creative commons mud pump oil and gas price

A mud pump (sometimes referred to as a mud drilling pump or drilling mud pump), is a reciprocating piston/plunger pump designed to circulate drilling fluid under high pressure (up to 7,500 psi or 52,000 kPa) down the drill string and back up the annulus. A mud pump is an important part of the equipment used for oil well drilling.

Mud pumps can be divided into single-acting pump and double-acting pump according to the completion times of the suction and drainage acting in one cycle of the piston"s reciprocating motion.

Mud pumps come in a variety of sizes and configurations but for the typical petroleum drilling rig, the triplex (three piston/plunger) mud pump is used. Duplex mud pumps (two piston/plungers) have generally been replaced by the triplex pump, but are still common in developing countries. Two later developments are the hex pump with six vertical pistons/plungers, and various quintuplexes with five horizontal piston/plungers. The advantages that these new pumps have over convention triplex pumps is a lower mud noise which assists with better measurement while drilling (MWD) and logging while drilling (LWD) decoding.

The fluid end produces the pumping process with valves, pistons, and liners. Because these components are high-wear items, modern pumps are designed to allow quick replacement of these parts.

To reduce severe vibration caused by the pumping process, these pumps incorporate both a suction and discharge pulsation dampener. These are connected to the inlet and outlet of the fluid end.

Displacement is calculated as discharged liters per minute. It is related to the drilling hole diameter and the return speed of drilling fluid from the bottom of the hole, i.e. the larger the diameter of drilling hole, the larger the desired displacement. The return speed of drilling fluid should wash away the debris and rock powder cut by the drill from the bottom of the hole in a timely manner, and reliably carry them to the earth"s surface. When drilling geological core, the speed is generally in range of 0.4 to 1.0 m^3/min.

The pressure of the pump depends on the depth of the drilling hole, the resistance of flushing fluid (drilling fluid) through the channel, as well as the nature of the conveying drilling fluid. The deeper the drilling hole and the greater the pipeline resistance, the higher the pressure needed.

With the changes of drilling hole diameter and depth, the displacement of the pump can be adjusted accordingly. In the mud pump mechanism, the gearbox or hydraulic motor is equipped to adjust its speed and displacement. In order to accurately measure the changes in pressure and displacement, a flow meter and pressure gauge are installed in the mud pump.

The construction department should have a special maintenance worker that is responsible for the maintenance and repair of the machine. Mud pumps and other mechanical equipment should be inspected and maintained on a scheduled and timely basis to find and address problems ahead of time, in order to avoid unscheduled shutdown. The worker should attend to the size of the sediment particles; if large particles are found, the mud pump parts should be checked frequently for wear, to see if they need to be repaired or replaced. The wearing parts for mud pumps include pump casing, bearings, impeller, piston, liner, etc. Advanced anti-wear measures should be adopted to increase the service life of the wearing parts, which can reduce the investment cost of the project, and improve production efficiency. At the same time, wearing parts and other mud pump parts should be repaired rather than replaced when possible.

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The 2,200-hp mud pump for offshore applications is a single-acting reciprocating triplex mud pump designed for high fluid flow rates, even at low operating speeds, and with a long stroke design. These features reduce the number of load reversals in critical components and increase the life of fluid end parts.

The pump’s critical components are strategically placed to make maintenance and inspection far easier and safer. The two-piece, quick-release piston rod lets you remove the piston without disturbing the liner, minimizing downtime when you’re replacing fluid parts.

The circulation system on the rig is the system that allows for circulation of the Drilling Fluid or Mud down through the hollow drill string and up through the annular space between the drill string and wellbore. It is a continuous system of pumps, distribution lines, storage tanks, storage pits, and cleansing units that allows the drilling fluid to fulfill its primary objectives (these will be discussed later in this lesson). The mud pumps of the circulation system and the drawworks of the hoisting systems are the two largest draws on the power from the power system

Drilling fluid is mixed in the mud pits and pumped by the mud pumps through the swivel, through the blow out preventer (not part of the circulation system) down the hollow drill pipe, through holes (Jet Nozzles) in the bit, up the annular space between drill pipe and wellbore (where it lifts the rock cuttings), to the surface, through the Solids Control Equipment (Shale Shaker, Desander, and Desilter), and back to the mud pits. A schematic of the circulation system is shown in Figure 9.05.

In this figure, fresh water-based drilling fluid (mud) is mixed with water from the Water Tank (not shown in Figure 9.05) and components from the Bulk Mud Components Storage (not shown in Figure 9.05) in the Mud Pit. The Mud Pumps then pump the mud through the swivel, kelly, kelly bushing, and rotary table down to the drill string.

The mud pumps on a typical drilling rig are either single-action or double-action Reciprocating (Positive Displacement) Pumps which may contain two pistons-cylinders (duplex pump) or three pistons-cylinders (triplex pump). Figure 9.06 shows schematics of a single piston-cylinder in (A) a single-action and (B) a double-action reciprocating pump.

In these pumps, the positive pressure and negative pressure (suction) in the cylinder cause the valves to open and close (note: the valves in the schematic are simple representations of the actual valves). Due to the high viscosity of the drilling fluid, the inlet side of the pump may require a Charge Pump to keep fluids moving into the cylinders at high pressures and to prevent Cavitation in the pump.

From the mud pumps, the drilling fluid goes to the swivel, through the blow out preventer, and down the hollow drill string and bottom-hole assembly. The drilling fluid then goes through jet nozzles in the drill bit; at which point, it begins its return to the surface. The drilling fluid travels up the annular space between the drill pipe and the wellbore, picking up and carrying the drill cuttings up the hole.

Once the drilling fluid reaches the surface, it goes through the mud return line to the gas-mud separator and the solids control equipment. The shale shaker is where the large cuttings from the returning drilling fluid are removed. The shale shaker is a set of vibrating mesh screens that allow the mud to pass through while filtering out cuttings of different size at screen screen mesh sizes. A Mudlogger or a Well-Site Geologist may be stationed at the shale shaker to analyze the cuttings to determine the lithology of the rock and the depth within the Stratigraphic Column at which the well is currently being drilled.

The drilling fluid then passes through the Desander and Desilter. These are hydrocyclones which use centrifugal forces to separate the smaller solids from the drilling fluid. The desander typically removes solids with a diameter in the range of 45 – 74 μm, while the desilter removes solids with a diameter in the range of 15 – 44 μm.

