The Herschel-Bulkley method is a two-term non-linear approach to mud and debris rheology. This method raises the strain to a user-selected power, which can be greater or less than 1. Unlike the Bingham method that raises strain to the power of 1 or O'Brien that uses a quadratic (raising strain to the powers of 1 and 2) Herschel-Bulkley can raise strain to non-integer powers greater or less than 1. This flexibility allows users to define a range of non-linear stress-strain relationships including shear-thickening and shear-thinning rheologies. A shear-thinning mixture becomes easier to deform under higher stresses. A shear-thinning viscosity decreases as stress increases. As shear stress increases, the rate of strain increases non-linearly, so each increment of additional stress causes more strain than the previous increment. Increased viscosity at higher shear stresses essentially means that the slope of the stress-strain relationship increases with stress, which can be confusing with plots like the figure above (or most of the rheological plots in this document with strain on the x-axis). Because depth and velocity are model results, and DebrisLib uses them to compute an internal stress, the numerical model considers strain the independent variable and stress the dependent variable. But stress is the independent variable in physical deformation, so shear thinning and thickening responses are inverted in these plots (e.g. the slope of the strain-stress curves decrease at higher strains for shear thinning). The "shear-thinning" terminology illustrates this relationship. As shear increases, the material "thins" or becomes easier to strain. The Herschel-Bulkley model simulates shear thinning relationships by raising strain to a power less than one (n<1).
Shear-thickening materials get more viscous under higher stresses. Stress has a negative feedback on strain, making the material more difficult to deform. The Herschel-Bulkley model simulates shear-thickening by raising strain to a power greater than 1 (n>1). The O'Brien Quadratic is a de-facto shear thickening model because it includes squared strain terms (n=2 n>1).
Setting the power of Herschel-Bulkley to one (n=1) collapses the model to the Bingham approach, because a linear stress-strain relationship with a yield stress is the definition of a Bingham Plastic. The Herschel-Bulkley model requires three parameters:
The Yield Stress in Herschel-Bulkley is the same as the previous methods, and can be computed with the same options. But the linear parameter in front of the Strain term is loses its viscosity units if strain is raised to a power other than 1. Therefore, K is no-longer viscosity when Herschel-Bulkley diverges from the Bingham model (n≠1). Both K and n are empirical user parameters.
The figure below includes screen shots of shear-thickening and shear-thinning simulations (Gibson et al., in revision) of Parsons et al's (2000) experiments that displayed these processes.
