On the dynamic mechanism of pumping steel ball in downhole tubing

 With the exploitation of unconventional oil and gas reservoirs, staged hydraulic fracturing technology has been widely used, especially in the re-fracturing of old Wells, which usually adopts the staged fracturing technology of tubing sliding sleeve.

The technology consists of tubing with multiple sleeves and packer tool strings, and is used to open and fracture the sleeves step by step by dropping balls of different sizes into the tubing.

In order to realize the rapid opening of sliding sleeve and sealing of steel ball seat, pumping operation is usually adopted in engineering.

However, the migration state of steel ball and the change law of fluid force in the process of pumping are rarely studied. In this paper, a sub domain solution algorithm based on the immersed boundary method is used to study the fluid force and the impact state of the tube wall in pumping steel balls.

Based on the rigid body dynamics theory, the dynamic formula of pumping steel ball in tubing is derived, and the program is written and adjusted to provide the calculation method and technical means for the field construction.

According to the stage fracturing process parameters of tubing sliding sleeve, the fluid-solid coupling numerical model of pumping steel balls in tubing in vertical, curvature and horizontal Wells was established based on the immersed boundary method of fluid-solid coupling.

The numerical examples of static sphere flow, free settlement of single spherical particle and oblique collision between particle and wall are given to verify the correctness of the numerical model and calculation method.

The dynamic state of steel ball pumping was numerically simulated by selecting part of the middle section of horizontal well, and the changing law of the movement displacement, velocity and fluid force of steel ball in different sections was obtained.

In order to obtain the calculation formula for the fluid force of steel balls pumped in tubing, fluid-solid coupling numerical simulation was performed in different conditions, including diameter ratio, ball initial velocity, eccentricity, fluid viscosity and pumping capacity.

Maximum flow rate of steel balls in tubing, drag between front and back, fluid force and lift were provided.

By extracting and regression processing the numerical simulation results, the modified calculation formulas of drag coefficient and lift coefficient are obtained.

Compared with the existing calculation results of drag coefficient and lift coefficient, the maximum difference is 16% and 3%, respectively, which provides a more reasonable calculation formula for the calculation of the fluid force of pumping steel balls in tubing.

It also provides a way to analyze the solid forces in the finite spatial dynamic watershed.

In order to obtain the normal and tangential recovery coefficient of the collision between steel ball and pipe wall, the orthogonal experiment method was used to study the change law of the recovery coefficient of five factors before and after the collision between steel ball and pipe wall, including the diameter ratio, the initial velocity of steel ball, the Angle between the external force and the flow velocity, the fluid viscosity and the flow velocity.

The following conclusions were obtained: The influence of all factors on the normal recovery coefficient is in the order of fluid viscosity, diameter ratio, velocity, Angle between force and velocity and initial velocity of the ball.

The influence of fluid viscosity is the largest, which is 2.14times of initial velocity of the ball, 1.41 times of the Angle between force and velocity, 1.32times of velocity and 1.23 times of diameter ratio.

The influences of all factor son tangential recovery coefficient are in the order of initial velocity of steel ball, fluid viscosity, Angle between external force and velocity, diameter ratio and velocity.

The influence of initial velocity of steel ball is the largest, which is 2.07 times of diameter ratio and velocity range, 1.59 times of the Angle between force and velocity, and 1.35 times of fluid viscosity.

From the perspective of single factor, the influence of diameter ratio, collision normal velocity, collision tangential velocity, fluid viscosity and flow velocity on the change rule of recovery coefficient is analyzed. The normal recovery coefficient is positively correlated with diameter ratio, collision tangential velocity, fluid viscosity and flow velocity, but has nothing to do with normal impact velocity.

Tangential recovery coefficient is positively correlated with collision normal velocity and flow velocity, and negatively correlated with collision tangential velocity and fluid viscosity, but independent of diameter ratio.

Besides the impact tangential velocity, other factors have a greater impact on the normal recovery coefficient, but a smaller impact on the tangential recovery coefficient.

The results of orthogonal experiment are treated by regression formula, and the fitting formulas of normal and tangential recovery coefficient sunder different influencing factors are given, which can provide reference for the value of normal and tangential recovery coefficients of finite space dynamic watershed problems.

Based on Newton’s second law and the idea of discrete element, the dynamics model and governing equation of the steel ball pumped in tubing are established, the theoretical calculation process is given, and the dynamics solution program is compiled by Delphi 7development platform.

Compared with the numerical simulation results, the calculation errors of ball migration time in vertical, curvature and horizontal sections are 7.5 %, 7.4%and10.5 %, respectively.

Taking the multi-stage fracturing operation of a well using soluble packer string in Daqing oilfield as an engineering example, the time of setting three throws is calculated by the calculation program.

The errors of three throws are 12.5 %, 10 %and7.6%respectively by comparing with the field construction feedback data.

It shows that the calculation method established in this paper and the calculation program developed in this paper, It can provide theoretical support and technical means for pumping steel balls in stage fracturing.