The drilling fluid is then sent through a degasser to remove any gas bubbles that have been picked up during the circulation. These gasses may include natural gas from the subsurface or air acquired during the solids control. Typically, the degasser is a piece of equipment that subjects the drilling fluid to slight vacuum to cause the gas to expand for extraction. The drilling fluid is then returned to the mud pit to start the circulation process over again.

We have discussed the mechanics of how the drilling fluid is circulated during the drilling process, but we have not discussed the role of the drilling fluid. The term “mud” is often used in oil and gas well drilling because historically the most common water-based drilling fluids were mixtures of water and finely ground, bentonite clays which, in fact, are muds.

control formation pore pressures to assure desired well control (apply hydrostatic and hydrodynamic pressures in excess of the formation pore pressures to prevent fluids from entering the wellbore);

allow for pressure signals from Logging While Drilling (LWD) or Measurement While Drilling (MWD) tools to be transmitted to the surface (LWD and MWD data are transmitted to the surface using pressure pulses in the drilling fluid);

As I stated earlier, historically drilling fluids were mixtures of bentonite clay, water, and certain additives to manipulate the properties of the mud (density, viscosity, fluid loss properties, gelling qualities, etc.). Today, there are several different options available for drilling fluids. These include:

Of the listed drilling fluids, the water-based muds and the oil-based muds are the most common; foam drilling and air drilling can only be used under specialized conditions. Of the two liquid based mud systems (water-based muds and oil-based muds), water-based muds are the most common mud system. They are more environmentally friendly and are used almost exclusively to drill the shallow portions of the well where fresh water aquifers exist to minimize any contamination to those aquifers. As this implies, drilling fluids can be – and often are – switched during the course of drilling operations in single well.

In addition, water-based muds are cheaper than oil-based muds, so they are used to reduce drilling costs and commonly represent the “default” selection for a drilling fluid. In other words, water-based muds are often used unless there is a specific reason to switch to an oil-based mud.

Oil-based muds are formulated with diesel oil, mineral oil, or synthetic oils as a continuous phase and water as a dispersed phase in an emulsion. In addition, additives such as emulsifiers and gelling agents are also used. They were specifically developed to address certain drilling problems encountered with water-based muds. The reasons for using an oil-based mud include:

drilling through shales that are susceptible to swelling (in particular, highly smectite-rich shales). Shales contain a large amount of clay material and when these clays come in contact with the water in a water-based mud system, the clays may swell causing the shales to collapse into the hole. Smectite-rich shale formations are often referred to as “Gumbo” or “Gumbo Clays” in the drilling industry;

reducing torque and drag problems in deviated wells. Since oil, a lubricant, is the continuous phase in the mud system, the torque and drag between the drill pipe and the wellbore is reduced with oil-based muds;

achieving greater thermal stability at greater depths. Oil-based muds have been found to retain their stability (retain their desired properties) at greater down hole temperatures;

achieving greater resistance to chemical contamination. Many substances found down-hole (salt, CO2, H2S, etc.) are soluble in water. The introduction of these substances into the water-based mud system may have a deleterious impact on different mud properties (density, viscosity, fluid loss properties, gelling properties, etc.). These substances are not soluble in oil and, therefore, have will not impact oil-based mud properties.

The first three bullet points in this list are becoming more common problems in the oil and gas industry. The shale boom in the U.S. has made long horizontal sections in shale reservoirs targets for drilling. In addition, deviated wells and deeper wells are also becoming more common. For these reasons, the use of oil-based muds is also becoming more common.

high initial costs. Often in an active drilling campaign, if certain depth intervals require an oil-based mud, the mud is stored and reused in different wells;

slow rates of penetration. Historically, the rate of penetration has been statistically slower for oil-based muds than it is for water-based muds. The rate of penetration is the speed at which the drilling process progresses (depth versus time) and is a function of many factors other than mud type, including: weight on bit, RPM, lithologies being drilled through, bit type, bit wear, etc.;

kick detection. If gas enters the wellbore (a Kick), it may go into solution in the oil in deeper, higher pressure sections of the well and come out of solution closer to the surface;

formation evaluation. Some readings from well logs or core analysis may be sensitive to oil entering the formation of interest (for example, if oil from the oil-based mud enters the reservoir in the near-well vicinity, then tools used to detect oil saturation may read artificially high).

Other drilling fluids currently in use that were listed earlier are foams and air. In the context of drilling fluids, foams have the consistency of shaving cream. Both foam and air drilling are used in hard rock regions, such as in the Rocky Mountains, where drill bits render the drill cuttings to dust. Thus, the foam or air only needs to lift this dust to the surface. Air drilling is always an environmentally friendly option if it is applicable because environmental contamination by air is never an issue.

Researchers have shown that mud pulse telemetry technologies have gained exploration and drilling application advantages by providing cost-effective real-time data transmission in closed-loop drilling operations. Given the inherited mud pulse operation difficulties, there have been numerous communication channel efforts to improve data rate speed and transmission distance in LWD operations. As discussed in “MPT systems signal impairments”, mud pulse signal pulse transmissions are subjected to mud pump noise signals, signal attenuation and dispersion, downhole random (electrical) noises, signal echoes and reflections, drillstring rock formation and gas effects, that demand complex surface signal detection and extraction processes. A number of enhanced signal processing techniques and methods to signal coding and decoding, data compression, noise cancellation and channel equalization have led to improved MPT performance in tests and field applications. This section discusses signal-processing techniques to minimize or eliminate signal impairments on mud pulse telemetry system.

At early stages of mud pulse telemetry applications, matched filter demonstrated the ability to detect mud pulse signals in the presence of simulated or real noise. Matched filter method eliminated the mud noise effects by calculating the self-correlation coefficients of received signal mixed with noise (Marsh et al. 1988). Sharp cutoff low-pass filter was proposed to remove mud pump high frequencies and improve surface signal detection. However, matched filter method was appropriate only for limited single frequency signal modulated by frequency-shift keying (FSK) with low transmission efficiency and could not work for frequency band signals modulated by phase shift keying (PSK) (Shen et al. 2013a).

