Dynamic study on particle throwing motion of pulsating negative pressure shale shaker

This research is based on a new drilling fluids circulation treatment device, which belongs to solid control system, namely pulsating negative pressure shale shaker. The purpose of this article is to better understand the operating principle and provide basis for setting working parameters and optimizing design. The above content depends on the dynamic analysis of the throwing motion of particles. To be specific, the motion equation of particle is established from the viewpoint of force state, and the influence of various parameters on particle conveyance velocity is discussed. Factors influencing the throwing index are also analyzed. Higher solid conveyance speed means higher handling capacity, but it also reduces the efficiency of particle dehydration; therefore, a trade-off should be considered among high removal efficiency, low energy consumption, and high handling capacity. The results show that negative pressure in pulsating form can effectively improve the disadvantage of low conveyance velocity of particles in constant negative pressure shale shaker.


Particle cluster diameter d s
Particle diameter D 0 Nominal throwing index of dry single particle (without negative pressure) D 1 Actual throwing index of single particle (without negative pressure) D 2 Nominal throwing index of dry single particle (with negative pressure) D 3 Actual throwing index of particle (with negative pressure) D ′ 3 Actual throwing index of particle cluster F f Friction force F D Drag force caused by airflow F x , F y Forces on particle in x and y directions Half of the maximum velocity of pulsating airflow v sx Average velocity of particle throwing motion Δẍ , Δÿ Relative acceleration component between drilling fluid and shale shaker Greek letters Screen slope m Surface tension of drilling fluid δ Angle between vibration direction and screen surface

Introduction
Environmental protection in drilling industry has been paid more and more attention. According to the American Petroleum Institute (API), it has been estimated that approximately 1.2 barrels of total drilling waste fluids are generated for every foot drilled (Newswire 2015). After drilling waste fluid is treated by traditional drilling fluid solid control system, the liquid content of drilling cuttings is still too large to be discharged directly. Realizing the recovery and harmless treatment of drilling waste fluid is an urgent problem to be solved (Pereira et al. 2014). The key equipment in solid control system is drilling fluid vibrating screen, also called shale shaker, where the purposes are to maximize the recovery of drilling fluid adhered to drilled cuttings and to maximize the removal of these solids from drilling fluid (Irawan et al. 2017). For more than 80 years, solid-liquid separation technology of shale shaker has remained essentially the same. Shale shaker works under open or semi-open working conditions, both above and below the screen are atmospheric pressure. Drilling fluid screening and solid particles drying are mainly realized by inertial force generated by vibration movement of screen and potential energy of the drilling fluid itself (Steinsvag et al. 2011;Galea et al. 2012). With rapid development of drilling new technology, the requirement for drilling fluid solids control is increasingly higher. The existing shale shaker has not been able to well meet the needs of modern drilling technology development.
In recent years, some companies and researchers have proposed to use negative pressure for solids control and fluid recovery. Norwegian-based Cubility has developed an enclosed, vacuum-based filtration system called Mudcube (Melhus et al. 2010). Drilling fluids are vacuumed through a rotating filter belt using high airflow to separate cuttings from fluid. The particles move toward discharge end together with screen, and the conveyance velocity is not affected by vacuum. For water-based drilling fluid, the liquid volume content in separated cuttings can be reduced to less than 40%, which is 50-60% of traditional shale shaker, but it has the disadvantages of complex mechanical structure, expensive price and large consumption of compressed air. MI SWACO invented screen pulse fluid and cuttings separator, it operates by means of compressed air jet creating suction under the last screen surface, pulling additional fluid off the cuttings. Although, for oilbased drilling fluid, it can reduce liquid weight content to less than 7%, but the large consumption of compressed air limits its application. Yongjun et al. (2018) proposed a negative pressure shale shaker, in which drilling fluid passes through screen under the joint action of vibration and negative pressure, and drilling cuttings are conveyed forward by vibration. In this process, although the negative pressure makes drilling cuttings become drier, it also makes conveyance velocity of particles decrease, and even appears to accumulate in screen surface, thus affecting the processing capacity of shale shaker. In order to give play to the advantages of negative pressure shale shaker in reducing the liquid content and improving conveyance ability of drilling cuttings, pulsating negative pressure shale shaker was proposed.
The capability of shale shaker is influenced greatly by many factors; most of research has been done on vibration principle, vibration pattern, kinematics and dynamics parameter of a traditional shale shaker (Zhao et al. 2009;Peng et al. 2018). Beyond that, the kinematic law of particles on shale shaker plays an important role in its working processes (Rotich et al. 2016;Weibing et al. 2013). Dorry and Varco (2010) conducted experiments using a fluid with similar physical properties to a drilling fluid in order to study the effect of the g-force on the screen capacity, solids conveyance, and dryness of the retained material. Raja (2012) evaluated the effects of g-force, solid content in the feed, fluid viscosity and angle of inclination on the screen capacity. Fowler and Lim (1959) investigated the effects of feed rate, frequency of vibration, angle of inclination and screen aperture size on the effectiveness of a vibrating screen. Lal and Hoberock (1988) have built the solids-conveyance model based on the physics of the problem. Chen and Tong (2009) presented a numerical model for the study of a particle screening process using the discrete element method (DEM) and effect of screen length and vibrating frequency regarding to screening efficiency was evaluated. Yongjun et al. (2018) established the conveyance model of shale shaker particles under constant negative pressure and discussed the influence of vibration parameters and air flow formed by negative pressure. Until now, there are few reports in the literature on the particle motion law under the joint action of vibration and pulsating negative pressure.
The purpose of this article is to make a thorough study on kinematic behavior of particle on the pulsating negative pressure shale shaker. Considering the shape and density, the single particle formula is modified to make it more similar to actual condition. At the same time, the influence of various parameters on conveyance velocity of particles is discussed. All these works can be used as the basis for computer simulation, and it can help in structure design and choice of basic parameters. Figure 1 shows the main components of pulsating negative pressure shale shaker, which is composed of vibration sorting system and pulsating airflow system. Screen box, exciting motor, screen, spring and base constitute the vibration sorting system. In working process, exciting force generated by exciting motor makes the screening device reciprocate in both horizontal and vertical directions. In the pulsating airflow generating system, the distributor and vertical pipe are intermittently connected to gas-liquid separator and vacuum pump. The gas-liquid separator is connected to a vacuum disk under screen by means of a vacuum hose. When the shaker works, vacuum pump runs continuously, and the distributor rotates at a certain speed, its internal channel makes vacuum pump connect to vertical pipe or atmospheric environment alternately, so as to form pulsating negative pressure under screen.

