Drilling Fluid Rheology Analyzer

Fann Viscometer Conversion

The Fann 35 viscometer with standard R1B1 rotor-bob geometry measures dial readings at six rotational speeds. These are converted to shear rate and shear stress using instrument-specific factors.

Shear Rate

Shear Rate (s⁻¹) = 1.7023 × RPM

Shear Stress

Shear Stress (lb/100ft²) = 1.0678 × Dial Reading

Standard Shear Rate Table

API Standard Derived Properties

These are the standard field calculations from the θ600 and θ300 readings, per API RP 13D.

Plastic Viscosity (PV)

PV = θ600 - θ300    (cp)

Reflects mechanical friction from solids in the mud. High PV usually indicates excessive fine solids (drilled solids, barite fines). Reducing PV improves hydraulics efficiency and lowers ECD.

Yield Point (YP)

YP = θ300 - PV    (lb/100ft²)

Indicates electrochemical forces between mud particles. YP is the primary indicator of hole cleaning capability in vertical wells. Higher YP improves cuttings transport but increases ECD.

Flow Behavior Index (n)

n = 3.32 × log&sub1;&sub0;(θ600 / θ300)    (dimensionless)

Describes shear-thinning behavior. n = 1 is Newtonian; most drilling fluids have n between 0.4 and 0.8. Lower n means more shear-thinning — the fluid thins dramatically under high shear (good for bit hydraulics) but maintains viscosity at low shear (good for suspension).

Consistency Index (K)

K = 510 × θ300 / 511n    (eq.cp)

The magnitude parameter in the Power Law model. Higher K means the fluid generates more shear stress at any given shear rate. Used with n in hydraulics calculations for pressure loss in pipe and annular flow.

Apparent Viscosity

AV = θ600 / 2    (cp)

Low-Shear Yield Point (LSYP)

LSYP = 2 × θ3 - θ6    (lb/100ft²)

Critical for evaluating barite sag potential in deviated wells. LSYP ≥ 7 lb/100ft² is generally recommended to prevent weighting material settling during static periods. Also correlates with hole cleaning in high-angle wells better than the standard YP.

Bingham Plastic Model

τ = τy + μp × γ̇

Strengths: Simple, familiar to field personnel, widely used in daily mud reports. Two parameters (PV, YP) from just θ600 and θ300.

Weaknesses: Assumes linear shear stress vs. shear rate, which most drilling fluids don't follow. Overestimates stress at low shear rates and underestimates at high shear rates. Poor fit for highly shear-thinning fluids.

Power Law Model

τ = K × γ̇n

Strengths: Captures shear-thinning behavior (n < 1). Appears as a straight line on log-log plot. Two parameters fitted from all six data points.

Weaknesses: No yield stress term — predicts zero shear stress at zero shear rate, which is incorrect for most drilling fluids. Overestimates flow at very low shear rates.

Herschel-Bulkley Model

τ = τ0 + K × γ̇m

Strengths: Most accurate model for drilling fluids. Combines yield stress (like Bingham) with shear-thinning (like Power Law). Three parameters give the best fit across the full shear rate range. Recommended by API RP 13D for hydraulics calculations.

Weaknesses: Requires nonlinear curve fitting (Levenberg-Marquardt algorithm). Three parameters can overfit with only six data points if the data has measurement noise. More complex to use in hand calculations.

Special Cases

• τ₀ = 0 → reduces to Power Law
• m = 1 → reduces to Bingham Plastic
• τ₀ = 0 and m = 1 → reduces to Newtonian

Fitting Method

Bingham Plastic: Least-squares linear regression on all six (shear rate, shear stress) data points. Slope = plastic viscosity, intercept = yield point.

Power Law: Log-log linear regression. Taking log of both sides: log(τ) = log(K) + n × log(γ̇). Linear regression in log-log space gives n (slope) and K (antilog of intercept).

Herschel-Bulkley: Levenberg-Marquardt nonlinear least squares. Minimizes the sum of squared residuals: Σ(τ_measured - τ₀ - Kγ̇^m)². Uses an analytical Jacobian for efficiency. Constrained: τ₀ ≥ 0, K > 0, 0 < m < 2. Initial guess from Power Law fit with τ₀ = 0.

R² (Coefficient of Determination)

R² = 1 - SSres / SStot

SSres = Σ(τmeasured - τpredicted)²
SStot = Σ(τmeasured - τ̄)²

R² ranges from 0 to 1 (perfect fit). The model with the highest R² best describes your fluid's behavior. For most drilling fluids, expect Herschel-Bulkley R² > 0.99, Bingham R² > 0.95, and Power Law R² > 0.90.

Worked Example

Given Fann readings: θ600 = 65, θ300 = 40, θ200 = 30, θ100 = 22, θ6 = 8, θ3 = 6

API Standard Properties

PV  = 65 - 40 = 25 cp
YP  = 40 - 25 = 15 lb/100ft²
n   = 3.32 × log(65/40) = 3.32 × 0.2109 = 0.700
K   = 510 × 40 / 5110.700 = 20,400 / 78.73 = 259.1 eq.cp
AV  = 65 / 2 = 32.5 cp
LSYP = 2(6) - 8 = 4 lb/100ft²

Model Fit Results

Interpretation: Herschel-Bulkley provides the best fit (R² = 0.998). The fluid shows moderate shear-thinning (m = 0.77) with a true yield stress of 6.2 lb/100ft². The Bingham model overpredicts stress at low shear rates and underpredicts at high shear rates, while the Power Law model ignores the yield stress entirely.

Valid Ranges & Typical Values

Practical Interpretation

• High PV, normal YP: Excessive drilled solids — run solids control equipment (shakers, centrifuges, dilution)
• High YP, normal PV: Chemical flocculation or contamination — check for cement, salt, or polymer issues
• Low LSYP (< 7): Risk of barite sag in deviated wells — add low-shear viscosifiers
• n < 0.5: Highly shear-thinning fluid — excellent bit hydraulics but verify low-shear suspension
• Bingham R² < 0.95: Fluid behavior is significantly non-linear — use Herschel-Bulkley for hydraulics calculations

When Model Selection Matters

For routine operations in conventional wells, the Bingham Plastic model is adequate. Model selection becomes critical in:

• Narrow-margin drilling: ECD must be predicted within 0.1–0.2 ppg; an inaccurate rheology model can cause lost circulation or kicks
• High-angle/horizontal wells: Hole cleaning depends heavily on low-shear behavior that Bingham cannot capture
• Deepwater/HPHT: Temperature and pressure effects on rheology make accurate modeling essential
• Managed Pressure Drilling (MPD): Precise ECD prediction is the foundation of MPD operations

References

1. API RP 13D — Recommended Practice for Rheology and Hydraulics of Oil-Well Drilling Fluids. American Petroleum Institute.
2. Bourgoyne, A.T. et al. (1986). Applied Drilling Engineering. SPE Textbook Series, Vol. 2.
3. Fann Instrument Company. Model 35 Viscometer Operating Instructions. Technical Bulletin.
4. Herschel, W.H. & Bulkley, R. (1926). "Measurement of Consistency as Applied to Rubber-Benzene Solutions." Proc. ASTM, 26(2), 621–633.
5. PetroWiki — Drilling fluid properties: https://petrowiki.spe.org/Drilling_fluid_properties