Wavelet transform method was developed and widely adopted and used in signal processing to overcome limitation of Fourier transform in time domain (Bultheel 2003). Although Fourier and its revised fast Fourier transforms are powerful mathematical tool, they are not very good at detecting rapid changes in signals such as seismic data and well test data in petroleum industry containing many structure of different scales (Multi-scale structures) (Guan et al. 2004). Fourier coefficients do not provide direct information about the signal local behavior (localization); but the average strength of that frequency in the full signal as the sine or cosine function keeps undulating to infinity. Wavelet transform analyzes the signal frequency components and time segment, and fine tune sampling of localized characters of time or frequency domain. Principles of wavelet transform and de-noising technique show that signal can be divided into space and scale (time and frequency) without losing any useful information of the original signal, hence ensuring the extraction of useful information from the noised signal (Li et al. 2007). Different wavelet base parameters constructed, such as haar, db, coif, sym, bior, rbio and dmey, are suitable for different signal processing requirements. The small the scale parameter is, the higher the resolving power in frequency, suitable for processing high frequency signals; conversely, the larger the scale is the higher resolving power suitable for low frequency signal.

In processing noise-contaminated mud pulse signals, longer vanishing moments are used, but takes longer time for wavelet transform. The main wavelet transform method challenges include effective selection of wavelet base, scale parameters and vanishing moment; the key determinants of signal correlation coefficients used to evaluate similarities between original and processed signals. Chen et al. (2010) researched on wavelet transform and de-noising technique to obtain mud pulse signals waveform shaping and signal extraction based on the pulse-code information processing to restore pulse signal and improve SNR. Simulated discrete wavelet transform showed effective de-noise technique, downhole signal was recovered and decoded with low error rate. Namuq et al. (2013) studied mud pulse signal detection and characterization technique of non-stationary continuous pressure pulses generated by the mud siren based on the continuous Morlet wavelet transformation. In this method, generated non-stationary sinusoidal pressure pulses with varying amplitudes and frequencies used ASK and FSK modulation schemes. Simulated wavelet technique showed appropriate results for dynamic signal characteristics analysis.

While Fourier coefficients provide average signal information in frequency domain and unable to reveal the non-stationary signal characteristics, wavelet transform can effectively eliminate MPT random noise when signal carrier frequency characteristics (periods, frequencies, and start and end time) are carefully analyzed.

As discussed in “MPT mud pump noises”, the often overlap of the mud pulses frequency spectra with the mud pump noise frequency components adds complexity to mud pulse signal detection and extraction. Real-time monitoring requirement and the non-stationary frequency characteristics made the utilization of traditional noise filtering techniques very difficult (Brandon et al. 1999). The MPT operations practical problem contains spurious frequency peaks or outliers that the standard filter design cannot effectively eliminate without the possibility of destroying some data. Therefore, to separate noise components from signal components, new filtering algorithms are compulsory.

Early development Brandon et al. (1999) proposed adaptive compensation method that use non-linear digital gain and signal averaging in the reference channel to eliminate the noise components in the primary channel. In this method, synthesized mud pulse signal and mud pump noise were generated and tested to examine the real-time digital adaptive compensation applicability. However, the method was not successfully applied due to complex noise signals where the power and the phases of the pump noises are not the same.

Jianhui et al. (2007) researched the use of two-step filtering algorithms to eliminate mud pulse signal direct current (DC) noise components and attenuate the high frequency noises. In the study, the low-pass finite impulse response (FIR) filter design was used as the DC estimator to get a zero mean signal from the received pressure waveforms while the band-pass filter was used to eliminate out-of-band mud pump frequency components. This method used center-of-gravity technique to obtain mud pulse positions of downhole signal modulated by pulse positioning modulation (PPM) scheme. Later Zhao et al. (2009) used the average filtering algorithm to decay DC noise components and a windowed limited impulse response (FIR) algorithm deployed to filter high frequency noise. Yuan and Gong (2011) studied the use of directional difference filter and band-pass filter methods to remove noise on the continuous mud pulse differential binary phase shift keying (DBPSK) modulated downhole signal. In this technique, the directional difference filter was used to eliminate mud pump and reflection noise signals in time domain while band-pass filter isolated out-of-band noise frequencies in frequency domain.

Other researchers implemented adaptive FIR digital filter using least mean square (LMS) evaluation criterion to realize the filter performances to eliminate random noise frequencies and reconstruct mud pulse signals. This technique was adopted to reduce mud pump noise and improve surface received telemetry signal detection and reliability. However, the quality of reconstructed signal depends on the signal distortion factor, which relates to the filter step-size factor. Reasonably, chosen filter step-size factor reduces the signal distortion quality. Li and Reckmann (2009) research used the reference signal fundamental frequencies and simulated mud pump harmonic frequencies passed through the LMS filter design to adaptively track pump noises. This method reduced the pump noise signals by subtracting the pump noise approximation from the received telemetry signal. Shen et al. (2013a) studied the impacts of filter step-size on signal-to-noise ratio (SNR) distortions. The study used the LMS control algorithm to adjust the adaptive filter weight coefficients on mud pulse signal modulated by differential phase shift keying (DPSK). In this technique, the same filter step-size factor numerical calculations showed that the distortion factor of reconstructed mud pressure QPSK signal is smaller than that of the mud pressure DPSK signal.

Study on electromagnetic LWD receiver’s ability to extract weak signals from large amounts of well site noise using the adaptive LMS iterative algorithm was done by (Liu 2016). Though the method is complex and not straightforward to implement, successive LMS adaptive iterations produced the LMS filter output that converges to an acceptable harmonic pump noise approximation. Researchers’ experimental and simulated results show that the modified LMS algorithm has faster convergence speed, smaller steady state and lower excess mean square error. Studies have shown that adaptive FIR LMS noise cancellation algorithm is a feasible effective technique to recover useful surface-decoded signal with reasonable information quantity and low error rate.

