Annular frictional pressure losses for drilling fluids

In the paper it is demonstrated how a Herschel-Bulkley fluid model, where the parameters are selected from relevant shear rate range of the flow and are parametrically independent, can be used for pressure loss calculations. The model is found to provide adequate pressure loss predictions for axial flow in an annulus where the inner cylinder does not rotate. It is described how one can simplify a slot model approximation of the annulus pressure loss using the Herschel-Bulkley fluid model (Founargiotakis model). This simplified model gives approximately the same accuracy as does the full Founargiotakis model. It is shown that use of such a parallel plate model gives reasonably good fit to measured data on laminar flow of oil-based drilling fluids if


INTRODUCTION Annular friction loss
The topic of annular frictional pressure loss modelling has been a challenge in drilling engineering. To be able to select drilling fluids with proper chemical composition to avoid un-wanted interaction with the formation, it is necessary to predict the annular frictional pressure losses with a reasonable accuracy. It is important that the down hole pressures never exceed the fracturing pressure or enter pressure areas vulnerable for hole instability issues. The accessible pressure window is estimated from geological analyses and are modified by a series of formation strength tests [1] as the drilling proceeds.
The Herschel-Bulkley model includes the Bingham and power-law models by selection of their parameters. The appearance of a non-zero yield stress makes all annular calculations complicated. A non-yielded region with constant velocity and two yield stress boundaries will be introduced, leading to a complex set of equations. Often, an iterative procedure is selected to be able to calculate the flow equations. Therefore, to simplify the calculations, it is common to use the power-law model for the fluid flow calculations. Here the parameters can be selected to be valid for a realistic range of shear rates. Models for annular frictional pressure losses have been treated in many drilling textbooks like for example by Ahmed and Miska [2] and Bird et al. [3]. Very seldom the models are coupled with the selection of viscosity data from measurements at the relevant shear rates. Ytrehus et al. [4] showed the importance of selecting shear stress data from the relevant shear rates in building the viscosity model for pressure loss calculations as shown in Figure 1 for a series of experiments simulating flow inside a cased hole. By selecting the shear stress data from shear rates less than 300 1/s, the accuracy of prediction was improved [4,5]. In this case the pressure loss prediction was obtained using the Founargiotakis model [6].

Effect of drillstring rotation
It is earlier shown how rotation of the inner string in an annulus can complicate the flow for example by establishment of Taylor vortices in full scale experiments [7], field applications [8] and theoretically [9]. There are currently no analytical methods to handle such flow. The effect of the vortices depends strongly on the fluid's composition in addition to the flow conditions. The practical way to handle these situations in offshore drilling is by "fingerprinting" during circulation, meaning comparing pressure losses directly to the pressure losses of a previously conducted similar operation at defined rotation rates.

ANNULAR FRICTION LOSS PREDICTIONS Simplified laminar flow model
Founargiotakis [6] presented a model for laminar, transitional and turbulent flow of Herschel-Bulkley fluids in concentric annuli, using the slot approximation.
For laminar flow the Founargiotakis model [6] is simply a determination of the shear rate at the actual flow rate given the Herschel-Bulkley fluid model. This shear rate is used for determination of a consistency parameter and a generalized flow index for an equivalent power-law model giving the equal shear stress at that particular shear rate.
Thereafter, this consistency parameter and the associated flow index is used to calculate the pressure loss of a unidirectional flow between two parallel plates, as these two plates are applied to approximate the annulus.
The hydraulic model of Founargiotakis et al. [6] is here applied to axial flow of Herschel-Bulkley fluids in an eccentric annulus. First, we rewrite the constitutive equation for the Herschel-Bulkley model using dimensionless shear rates [5] as shown in Equation 1. (1) where s=-y at the shear rate = .
Following Founargiotakis et al. [6], and by using the traditional terminology for (2) where the relative importance of the yield stress, , is defined using Equation 3 and dp/dL is the frictional pressure gradient. For simplicity in the equations, the inverse of the Herschel-Bulkley curvature index, m, is also used: m=1/n.
In the slot approximation, the wall shear stress is related to the pressure loss using . Then Equation 5 is re-written as Equation 6.
This equation is solved with respect to wall shear stress for any given velocity U. The troublesome part is that the relative importance of the yield stress, , is dependent on the wall shear stress, w making the equation implicit in w.
A possible simplification of this model is to make an estimation of the relative importance of the yield stress. The sum of the yield stress and the surplus stress will approximate the wall shear stress if the surplus stress is selected in a relevant shear rate region for the flow case. Hence, we approximate the wall shear stress as shown in Equation 7.  are made using the same measurement data set, these curves must be equal also at another shear rate. It is possible to select this other shear rate by adjustment of the index n'. This is obtained by requiring that (11) where is the dimensionless shear rate at which the value of n' shall be determined.
The yield stress will be a main contributor to the shear thinning of the Power-Law function. Hence, n' will depend on the yield stress.
To calculate the turbulent frictional pressure loss, the friction factor concept will be used. The friction factor is defined in Equation 12. This friction factor depends on the Reynolds number. The Reynolds number is defined as shown in Equation 13.
In the Reynolds number an effective, or apparent viscosity, e, is used. This viscosity can only be approximated. A practical way to approximate this is by using the definition of viscosity at the wall assuming a laminar flow as shown in Equation 14.
where the wall stress is presented using Equation 7 and the wall shear rate is given as presented in Equation 15. This equation can be found in most textbooks on flow of complex fluids, like for example Guillot [10].
By solving Equation 16, it is now possible to calculate the friction factor, fturb. (16) The turbulent frictional pressure loss can now be calculated using Equation 12, valid for smooth pipe. There is however, a fairly large transition zone from laminar to fully turbulent flow. It is normal to assume two boundaries as shown in Equation 17.
These two boundaries are dependent on the shear thinning characteristics of the drilling fluid: where and [11]. The friction factor is calculated as 24/ReMRa for ReMRa < Re1, and for ReMRa > Re2, the the friction factor from Equation 16 is used in Equation 12. In the transition zone between Re1 and Re2 a linear interpolation between laminar and turbulent friction factors is often used:

