Increasing the speed of drilling operations is of commercial and military interest for transportation infrastructure as well as rapid installation of underground utilities in urban settings and over long distances. A significant challenge to increasing speed in horizontal directional drilling is pressure and flow rate management of drilling fluids circulating into and out of the borehole, removing solids cut free by the drill bit. The mixture of solids and drilling fluid results in a highly complex fluid dispersion, typically with a shear-thinning continuum. It is challenging to characterize the viscometric behavior of these dispersions, and such data are limited in the literature. It is increasingly important to understand and accurately model the viscosity of these dispersions since high drilling speeds increase the drilling fluid flow rate, approaching the pressure limits that borehole walls can withstand before failure. In this work, we characterize the viscometric properties of a drill test and model drilling fluid dispersion in a custom-built flow loop with solid concentrations up to 45 wt. %. The fluid viscosity is reported in terms of power-law parameters, which can be used to predict the pressure drop during real drilling conditions. We found a significant difference in the viscometric response between the drill test and model drilling fluid dispersions. The Shields parameter can capture the influence of solids settling on the measurable pressure losses. An important conclusion is that even model drilling fluid dispersions prepared with geotechnical data from a drill site may have significantly different viscometric characteristics than those relevant during a drilling operation.

## INTRODUCTION

Horizontal directional drilling (HDD) is a developing technology for precisely controlled shallow tunnel (<30 m) construction, but it operates at slow operational speeds of 0.0025–0.01 m/s.^{1–5} Current slow drilling speeds result in additional costs and labor, making them more costly than open-cut methods of excavating trenches,^{6} but if they can be improved there are military and economic advantages. Directional drilling is a high-powered technology focused on speed rather than maneuverability, reaching high-speeds up to 0.08–0.19 m/s. To increase the drilling speed of HDD while maintaining its nimble movement, directional drilling technology has to be integrated with HDD. Drilling at higher speeds means that cuttings from the underground formation need to be evacuated more quickly and high solids content drilling fluid viscosity will need to be better understood. In underground drilling, drilling fluids are pumped into boreholes through the drill rig to remove sand, clay, sediment, rock, etc., broken down by the drill bit.^{7} This complex mixture of solids and drilling fluids is termed “cuttings returns.”

This work focuses on the viscosity of cuttings returns required to make high-speed HDD possible. One major constraint to high-speed HDD is removing cuttings and excavating the formation at high speeds. Increasing these speeds an order of magnitude can also mean an order of magnitude reduction in costs by shortening the time of operation. The term rate of penetration (ROP) is used to represent the linear speed the equipment advances forward during drilling, dependent on formation excavation and cuttings removal. Increasing ROP requires knowledge of drilling fluid viscosity before and after cuttings loading to predict the expected pressure drop of the system and ultimately fluid pumping speeds that can be utilized. The drilling fluid is bentonite clay dispersed in an aqueous medium, typically water. These fluids are shear-thinning due to the presence of the platelet-shaped particles,^{8} a desirable characteristic for underground drilling. Shear-thinning fluids slow the settling of solids when the fluid is at rest, while reducing pressure losses at high flow rates compared with an equivalent Newtonian fluid.^{9}

Viscometric properties of the cuttings returns are a major factor influencing the pressures near the drill bit. Drilling at higher speeds requires greater fluid pressure to match the drilling fluid flow rate with the removal rate of cuttings. Cuttings returns pressure will decrease as they flow back to the hole entrance due to friction losses.^{10,11} Typically, the pressure near the drill bit will be the largest pressure the borehole wall experiences, and if the pressure near the drill bit exceeds the strength of the surrounding formation, the hole walls fracture and cuttings are no longer removed from the borehole. Once a fracture is initiated, it becomes a path of least resistance for the cuttings returns, and the flow of cuttings returns to the hole entrance reduces or ceases entirely. The loss of cuttings returns has at least two important consequences: a significant increase in the risk of the entire drill string becoming stuck, and an inability to recover and reuse the drilling fluid after it returns to the surface in the cuttings returns.^{12} Continuous loss of drilling fluid at large volumes is prohibitive from both a cost and logistics standpoint when considering borehole lengths at the kilometer scale. These viscometric challenges are specific to shallow drilling in soft ground, whereas in petroleum and geothermal applications are typically in hard ground dealing with rock formations that will present their own unique challenges, falling outside the scope of this manuscript.