Different techniques which utilize two pressure sensors have been proposed to reduce or eliminate mud pump noises and recover downhole telemetry signals. During mud pressure signal generation, activated pulsar provides an uplink signal at the downhole location and the at least two sensor measurements are used to estimate the mud channel transfer function (Reckmann 2008). The telemetry signal and the first signal (pressure signal or flow rate signal) are used to activate the pulsar and provide an uplink signal at the downhole location; second signal received at the surface detectors is processed to estimate the telemetry signal; a third signal responsive to the uplink signal at a location near the downhole location is measured (Brackel 2016; Brooks 2015; Reckmann 2008, 2014). The filtering process uses the time delay between first and third signals to estimate the two signal cross-correlation (Reckmann 2014). In this method, the derived filter estimates the transfer function of the communication channel between the pressure sensor locations proximate to the mud pump noise source signals. The digital pump stroke is used to generate pump noise signal source at a sampling rate that is less than the selected receiver signal (Brackel 2016). This technique is complex as it is difficult to estimate accurately the phase difference required to give quantifiable time delay between the pump sensor and pressure sensor signals.

As mud pulse frequencies coincide with pump noise frequency in the MPT 1–20 Hz frequency operations, applications of narrow-band filter cannot effectively eliminate pump noises. Shao et al. (2017) proposed continuous mud pulse signal extraction method using dual sensor differential signal algorithm; the signal was modulated by the binary frequency-shift keying (BFSK). Based on opposite propagation direction between the downhole mud pulses and pump noises, analysis of signal convolution and Fourier transform theory signal processing methods can cancel pump noise signals using Eqs. 3 and 4. The extracted mud pulse telemetry signal in frequency domain is given by Eqs. 3 and 4 and its inverse Fourier transformation by Eq. 4. The method is feasible to solve the problem of signal extraction from pump noise,

\(H(\omega )={f^{ - 1}}h(t)=G(\omega ){e^{ - j\omega \tau }}\) is the Fourier transformed impulse response, \(h(t)\), data transmission between sensor A and sensor B.

These researches provide a novel mud pulse signal detection and extraction techniques submerged into mud pump noise, attenuation, reflections, and other noise signals as it moves through the drilling mud.

Copyright © 2013 Yuanhua Lin 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.

Investigation of propagation characteristics of a pressure wave is of great significance to the solution of the transient pressure problem caused by unsteady operations during management pressure drilling operations. With consideration of the important factors such as virtual mass force, drag force, angular frequency, gas influx rate, pressure, temperature, and well depth, a united wave velocity model has been proposed based on pressure gradient equations in drilling operations, gas-liquid two-fluid model, the gas-drilling mud equations of state, and small perturbation theory. Solved by adopting the Runge-Kutta method, calculation results indicate that the wave velocity and void fraction have different values with respect to well depth. In the annulus, the drop of pressure causes an increase in void fraction along the flow direction. The void fraction increases first slightly and then sharply; correspondinglythe wave velocity first gradually decreases and then slightly increases. In general, the wave velocity tends to increase with the increase in back pressure and the decrease of gas influx rate and angular frequency, significantly in low range. Taking the virtual mass force into account, the dispersion characteristic of the pressure wave weakens obviously, especially at the position close to the wellhead.

One of the future trends of the petroleum industry is the exploration and development of high pressure, low permeability reservoirs [1]. Drilling-related issues such as excessive mud cost, wellbore ballooning/breathing, kick-detection limitations, difficulty in avoiding gross overbalance conditions, differentially stuck pipe, and resulting well-control issues together contribute to the application of managed pressure drilling (MPD) technology [2]. MPD technology has the ability to quickly react to the expected drilling problems and formations pressures uncertainties, reduce nonproductive time and mitigating drilling hazards, and offer a considerable amount of tangible benefits while drilling in extremely narrow fracture/pore pressure windows [3]. It allows drilling operations to proceed where conventional drilling is easy to cause formation damage or considered uneconomical,

of high risk, or even impossible [4]. Although drilling operators try to avoid the risk of influxes, occasionally there are influxes for various reasons. Gas influx occurs whenever the pressure of a gas-bearing formation exceeds the pressure at the bottom of a wellbore. Since the subsequent intrusion of gas displaces drilling mud, it decreases the pressure in the wellbore and makes gas enter even faster [5, 6]. If it is not counteracted in time, the unstable effect can escalate into a blowout creating severe financial losses, environmental contamination, and potential loss of human lives. The basic principle of MPD well control is to keep the bottomhole pressure (BHP) as constant as possible at a value that is at least equal to the formation pressure [7]. MPD is a class of techniques that allow precise management of BHP under both static and dynamic conditions through a combination of controlling the flow rate, mud density, and back pressure (or wellhead pressure) on the fluid returns (choke manifold) of

the enclosed and pressurized fluid system [8, 9]. As a result of the ability to manipulate the back pressure, MPD offers the capability to improve safety and well control through early detection of influxes and losses in a microflux level, reduces the risk of influx and thus the chance of blowouts, and controls an influx dynamically without conventionally shutting-in [10].

During the managed pressure drilling process, all the unsteady operations such as adjustment of choke, wellhead backpressure controls [11], tripping in/out [12], shutting-in [13], and mud pump rate changes [14] will cause generation and propagation of pressure waves, which would threaten the whole drilling system from the wellhead facilities to the bottomhole drilling equipment and the formation [15]. In the analysis of an influx well and formulation of well control scheme, the dynamic effects of these operations, appeared as pressure fluctuations, should be accounted for [16]. As a basic parameter of pressure fluctuation, the pressure wave velocity has close relevance with the determination of transient pressure and safe operating parameters for well control. However, the gas influx is more troublesome for the higher compressibility and lower density of influx gas than the single phase drilling mud [17]. The works of Bacon et al. [18] had demonstrated that compressibility effects of a gas influx can be significant during an applied-back-pressure, dynamic, MPD well control response and can impact the well control process. Hence, this paper considered the propagation behavior of pressure wave in gas-drilling mud two-phase flow in the annulus to provide reference for the MPD operations.

Well control includes not only the handling but also early detection of a gas influx. Besides the transient pressure problem mentioned previously, investigation of the propagation characteristics of pressure wave is of great significance to the early detection of gas influx [19]. Many scholars believe that the pressure fluctuations procedure contains a wealth of information about the flow. Thus, characteristics of pressure wave can be easily used to measure some important parameters in the two-phase flow [20]. In the late 1970s, the former Soviet All-Union Drilling Technology Research Institute began to study characteristics of pressure wave propagation velocity in gas-liquid two-phase flow to detect early gas influx and achieved some important results [21]. According to the functional relationship between the pressure wave propagation velocity in gas-liquid two-phase flow and the gas void fraction, Li et al. [22] presented a method of detecting the gas influxes rate and the height of gas migration early after gas influxes into the wellbore. Furthermore, mud pulse telemetry is the most common method of data transmission used by measurement-while-drilling, and the transmission velocity of the pulse is a basic parameter for this kind of data transmission mode [23].

