A simplified two-phase flow model using a quasi-equilibrium momentum balance

https://doi.org/10.1016/j.ijmultiphaseflow.2016.03.017Get rights and content

Highlights

  • Transient two phase flow is described with a single dynamic PDE.

  • Closure relations are obtained assuming a quasi-equilibrium momentum balance.

  • Approximations make closure relations explicit, allowing for fast computation.

  • The result has applications in model-based estimation and control algorithms.

Abstract

We propose a simple model of two-phase gas–liquid flow by imposing a quasi-equilibrium on the mixture momentum balance of the classical transient drift-flux model. This reduces the model to a single hyperbolic PDE, describing the void wave, coupled with two static relations giving the void wave velocity from the now static momentum balance. Exploiting this, the new model uses a single distributed state, the void fraction, and with a suggested approximation of the two remaining static relations, all closure relations are given explicitly in, or as quadrature of functions of, the void fraction and exogenous variables. This makes model implementation, simulation and analysis very fast, simple and robust. Consequently, the proposed model is well-suited for model-based control and estimation applications concerning two-phase gas–liquid flow.

Introduction

Multi-phase flow simulation models have evolved significantly over the last couple of decades. With the increase in computational power and sophistication of numerical schemes, simulating two-phase pipe flow no longer suffers the same limitations on computational size, and state of the art high-fidelity models such as OLGA (Bendiksen et al., 1991) and LedaFlow (Danielson et al., 2011) typically run many times faster than real-time on a standard desktop computer.

Before this development, however, significant efforts were devoted to obtaining simplifications of multi-phase flow models which could ease implementation and increase their simulation speed. The Drift Flux Model (DFM) (Ishii, 1977) was first proposed by Zuber and Findlay (1965) as a correlation for predicting steady-state void-fraction profiles and later used in transient representations of two-phase flow (Pauchon and Dhulesia, 1994). In this form it is a simplification of the transient two-fluid model obtained by relaxing (i.e. imposing immediate steady-state on (Flåtten and Lund, 2011)) the dynamic momentum equation of each phase, replacing them with a mixture momentum equation and a static relation typically called a slip law.

Further simplification can be achieved by using a similar approach to other parts of the dynamics deemed insignificant for the application at hand. Specifically, by imposing steady state on the momentum balance, the pressure wave dynamics are neglected, yielding so-called “No Pressure Wave” (NPW) models or “Reduced DFMs”. This simplification is motivated by applications for which slow gas propagation dynamics are more critical than fast pressure wave propagation. Furthermore, it has been shown that the validity of the drift-flux models representation of the fast pressure dynamics is imprecise in many scenarios due to the relaxations involved in obtaining the DFM from the full formulation of Baer and Nunziato (1986), which lowers the sonic velocity (Flåtten, Lund, 2011, Linga). Thus, if the “medium” complexity DFM representation of the pressure waves is imprecise, the argument can be made that they could be discarded.

This approach was used by Taitel et al. (1989) where the resulting model was described by a single transient PDE of the liquid continuity, obtained by assuming incompressible liquid, and a set of steady-state relations. The resulting model was further investigated by Minami and Shoham (1994) where it was found to be amenable for certain scenarios. The approach was expanded upon by Taitel and Barnea (1997), where the assumption of incompressible liquid was dropped, yielding two transient equations. A similar model was investigated by Masella et al. (1998), here called the “No Pressure Wave” (NPW) model. More recent additions to the literature on models using quasi-equilibrium momentum balance include (Choi, Pereyra, Sarica, Lee, Jang, Kang, 2013, Aarsnes, Ambrus, Karimi Vajargah, Aamo, van Oort, 2015, Ambrus, Aarsnes, Karimi Vajargah, Akbari, van Oort, 2015).

Interestingly, many of these recent studies have not been motivated by the desire to reduce computational complexity. Rather, the advent of computerized automation and optimization in the oil and gas industry has created new applications for various forms of simplified models, causing renewed interest in these models.