Solids concentration in cuttings returns depends on many variables, but the influence of solids concentration on cuttings returns viscosity is generally unknown. In practice, the concentration of solids is typically maintained within a window between 5 and 15 vol. %, but fluctuates with changes in ROP, drilling resistances, and drilling fluid pressures.^{13,14} At low solids concentrations, the cuttings returns viscosity will be lower with higher drilling fluid flow rates and high pressures. At high solids concentrations, the cuttings returns viscosity will be higher with lower drilling fluid flow rates and high pressures. A minimum in the pressure is obtained at a particular solid concentration for any given combination of drilling variables. Since cuttings returns pressure is a limiting factor in high-speed drilling applications, it is critical to understand what conditions will lead to this minimum pressure. A formidable challenge to identifying these conditions is in understanding the relationship between the cuttings returns viscosity and the solids concentration. Modeling the pressure vs flow rate relationship requires this understanding, but identifying flow properties of dispersions of clay, sand, silt, and rock under turbulent conditions has multiple challenges. First, conventional rheometers are generally inadequate for these measurements, where rapid settling of highly heterogeneous solids at high concentrations leads to non-viscometric flows, and torque transducers with high limits.^{15} Second, the complexity of particle size, shape, and density, along with the inclusion of clay dispersions, generally forbids application of theory derived from first principals for colloidal suspensions to date. Empirical measurements of cuttings returns viscosity to obtain these relationships are warranted in these instances, but there is extensive variability in formation composition in even a single drill site that makes it difficult to know the applicability of any one measurement of cuttings returns viscosity on any given ground conditions.

A flow loop is a valuable tool to understand how high concentrations of solids influence the viscosity of cuttings returns and to extract apparent viscometric material functions under flow conditions similar to drilling operations. In underground drilling, the complex fluid flows in an annular space, but material properties are independent of geometry so a flow loop with simple pipe flow is acceptable and easy to use, removing additional geometric influences that could add noise to the complex fluid characterization. A flow loop can also be used to identify conditions of solids settling and to understand the problems caused by settling on the experimental measurements. In general, accurate and consistent results for turbulent, non-Newtonian flows have been challenging to produce.^{16–20} There have been multiple attempts to derive analytical, semi-theoretical, and empirical correlations for pressure and flow rate relationships, but wide variations exist when comparing between the many developed correlations,^{21} and the source of the variation is not well understood.^{20,22,23} Various explanations for these variations have been suggested, including pipe roughness by Szilas *et al.,*^{18} assuming steady-state conditions rather than unsteady-state by Tuoc,^{24} and the effects of elasticity in emulsions by Werner *et al.*^{25} The experimental approach described here can be used with any correlation relating pressure drop and flow rate developed for non-Newtonian fluids, although the focus is characterization of the shear-thinning nature of cuttings returns.

In this work, we describe the construction of a custom flow loop for measuring viscometric properties of high-solid concentration cuttings in a bentonite-based drilling fluid and use it to characterize viscometric properties. The validity of using measurements of viscometric properties of a “model cuttings returns sample” based on approximate formation composition of an actual drill site are also investigated by comparing a model cuttings returns sample with actual cuttings returns obtained from a drill site. A power-law model^{26,27} with solids-concentration dependent parameters is sufficient to characterize the viscometric properties. These clay dispersions have yield stresses that are orders of magnitude smaller than stresses originating from their plastic viscosity [supplementary material, SI 4, SI 5, SI 6, and Eq. (1)]. Contributions of yield stress to the measured pressure loss at the flow rates used are small, making a power-law model sufficient rather than a more complicated model including a yield stress. Observations of solids settling in the flow loop agree well with theoretical analysis based on the dimensionless Shields parameter. Critical findings include that the viscosity of model cuttings returns samples cannot be trusted to represent real cuttings returns samples based on simple sand-to-kaolinite clay ratio approximations and that fluidized solids regimes must be identified in flow loop measurement of these systems to properly report viscometric properties.

## EXPERIMENTAL

### Material preparation and characterization

Two types of formation cuttings returns samples were prepared, one a lab-made model fluid and the other a field-test concentrate. Each sample required their own preparation procedures. The model fluid is made by mixing (1) C144 mason *sand* from Jones Fine Sand in Denver, CO (70% of sand greater than 0.4 mm, particle size distribution in supplementary material, SI 7 and SI 8); (2) PONDSEAL-200 natural unaltered high swelling sodium *bentonite* from Redmond Minerals, Inc. in Grand Junction, CO; (3) ASP 600 hydrous aluminum silicate (*kaolinite*) from BASF Corporation in Charlotte, NC; and (4) tap *water*. These components will be referred to as sand, bentonite, kaolinite, and water. The model fluid was mixed based on expected formation cuttings returns with solid concentrations from 3 to 35 wt. % composed of 3:1 sand to kaolinite clay based on geotechnical data from the expected drill site, and 3 wt. % bentonite in water was the drilling fluid. 3 wt. % bentonite will now be referred to as bentonite slurry. Clay is present as kaolinite and bentonite, kaolinite is found in the formation and bentonite is used as the drilling fluid. Table I details the composition of the model fluid for the various samples tested.