The pressure wave discussed in this paper can be transmitted as a pressure perturbation along the direction of flow in wellbore, which propagates with the speed of sound in the mud and gas two-phase drilling fluid. Due to the compressibility of the gas phase, the changes in interface between the gas and drilling mud, and the momentum and energy transfer between two phases, it is complicated to

predict the pressure wave velocity in gas and drilling mud two-phase flow. Since the 1940s, many experimental and theoretical studies have been performed. Experimental tests were conducted to inspect the contributions of fluctuation and flow characteristics on pressure wave. Ruggles et al. [24] firstly performed the experimental investigation on the dispersion property of pressure wave propagation in air-water bubbly flow. It was demonstrated that the propagation speed of pressure wave varies over a range of values for the given state, depending on the angular frequency of the pressure wave. Legius et al. [25] tested the propagation of pressure waves in bubbly and slug flow. The experimental result is similar to the calculation result of the Nguyen model and simulation result of the Sophy-2 package. Concluded form Miyazaki and Nakajima experiments [26] in Nitrogen-Mercury two-phase system, the slip between the phases plays a very important role in the mechanism of pressure wave propagation. From experimental investigation, Bai [27] found that the fractal dimension, correlation dimension, and the Kolmogorov entropy have close relationship with flow regime, and the fractal dimension will be greater than 1.5 when the flow is annular with high gas velocity. The characteristics of pressure wave propagation in bubbly and slug flow in a vertical pipe were investigated experimentally by Huang et al. in detail [28]. It confirmed that the propagation velocity is greatly affected by the gas void fraction and angular frequency of the pressure disturbance, and the superficial velocity of flowing medium has almost no effect on the propagation velocity. Also, there are some widely accepted models including elasticity model, homogeneous mixture model, and continuum model for pressure wave propagation in gas-liquid two-phase flow. Tangren et al. [29] took the two-phase media as a homogeneous fluid and presented the solution model concerning the problem of pressure wave velocity in two-phase flow at low gas void fraction. Wallis [30] studied the propagation mechanism of pressure waves and derived the propagation velocity in a bubbly flow and separated flow using the homogeneous model, in which the two-phase mixture is treated as a compressible fluid with suitably averaged properties. Nguyen et al. [31] applied the elastic theory to predict the propagation velocity of pressure waves in several different flow regimes. The comparison between the calculation results and available experimental data suggests its success at low void fraction. Mecredy and Hamilton [32] derived a detailed continuum model for sound wave propagation in gas-liquid flow by using six separate conservation equations to describe the flow of the vapor and liquid phases. This so-called two-fluid representation allows for nonequilibrium mass, heat, and momentum transfer between the phases. Results indicated that in a bubbly flow, high angular frequency waves travelled an order of magnitude faster than low angular frequency waves. With the development of hydrodynamics, the two-fluid model is widely used to determine the propagation velocity of the pressure wave in two-phase flow as it can provide a general dispersion relation valid for arbitrary flow regimes including effects of the interphase mass, heat, and momentum transfer. Ardron and Duffey [33] developed a model for sound-wave propagation in nonequilibrium vapor-liquid flows which

predicts sound speeds and wave attenuations dependent only on measurable flow properties on the basis of two-fluid conservation equations. Ruggles et al. [24], Xu and Chen [34], Chung et al. [35], Huang et al. [28], and Bai et al. [27] investigated the propagation velocity behavior of pressure wave via two-fluid model and small perturbation theory, and the predicted results show good agreement with the experimental data. In recent years, some new researches and models that are especially important in this area were developed. Xu and Gong [36] proposed a united model to predict wave velocity for different flow patterns. In this united model, the effect of a virtual mass coefficient was taken into consideration. The propagation of pressure wave during the condensation of R404A and R134a refrigerants in pipe minichannels that undergo periodic hydrodynamic disturbances was given by Kuczynski [37, 38]. Li et al. [39] simulated the condensation of gas oxygen in subcooled liquid oxygen and the corresponding mixing process in pump pipeline with the application of thermal phase change model in Computational Fluid Dynamics code CFX and investigated the pressure wave propagation characteristics in two-phase flow pipeline for liquid-propellant rocket based on a proposed pressure wave propagation model and the predicted flow parameters. Based on the unified theory of Kanagawa et al. [40], the nonlinear wave equation for pressure wave propagation in liquids containing gas bubbles is rederived. On the basis of numerical simulation of the gaseous oxygen and liquid oxygen condensation process with the thermal phase model in ANSYS CFX, Chen et al. [41] solved the pressure wave propagation velocity in pump pipeline via the dispersion equation derived from ensemble average two-fluid model. Li and He [42] developed an improved slug flow tracking model and analyzed the variation rule of the pressure wave along the pipeline and influence of the variation of initial inlet liquid flow rate and gas flow rate in horizontal air-water slug flow with transient air flow rate. Meanwhile, the compressibility effects of gas had been noticed in the research field of propagation of pressure waves in the drilling industry, and some efforts have been made. Li et al. [22] established the relationship between wave velocity and gas void fraction according to the empirical formula of the homogeneous mixture model presented by Martin and Padmanatbhan [43] and frequency response model presented by Henry [44]. By applying the unsteady flow dynamic theory, Liu et al. [45] derived the pressure wave velocity calculation formula for gas-drilling mud-solid three-phase flow based on the continuity equation. Wang and Zhang [46] studied the pressure pulsation in mud and set up a model for calculating the amplitude of pressure pulsation when pressure wave is transmitted in drilling-fluid channel especially drilling hose with different inside diameters. However some efforts have been made; the pressure wave velocity is usually determined by empirical formula. In the past researches, the influencing factors for pressure wave propagation were simulated and analyzed with the mathematical model; however, the variation of wave velocity and gas void at different depth of wellbore was not considered. In addition, the current researches are limited in their assumption and neglect the flow pattern translation and interphase forces along the annulus. Up to now, no complete

mathematical model of pressure wave in an annulus with variations of gas void, flow pattern, temperature, and back pressure during MPD operations has been derived.