Modern advances in the theory of dynamic systems have the potential of improving robustness and performance in the monitoring, optimization and control of dynamic processes which can be described by an amenable mathematical model. By intelligently combining predictions from the mathematical model with information from multiple sensors one can estimate unmeasured quantities, optimize automatic control procedures, predict future behavior, and plan countermeasures for unwanted incidents. Such design techniques, often referred to as model-based estimation and control (Åström, Murray, 2010, Anderson, Moore, 1990), require a mathematical model with the proper balance between complexity and fidelity, i.e. the complexity must be limited to facilitate the use of established mathematical analysis and design techniques, while the qualitative response of the process is retained.

Models that achieve such balance between complexity and fidelity are sometimes referred to as fit-for-purpose models. Obtaining such models often proves difficult for gas–liquid two-phase dynamics due to the significant complexity and distributed nature of multi-phase pipe flow (Aarsnes, Di Meglio, Evje, Aamo, 2014, Aarsnes, Meglio, Graham, Aamo, 2016).

If the appropriate model can be developed, however, it could see a wide range of uses in model-based control and estimation applications where two-phase pipe flow is encountered, such as underbalanced drilling of oil and gas wells (Pedersen et al., 2015), well control (both in conventional and Managed Pressure Drilling) (Carlsen et al., 2008), riser gas handling (Hauge et al., 2015), hydrocarbon production monitoring (Bloemen, Belfroid, Sturm, Verhelst, 2006, Teixeira, Castro, Teixeira, Aguirre, 2014) and mitigating severe slugging during hydrocarbon production (Eikrem, Aamo, Foss, 2008, Esmaeil, Skogestad, 2011, Di Meglio, Kaasa, Petit, Alstad, 2010).

Section snippets

The drift flux model

A popular model for representing one-dimensional two-phase flow dynamics in drilling and production at an acceptable fidelity is the classical three-state transient Drift Flux Model (DFM), see e.g. (Lage, Time, 2000, Fjelde, Rommetveit, Merlo, Lage, 2003, Aarsnes, Di Meglio, Evje, Aamo, 2014).

For certain boundary conditions, the existence of solutions has been proven (Evje, Wen, 2013, Evje, Wen, 2015), and it is well known that the DFM is, in most practical situations, hyperbolic, with three

Derivation of the new formulation

The full three-state drift-flux model is too complicated for most model-based estimation and control approaches (Aarsnes, 2016), hence simplification is desirable. Based on the analysis of the previous section we argue that when relaxing the fast pressure dynamics, it should be possible to reduce the model description to a first-order PDE, while still retaining the qualitative dynamics of the system.

For this derivation we will again start with the classical Drift Flux formulation (1)–(7). First

Some numerical examples

In this section two numerical examples are considered. The first one highlights the effect of removing the pressure dynamics, while the second numerical example illustrates the feasibility of employing the model to a typical scenario from underbalanced drilling (described in the following).

For both scenarios we consider a 1000 m long domain with cG=300m/s,ρL=1000kg/m3,v=αL*=0. The full Drift-Flux model Eqs. (1)–(3) are implemented with the AUSM scheme of Evje and Fjelde (2002) and time step Δt=

Summary and conclusions

In this paper we have presented a simplified two-phase flow model obtained by relaxing the distributed pressure dynamics, equivalent to using a quasi-steady momentum balance. The resulting model is a transport equation, with void fraction as the distributed state. The gas travels with an exponentially increasing (for negative pressure gradient), quasi-steady velocity driven by the gas expansion, which is modeled as a source term in the transport equation. The closure relations can be

Acknowledgment

This work was supported by Statoil ASA, the Research Council of Norway (NFR project 210432/E30 Intelligent Drilling) and the Rig Automation and Performance Improvement in Drilling (RAPID) sponsor group (ExxonMobil, Sinopec, Baker Hughes and NOV). The work of the first author was also supported by the Research Council of Norway, ConocoPhillips, Det norske oljeselskap, Lundin, Statoil and Wintershall through the research center DrillWell (Drilling and Well Centre for Improved Recovery) at IRIS.

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