Sample no. . | Water mass (kg) . | Bentonite mass (kg) . | Kaolinite mass (kg) . | Sand mass (kg) . | Total mass (kg) . | Solid fraction (wt/wt) . |
---|---|---|---|---|---|---|

1 | 211 | 6.53 | 0 | 0 | 218 | 0.03 |

2 | 211 | 6.53 | 3.38 | 13.5 | 234 | 0.10 |

3 | 211 | 6.53 | 6.30 | 25.2 | 250 | 0.15 |

4 | 211 | 6.53 | 9.45 | 37.8 | 265 | 0.20 |

5 | 211 | 6.53 | 13.1 | 52.2 | 283 | 0.25 |

6 | 211 | 6.53 | 16.7 | 66.6 | 301 | 0.30 |

7 | 211 | 6.53 | 21.6 | 86.4 | 326 | 0.35 |

Sample no. . | Water mass (kg) . | Bentonite mass (kg) . | Kaolinite mass (kg) . | Sand mass (kg) . | Total mass (kg) . | Solid fraction (wt/wt) . |
---|---|---|---|---|---|---|

1 | 211 | 6.53 | 0 | 0 | 218 | 0.03 |

2 | 211 | 6.53 | 3.38 | 13.5 | 234 | 0.10 |

3 | 211 | 6.53 | 6.30 | 25.2 | 250 | 0.15 |

4 | 211 | 6.53 | 9.45 | 37.8 | 265 | 0.20 |

5 | 211 | 6.53 | 13.1 | 52.2 | 283 | 0.25 |

6 | 211 | 6.53 | 16.7 | 66.6 | 301 | 0.30 |

7 | 211 | 6.53 | 21.6 | 86.4 | 326 | 0.35 |

The field-test concentrate was collected from horizontal directional drilling in Central Louisiana. The concentrate was collected with a 76 wt. % solids content (solids weight, both dispersed and settled, as a percent of the total sample weight) that includes sand, kaolinite, and bentonite. A notable difference between the model and field-test concentrate cuttings returns is that the drilling fluid used for the field test was an industry provided bentonite. Bentonite slurry was used to dilute the field-test concentrate, which is PONDSEAL-200, not the industry bentonite. Table II includes composition data for the cuttings returns samples prepared and tested from the field-test concentrate.

Sample no. . | Water mass (kg) . | Bentonite mass (kg) . | Sand and kaolinite mass (kg) . | Total mass (kg) . | Solid fraction (wt/wt) . |
---|---|---|---|---|---|

8 | 106.62 | 2.40 | 100.38 | 209.40 | 0.49 |

9 | 129.32 | 3.10 | 100.38 | 232.80 | 0.44 |

10 | 159.52 | 4.05 | 100.38 | 263.95 | 0.40 |

11 | 197.32 | 5.25 | 100.38 | 302.95 | 0.35 |

12 | 250.22 | 6.90 | 100.38 | 357.50 | 0.30 |

Sample no. . | Water mass (kg) . | Bentonite mass (kg) . | Sand and kaolinite mass (kg) . | Total mass (kg) . | Solid fraction (wt/wt) . |
---|---|---|---|---|---|

8 | 106.62 | 2.40 | 100.38 | 209.40 | 0.49 |

9 | 129.32 | 3.10 | 100.38 | 232.80 | 0.44 |

10 | 159.52 | 4.05 | 100.38 | 263.95 | 0.40 |

11 | 197.32 | 5.25 | 100.38 | 302.95 | 0.35 |

12 | 250.22 | 6.90 | 100.38 | 357.50 | 0.30 |

Solid contents were measured by drying samples. Samples were mixed for 30 s and dispensed into 100 ml open-top containers in approximately 50 ml volumes. The mass of the samples was weighed before and after drying to obtain the measured solid concentrations. Drying was performed in a thermo scientific gravity convection oven at 80 °C over 48 h.

Density is a critical input in the correlations to determine the viscometric properties of our cuttings returns samples. Density was obtained by measuring the mass of known volumes using a 250 ml volumetric flask and drying samples in an oven at 80 °C overnight. The densities are highly correlated with the solid type and concentration. Table III includes the densities of each cuttings returns sample tested.

Sample no. . | Sample type . | Solids fraction . | Density(g/ml) . |
---|---|---|---|

1 | Model | 0.03 | 1.018 |

2 | Model | 0.10 | 1.033 |

3 | Model | 0.15 | 1.042 |

4 | Model | 0.20 | 1.076 |

5 | Model | 0.25 | 1.114 |

6 | Model | 0.30 | 1.216 |

7 | Model | 0.35 | 1.343 |

8 | Field-test concentrate | 0.49 | 1.445 |

9 | Field-test concentrate | 0.44 | 1.387 |

10 | Field-test concentrate | 0.40 | 1.330 |

11 | Field-test concentrate | 0.35 | 1.279 |

12 | Field-test Concentrate | 0.30 | 1.230 |

Sample no. . | Sample type . | Solids fraction . | Density(g/ml) . |
---|---|---|---|