The object of the present work is to study the velocity for the transmission of pressure disturbance in the two-phase drilling fluid in the form of a pressure wave in annulus during MPD operations. In this paper, in addition to the pressure, temperature, and the void fraction in the annulus, the compressibility of the gas phase, the virtual mass force, and the changes of interface in two phases are also taken into consideration. By introducing the pressure gradient equations in MPD operations, gas-liquid two-fluid model, the gas-drilling mud equations of state (EOS), and small perturbation theory, a united model for predicting pressure wave velocity in gas and drilling mud in an annulus is developed. The model can be used to predict the wave velocity of various annulus positions at different influx rates, applied back pressures, and angular frequencies with a full consideration of drilling mud compressibility and interphase forces.

2.1. The Basic Equation. In this paper, the two-fluid model and the pressure gradient equation along the flow direction in the annulus are combined to study the pressure wave velocity in MPD operations. Drilling fluid contains clay, cuttings, barite, other solids, and so forth. The solid particles are small and uniformly distributed; therefore, drilling fluid is considered to be a pseudohomogeneous liquid, and the influx natural gas is considered to be the gas phase. The following assumptions are made:

As shown in Figure 1, the gas and drilling mud two-phase fluid travels along the annulus in the drilling process. The fluid flows along the annulus in "-z" direction, and the annulus is formed by the casing and drill string.

The interphase forces include virtual mass force, drag force, and the wall shear stress. The transfer of momentum between the gas and drilling mud phases MGi and MLi can be written as follows:

The proportion of gas phase in the interface between the gas and drilling mud is rather small in that the pressure difference between the gas interface and gas is not very high. Omitting the pressure difference, the gas interface pressure

In fact, the shear stress and the interphase shear stress are very small relative to the Reynolds stress. Also, the Reynolds stress in gas phase can be omitted relative to the interphase force. Hence, it can be described as

The formula presented by Dranchuk and Abou-Kassem has been used to solve the gas deviation factor under the condition of low and medium pressure (p < 35 MPa) [49]:

The formula presented by Yarborough and Hall has been adopted to solve the gas deviation factor under the condition of high pressure (p > 35 MPa) [50]:

2.2.2. Equations of State for Liquid. Under different temperatures and pressures, the density of drilling mud can be obtained by the empirical formulas.

2.2.3. Correlation of Temperature Distribution. The temperature of the drilling mud at different depths of the annulus can be determined by the relationship presented by Hasan and Kabir [51]:

2.3. Flow Pattern Analysis. Based on the analysis of flow characteristics in the enclosed drilling system, it can be safely assumedthatthe flowpattern in an annulusiseitherbubblyor slug flow [52]. The pattern transition criteria for bubbly flow and slug flow given by Orkiszewski are used to judge the flow pattern in the gas-drilling mud two-phase flow [53].

the simplified EOS ((18) and (19)) which neglects the thermal expansion of liquid. The sonic speed of gas phase cG and that of liquid phase cL can be presented in the following form:

where the D; and D0 are the diameters of inner pipe and outer pipe, respectively; the k0 and kt are the roughnesses of outer pipe and the inner pipe, respectively.

The total pressure drop gradient is the sum of pressure drop gradients due to potential energy change and kinetic energy and frictional loss. From (1), the equation used to calculate the pressure gradient of gas and drilling mud flow within the annulus can be written as

Assuming the compressibility of the gas is only related to the pressure in the annulus, the kinetic energy or acceleration term in the previous equation can be simplified to

where A is the matrix of parameters considered in relation to time, B is the matrix of parameters considered in relation to the spatial coordinate, and C is the vector of extractions.

In (44), k is the wave number, and w is the angular frequency of the disturbances. Substituting (44) into (43) gives homogenous linear equations concerning the expression (

By solving the dispersion equation indicated previously, four roots can be obtained. Because the wavelengths associated with two of the roots are too short to allow the two-fluid medium to be treated as a continuum, the two roots should be omitted. As for the two remaining roots, one of them represents a pressure wave that transmited along the axis z, and the other represents a pressure wave transmits in the opposite direction in accordance with the direction of the flow along the annulus. The real value of wave number can be used to determine the wave velocity c.Thewavevelocityinthe two-phase fluid can be determined by the following model:

Obtaining the analytical solution of the mathematical models concerned with flow pattern, void fraction, characteristic parameters, and pressure drop gradient is generally impossible for two-phase flow. In this paper, the Runge-Kutta method (R-K4) is used to discretize the theoretical model.

We can obtain pressure, temperature, gas velocity, drilling mud velocity, and void fraction at different annulus depths by R-K4. The solution of pressure drop gradient equation (41) can be seen as an initial-value problem of the ordinary differential equation:

In the present work, the mathematical model and pressure wave velocity calculation model are solved by a personally compiled code on VB.NET (Version 2010). The solution procedure for the wave velocity in the annulus is shown in Figure 1. At initial time, the wellhead back pressure, wellhead temperature, wellbore structure, well depth, gas and drilling mud properties, and so forth, are known. On the node i, the pressure gradient, temperature, and the void fraction can be obtained by adopting R-K4. Then, the determinant (45) is calculated based on the calculated parameters. Omitting the two unreasonable roots, the pressure wave velocity at different depths of the annulus in MPD operations can be solved by (47). The process is repeated until the pressure wave velocity of every position in the wellbore is obtained as shown in Figure 2.

The developed model takes full consideration of the interfacial interaction and the virtual mass force. Owing to the complex conditions of the annulus in MPD operations, measurement of wave velocity in the actual drilling process is very difficult. In order to verify the united model, the predicted pressure waves are compared with the results of previous simulated experimental investigations presented for gas and drilling mud by Liu et al. [45] in Figure 3(a) and by Li et al. [22] in Figure 3(b). The lines represent the calculation results, and the points represent the experimental data.

The comparisons reveal that the developed united model fits well with the experimental data. Thus, the united model can be used to accurately predict wave velocity at different wellhead back pressures and gas influx rates (the influx rate of gas in the bottomhole) in MPD operations.

