Mathematical Problems in Engineering

Volume 2015, Article ID 134102, 17 pages

http://dx.doi.org/10.1155/2015/134102

## A Novel Dynamic Model for Predicting Pressure Wave Velocity in Four-Phase Fluid Flowing along the Drilling Annulus

^{1}School of Chemistry and Chemical Engineering, Daqing Normal University, Daqing, Heilongjiang 163712, China^{2}State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu,
Sichuan 610500, China

Received 22 March 2014; Accepted 27 July 2014

Academic Editor: Alexei Mailybaev

Copyright © 2015 Xiangwei Kong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A dynamic pressure wave velocity model is presented based on momentum equation, mass-balance equation, equation of state, and small perturbation theory. Simultaneously, the drift model was used to analyze the flow characteristics of oil, gas, water, and drilling fluid multiphase flow. In addition, the dynamic model considers the gas dissolution, virtual mass force, drag force, and relative motion of the interphase as well. Finite difference and Newton-Raphson iterative are introduced to the numerical simulation of the dynamic model. The calculation results indicate that the wave velocity is more sensitive to the increase of gas influx rate than the increase of oil/water influx rate. Wave velocity decreases significantly with the increase of gas influx. Influenced by the pressure drop of four-phase fluid flowing along the annulus, wave velocity tends to increase with respect to well depth, contrary to the gradual reduction of gas void fraction at different depths with the increase of backpressure (BP). Analysis also found that the growth of angular frequency will lead to an increase of wave velocity at low range. Comparison with the calculation results without considering virtual mass force demonstrates that the calculated wave velocity is relatively bigger by using the presented model.

#### 1. Introduction

In petroleum industry, managed pressure drilling (MPD) is considered to be one of the most important techniques, which allows accurate control of bottom hole pressure (BHP) by controlling the flow rate, drilling mud density, and back pressure (BP) at the wellhead [1]. As extensively used in the drilling of huge risk, uneconomical, or even abnormal formation, MPD has been of interest in the literature for at least the past decade, especially the related issues about pressure control inducing the dynamic well and kick detection [2]. Due to those efforts, research of the dynamic pressure wave velocity is of great significance to the detection of gas influx and effective control of the pressure at the bottom of well [3]. During the drilling operation of the so-called “microflux control,” an MPD technique developed by Santos et al. [4], the return flow is monitored and adjusted to control fluid loss or gain. In the light of control principal, simulation studies were performed to determine the most appropriate initial response to kicks arising due to MPD specific complications caused by BHP fluctuations [5]. During the MPD operations, all unsteady operating, such as changing of pumping rate, adjustment of choke, and controls of BP at wellhead, will generate a pressure wave and threaten the drilling equipment [6]. For the same reason, while tripping out of a drill string in the wellbore, bottom hole is submitted to a suddenly decrease in pressure, leading to fluid expansion and movement out of the annulus. The rapidly expanding fluids and dynamic pressure fluctuations can also lead to rock instability in a reservoir [7]. Particularly, the effects are more important in systems in which multiphase flows occur. New kick-detection tools are now available that are based on acoustic principles, which are of great benefit to potentially earlier and more sensitive detection of a gas influx than pit-gain or paddle flow measurements [8]. The study of propagation of pressure wave is also relevant to control of downhole tool, such as intelligent well downhole control valves applied in different field for many purposes. With further development of oilfield, downhole tool technique for special casing wells is receiving much more attention [9]. At the meantime, an important problem in deep drilling is the propagation of measurement-while-drilling (MWD) mud pulse, transmitting real-time various data from sensors located down hole near the drill bit. Hence, the propagation behavior of pressure wave is considered to provide reference for the MPD operations.

However, we also noticed that the conventional theories present are difficult to be employed in a systematic and accurate prediction when influxes generates. As the influxes fluid including gas, oil, and water will lead to variations of physic characteristics parameter of fluid in the annulus, the distribution of pressure wave velocity in gas, oil, water, and drilling mud four-phase flow along the annulus will be dynamic changing with respect to time and well depth. These effects may influence the safe operation of devices.