1 | Model | 0.03 | 1.018 |

2 | Model | 0.10 | 1.033 |

3 | Model | 0.15 | 1.042 |

4 | Model | 0.20 | 1.076 |

5 | Model | 0.25 | 1.114 |

6 | Model | 0.30 | 1.216 |

7 | Model | 0.35 | 1.343 |

8 | Field-test concentrate | 0.49 | 1.445 |

9 | Field-test concentrate | 0.44 | 1.387 |

10 | Field-test concentrate | 0.40 | 1.330 |

11 | Field-test concentrate | 0.35 | 1.279 |

12 | Field-test Concentrate | 0.30 | 1.230 |

### Methods

Correlations are used to determine power-law parameters from the measured data of pressure drop and flow rate from the flow loop and from density measurements. A flow loop was selected rather than a conventional rheometer because: (1) the cuttings returns samples contain large particles (e.g., sand, gravel) that exceed the measurement gap of conventional rheometers and (2) the flow loop mimics the actual drilling process with similar flow rates, pipe dimensions, and flow regimes (turbulent flow), while conventional rheometers are best suited for measurements under laminar flow.

The flow loop was designed and built to reach flow rates above 600 gallons-per-minute (0.038 m^{3}/s) in 4-in. schedule 40 PVC piping. The average shear-rates assuming a Newtonian parabolic profile spanned 98–367 s^{−1} for flow rates from 160 to 600 GPM. An illustration of the flow loop is shown in Fig. 1. From left to right, the flow loop consists of a holding tank (8), open to atmosphere, that feeds a pump (1), controlled by a variable frequency drive dictating the flow rate. The pump is a self-priming centrifugal pump from Gormon-Rupp, model T4A60S-B, with 4-in. suction and discharge, designed to handle up to 3-in. diameter solids. The electric motor is 3-phase, from Baldor, model ECP4110T, that can supply 40 HP and 1775 RPM. The variable frequency drive is from DURAPULSE, model GS3–2040. The pump circulates cuttings returns through the flow loop where pressure and flow rate are measured at the top (4, 5, 6, 7) by absolute and differential pressure transducers and a Doppler flowmeter. The cuttings returns can be cycled in an open or closed loop configuration by either directing cuttings returns back to the holding tank (close valve 4, open valve 6), or by bypassing the holding tank (close valve 6, open valves 4 and 5). A National Instruments data acquisition system takes in the flow rate and pressure data to catalog in unison. The schematic illustration is also shown in Fig. 1.

Detailed information of each sensor is included in Table IV. Due to the high solid concentration of the cuttings returns samples, neither the paddlewheel flow meter nor the electromagnetic flow meter provided reliable measurements; thus all flow rate data were obtained from the Doppler flowmeter and the paddle wheel sensor was far upstream (∼30 pipe diameters) from the measurement section of the flow loop, as shown in Fig. 1. Pressure loss data used in calculating fluid properties came from the differential pressure transducer rather than the absolute pressure transducers because of the superior resolution of the differential sensor.

Sensor type . | Sensor . | Supplier . | Range . | Accuracy (full scale range) . | Output signal . |
---|---|---|---|---|---|

Pressure | Absolute pressuretransducer, Model PX409-050GV | Omega Engineering Inc. | 0–344 kPa | ±0.25% (±0.86 kPa) | Voltage 0–5 VDC |

Pressure | Differential pressure transducer, Model PX409-001DWUV | Omega Engineering Inc. | 0–6.9 kPa | ±0.08% (±5.5 × 10^{−3} kPa) | Digital |

Volumetric flow | Doppler flowmeter, DFM 6.1 | Greyline | 0–12 m/s (0–0.099 m^{3}/s) | ±2% (±2.0 × 10^{−3} m^{3}/s) | Voltage 0–5 VDC |

Sensor type . | Sensor . | Supplier . | Range . | Accuracy (full scale range) . | Output signal . |
---|---|---|---|---|---|

Pressure | Absolute pressuretransducer, Model PX409-050GV | Omega Engineering Inc. | 0–344 kPa | ±0.25% (±0.86 kPa) | Voltage 0–5 VDC |

Pressure | Differential pressure transducer, Model PX409-001DWUV | Omega Engineering Inc. | 0–6.9 kPa | ±0.08% (±5.5 × 10^{−3} kPa) | Digital |

Volumetric flow | Doppler flowmeter, DFM 6.1 | Greyline | 0–12 m/s (0–0.099 m^{3}/s) | ±2% (±2.0 × 10^{−3} m^{3}/s) | Voltage 0–5 VDC |