Operating principle
Drilling fluid discharged from wellhead enters shale shaker and is subjected to vibration to obtain a certain degree of screening capability. At the same time, vacuum pump works, and forms negative pressure in the vacuum disk, so that drilling fluid can obtain additional screening capability. Drilling fluid moves through the screen under the action of vibration and pressure difference, and drilling cuttings contact and collide with screen and move toward outlet. Some of air and drilling fluid on screen surface is drawn into vacuum hose under the action of differential pressure, and they are separated in gas-liquid separator. The separated drilling fluid enters circulation tank for recycling, while the gas flows through vertical pipe into vacuum pump and then out through the vent.

Particle force analysis
When drilling fluid containing drilled solids flows onto shale shaker, the liquid/solid separation process begins. The drilling fluid gradually passes through screen, and particles convey toward the discharge end. The liquid along screen disappears through screen cloth as it reaches a point, called liquid endpoint. Once shale shaker has reached a stable state, solid phase migration is consistent in the submerged and non-submerged states. For simplicity, this paper analyzes the dynamics of solid phase migration in non-submerged state.
After solids pass the liquid endpoint, they have four types of kinematic modes: sliding backward or forward on screen, throwing motion, remaining in contact with screen and staying stationary. Because of the periodic nature of shale shaker, the solid continues to enter these modes during various cycles until the steady state is established. Due to the action of pulsating airflow in pulsating negative pressure shale shaker, the existing mechanical model is no longer suitable for analyzing the kinematic law of particles on it.
In non-submerged state, due to the adhesion of drilling fluid between the particles and the adsorption effect produced by negative pressure system, particles rarely appear as a single particle state, but as a group of particles. The particle cluster is oblate and approximately cylindrical, with diameter of d ′ s and height of h. As shown in Fig. 2, forces acting on the particle include the mass gravity G, the viscous resistance of drilling fluid R, the force of inertia P, the friction force F f , the drag force caused by airflow through screen F D , the normal pressure when contacting with the screen N, Other forces on particles such as virtual mass force and Basset force are negligible because the density ratio of air versus particle is too small. F x , F y are the forces on particle in x and y directions, can be written as: where is screen slope.
The force of inertia as follows: where m is the mass of the particle, a x is acceleration component of particle parallel with screen surface, a y is acceleration component of screen normal to screen surface, Δẍ and Δÿ are relative acceleration component between drilling fluid and shale shaker, respectively.
The negative pressure under screen is generated by vacuum pump pumping air, during this process, air on screen surface is constantly inhaled, which creates drag force on the particle. The drag force F D can be written in terms of the drag coefficient C D C D as follows (Haider and Levenspiel 1989;Ishii and Zuber 2010).
where subscript s represents the property of particles, s is the density of particle, d s is the particle diameter, is the viscosity of fluid, and R e is the relative Reynolds number. (1) Considering the particle shape, the drag coefficient C D is defined by the correlations developed by Haider and Levenspiel, v and v sy are the airflow velocity and the particles velocity along the coordinate y , respectively.
The relative Reynolds number R e is defined by: where is air density.

