1. Introduction
In the process of oil and gas drilling, using mud (non-Newtonian fluid) to remove solids (cuttings) of different shapes in the wellbore is an essential step [
1,
2,
3,
4]. However, due to operational limitations (e.g., bit change, single operation) or emergencies, the drilling operation must be stopped briefly, and the circulation of drilling fluid in the annulus will also be interrupted briefly. The cuttings initially transported to the wellhead through drilling fluid circulation begin to settle and deposit at the bottom of the well, causing hole cleaning problems [
5,
6]. The settling velocity can be used to estimate the formation depth of cuttings. Through cuttings analysis, important formation information such as lithology, porosity, permeability, and pore pressure can be obtained. The settling velocity can also be used as an input parameter for drilling optimization and design. For example, given the condition that the upward velocity of drilling fluid in the annulus needs to be greater than the settling velocity of cuttings, it can guide the mud pump displacement required for the drilling fluid to effectively carry cuttings. The settling velocity of cuttings can usually be calculated according to the prediction model of the particle settling drag coefficient. The in-depth study of the settlement law of cuttings in the mud drilling fluid of horizontal wells and accurate calculation of the settlement drag coefficient is the basic work needed to understand the hole cleaning problem of horizontal wells, which can provide a basis for optimizing the rheological properties and circulation speed of drilling fluid [
7,
8].
Revealing the law of cuttings settlement can start from the study of spherical particle settlement. Since Stokes (Stokes, 1851,
CD = 24/
Res), researchers have carried out many studies on using the
CD-
Res relationship to accurately predict the drag coefficient. The drag coefficient (
CD) is defined as the ratio of viscous drag to kinetic energy acting on particles [
6,
7], and its expression is:
where
ρp is the density of particles, kg/m
3;
ρl is the density of fluid, kg/m
3;
g is the acceleration of gravity, m/s
2;
de is particle diameter, m; and
Vt is the settling velocity, m/s. If the particle is non-spherical, then
de is the equivalent diameter of the particle, and its expression is
, where
mp is the mass of the particle, kg. The particle Reynolds number (
Res) is defined as:
where
μ is the Newton viscosity of the fluid, Pa·s.
There is much research on particles’ drag coefficient in Newtonian and non-Newtonian fluids [
8,
9,
10,
11,
12,
13,
14,
15,
16,
17,
18,
19,
20,
21,
22,
23,
24,
25,
26,
27,
28,
29,
30]. For example, Shah [
21] proposed a method using a drag coefficient to analyze particle settling velocity in the power-law non-Newtonian fluid. This method takes into account the influence of fluid flow index n and establishes the functional relationship between particle Reynolds number
Res and
CD2−n. In data processing, Shah fitted
CD2−n and
Res to a straight line on the coordinate diagram. At a low Reynolds number, the particle settling velocity is greatly affected by the fluid flow index, which decreases with the increase in the Reynolds number. Especially when
n = 1, the drag coefficient decreases to the Newton curve of
CD and
Res. The method has been tested with the data of previous studies, and the results show that the method has high prediction accuracy.
The settling velocity of particles in a liquid is affected by fluid properties. The non-Newtonian characteristics of fluid have a great influence on the settlement rate, especially the non-Newtonian fluid with yield stress (
τ0). Dedegil [
31] established a prediction model of the sedimentation drag coefficient of particles in Herschel–Bulkley fluid, considering viscous resistance, gravity and buoyancy. The yield stress is included in the influencing conditions of the settling velocity of the model, which is verified by the published clay suspension data.