The drilling system described is an enclosed system. The schematic diagram of the gas influx process is illustrated in Figure 4. The drilling mud is pumped from surface storage down the drill pipe. Returns from the wellbore annulus travel back through surface processing, where drilling solids are removed, to surface storage. The key equipments include the following.

(i)The Rotating Control Device (RCD). The rotating control device provides the seal between atmosphere and wellbore, while allowing pipe movement and diverting returns flow. In conjunction with the flow control choke, the RCD provides the ability to apply back pressure on the annulus during an MPD operation.

The gas and drilling mud flow rate measured by the Cori-olis meter and the back pressure measured by the pressure sensor are the initial data for annulus pressure calculation. The well used for calculation is a gas well in Xinjiang Uygur Autonomous Region, Northwest China. The wellbore structure, well design parameters (depths and diameters), gas and drilling mud properties (density and viscosity), and operational conditions of calculation well are displayed in Table 1.

The drilling mud mixed with gas is taken as a two-phase flow medium. The propagation velocity of pressure wave in the gas-drilling mud flow is calculated and discussed by using the established model and well parameters.

pressurized system. The pressure at different depths of the annulus varied with the change of back pressure. According to the EOS of gas and drilling mud, the influence of pressure on gas volume is much greater than that of drilling mud for the greater compressibility of gas. So, the gas void fraction changes with the variation of gas volume at the nearly constant flow rate of drilling mud at different annular pressures. Meanwhile, the pressure wave velocity is sensitive

to the gas void fraction. As a result, when the back pressure at the wellhead is changed by adjusting the choke, the wave velocity in the two-phase drilling fluid and the distribution of void fraction at different depth of the annulus will diverge. The calculation results affected by the back pressure are presented.

Figures 5 and 6 show the distributions of void fraction and variations of wave velocity along the flow direction in the annulus when the back pressure at the wellhead is 0.1 MPa, 0.8 MPa, 1.5 MPa, 2.5 MPa, 5.0 MPa, and 6.5 MPa,

respectively. It can be seen that the void fraction significantly increases along the flow direction in the annulus. Conversely, the wave velocity shows a remarkable decrease tendency. For instance, BP = 1.5 MPa, at the wellhead, the wave velocity is 90.12 m/s, and the void fraction is 0.611, while in the bottomhole, the wave velocity reaches 731.42 m/s, and void fraction is reduced to 0.059. However, a sudden increase in wave velocity is observed in Figure 5 when the back pressure is 0.1 MPa at the position close to wellhead as the void fraction is further increased. Moreover, the influence of backpressure

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 bottomhole, the density of the drilling mud has little variation while the compressibility increases obviously, which makes the wave velocity decreased. Then, the two-phase drilling mud flow from the bottomhole to the wellhead along the annulus with a drop of pressure caused by, potential energy change, kinetic energy and frictional loss, which leads to an increase in void fraction. When the void fraction is increased, both the density and the compressibility of the two-phase fluid change slightly, resulting in a flat decrease in wave velocity along the flow direction. With the further increasing in void fraction, the bubbly flow turns into slug flow according to the flow pattern transient criteria of Orkiszewski which shows that the flow pattern is dependent on the void fraction. Generally speaking, the gas-drilling mud slug flow is composed of liquid and gas slugs. In the preliminary slug flow, the liquid slugs are much longer than the gas slugs, so the wave velocity is determined primarily by the wave velocities in the liquid slugs which are hardly affected by the void fraction. At the position close to the wellhead, the pressure of two-phase flow fluid can be reduced to a low value which approaches the back pressure. The compressibility of gas will be improved at the low pressure. It results in significant increase in void

fraction. As the void fraction increases greatly at the position close to the wellhead, gas slugs become much longer than the liquid slugs. It is assumed that the actual wave velocity in slug flow happens to play a leading role when the void fraction increases to some extent. For the decrease in liquid holdup in the gas slug with the increasing in void fraction, the wave velocity in the gas slug is infinitely close to that in pure gas, which is equivalent to an increase in wave velocity in the gas slug. Meanwhile, the gas content in liquid slug also increases and results in a lower wave velocity in the liquid slug. As a result, a slight increase in wave velocity appears.

Figures 7 and 8 present the influence of back pressure on the void fraction and wave velocity with respect to the parameters of well depth. As the back pressure increases, the void fraction at different depth of the annulus is reduced gradually, while the wave velocity in the two-phase flow tends to increase. Analytical results show that the increased back pressure is equivalent to be applied to the entire enclosed drilling fluid cyclical system. The pressure transmits from the wellhead to the bottomhole; therefore, the annular pressure in the entire wellbore is increased. According to the EOS, the density of gas increases and the compressibility of gas decreases with the increasing of gas pressure. So, the loss of interphase momentum and energy exchange is reduced and the interphase momentum exchange is promoted. It contributes to the increase in wave velocity with the increase in gas pressure. In addition, due to lower compressibility of two-phase flow medium under high pressure, the increase tendency of pressure wave velocity and the decrease tendency of void fraction are slowed down in the high back pressure range.

5.2. Effect of Gas Influx Rate on Wave Velocity. Figures 9 and 10 graphically interpret the distributions of void fraction and variations of wave velocity along the flow direction in the annulus. When gas influx occurs in the bottomhole, gas invades into the wellbore and migrates from the bottomhole to the wellhead along the flow direction. At a low gas influx rate, it is extremely obvious that the void fraction and wave velocity first slightly change in a comparatively smooth value then change sharply. It is because of the rapid expansion of gas volume with the decreasing in pressure near the wellhead that the void fraction increases sharply, and the wave velocity decreases obviously at the same time. But under the high bottomhole pressure (up to 50 MPa), the compressibility of the gas is low. This results in a slight change in void fraction and wave velocity at the position far away from the wellhead. Since the compressible component increases with the increase in the gas influx rate, the compressibility of the gas and drilling mud two-phase fluid is improved. So the variations of void fraction and wave velocity become more prominent. Also, the void fraction still shows an increase tendency that is steady first and then sharp. It acts in accordance with the variation of void fraction at a low gas influx rate. At a low gas influx rate, if the void fraction at the position close to the wellhead can not increase to a high extent, the wave velocity always shows a decrease tendency. At a high gas influx rate, such as the Q = 8.312 m /h, the wave velocity tends to increase because the void fraction in the wellhead is increased to a high extent. In conclusion, the wave pressure is sensitive to the void fraction, and the void fraction is dominated by influx rate and pressure in the annulus, especially the influx rate.

