Basics of Multiphase Flow - III

As we have mentioned in the previous article, pressure gradient for non-homogeneous flow follows a procedure that consists of: 1) Phase properties calculation. 2) Flow pattern prediction. 3) Liquid hold up and mixture properties calculation. 4) Pressure gradient determination. After discussing flow pattern types, and mixture properties, hereby we divide this article into 3 parts: I) Mathematical concepts in vertical wells. II) Basics of flow through restrictions. III) The advances in machine learning.?

I) Mathematical Concepts in Vertical Wells

A. Empirical Correlations:

1. Flow Patterns Identification: A common approach is to use Ros (1961) correlations to predict vertical flow patterns, while Beggs and Brill correlations (1973) are used to predict liquid hold-up and pressure gradients. Ros introduced dimensionless groups including liquid velocity number (NLV), and gas velocity number (Ngv), which are the coordinates of vertical flow pattern map.
2. Holdup Prediction: Beggs and Brill (1973) calculate liquid holdup based on no-slip assumption. Their horizontal correlations are assumed to be valid for vertical wells, then a correction factor is used to account for inclination.
3. Pressure gradient: it is then calculated using the general energy equation.

B. Mechanistic Models:

1. Flow Patterns Identification: A popular approach is Ansari et. al. model, based on Taitel et. al. (1980). We first identify transition boundaries between the different patterns, then generate a map for transitions using Vsl and Vsg, for example and determine the flow pattern. This could be summarized by first checking for Annular/Slug transition, if not, check whether it could be dispersed bubble flow/bubble flow transition, if not, check for bubble/slug transition. A) Check for bubble flow existence: if pipe diameter is less than a minimum diameter (dmin), small bubbles merge into Taylor bubbles whose velocity is lower than bubble rise velocity, and bubble flow would not exist. Inclination also have a critical value below which bubble flow will not exist as bubbles would coalesce. B) Bubble/Slug Transition: it is governed by bubbles collisions and coalescence. If the gas void fraction (alpha) or gas hold-up or the distance between two bubbles exceeds 0.25 of the bubble diameter, a transition to slug flow occurs. Inclination angle should be accounted for in deviated wells. C) Bubble/Dispersed bubble Transition: if bubbles surface tension is greater than liquid turbulence force, bubbles will retain their shape and bubble flow occurs. This happens when bubbles size is greater than agglomeration diameter (dc). D) Slug/Dispersed Bubble Transition: if gas void fraction exceeds 0.25, yet turbulence forces in liquid are large enough to break Taylor bubbles, and no-slip gas void fraction is less than 0.76, dispersed bubble flow would exist. E) Slug/Annular Transition: it requires three conditions to occur; a) gas flow rate should be high enough to transport liquid droplets upward preventing liquid fallback i.e. gas velocity (vg) greater than liquid terminal velocity (vt). b) Liquid hold up of each film is small enough to avoid liquid film bridging (HL < 0.12). c) Liquid film thickness is small enough to achieve stability and upward flow against the gravity of liquid.
2. Liquid Hold-up Calculations: it primarily depends on the flow pattern: A) dispersed bubble flow: it is equal to no-slip liquid holdup. B) bubble flow: liquid hold-up is a function of slip velocity, densities, surface tension and inclination angle. C) slug flow: liquid hold-up is the weighted average HL of two regions (slug region – liquid surrounding Taylor bubble region) assuming steady state and constant lengths. HL calculation is an iterative method. D) annular flow: liquid hold-up is the weighted average HL in the film and gas core.
3. Pressure gradient: it is calculated using the general energy equation accordingly.

II) Flow through Restrictions

Restrictions or changes in pipe geometry usually represent a sudden decrease in flow area, which results in an increased fluid velocity (kinetic energy) and a decreased static pressure (mechanical energy) for the sake of mass conservation. Pressure recovers to a lower value after the flow exists this restriction, and this energy loss results in any of the following: 1) fluid expansion 2) temperature drop leading to probable hydrate formation 3) freeing some gas or condensing wet gas due to pressure drop. Flow pattern is predicted based on the critical and subcritical boundary and comparing it to pressure ratio (Pdownstream / Pupstream) as each type of flow has different behavior. If pressure ratio is less than the boundary, then it is critical. For piping components that result in turbulent flow, they are accounted for using additional equivalent length of pipe that is added straight to pip original length.?