Kinematic equation of shaker
In this paper, a linear shale shaker is taken as an example to discuss relevant problems. Suppose the movement of screen is stable, each point has the same trajectory as the center of mass (COM). Hence, the trajectory equation of pulsating negative pressure linear shale shaker takes the following form: where A x and A y are the amplitude components of the shaker/screen in the vibration direction. The angle between vibration direction and screen surface is δ , the distance component parallel with screen surface S x and normal to screen surface S y can be written as: where is the amplitude of screen in the direction of vibration.
From the above equation, we can get the velocity component v x , v y and the acceleration component a x , a y .

Throwing index
During the working process of shale shaker, throwing motion of particles can not only accelerate solid/liquid separation, but also improve the conveyance velocity. When particles are thrown away from screen surface, they will fly in the air for a certain distance, so as to increase the penetration area of drilling fluid on screen surface.
(4) R e = |v−vsy|ds cos cos t v y = sin cos t (8) a x = − 2 cos sin t a y = − 2 sin sin t Fig. 2 Graph of the acting forces of particle in the non-submersed condition In theory, once the structure and working parameters have been determined, the throw index D of shale shaker is a constant value. And it is only when D 0 is greater than 1 that throwing motion occurs. It can be defined as: D 0 is the nominal throwing index of dry single particle (without negative pressure), is rotational angular velocity of the vibration exciter, and g is gravitational acceleration.
Considering viscous resistance, the actual throwing index of single particle (without negative pressure) is: where d is the phase angle between throwing direction of particle and screen surface.
Considering drag force, the nominal throwing index of dry single particle (with negative pressure) is: If N = 0 , the particle leaves screen cloth, and enter a flight mode. At the beginning, acceleration of particle is identical to screen surface, i.e., Δÿ = 0 , so the term of flight motion for solids is as shown in Eq. (12).
When N = 0 , non-submerged particle tends to leave screen. The particle does not enter flight mode immediately because there is drilling fluid on the surface of particles. When particles are about to jump in non-submerged state, their movement is hindered by the static shear force d 2 0 ∕4 . This process is extremely short, so it can be considered that the viscous resistance exerted by solid particles moving in drilling fluid is a combination of static shear force and surface tension.
where 0 is dynamic shear stress and m is surface tension of drilling fluid.
According to the above analysis, the actual throwing index of particle (with negative pressure) is defined as: The density and shape need to be modified. Generally, the density of a single particle is s = 2.6g∕cm 3 , and the particle mass contains water, which is about 20-35%. Therefore, the density of particle mass is modified to be � s = 2.04 − 2.28 g/cm 3 . As mentioned above, particle cluster (9) D 0 = 2 sin g cos (10) is oblate and approximately cylindrical, with the diameter of d ′ s and the height of h = (1∕7 − 1∕5)d � s , so the mass of particle cluster is: Change the correction coefficient K 1 to make it conform to the properties of particle cluster, and the actual throwing index of particle cluster has the similar form to Eq. (14).
As mentioned before, K � 1 + K 2 − 1 is absolutely more than 1, so the nominal throwing index is always greater than the actual throwing index. The influence of other factors on the throwing index is as follows. Fig. 3a, for particle of 0.05 cm,K 1 = 32 , D 0 = D 3 K 1 + K 2 − 1 > D 3 K 1 , D 3 greater than 1 means D 0 is required to be greater than 32 at least if throwing motion is made, which cannot be realized due to the limitation of vibration strength in practice. While for particle of 0.5 cm, the throwing index is about 3, and the relatively ideal throwing motion can be achieved. From Fig. 3b, the smaller particle size d ′ s and density ′ s are, the larger K ′ 1 is, that is, the smaller and lighter particle is, the harder it is to jump up. K ′ 1 is smaller as d ′ s squared increases, so the influence of d ′ s is more significant. When the size of particle cluster is 1-2 cm, the nominal throwing index is about 6, and the rapid enhance of throwing motion can be achieved. When the nominal throwing index is about 7, the actual throwing index is about 3, and a more ideal throwing movement can be achieved. 2. The screen slope only affects the value of nominal throwing index. However, it should be noted that if is too large, mud will flow easily along screen surface, increasing submerged area of screen surface, which is not conducive to the removal of solid phase and dehydration process. 3. There are two main parameters that affect the throw index in pulsating negative pressure system, namely negative pressure peak and pulsation frequency. Figure 3c shows calculation results of the actual throwing index of different airflow velocities. It can be seen that the throwing index decreases with the increase of airflow velocity in a certain range, so more exciting forces are required for particles to achieve throwing motion. Figure 3d shows the effect of pulsation frequency f′. Higher pulsation frequency means that pulsation airflow has less time to operate and particles are easier to jump up.

