocity need to be characterized to accurately determine the flow rates of the individual phases in a twophase flow.

An intuitive and simple method to measure the instantaneous liquid holdup would be to use the MR signal amplitude. However, for samples flowing through an industrial MR system this signal amplitude may depend on other factors than the liquid holdup alone. For instance, consider the CPMG pulse sequence [14,15] measurements presented in Figure 3 for a water-air slug flow-like flow regime. Both the amplitude at *t* = 0 s and the signal amplitude decay time of the CPMG signals vary considerably between measurements. Since air does not contribute to the CPMG signal in this experiment and water relaxation (*T*<sup>2</sup> ~ 2–3 s) is slow compared to the signal decay time, the signal decay time correlates with flow velocity *u*liq [9]. The signal amplitude correlates strongly with *h*liq, although the spin residence time in the polarizing magnetic field influences the observed signal amplitude as well, complicating the direct conversion of signal amplitudes to liquid holdups. The four CPMG signals highlighted in red in Figure 3 show that a given signal decay time or liquid flow velocity, can be observed

for multiple signal amplitudes or liquid holdups. The major complication in multiphase flow measurement consequently is that the instantaneous flow rate, i.e.,

$$\mathbf{Q}\_{\text{liq}}(t) = \boldsymbol{\mu}\_{\text{liq}}(t)\boldsymbol{h}\_{\text{liq}}(t)\boldsymbol{A}\_{\text{pipe}} \tag{2}$$

needs to be sampled in a way that ensures representative sampling of all characteristic flow events. When representative sampling is achieved, the average liquid flow rate <Qliq> during a given time interval is given by the mean of the discrete set of flow events sampled during that time interval, i.e.,

$$\mathbf{Q}\_{\rm liq} = \mathfrak{u}\_{\rm liq}(t) h\_{\rm liq}(t) A\_{\rm pipe} \cong \mathfrak{u}\_{\rm liq} j\_{\rm liq} \mathcal{A}\_{\rm pipe}.\tag{3}$$

**Figure 3.** CPMG signal amplitude as a function of time shown for a set of measurements performed on a slug flow-like water-air multiphase flow. The two pairs of CPMG signals printed in red show that different signal amplitudes at *t* = 0 s can occur for the same signal amplitude decay time.

MR imaging (MRI) techniques may provide a direct measurement of liquid holdup, which is, for example, independent of the magnetic history of the sample. When MRI sequences are implemented, a spatially varying magnetic field or gradient **G**, is introduced in addition to the main magnetic field **B**0. The effect of introducing the gradient **G** is that the resonance frequency of the nuclear spins varies with the position. The resonance frequency can thus be used to encode the position of nuclear spins. When MRI is applied in presence of flow, translational motion information can be extracted combining an imaging sequence with a spatially resolved measurement of molecular displacement. In many flow MRI studies, the velocity of a fluid media is measured by time-of-flight (TOF) [16–18] and phase shift methods [19]. A comprehensive review of non-medical flow MRI methods can be found in the articles by Gladden and Sederman [19,20]. The principles and relevant theory of flow MRI can be found in the books by Callaghan [21,22]. The fundamental concepts of MRI are discussed in an intuitive manner by McRobbie et al. [23].

Here, we focus on obtaining the bulk liquid flow velocity *u*liq from the convective amplitude decay of the CPMG signals that is induced by the outflow of the excited sample volume [9]. We combine the CPMG pulse sequence with an external gradient *G*z applied in the transversal vertical direction of the pipe to obtain a one-dimensional spatial distribution of the liquid, which we refer to as an one-dimensional (1D) distribution image. There are several techniques that combine the CPMG pulse sequence with an imaging sequence for spatial encoding [24]. As typical pulse sequences based on phase encoding gradients may increase the total acquisition time [24], we use frequency encoding to spatially encode all the points simultaneously during one CPMG spin-echo train. Since pulsed or modulated gradients require highly complex power electronics and gradient coils design that in an industrial application are translated into complexity for manufacturing, we use a constant-gradient CPMG [25]. This implementation additionally provides the advantage of short echo spacing for the convective amplitude decay velocity measurement. The frequency encoded spatially resolved 1D distribution image, can be obtained from the Fourier transform of each individual spin-echo signal [25].

To maximize the resolution and minimize the blurring effect due to inhomogeneities by spatial variations in **B**<sup>0</sup> [26], we want to apply the maximum gradient strength available. During the application of the constant gradient *G*z, the spectral width of the RF-pulses, ΔνRF, must be larger or equal to the spectral width of the sample, Δνsample. The spectral width of the sample is given by Δνsample = γ*G*z*D*/2π, where *γ* is the gyromagnetic ratio of the proton, and *D* is the pipe diameter [25]. The spectral width of a rectangular RF pulse can be approximated by ΔνRF ≈ 1/*t*pulse, where *t*pulse is the RF pulse duration. The pulse sequence design relation between RF pulse length and applied gradient strength may thus be written as

$$t\_{\text{pulse}} \le \frac{2\pi}{\gamma G\_\text{x} D}.\tag{4}$$

When we apply the maximum gradient strength *G*<sup>z</sup> in our application, both 90◦ excitation and 180◦ refocusing pulses as used in a standard CPMG pulse sequence do not fulfill Equation (4). In other words, a standard CPMG would have limited bandwidth and cannot be used to excite and monitor the convective amplitude decay over the full pipe cross-section. This limitation was overcome by using a low flip angle CPMG (LFA-CPMG) [27], where all RF pulses are substituted by short duration pulses. This way the LFA-CPMG pulse sequence allows the instantaneous liquid holdup *h*liq to be derived from the 1D liquid distribution image obtained from frequency encoded spin-echo signals, while the instant liquid flow velocity *u*liq can be determined from the effective convective amplitude decay of the LFA-CPMG signals.

In this article, we will show that low-field MR technology applied to multiphase flow has the capability to measure the instantaneous liquid holdup and liquid flow velocity using the constant gradient LFA-CPMG pulse sequence. To this end, we applied the LFA-CPMG to study two-phase air-water flow experiments. The details of the experimental method and setup are presented in Section 4. In the following section it will be shown that LFA-CPMG allows the correlations between liquid holdup and liquid flow velocity to be determined, and it is shown that flow profiles can be identified based on these correlations. In addition, we show that flow calculations based on these correlations allow liquid flow rate determination with an accuracy that is independent of the multiphase flow profile observed in a horizontal pipe flow for a wide dynamic range in terms of the average gas and liquid flow rates.