Measurements of flow rate and pressure were performed along the highest horizontal section of the flow loop. The upstream 90°-bend leading into this section, labeled “Tee 5” in Fig. 1, introduces eddies in the flow that propagate downstream and potentially influence pressure measurements.^{28–32} Such flow disturbances, or “pipe swirl,” decay with distance, and it is important to understand the effect on measured pressure as a function of distance from Tee 5. Multiple sensor ports were installed downstream of Tee 5 where pressure sensor connections can be made to measure the influence of pipe swirl. Pressure loss measurements with the differential pressure sensor (ΔP = P_{1} − P_{2}) were made at the 5 different upstream sensor port locations (the downstream port location, port 10, was held constant) for six different flow rates of water at approximately 25 °C, measured with an infrared temperature sensor. Results of those measurements are shown in Fig. 2. The pressure differential measured is normalized by the distance between the upstream and downstream ports and is plotted as a function of the normalized distance of the upstream port from Tee 5. The distance is normalized by the pipe inner diameter, and the zero position is taken as the center of the vertical section of pipe between Tee 4 and Tee 5. Error bars show a single standard deviation on a time average over 60 s at a 5 Hz acquisition rate. Ports 1 and 2 correspond to L/D distances from Tee 5 of 3.9 and 5.3, respectively, and results from those ports differ considerably from results obtained from the three ports further downstream, suggesting the influence of pipe swirl on the measurements. Port 3, with an L/D of 7, provides pressure loss values similar to those from ports 4 and 5 within the error, suggesting that this is sufficiently far enough downstream to avoid measurable interference from upstream eddies through at least a flow rate of 0.034 m^{3}/s (540 GPM). Port 3 was chosen for experiments based on these results and the need to maximize the distance between upstream and downstream pressure measurement locations to maximize measurement resolution. The distance between port 3 and port 10, the two connections of the differential transducer, was 1.89 m.

### Viscometric characterization

The cuttings returns samples are generally considered as non-Newtonian fluids, so the power law model was selected to describe their viscometric behavior. The power law model can be written as

where $\gamma \u0307$ is the shear rate, $\eta $ is viscosity, and *n* and *m* are power law indices that depend on the viscometric properties of the fluid. *n* is the dimensionless flow behavior index that equals 1 for Newtonian fluid, *n *<1 for shear-thinning fluid, and *n *>1 for shear thickening fluid. *m* is the flow consistency index that can range from 0.001 for water and beyond 10^{6} for malleable bentonite clays with water content beyond their liquid limit (the unit of *m* depends on the value of *n*).

Determining these power law indices, and how they vary with increasing solid concentration in drilling fluid samples, is the primary objective of these flow loop measurements. To achieve this objective, three parameters need to be measured: (1) the density of drilling fluids with various cuttings returns concentrations, (2) the average velocity (or flow rate) of the flow, and (3) the pressure drop when the drilling fluids flow through a section of straight pipe.

The flow rate and the pressure drop can be used to obtain the friction factor, *f*, using^{33,34}

*D* is the pipe diameter, *ΔP* is the pressure differential, *L* is the distance between the pressure transducers. *ρ* is the density of fluid, and *v* is the average velocity, which can be calculated from the flow rate.

Another important parameter is the Reynolds number (Re) of the flow. Since the fluids are non-Newtonian, the expression for *Re* is^{21}

where *ρ* is the density of fluid, *v* is the average fluid velocity, and *n* and *m* are the power law indices in Eq. (1).

Based on Eqs. (2) and (3), a relation between the friction factor and Reynolds number is required to determine *n* and *m* from the flow loop tests. For laminar flow, the relation is^{21}

There are many models that describe the relationship between the friction factor and power law indices for turbulent flow of a power law fluid. Many of these models can be found in Heywood and Cheng 1984.^{21} In this study, two classic models were used: the Dodge and Metzner (D&M) model, and the Bobok, Navratil, and Szilas (BNS) model.^{18} Of the many empirical correlations relating pressure drop and flow rate to fluid power-law parameters, the D&M model is the first, and a classic model used for describing a power-law fluid in turbulent flow, and the BNS model has been shown to be one of the most accurate.^{35} Despite the highly nonlinear nature of both, they can be readily fit to data with a nonlinear regression. The D&M model can be written as

The BNS model is analytically deduced for non-Newtonian fluids^{18} and can be written as

In this work, all flow was turbulent during testing; thus Eqs. (2), (3), (5), and (6) were applicable. Once the flow rate and pressure drop data were obtained, the power-law indices (*n*) and (*m*) were calculated from the nonlinear equation system above. The nonlinear regression was performed with a custom Python script. The code is included as supplementary material.