Certain yield stress is a unique feature of Herschel–Bulkley fluid. When the vertical force acting on it reaches equilibrium, a single particle can be completely suspended. If other parameters remain unchanged, the net downward force increases with the increase in particle size. The fluid used in the petroleum industry not only has yield stress but also has time-dependent and viscoelastic rheological behavior [
29]. Through the study of particle settlement in a viscoelastic fluid, it can be found that fluid viscoelasticity will significantly affect the settling velocity in the transient region and has no effect on the settling velocity in the laminar region. The experimental phenomenon reveals that the drag coefficient has also changed substantially due to the changes in particle Reynolds number (
Res), particle shape, and viscoelastic fluid parameters. Since the Herschel–Bulkley fluid has certain yield stress, and the settlement law of particles in a non-Newtonian fluid is also different from that in a Newtonian fluid, and to ensure that the settlement results conform to the distribution form of the standard Newton drag coefficient curve, it is necessary to modify the definition of the Reynolds number and apparent viscosity. The modified particle Reynolds number
Res in the Herschel–Bulkley fluid is expressed as [
32];
where
τ0 is the yield stress, Pa;
n is the fluid behavior index, dimensionless; and
K is the Herschel–Bulkley fluid consistency index, Pa·s
n. The fluid behavior index is the indicator of shear behavior, and the consistency coefficient is the indicator of fluid viscosity, both of which are used to define the rheological parameters of Herschel–Bulkley fluid. The expression is as follows:
where
τ is the shear stress, Pa, and
is the shear rate, 1/s.
The shape characteristics of the particles are one of the factors affecting the settlement behavior of irregular particles in the fluid. Drill cuttings are irregularly shaped non-spherical particles. It can be seen from the public literature that researchers have established many correlations to predict the drag coefficient of irregular particles, indicating that the shape factor has an impact on the drag coefficient. However, few studies have developed drag coefficient models for irregular particles in Herschel–Bulkley fluids.
Among the parameters describing the shape characteristics of particles in the drag coefficient prediction model, sphericity is a very important one [
33,
34]. However, it is not easy to measure the surface area of irregular particles, especially for highly irregular particles such as shale cuttings, and it is also difficult to operate in the field. Therefore, other particle shape characteristic parameters, such as circularity
c [
35,
36], can be applied. Circularity
c is the ratio of the circumference of the maximum projection surface of a particle to the rim of its equivalent circle. Since it is a two-dimensional shape parameter sensitive to the irregularity of the particle contour, it is relatively easy to measure [
37]. Its definition is as follows:
where
dA is the diameter of the equivalent circle of the maximum projection surface of the particle, m;
P is the entire projection perimeter of the particle, m.
This paper studies the deposition characteristics of shale cuttings in Herschel–Bulkley fluid and verifies the correctness of the experimental process by using particle deposition experiments in Newtonian fluid. By introducing the description parameters representing the cutting morphology, and based on the regression analysis of experimental data, a prediction model for the settlement drag coefficient of spherical particles and shale cuttings in Herschel–Bulkley fluid is proposed. The iterative method was used to calculate the particle settling velocity, which was compared with the particle settling velocity measured in the experiment. The results show that the model has good prediction accuracy for drag coefficient and settling velocity of the Herschel–Bulkley fluid of spherical particles and shale cuttings. These results have important guiding significance for clean hydraulic parameters of shale oil wells in the petroleum industry.
4. Discussion on the Limitations of the Model
The limitation of the prediction model of shale cuttings’ drag coefficient proposed in this study is that the range of particle Reynolds number is relatively small (0.08~15). With increasing the Reynolds number of the particles, either by increasing the particle size or density or by reducing the viscosity of the fluid, particles collide frequently with the column wall during its settlement, even if the inner diameter of the resin glass column we studied is 100 mm. Therefore, these experimental data are invalid because the velocity after particle collision is very different from the original velocity.
We plan to conduct further experiments after increasing the wellbore diameter. With the discovery of relevant fields, the model will continue to evolve, and with more additional data, the rationality and application scope of our results will gradually improve.
In summary, for the test of spherical and shale cuttings, the prediction model proposed in this paper is effective in the range of particle Reynolds numbers and roundness. However, the use of these prediction models outside these ranges may lead to inaccurate results. Therefore, the correlation needs to be confirmed by additional experimental data, such as large particle Reynolds numbers, which will be pursued in our future work.