With the influx of gas, the mixing of gas and drilling mud occurs in the annulus and the corresponding interfacial transfer of momentum and mass causes an increase in gas phase void fraction and a decrease in pressure wave velocity, as shown in Figures 11 and 12. Within the range of low gas influx rate, the wave velocity decreases significantly. It is because of this that the compressibility of the gas increases remarkably, and the medium appears to be of high elasticity, though the density of gas-drilling mud two-phase flow changes slightly. With the increase in the gas influx rate and corresponding increase in the void fraction in the annulus, the compressibility of the two-phase unceasingly increases, which promotes the momentum and energy exchange in the interface. So, the pressure wave continuously decreases. When the void fraction increases to some extent following the increase in the gas influx rate, the decrease in wave velocity in the liquid slug is gradually less than the increase in wave velocity in the gas slug; thus, the decrease of wave velocity is slowed down for the growth of gas slug. Especially in the wellhead, a slight increase in wave velocity is observed.

5.3. Effect of Virtual Mass Force on Wave Velocity. Figure 13 illustrates the effect of virtual mass force on the wave velocity with the parameter of gas influx rate. As shown in Figure 15, under the same gas influx rate (0.594 m3/h, 1.098 m3/h, and 1.768 m /h, resp.) the wave velocity curve of Cvm = Re diverges from the curve of Cvm = 0 at the position close to the wellhead, whereas the wave velocity curve of the two types is almost coincided below the position of H = 500 m. This divergence between the two types of curves is connected with the fact that significance of influence of interphase virtual

mass forces increases together with the increase of the relative motion in the interphase. It is evident that the bottomhole pressure is hundreds of times higher than the wellhead pressure. As a result, the void fraction at the position close to the wellhead increases sharply, meanwhile the interphase momentum and energy exchanges are promoted. The virtual mass force can be described as the transfer of momentum between the gas phase and drilling mud phase caused by the relative motion in the interphase. If the relative motion in the interphase is quite weak, the value of virtual mass force will intend to approach 0, and the influence on the wave velocity can be ignored. However, if the relative motion is rather intense, the effect of virtual mass force on the wave velocity should not be ignored. Furthermore, taking the virtual mass force into account, the dispersion characteristic of the pressure wave weakens obviously. Compared with the pressure wave velocity calculated by ignoring the effect of virtual mass force, the calculated pressure wave velocity with a consideration of the virtual mass force is lower. Therefore, it is necessary to analyze the effect of the virtual mass force on the wave velocity at the position close to the wellhead in MPD operations.

5.4. Effect of Disturbance Angular Frequency on Wave Velocity. Figure 14 presents the pressure wave velocity in the annulus at different angular frequencies. According to Figures 5 and 9, the void fraction first slightly increases in the bottomhole and then sharply increases along the flow direction at the position close to the wellhead. At a fixed angular frequency, this phenomenon results in an overall decrease in the pressure wave velocity except for the section near the wellhead. At the

Figure 15 shows the effect of frequency on the wave velocity in the gas-drilling mud flow (Cvm = 0). The curves of the wave velocity of different positions reveal that the propagation velocity of pressure disturbances increases together with the growth of the angular frequency (0 < w < 500 Hz). It proves that the pressure wave has an obvious dispersion characteristic in the two-phase flow. As the angular frequency increases in the range of less than 500 Hz, there is sufficient time to carry out the exchange of energy and momentum between two phases. It achieves a state of mechanical and thermodynamic equilibrium between the two phases. So the wave velocity increases gradually with the increase in the angular frequency at different depths of the annulus. It is considered that the wave velocity is mainly affected by the interphase mechanical and thermodynamic equilibrium at low anglular frequencies. When the angular frequency reaches the value of w = 500 Hz, the pressure wave velocity achieves a constant value and remains on this level regardless of the further growth in angular frequency w. With the increase in angular frequency, there is not enough time for energy and momentum exchange between the gas-drilling mud two phases to reach the mechanical and thermodynamic

equilibrium state, and thus the wave dispersion does not exist. At a high angular frequency, the wave velocity is mainly dominated by the mechanical and thermal nonequilibrium in the flow and keeps almost unchanged when the angular frequency is further increased. This is consistent with the influences of the angular frequency in the horizontal pipe [28]. Moreover, the effect of virtual mass force is also shown in this figure (Cvm = Re). It is observed that the wave velocity is significantly reduced when the virtual mass force is taken into consideration in the calculation of the wave velocity at the position close to the wellhead such as H = 0 m. It can be explained by the reduction of dispersion characteristic of a pressure wave. Along the wellbore from the wellhead to the bottomhole, the distinction gradually decreases.

With full consideration of the important factors such as virtual mass force, drag force, gas void fraction, pressure, temperature, and angular frequency, a united wave velocity model has been proposed based on pressure drop gradient equations in MPD operations, gas-liquid two-fluid model, the gas-drilling mud equations of state, and small perturbation theory. Solved by the fourth-order explicit Runge-Kutta method, the model is used to predict wave velocity for different back pressures and gas influx rates in MPD operations. The main conclusions can be summarized as follows.

(1) With the introduction of virtual mass force and drag force, the united model agrees well with the previous experimental data. The united model can be used to

accurately calculate the wave velocity in the annulus. The application of the model will be beneficial to further study the wave velocity at different gas influx rates and back pressures in MPD operations, reduce nonproductive times, and provide a reference for the drilling operations in extremely narrow pore/fracture windows existing conditions.

(2)The wave velocity and void fraction have different values with respect to well depth. In the annulus, the drop of pressure causes an increase in the void fraction along the flow direction. The void fraction increases first slightly and then sharply. Correspondingly, the wave velocity first gradually decreases in th bubbly flow and preliminary slug flow, and then the wave velocity slightly increases accompanying with the increase in the relative length ratio of gas slug to the liquid slug for the continuous increase in void fraction at the position close to the wellhead. The minimum wave velocity appears in the long gas slug flow.

(3)When the back pressure in the MPD operations increases, the void fraction at different depths of the annulus is reduced gradually, while the wave velocity in the two-phase flow tends to increase. Mo