Pressure waves are disturbances that transmit energy and momentum along the wellbore through drilling fluid without significant displacement of matter. As migration of dispersed gas, water, and oil towards wellhead is quite complicated, the fundamental characteristics of the four-phase flow are still unknown and modeling results of pressure wave velocity are questionable. The proper treatment of propagation behavior of pressure wave in the four-phase flow in the annulus requires knowledge of description of influxes generation, development, and pressure wave propagation model in a two-phase mixture.

In gas/liquid two-phase flow, two-phase medium interaction greatly changes the structure characteristic of flowing fluid, which results in greater compressibility of two-component flow than single-phase gas or liquid and further causes pressure waves propagating speed to be greatly reduced. This can be explained from the viewpoints of mixture density and compressibility of two-phase fluid and the pressure drop along the flow direction in the wellbore. In the low void fraction range, the gas phase is dispersed in the liquid as bubble, so the wave velocity is influenced greatly by the added gas phase. According to the EOS, if gas invades into the wellbore with a small amount in the bottom hole, the density of the drilling mud has little variation while the compressibility increases obviously, which makes the wave velocity be decreased. This phenomenon was first proposed by Mallock [10] and attracts much attention for its important role in the development of science and technology applications. Extensive investigations involved in the issue of pressure wave propagation have been taken and some significant achievements associated with the theories have also been made in the researches. In early stage researching, Wood [11] extended the researches of Mallock and presented a succinct formula by assuming that the compressibility of two-component fluid is a function related to single-phase compressibility and elastic modulus* E*. Carstensen and Flody [12] measured the velocity of pressure wave under a lake. A dispersion relation for pressure waves propagating through a bubbly fluid was derived by using a linear scattering theory developed by Foldy. Under the hypothesis of homogeneous and adiabatic laminar flow, Thuraisingham [13] took the two-phase media as a homogeneous fluid and derived the solution model concerning the problem of wave velocity model in two-phase flow at low gas void fraction according to the analysis of state equation of mixture. According to the early stage investigations, Hsieh and Plesset [14] and Murray [15] researched the influence of thermal conductivity and viscosity coefficient on pressure wave propagation. Wallis [16] firstly studied the propagation mechanism of pressure wave and derived the propagation velocity in bubbly flow and separated flow using the homogeneous model. The proposed model is based on the respective compressibility of the vapor and the liquid. Also, the two-phase mixture is treated as a compressible fluid with suitably averaged properties. In that expression, mass and heat transfers are neglected, so it can be applied to any gas/liquid mixture, in so far as no major effect due to vaporization or condensation must be taken into account. It may not be valid for such complex mixtures that never reach equilibrium. Assuming that no evaporation or condensation occurs when pressure wave is transporting, Moody [17] developed a simple acoustic wave model for bubbly flow and annular flow and established a relationship between sonic velocity and two-phase critical flow. Similar model was developed by D’Arcy [18]. In the researches of Mcwilliam and Duggins [19], surface tension and compressibility of liquid phase were also considered. Henry et al. [20] calculated the velocity as a function of void fraction using a correlation to account for the change in bubble shape with void fraction. Martin and Padmanabhan [21] extended the simple model proposed by Henry by considering wave reflection and wave transmission at gas-liquid interfaces. Researches of Mori et al. [22, 23] suggest that impact pipe elasticity on pressure wave propagation velocity is limited to the range of gas void of less than 1%. At high gas void range, the pressure wave velocity is in between the two velocities of single-phase. Nguyen et al. [24] proposed another type of model for bubbly flow (diluted gas phase in the liquid). The simple relations for prediction of the propagation of pressure disturbances in liquid-gaseous two-phase systems are presented. The model makes use of the well-known physical behavior that the wave velocity in a single-phase fluid is influenced by the elasticity of the confining walls. The interface of the one phase is considered to act as the elastic wall of the other phase and vice versa. Mecredy and Hamilton [25] used the two-fluid model to predict the pressure wave propagation in vapor-liquid flow in detail. However, the analysis contained the important assumption that the evaporation or condensation was governed by kinetic theory. Michaelides and Zissis [26] developed a computational method which yields the sound velocity in terms of the thermodynamic coordinates of the substance without the use of diagrams. Corresponding velocities of sound for the four substances considered exhibit a certain similarity which is examined statistically. The relationship between the sound velocity and the critical mass flux is also investigated. Thuraisingham [13] studied the wave velocity in bubbly water at megahertz frequencies (1~10 MHZ). Numerical analytical results indicate that volume concentrations and the radius of the bubble relative to the incident wavelength of sound are the important parameters which determine the deviation of sound speed form that of bubble-free water.

