Bingham Plastic Model: PV, YP, Shear Stress, and Drilling Fluid Hydraulics

Key Takeaways

• PV and YP from the Fann viscometer: field measurement and interpretation: The Fann Model 35 viscometer operates at six standardized shear rates corresponding to rotor speeds of 600, 300, 200, 100, 6, and 3 RPM, with the bob-and-sleeve geometry calibrated so that the dial reading at 600 RPM approximates the shear stress in lb/100 ft² at a shear rate of 1,022 s⁻¹. For the Bingham plastic calculation, only the 600 and 300 RPM readings are required: a mud with θ600 = 55 and θ300 = 35 has PV = 55 - 35 = 20 cP and YP = 35 - 20 = 15 lb/100 ft². The physical significance: PV (20 cP) reflects the viscous drag of the suspended solids — higher drilled solids content, finer solids, or higher base fluid viscosity all raise PV; in a properly maintained WCSB bentonite surface hole mud, target PV is typically 15-25 cP. YP (15 lb/100 ft²) reflects the strength of the electrostatic force network (gel structure) between bentonite clay platelets — higher YP provides better cuttings carrying capacity in deviated wells and better suspension of barite in high-density muds, but too-high YP increases ECD and annular pressure loss. Typical WCSB targets: surface hole bentonite mud YP 10-20 lb/100 ft²; intermediate hole polymer/bentonite hybrid YP 8-15 lb/100 ft²; horizontal Montney KCl-polymer WBM YP 5-12 lb/100 ft²; OBM for Duvernay YP 8-15 lb/100 ft² (measured as a direct dial reading equivalent for OBM using the same instrument). Gel strengths (measured at 10 seconds and 10 minutes after stopping rotation) are reported separately from PV/YP: 10-second and 10-minute gels provide information about thixotropic structure that the Bingham model does not capture.
• Bingham plastic ECD calculation in the annulus: The equivalent circulating density (ECD) in the annulus during drilling is the sum of the static mud weight and the annular pressure loss per unit depth, expressed in kg/m³ or ppg. The Bingham plastic annular pressure loss (dP/dL) per unit length is calculated using the simplified Bingham slot-flow equation: dP/dL (Pa/m) = [48 × PV × v / (D_a² - D_p²)] + [6 × YP / (D_a - D_p)], where v is the average annular velocity (m/s), D_a is the annular diameter (m), and D_p is the drill pipe/collar OD (m), with PV in Pa·s (divide cP by 1,000) and YP in Pa (multiply lb/100 ft² by 0.4788). For a typical WCSB Montney horizontal well: annular geometry 8-1/2 inch hole (0.216 m) with 5-inch drill pipe (0.127 m), PV = 18 cP (0.018 Pa·s), YP = 8 lb/100 ft² (3.83 Pa), annular velocity v = 0.65 m/s (calculated from pump rate and annular cross section): dP/dL = [48 × 0.018 × 0.65 / (0.216² - 0.127²)] + [6 × 3.83 / (0.216 - 0.127)] = [0.5616 / (0.0467 - 0.0161)] + [22.98 / 0.089] = [0.5616 / 0.0306] + 258 = 18.3 + 258 = 276 Pa/m = 0.276 kPa/m. For a 4,000 m horizontal well (2,000 m vertical + 2,000 m horizontal), annular pressure loss in the horizontal section ≈ 0.276 × 2,000 = 552 kPa = 80 psi, representing an ECD increase of approximately 0.027 sg (2.7 kg/m³) above the static mud weight — a material contribution to ECD management in narrow-window Duvernay drilling where the safe-drilling density window may be only 0.05-0.10 sg between pore pressure and fracture gradient.
• Limitations of the Bingham plastic model: shear-thinning and Power Law fluids: The Bingham plastic model assumes a linear shear stress versus shear rate relationship above the yield point, implying that the apparent viscosity (shear stress / shear rate) is constant above YP. In reality, most drilling fluids are pseudoplastic (shear-thinning): apparent viscosity decreases as shear rate increases, meaning the fluid is thicker at the low shear rates in the annulus (where cuttings suspension is desired) and thinner at the high shear rates through the bit nozzles (where pressure loss should be minimized). The Power Law model (τ = K × γ̇ⁿ) captures shear-thinning behavior through the flow behavior index n (n < 1 for shear-thinning, n = 1 for Newtonian) and consistency index K, but lacks a yield point term and therefore cannot describe the initial gel strength of structured drilling fluids. The Herschel-Bulkley model (τ = τ_y + K × γ̇ⁿ), which combines a yield point with Power Law shear-thinning behavior, is the most accurate three-parameter model for most drilling fluids, but requires three viscometer readings (typically θ300, θ100, θ3) for parameter estimation and is more complex to apply in field ECD calculations. For most WCSB well applications, the Bingham plastic model provides adequate hydraulics design accuracy (within 10-15% of actual measured annular pressure) and the simplicity of two parameters directly readable from two Fann dial readings makes it the universal practical standard despite its theoretical limitations.
• Bingham model contamination detection: PV and YP as diagnostic tools: Changes in PV and YP relative to the baseline mud report values are among the earliest and most sensitive indicators of mud contamination events in WCSB drilling. Bicarbonate contamination (CO2 influx from calcareous formations) raises YP disproportionately to PV — a YP increase from 12 to 28 lb/100 ft² with PV relatively stable (18 to 20 cP) indicates pH depression and clay flocculation from CO2/HCO3⁻ contamination, requiring lime treatment. Calcium contamination (from cement filtrate or anhydrite dissolution) raises both PV and YP simultaneously — PV rising from 20 to 32 cP (solids increase from calcium carbonate precipitation) and YP from 14 to 35 lb/100 ft² — indicating severe clay flocculation requiring soda ash (Na2CO3) treatment. Salt contamination (NaCl from evaporite beds) depresses both PV and YP initially as sodium ions deflocculate the bentonite, followed by a rise in PV as the mud thins and liquid phase salinity increases, recognized by the characteristic drop in 10-second gel strength to near zero (complete deflocculation). Mud engineers on WCSB wells monitor the daily PV/YP trend chart alongside pH and alkalinity to diagnose contamination events within one circulation cycle (lag time to surface) — the faster the PV/YP changes relative to the baseline trend, the more severe and recent the contamination event.
• Bingham plastic parameters in cement slurry design: Cement slurries used in WCSB casing programs are also described by Bingham plastic parameters (PV, YP) measured at downhole temperature using the atmospheric consistometer or pressurized consistometer per API RP 10B-2. Cement slurry PV typically ranges 15-50 cP at circulating temperature (BHCT); YP typically ranges 5-20 lb/100 ft². The cement annular pressure calculations use the same Bingham slot-flow equations as for drilling mud, but the interpretation differs: cement YP must be high enough to prevent U-tubing (cement falling back due to density difference after pumping stops) but low enough that placement is possible at reasonable pump pressures without exceeding the fracture gradient of weak formations ahead of the shoe. For WCSB surface casing cementation (13-3/8 inch or 10-3/4 inch, typical depths 300-700 m), the lead cement slurry YP is typically 6-10 lb/100 ft² (low YP, thin slurry for efficient mud displacement) while the tail slurry (placed across the formation top of cement target zone) uses YP 12-18 lb/100 ft² to resist U-tubing during WOC (wait on cement). The Bingham plastic model governs cement placement hydraulics in the AER Directive 009 cementing program design, which requires the operator to demonstrate that the cement placement pressure will remain below 80% of the fracture gradient throughout the cemented interval.