## RESULTS AND DISCUSSION

### Comparison between model and field-test cuttings returns samples

There are notable viscometric differences between the model and field-test cuttings returns. In Fig. 3, the model cuttings returns and the field-test cuttings returns data are plotted as normalized pressure drop vs flow velocity for direct comparison. Field-test cuttings returns yield a significantly higher pressure drop compared to the model cuttings returns at the same flow speed and solids concentration. Although the densities for both types of cuttings returns samples are similar at each concentration, the components (kaolinite clay, bentonite clay, sand, silt, etc.) originated from different sources and will have subtle, but significant differences in particle size, chemistry, and composition that can yield measurable differences in viscometric properties. This is captured in the data in Table V where the consistency index, *m*, of the field-test cuttings returns is significantly higher than the model cuttings returns samples. It has been well established in the literature that subtle differences in electrolyte content, pH, exchangeable cations, and clay composition can account for this offset in pressure drop values related to a difference in the consistency index.^{36,37} Even though these variations are well known, the characterization of these differences is seldom reported along with their respective power-law parameters. Since clay platelet interactions are known to exhibit wide variations in viscometric properties, we suspect that the bentonite clay variation in our model and field-test cuttings returns and the differences in sand particle size distribution are responsible for the differences between the two trends in Fig. 3. The particle size distribution of the sand between samples indicates that the field-test cuttings are finer than the model cuttings (see supplementary material, SI 7 and SI 8). It is also known in the literature that particle size distribution can affect the pressure drop in pipe flow.^{38} The two cuttings returns samples are also both heterogeneous mixtures of sand and dispersed clay, so individually or together mineralogy and particle size distribution is the root cause rather than clay agglomeration. Powder x-ray diffraction (XRD) confirms the mineralogy differences between the two bentonite clays, indicating that silica oxide is the major compound in the two cuttings returns, but it exists as different SiO_{2}-polymorphs between the two fluids (supplementary material, SI 9 and SI 10). The field-test cuttings contain an abundant amount of cristobalite in addition to quartz which is the dominant SiO_{2}-polymorph in the model cuttings sample. Cristobalite is known to form crystals <2 *μ*m that has the potential to affect gel formation and alter the viscosity of the bentonite dispersions.^{39} The field-test cuttings returns bentonite clay fluid also contains additional silicate minerals not found in the model cuttings returns (see supplementary material, SI 9 and SI 10). The difference in bentonite originates from the industry bentonite used during the field test drilling compared to the bentonite used for model cuttings returns.

Sample no. . | D&M n . | D&M m . | BNS n . | BNS m . |
---|---|---|---|---|

1 | 0.99 | 0.0006 | 0.97 | 0.0008 |

2 | 0.78 | 0.0132 | 0.74 | 0.0272 |

3 | 0.98 | 0.0009 | 0.96 | 0.0011 |

4 | 0.91 | 0.0029 | 0.84 | 0.0080 |

5 | 0.84 | 0.0100 | 0.81 | 0.0180 |

6 | 0.51 | 0.4355 | 0.47 | 1.8470 |

7 | 0.41 | 2.1342 | 0.44 | 2.4496 |

8 | 0.65 | 0.7704 | 0.65 | 1.0311 |

9 | 0.74 | 0.3402 | 0.74 | 0.4026 |

10 | 0.54 | 2.3015 | 0.54 | 4.0956 |

11 | 0.57 | 1.5926 | 0.57 | 2.5542 |

12 | 0.56 | 1.7236 | 0.56 | 2.8533 |

Sample no. . | D&M n . | D&M m . | BNS n . | BNS m . |
---|---|---|---|---|

1 | 0.99 | 0.0006 | 0.97 | 0.0008 |

2 | 0.78 | 0.0132 | 0.74 | 0.0272 |

3 | 0.98 | 0.0009 | 0.96 | 0.0011 |

4 | 0.91 | 0.0029 | 0.84 | 0.0080 |

5 | 0.84 | 0.0100 | 0.81 | 0.0180 |

6 | 0.51 | 0.4355 | 0.47 | 1.8470 |

7 | 0.41 | 2.1342 | 0.44 | 2.4496 |

8 | 0.65 | 0.7704 | 0.65 | 1.0311 |

9 | 0.74 | 0.3402 | 0.74 | 0.4026 |

10 | 0.54 | 2.3015 | 0.54 | 4.0956 |

11 | 0.57 | 1.5926 | 0.57 | 2.5542 |

12 | 0.56 | 1.7236 | 0.56 | 2.8533 |

The trendline seen in Fig. 3 is for turbulent pipe flow of water, using a Newtonian expression.^{34} The model cuttings returns align closely with the trendline. The steady-shear flow sweep data of the continuum bentonite clay fluid between each sample further support the assertion that mineralogy between the two samples could be one cause for this offset in pressure drop values across the same flow rate. The industry bentonite clay fluid of the field-test sample had multiple orders of magnitude greater viscosity than the model bentonite clay fluid across the same flow sweep from 1 to 200 1/s characterized in a cup and bob geometry on a TA Instruments DHR-3 rheometer (see supplementary material, SI 1 and SI 4).

The model cuttings returns samples were prepared with geotechnical data from the drill site, suggesting a composition of sand and kaolinite of approximately 70:30 mass ratio, so the conclusion of Fig. 3 is that only knowledge of sand and kaolinite ratios is insufficient to model viscometric behavior of cuttings returns in practice.

### Flow and pressure drop results

Flow rate and pressure drop are needed in the empirical correlations and were measured in the cuttings returns samples at flow rates from 3.97 to 4.53 m/s. Figure 4 includes the results of the flow loop experiments for the model cuttings returns samples, and Fig. 5 presents the results of the flow loop experiments for the cuttings returns samples obtained from field-test drilling.