Currently, the two-fluid continuum model is the most common and reliable method to describe the gas/liquid two-phase flow phenomenon [27]. In the model, the governing equation and phase interface relationship is established based on the assumption that each phase satisfies the continuum conditions. To obtain the practical flow equations, reasonable assumptions and constitutive equations should be introduced. In consequence, the predicted wave velocities were found to depend strongly on the introduced assumptions and equations. In recent years, the two-fluid model was applied in determining the pressure wave propagation characteristics [28]. Ruggles et al. [29, 30] firstly performed the experimental investigation on the dispersion of pressure wave propagation in air-water bubbly flow and studied the propagation of pressure disturbance based on the two-fluid model small perturbation analysis method. Through the comparison with experimental data, they found that the virtual mass force coefficient is a function of gas void. Chung et al. [31] calculated the sonic velocity versus angular frequency form the concept of bubble compressibility in a two-component bubbly flow regime. He also extended such a model to predict the sonic velocity of a vapor-liquid system. Lee et al. [32] constructed the two fluids model to determine the pressure wave propagation speed for two-phase bubbly flow, slug flow, and stratified flow by using pressure disturbance instead of virtual mass and other phase interfacial terms. The results fit well with the experimental in steam water and air water of Henry and theoretical analysis of Nguyen. Zhao and Li [33] derived the general formula of sonic velocity in gas-liquid two-phase flow linear analysis using the linear analysis of the closed fundamental equations of compressible gas-liquid two-phase flow. It is proposed that the appropriate formula for calculating sonic velocity in gas-liquid two-phase flows under usual conditions may be Wood adiabatic sonic velocity formula. By linearizing the conservation equations of two-fluid model, Liu [34] derived a wave number equation of pressure wave for adiabatic gas-liquid two-phase flow. The effects of drag force and virtual mass force on propagation and dispersion of pressure wave were investigated. Xu and Chen [35] used the transient two-fluid model to develop a general relation for acoustic waves with steam-water two-phase mixture in one-dimensional flowing system. Both the mechanical and thermal nonequilibrium are considered. Brennen [36] taken mass and heat exchanges into account and proposed more complete expressions of the speed of sound in two-phase mixture. However, calibration of the mass and heat exchanges requires some further experimental investigations. Yeom and Chang [37] numerically investigated the wave propagation in the two-phase flows. An assessment was made on the effect of interfacial friction terms. Zhang et al. [27] investigated the propagation of the pressure wave in the water-gas two-phase bubbly flow with a one-dimensional two-fluid model and employing small perturbation analysis. The governing equations are simplified and closured according to high-speed aerated flow characteristics in hydraulic engineering. The effects of aerated concentration, liquid pressure, perturbation frequency, and interfacial forces on the acoustic wave velocity and its attenuation in the aerated flow are also explored. With the application of thermal phase change model in computational fluid dynamics code CFX, Li et al. [38] proposed a pressure wave propagation model and investigated the pressure wave propagation characteristics in two-phase fuel systems of liquid-propellant rocket. The propagation of pressure wave during the condensation of R404A and R134A refrigerants in pipe minichannels was given by Kuczyński [39]. Heat exchange between the phases in the condensation process was calculated by using the one-dimensional form of Fourier’s equation.

