Herschel-Bulkley Fluid | Oil Authority

A Herschel-Bulkley fluid is a non-Newtonian fluid whose rheological behavior is described by a three-parameter model incorporating a yield stress, a consistency index, and a flow behavior index. Mathematically the model is expressed as tau = tau_y + K (gamma_dot)^n, where tau is the shear stress, tau_y is the yield stress (the minimum stress required to initiate flow), K is the consistency index, gamma_dot is the shear rate, and n is the flow behavior index. Because it combines the yield stress of the Bingham Plastic model with the power-law shear-thinning of the Power Law model, the Herschel-Bulkley equation produces more accurate representations of drilling fluid rheology across the full range of shear rates encountered in a wellbore, from the high shear rates inside the drill string to the low shear rates in the annulus and during static periods.

Relationship to Simpler Rheological Models

The Herschel-Bulkley model is a generalization that subsumes two simpler industry-standard models as special cases:

Bingham Plastic (n = 1): When the flow behavior index equals 1, the Herschel-Bulkley equation reduces to tau = tau_y + mu_p (gamma_dot), where mu_p is the plastic viscosity. The Bingham Plastic model is linear in shear rate above the yield stress, which works well at moderate shear rates but overestimates viscosity (and therefore pressure loss) at low shear rates in the annulus, leading to conservative ECD predictions.
Power Law (tau_y = 0): When the yield stress is set to zero, the equation reduces to tau = K (gamma_dot)^n. The Power Law model captures shear-thinning accurately at intermediate shear rates but predicts zero viscosity at rest, meaning it cannot describe the gel strength or static suspension of cuttings and weighting materials that real drilling fluids exhibit.

The Herschel-Bulkley model retains the yield stress that prevents the fluid from flowing below a critical stress threshold, while the power-law term accurately captures shear-thinning (n less than 1) or shear-thickening (n greater than 1) behavior above yield. For virtually all field drilling fluids, n is less than 1, indicating shear-thinning. This combination makes the model more accurate than either predecessor when adequate experimental data spanning the full shear rate range are available from a viscometer.

Parameter Determination from Viscometer Data

The three Herschel-Bulkley parameters are determined by fitting the model to rotational viscometer readings taken at multiple speeds. The standard Fann VG meter provides readings at 3, 6, 100, 200, 300, and 600 rpm, which correspond to shear rates of approximately 5.1, 10.2, 170, 340, 511, and 1,022 reciprocal seconds. A least-squares regression of the shear stress versus shear rate data yields tau_y, K, and n simultaneously. Determining the correct yield stress is the most challenging step because the 3 rpm and 6 rpm readings are close together and dominated by gel strength effects at low shear; some analysts apply a simplified graphical method using three data points while others use full nonlinear regression across all readings. HPHT viscometer data extend this exercise to elevated temperature and pressure, generating Herschel-Bulkley parameter sets that vary with depth and temperature profile for use in hydraulics simulators.

Application to Hydraulics and ECD Calculations

The equivalent circulating density (ECD) in the annulus depends directly on annular pressure loss, which in turn requires an accurate viscosity model at the low shear rates characteristic of upward fluid flow around the drill string. Shear rates in the drill pipe typically range from 500 to 1,500 reciprocal seconds depending on flow rate and pipe ID, where all three models give similar results. In the annulus, shear rates often fall between 20 and 200 reciprocal seconds, and at times approach zero near casing walls. In this zone the Bingham Plastic model overpredicts apparent viscosity because it assumes linear behavior all the way to the yield stress, while the Power Law underpredicts it near zero shear because the curve passes through the origin. The Herschel-Bulkley model matches measured pressure losses in the annulus with significantly better accuracy and is the preferred model in advanced hydraulics software packages used for deepwater, extended-reach, and HPHT well design. The practical consequence is a more reliable ECD prediction that allows engineers to design mud weight programs with tighter margins in narrow-window wells.

Cuttings Transport and Hole Cleaning

Hole cleaning depends on the annular velocity and the carrying capacity of the fluid, which is controlled by the apparent viscosity at the low shear rates prevailing in the annulus and during slow rotational pipe movement. The Herschel-Bulkley yield stress term is particularly important for cuttings suspension during connections, when circulation stops and gravity acts on cuttings that are not yet at surface. A fluid with an adequate yield stress (typically 10 to 20 lb/100 ft2 or higher for deviated wells) holds cuttings in suspension; one without a yield stress allows immediate settling. Cuttings transport models based on the Herschel-Bulkley equation predict settling velocities more accurately than Bingham or Power Law models and allow engineers to specify minimum flow rates and rotational speeds for acceptable hole cleaning in high-angle and horizontal wells.

Limitations and Alternative Models

Despite its advantages, the Herschel-Bulkley model has practical limitations. It requires at least three viscometer readings at well-separated shear rates to fit reliably; poorly distributed data or noisy low-shear readings produce large uncertainties in tau_y and n that can make the model less accurate than a simpler two-parameter fit. The model also does not capture time-dependent behavior such as thixotropy (gel strength development at rest) or viscoelasticity, which affect surge and swab pressures during pipe movement. For these phenomena, more complex constitutive models or direct measurement of gel strength profiles is required. The Robertson-Stiff model is an alternative three-parameter model sometimes preferred because its parameters can be determined analytically from three shear-rate measurements without iterative regression, which simplifies field application. Nonetheless, the Herschel-Bulkley model remains the industry standard for drilling fluid hydraulics in technically demanding wells.

Key Takeaways

The Herschel-Bulkley model is a three-parameter equation (tau = tau_y + K (gamma_dot)^n) that combines a yield stress with power-law shear-thinning to describe drilling fluid rheology across the full shear rate range.
It reduces to the Bingham Plastic when n equals 1 and to the Power Law when tau_y equals zero, making it a general framework that subsumes both simpler models.
The model produces more accurate ECD predictions than Bingham Plastic or Power Law models, particularly in the low-shear-rate annular flow regime critical for narrow-window well design.
The yield stress term governs cuttings suspension during static periods and is essential for predicting hole cleaning performance in deviated and horizontal wells.
Parameters are determined by nonlinear least-squares regression against multi-speed viscometer data; HPHT viscometer measurements extend this across the downhole temperature-pressure profile.
The model does not capture time-dependent gel strength development or viscoelasticity, which must be addressed separately for surge, swab, and equivalent circulating density transients.