Corrections for eccentricity
Haciislamoglu [12] and Haciislamoglu and Cartalos [13] presented models for pressure loss corrections for eccentric annuli in laminar and turbulent flow. These models are presented in Equations 17 and 18 for laminar flow and turbulent flow, respectively. In their work they used the Power-Law consistency factor, K. This is replaced by , using the method by Saasen and Ytrehus [14,5] to present the Herschel-Bulkley model with parametrically independent fluid parameters, obtained using dimensionless shear rates. (20)

PREDICTIONS COMPARED WITH EXPERIMENTS
The shear rate where the surplus stress is measured should be at a large relevant shear rate for the flow problem. Typical shear rates are tabulated by for example Guillot [10].  Three different field applied oil-based drilling fluids were used in a set of experiments where the scope was to compare the different models with experimental data. OBM A, B and C are labelled such that the shear stress and thus the viscosity of OBM A is less than those of OBM B, which again is less than those of OBM C at all the measured shear rates up to 300 1/s. The resulting viscosity data is shown in Table 1 both for a = 0.5 and a = 0.75. The frictional pressure losses as function of the flow rate for these fluids are shown in Figures 3, 4 and 5 for the OBMs A, B and C, respectively.
where fN,r and fN,s are the friction factors for Newtonian fluid for rough and smooth wall, respectively. Here we represent fN,r using the Haaland friction factor [15] 1.11 10 1 6.9 3.6 log Re where k is the absolute wall roughness.
All the smooth pipe results give nearly equal predictions. A roughness of k=200 micron gives an increased predicted pressure loss. Still, this predicted pressure loss is far from the measured values, especially at higher flow velocities. Now, the junctions between the inserts will introduce some irregularities each 20cm. These irregularities could perhaps introduce a local roughness around one millimetre or so. Therefore, a prediction with 1mm roughness was calculated. This calculation seems to predict the pressure losses with reasonable accuracy. Hence, it is possible to conclude that use of simplified viscosity models gives similar results as the use of full Founargiotakis model.  give any significant difference. Although not of importance, the results from selecting n at 75% of the shear rate of the surplus stress, seems to be marginally better than if n was selected at 50% of that shear rate.
For laminar flow, it may be possible to use Equation 7 directly to calculate the laminar frictional pressure loss. If this is used for OBM-A, a better correlation with the measurements is obtained than if the prediction is conducted with any of the other models. For the other two series of measurements, this correlation is not that good.
For OBM-C it is 10-15% overestimation and for OBM-B it is 20-25% overestimation if only Equation 7 is used to predict the pressure losses. OBM-A is only slightly shear thinning while both OBMS B and C are very shear thinning with OBM-B as the most shear thinning. This indicates that the degree of shear thinning complicates the pressure loss predictions. Also, the yield stress of OBM A is smaller than that of the other two fluids.
The parallel plate slot model with pre-set values seem to be only slightly less accurate than the more complicated models with multi-steps to predict laminar flow.
Hence, this model may be more useful for digitalization than the other models.

CONCLUSION
An analysis of annular frictional pressure loss calculations with simplified models has been conducted. A pressure drop model based on parametrically independent fluid parameters is well suited to improve applications in the digitalization development currently prioritized in the drilling industry.
• Use of a parallel plate model gives reasonably good fit to measured data on laminar flow of oil-based drilling fluids if the viscous data are measured at relevant shear rates for the flow.
• Use of pressure drop estimation solely based on shear stress predicted for the anticipated shear rate of the flow will generally overestimate the predictions.
• Strong indications for an increased difficulty of predicting pressure losses if the degree of shear thinning is increased.
• Laboratory measurements indicate that use of the simplified pressure loss model is also valid for turbulent flow. However, the predictions should be adjusted for the surface roughness in the well.  Table Caption List   Table 1 Drilling fluid viscosity parameters measured using Anton Paar rheometer.
The surplus stress shear rate is 198 1/s. All stresses are in Pascals (Pa) Table 1 Model