In drilling industry, some scholars have been devoted to this aspect. In the late 1970s, the former Soviet All-Union Drilling Technology Research Institute [40] began to study characteristics of pressure wave velocity in gas-liquid two-phase flow to detect early gas influx and achieved some important results. To study relationship between pressure wave velocity and gas void fraction, Li et al. [41] launched a gas-drilling mud two-phase flow simulation experiment in vertical annulus. It is proved that the two phase flow of gas-liquid patterns and the velocity of gas migration can be determined if the well depth, mud properties, and void fraction in bottom are given The method which is faster in detection time than the method of conventional kick detection was proposed. Starting with the analysis of transient flow, combining with the theory of transmission line, Wang [42] obtained the calculating model of frequency domain for the pulse velocity in drilling fluid built and the impendence and transmission operator of drilling fluid. Alternative initial responses to kicks for various well scenarios during MPD operations were also explored by Davoudi et al. [43]. Li et al. [44] proposed a mathematical model for predicting the attenuation and propagation velocity of measurement while drilling (MWD) pressure pulses in aerated drilling using the two-phase flow model and considering the momentum and energy exchange at the phase interface, gravity of each phase, viscous pipe shear, and other closing conditions. According to the theory of unsteady flow, Xiushan [45] developed the formulas of transmission velocity for mud pulse signal. The formulas which cover all kinds of boundary conditions, including thin wall pipes and thick ones and interaction influence of gas content and solid content on transmission velocity are suitable for positive and negative mud pulse and accord with drilling practice. In a previous work, we proposed a united wave velocity model to predict the pressure wave velocity in gas-drilling mud two-phase steady flow. The effect of well depth, back pressure, gas influx rate, virtual mass force, and angular frequency are all considered. However, under the effects of buoyancy and complicated turbulence interaction, the existing theoretical solutions are not involved in the dynamic model for predicting pressure wave velocity in four-phase fluid flowing along the drilling annulus when influx fluid migrate towards the top of wellbore.

In this paper, the drift model was used to analyze the flow characteristics of oil, gas, water, and drilling fluid multiphase flow. As the important characteristics of influx development, the relative motion of the interphase, such as slippage of gas phase and oil phase, is considered. Moreover, to predict pressure wave velocity in gas-oil-water-drilling mud four-phase flow in the annulus during MPD operations, a dynamic mathematical model is presented. By computing, the influence factors of pressure wave velocity, such as back pressure, gas void fraction, oil void fraction, influx time, influx rate, disturbance angular frequency, and virtual mass force, are analyzed.

#### 2. Mathematical Model

Before introducing the new dynamic model and to make this point clear, this paper reviews the hydraulic system in MPD operations. The drilling system is a closed circulation with BP at wellhead. The key equipments include the rotating control device, dynamic well control system, conventional pressure control system, industrial personal computer, Coriolis meter, choke, and pressure sensor. First the drilling mud begins to circulate from mud tank, down the drill pipe, and the drill string and returns from the annulus travel back through mud pit where drilling solids are taken away and then to surface mud tank. An important function of the drilling fluid is to provide pressure support to the wellbore wall. The rock formation drilled through has some form of porosity filled with formation fluids. These fluids can be water, or in the case of a reservoir, hydrocarbons. The pressure in these fluids is referred to as the pore pressure. If the pore spaces are connected, these formations will also have permeability. Fluids can flow through them in response to a pressure gradient. The pressure in the annulus is controlled by varying BP to operate the fluid pressure in the wellbore. The aim in MPD is thus to maintain the pressure in the annulus between the two limits of pore pressure and fracture pressure [1].

##### 2.1. The United Dynamic Model

When the bottom hole pressure is below the formation pressure, formation fluid will invade into the wellbore and the four-phase flow emerges in the annulus constituted by the drill string and wellbore. As seen in Figure 1, take any cross section of the wellbore as an infinitesimal control volume. In the infinitesimal control volume, the four-phase drilling fluid is consisted by drilling mud (considered as a pseudohomogeneous liquid), influx oil (considered as oil phase), influx natural gas (considered as gas phase), and influx water (considered as water phase). Appropriate assumptions and governing equations are critical to simulate realistic four-phase well-control operations. The four-phase model was established based on the following assumptions:(1)it is unsteady-state four-phase flow;(2)the flow along the flow path is one-dimensional;(3)the drilling mud is water-based;(4)drilling mud is incompressible.