A higher flow rate and larger solids fractions result in larger pressure drops in the fluidized regime where flow rates are greater than 1.7 and 2.6 m/s for the field-test cuttings returns and the model cuttings returns, respectively. In Figs. 4 and 5, there is a minimum in the pressure drop data for solids fractions above 20 wt. % that occurs at approximately 1.7 m/s. In this region of flow, extensive settling of solids within the pipe occurred and significantly reduced the cross-sectional area available to flow. With increasing flow rate above 1.7 m/s, settled solids were increasingly entrained in the flow and the cross-sectional area of flow increased. This minimum in the data is evident for all solids fractions, but with a greater solid fraction the minimum shifts to the right, to higher flow velocities, where a greater flow rate is needed to suspend cuttings returns with a greater solids content. No minimum is seen in Fig. 4 in samples 1–3 where the solid content was less than 15 wt. %. Figure 6 contains an image showing visible solids settling in the 35 wt. % model fluid sample (7) during flow loop operation at a flow velocity of 1.73 m/s. The light gray is the flowing fluid in the upper part of the horizontal pipe, while the darker bottom part of the pipe contains settled solids.

In Fig. 7, all data for the model and field-test concentrate cuttings returns are included and plotted as the dimensionless Shields parameter as a function of flow velocity.^{40} Shields parameter is a ratio between wall shear stress and buoyancy forces, the magnitude indicating the tendency for solids to suspend in a flow or settle. In Eq. (7), Shields parameter ($\theta $) is expressed as the ratio between the wall shear stress and the stress caused by settling. Pressure drop is converted to wall shear stress by Eq. (8),

When the Shields parameter is below 1, the driving stresses from gravitational settling dominate, and above 1 the flow stresses leading to solid dispersion are dominant. At low flow rates, the effects of settling are seen in the local minimum in pressure drop and can also be visually observed. Based on this analysis, all power-law parameter estimations were done with data above a flow rate of 1.7 m/s for the field-test concentrate and 2.6 m/s for the model cuttings returns to minimize the influence of solids settling on the correlation fits to the data.

Two important conclusions result from this analysis of Shields parameter and settling observations during flow loop measurements. The first is that a delineating line represented by a Shields parameter of 1 can be used to identify regimes of flow where viscometric modeling of fully suspended flow can be relied on to provide pressure drop calculations, and where solids settling and flow partitioning prohibits the use of those models. The second conclusion is that observations of settling in flow loops can be reliably connected to settling dynamics through the Shields parameter in such a way that flow regimes that lead to solids-settling (and thus plugging concerns during drilling) can be empirically identified.

### Power law indices analysis

By first determining cuttings returns sample densities and the applicable pressure drop and flow rate data, power-law indices can now be determined with empirical correlations. The applicable pressure drop–flow rate data were determined by the Shields parameter analysis to identify a regime where solids settling is no longer dominate, at flow rates greater than 1.7 and 2.6 m/s for the field-test cuttings returns and the model cuttings returns, respectively. If we were interested in the settled/partially settled regime, the extent of settling could be used to semi-quantitatively characterize the weight percent of solids suspended under each flow condition. By employing Eqs. (2), (3), (5), and (6), a nonlinear equation system with “n” and “m” as the unknown parameters can be developed. An in-house nonlinear equation solver programed in Python is used to determine power-law and consistency indices that are shear-thinning and increase in viscosity with solid content.

Using the nonlinear solver, “n” and “m” values for each cuttings returns sample are presented in Table V. In the model cuttings returns sample, *n* decreases with increasing solids fraction indicating greater shear-thinning behavior with a greater solids content. The 3 wt. % model fluid sample (1), which is pure bentonite in water at a concentration of 3 wt. %, has a *n* value of 0.99, indicating near Newtonian behavior. With increasing solid fraction, the value of *n* decreased incrementally to a value of 0.41 for the 35 wt. % model fluid sample (7), indicating increasingly shear-thinning behavior as would be expected for a particulate dispersion. In the field-test cuttings returns samples, there was no obvious trend in power-law index with solid fraction. Shear-thinning is still present in these samples, with *n* values in the range of 0.55–0.75. The model cuttings returns samples tended to have a smaller value of *n* at a given solid concentration, potentially because of differences in particle size, clay concentration, and type of clay between the field-test concentrate and the model cuttings returns, although the exact reason for this is unknown.

The *n* values obtained by the D&M and BNS models match closely, but values of *m* differed more. The *m* values from the BNS correlation are larger than from the D&M correlation. The expectation is that the value of *m* increases as the solid concentration increases, but this was not always the case for the field-test concentrate cuttings returns samples. The reason for this is still unclear, although the highly nonlinear nature of both correlations means that fitting the correlation to the data can yield multiple solution values for *n* and *m*.