Average velocity
Pulsating airflow is step type in theory, but due to the hysteresis of flow controlled by valve, the pulsating flow waveform is simplified in actual calculation, and the velocity of periodic pulsating air flow at the entrance is approximately considered to conform to the form of sine wave.
where v 0 is half of the maximum velocity of pulsating airflow and f ′ is pulsating frequency. After particle is thrown (left screen surface), its relative acceleration along screen surface is: The relative displacement can be calculated by integrating Δẍ with respect to time twice and taking the phase angle as the limit of integral.
where z is the phase angle when throwing motion ends.
The theoretical average velocity of particle throwing motion is equal to the ratio of the relative displacement of each throwing motion to vibration period.
(18) Δẍ = g sin + 2 cos sin t Fig. 3 Correction coefficient and actual throwing index. aK 1 and D 3 under different particle sizes. bK ′ 1 and D ′ 3 under different particle cluster sizes. cK 2 and D ′ 3 under different airflow velocity. dK 2 and D ′ 3 under different pulsation frequency

Sliding motion
When particle remains in contact with screen and stays stationary, it has the same kinematic trail as shale shaker. When it begins sliding motion, the direction of friction force is contrary to the direction of relative velocity. The limiting friction force on the particle is: where f is friction coefficient, f = tan , is static friction angle, " − " indicates that particles slide in the positive direction along x axis, " + " indicates that particles slide in the opposite direction along x axis. So, the initial motion equation of sliding movement takes the following form.
From Eq. (22), the particle forward and backward sliding index D k and D q can be expressed as: The angle between vibration direction and screen surface δ can be written as: The amplitude of shale shaker can be calculated as the following form.

Vibration frequency
Long-term industrial practice shows that vibration frequency is an important parameter that affects the performance of shale shaker. The variation range of vibration frequency during the working of shale shaker is 16-26 Hz. The conveyance velocity is calculated when vibration frequency is 16 Hz,18 Hz,20 Hz,22 Hz,24 Hz,26 Hz, respectively. At the same time, particles conveyance velocity corresponding .
to each frequency in the case of no negative pressure and constant pressure is calculated. Figure 4 shows that conveyance velocity increased with vibration frequency and they have an approximately linearly relationship. With vibration frequency increased from 16 to 26 Hz, particles conveyance velocity of system with no negative pressure, constant negative pressure and pulsating negative pressure increased by 57.7%, 32.6% and 43.2%, respectively. As frequency increases, the moving activity of particles will enhance, particles have more chances to touch the screen to get energy and complete throwing motion. In addition, conveyance velocity of particles in two shale shakers with vacuum system is lower than that in traditional shale shaker, which is beneficial to reduce the liquid content of drilling cuttings to some extent. When vibration frequency is 16 Hz, particles conveyance velocity of pulsating negative pressure shale shaker is 8.0% higher than that of constant negative pressure one, at 26 Hz, this value is 10.8%. Particles conveyance velocity of the pulsating negative pressure shale shaker under the same vibration frequency is obviously higher than that of the constant negative pressure shale shaker, indicating that the pulsating negative pressure system can effectively improve the particle conveyance state.