Critical flow: it occurs when the flowing fluid velocity at restriction is equal to sonic velocity in the fluid at in-situ conditions. Hence, the flow rate will only depend on restriction upstream condition and will not be affected by changes in downstream pressure. On the other hand, if fluid velocity is less than that threshold (subcritical flow), flow rate would be affected by both upstream and downstream conditions, so a decrease in downstream pressure would increase the flow rate through the restriction. It is always desirable to have the well producing in critical flow, to avoid any back pressure or disturbance that could occur by surface facilities were it subcritical flow.

For A) single-phase dry gas flow: Critical boundary is calculated in terms of the specific heat ratio for gas, which is a function of pressure and temperature. A rule of thumb for gas flow is assuming the well to be critical when the pressure ratio is less than a half. Oil or wet gas producers usually exist as a multiphase mixture across chokes. B) Homogeneous multiphase flow: Ashford and Pierce model is a commonly used model which assumes isentropic low, incompressible fluid, and no slippage. C) Non-homogeneous flow: the existing models are iterative, which assume that acceleration term is the predominant one for a 1D flow, where fluid properties are calculated upstream.

III) Machine Learning Applications

1. Kanin et. al. (2019) mentioned that mechanistic models are limited due to the dependency of empirical relations they are based on. They proposed using three surrogate machine-learning models, nested into each other, trained on lab experimental data. The iterative model could predict 1) liquid hold-up, 2) flow pattern, and 3) pressure gradient for each divided segmented interval, assuming they are homogeneous. They revealed that the alternative approach that predicts pressure gradient directly from the data using the model trained on lab data achieves low scores on field data, due to the complexity of gravitational, frictional and acceleration terms. They optimized their model using 1300 points, and utilized (Random Forest, Gradient Boosting Decision Trees, Support Vector Machines and Artificial Neural Networks). They validated the results on filed cases
2. Mask et. al. (2019) analyzed twelve experiments (8,846 points) to develop a machine-learning model (Boosted-Trees) that could predict multiphase flow pattern with accuracy higher than 90% (10% higher than semi-analytical methods). They derived three independent dimensionless numbers, which represent seven physical properties, using “The Buckingham- Theorem” to upscale experimental data to real application, and stated that they improve model performance rather than using the original attributes.
3. Gómez et. al. (2019) introduced the classification of flow patterns utilizing a) a probabilistic approach to identify flow pattern in pipes and b) supervised classification. They revealed a new technique to map flow patterns using experimental and synthetic generated data, as shown below. They introduced translucent flow pattern map with contour lines to better identify transition zones regardless of overlapping zones, and then probabilistic approach is used to deal with transition zones.

References:

1. H. Dale Beggs (1991). Production Optimization Using NODAL Analysis. OGCI Publications.
2. Al-Safran, E. M., & Brill, J. P. (2017). Applied multiphase flow in pipes and flow assurance: Oil and Gas Production. Society of Petroleum Engineers.
3. Kanin, E. A., Osiptsov, A. A., Vainshtein, A. L., & Burnaev, E. V. (2019). A predictive model for steady-state multiphase pipe flow: Machine learning on lab data.?Journal of Petroleum Science and Engineering,?180, 727-746.
4. Mask, G., Wu, X., & Ling, K. (2019). An improved model for gas-liquid flow pattern prediction based on machine learning.?Journal of Petroleum Science and Engineering,?183, 106370.
5. Amaya-Gómez, R., López, J., Pineda, H., Urbano-Caguasango, D., Pinilla, J., Ratkovich, N., & Mu?oz, F. (2019). Probabilistic approach of a flow pattern map for horizontal, vertical, and inclined pipes.?Oil & Gas Science and Technology–Revue d’IFP Energies nouvelles,?74, 67.