To assess the validity of the calculated *n* and *m* indices they are used to recreate the pressure drop and flow rate curves seen in Figs. 4 and 5. Figures 8 and 9 include the predictions by the D&M and BNS models overlaid on the experimental data. The pressure drop vs flow velocity curves match the experimental data well, especially at high flow velocities. The power law indices estimated from the data can be used to calculate the expected pressure drop for the drill rig at data points with Shields parameter values greater than one.

Pressure drop data from drilling pipes have been used for over a decade to parameterize hole cleaning and quantify drilling fluid viscosity under *in situ* temperature and pressure conditions.^{41–43} Flow loops have also been used to explore the major parameters driving hole-cleaning^{38,44–46} but both hole-cleaning and characterizing cuttings returns viscosity has not been done together. Hole cleaning studies are typically fundamental and attempt to understand the drivers behind dispersion and movement of granular material.^{47–49} It is also well known that granular materials affect the overall viscosity of the complex fluid as seen in variations of pressure drop when they are included in flow,^{38} but the viscometric characterization of these complex fluids with dispersion is either left out or smeared by the addition of concentric or eccentric pipes that contribute to the pressure drop.^{43,50,51} This paper provides an experimental approach to characterize the fluid properties of drilling fluids with solids. This work is focused on particles fully dispersed in the drilling fluid but can be extended to determine the power-law parameters of complex fluids with partially dispersed particles or bed formation. Power-law parameters are sufficient to describe the viscometric properties of the fluid because the effects of yield stress to pressure drop of the bentonite clay fluid was found to be negligible [see the supplementary material, SI 4, SI 5, SI 6, and Eq. (1)]. Mixer-type rheometers have also been used to characterize dispersion viscosity,^{50} but they often ignore the extent of solids settling and would need to introduce additional flow streams to suspend solids that would require a calibration procedure not currently implemented.^{52}

## CONCLUSIONS

Viscometric characterization of drilling fluids containing solids in concentrations ranging from 3 to 45 wt. % was performed in a custom flow loop apparatus. Measurements of pressure loss as a function of flow rate were performed on two different materials: a model cuttings returns material prepared from sand, kaolinite, bentonite, and water, in ratios intended to approximate the formation conditions at a drill site, and a material comprised of drilling fluid and solids collected from drill tests at the drill site. Two different correlations relating friction factors to flow rate for power-law fluids in turbulent flow were fit to the flow loop data to obtain estimates of power law parameters. The results indicate that a power law constitutive model is suitable for the materials. Increasing solids content led to a decrease in the power-law index, n, and an increase in the consistency index, m, indicating an increase in both viscosity and shear-thinning. The data provide a relationship between power-law parameters and solids concentration that is not present in the literature, but which is critical for use in computational modeling of pressure losses during drilling. A surprising result was the significant difference in pressure loss values between the model material and the material collected at the drill site, approximately a factor of two difference. Considering the limited data available on such materials, an important finding is that a model cuttings returns material prepared with a simple sand-to-kaolinite clay ratio from geotechnical data from a particular drill site cannot be trusted in experiments to provide representative viscometric parameters of materials encountered in field testing. A second important finding was the observation that the Shields parameter, a ratio of viscous stresses to gravitational stresses, was consistent with pressure loss measurements that indicated a transition at particular flow rates from fully suspended, homogeneous flow to a bifurcated flow with settled solids. The significance is that the Shields parameter can be a valuable quantitative tool in predicting flow rates where the onset of flow problems associated with solids settling can be expected.

## SUPPLEMENTRARY MATERIAL

See the supplementary material for yield stress characterization of the clay fluid in the cuttings returns samples, XRD characterization of the clay fluids from the model and field-test cuttings returns, particle size distribution of the model and field-test cuttings returns, and the in-house Python code used to determine power-law indices using a nonlinear regression.

## ACKNOWLEDGMENTS

The authors are grateful to Jon Peters, Bryant Robbins, and Minsu Cha for their contributions in fabricating and assembling the flow loop and William Smith for their contribution in analyzing XRD data.

This research was developed with funding from the Defense Advanced Research Projects Agency (DARPA, No. HR0011-20-C-0029). The views, opinions, and/or findings expressed are those of the author and should not be interpreted as representing the official views or policies of the Department of Defense or the U.S. Government.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

J.Y. and B.A.A. both contributed equally to the experiments and writing. J.Y., B.A.A., M.A.M., and J.R.S. conceived and designed the experiments. J.Y. and B.A.A. performed the experiments. J.Y. and B.A.A. analyzed the data. J.Y., B.A.A., and J.R.S. wrote the paper. J.Y., B.A.A., M.A.M., and J.R.S. contributed to manuscript revisions. All authors read and approved the final manuscript.

## DATA AVAILABILITY

The data that support the findings of this study are available within the article and its supplementary material.

## References

^{®}as Alternative to HDD in Permeable Soil Conditions

*30 August-3 September 1997*(SPE,