Peak negative pressure
Negative pressure under screen in pulsating negative pressure shale shaker is generated by vacuum pump pumping air, the peak of pulsating negative pressure is determined by velocity of airflow through screen cloth. In essence, discussing the influence of peak negative pressure on particle movement is to study the influence of airflow velocity.
As shown in Fig. 5, the conveyance velocity of particles on pulsating negative pressure shale shaker shows a The velocity of particles decreases with the increase of airflow velocity. When airflow velocity is less than 0.6 m/s, particles conveyance velocity decreases with a large range as it increases. For example, airflow velocity increases from 0.2 to 0.6 m/s, and the corresponding particles conveyance velocity decreases from 0.32 to 0.23 m/s, a reduction of 28.2%. When airflow velocity continues to increase, particles conveyance velocity gradually tends to the minimum value in a certain range. For example, when airflow velocity is 0.7-0.8 m/s, the corresponding particles conveyance velocity is about 0.22 m/s, which is the minimum value within the calculated range. In addition, it can be predicted that when airflow velocity increases further, particles conveyance velocity may be small or even 0. The reason is that when airflow velocity is too high, the resistance of particles movement is also greater. When the force generated by vibration to make particles move forward is less than the resistance, it is difficult for particles to move forward.
Particles conveyance velocity of pulsating negative pressure shale shaker under the same airflow velocity is obviously higher than that of the constant negative pressure shale shaker, indicating that pulsating negative pressure system can effectively improve the particle conveyance state. In terms of liquid content of drilling cuttings alone, pulsating negative pressure shale shaker can use vacuum pump with larger power to produce larger airflow velocity. Considering the influence of airflow on particle conveyance velocity and screening efficiency, however, it should not be too large.

Pulsation frequency
When pulsating negative pressure shale shaker works, the velocity of distributor's rotation determines the pulsation frequency of negative pressure in vacuum disk. The variation range of pulsation frequency is 0.1-0.5 Hz, and the ratio of opening and closing time of distributor is 1, that is, the operating time of pulsation airflow is half of the whole pulsation cycle.
As can be seen from Fig. 6, particles conveyance velocity of pulsating negative pressure shale shaker increases with the increase of pulsation frequency. The pulsation frequency increased from 0.1 to 0.5 Hz, and the corresponding particles conveyance velocity increased from 0.226 to 0.24 m/s, an increase of 6.2%. Higher pulsating frequencies mean less time for the air to act on particles, allowing them to move further after being thrown. It is worth noting that compared with airflow velocity, pulsation frequency of the airflow has less influence on the particles conveyance velocity. The main reason for using pulsating airflow is to avoid flow acting on the particles all the time, which makes it difficult to transport them forward. In practical engineering applications, pulsation frequency can be adjusted as needed, such as considering mechanical characteristics of the distributor.

Conclusions
This study has investigated particle movement of pulsating negative pressure shale shaker using revised classical mechanical method. Based on analysis, the following conclusions are obtained.
1. The motion of particles on pulsating negative pressure shale shaker is similar to that of traditional shaker, which depends on vibration to complete throwing motion. In general, for pulsating negative pressure shaker, to make the particles on it to achieve a more ideal throwing motion, that is, the throwing index is large enough, it needs to provide more exciting force. 2. The velocity of particles conveyance is mainly influenced by vibration parameters and pulsating airflow parameters, among which the vibration parameters account for a larger proportion. 3. The particles conveyance velocity of shale shaker with vacuum system, whether constant negative pressure type or pulsating negative pressure type, is less than that of conventional shale shaker with the same vibration parameters, which is beneficial to reduce the liquid content of cuttings. 4. In general, pulsating negative pressure shaker works with smaller amplitude and vibration frequency, larger airflow speeds, and pulsating frequency that match the properties of drilling fluid.
Drilling fluid treatment is actually the process of liquid phase screening and solid phase migration. Pulsating negative pressure shale shaker can better exert the effect of vibration on solid phase conveyance and loose, as well as the separation ability of negative pressure on liquid phase. It is worth noting that the dynamic study carried out in this paper is based on the force analysis of particles and the derivation of theoretical equations, focusing on the superiority of negative pressure in the form of pulsation. Limited by the test equipment, this paper lacks experimental data to compare with it, but the conclusions obtained can still provide support for new shale shaker design, along with developments in screening theory.