Limiting the Quemada Parameters

The Quemada model is a complex model with several parameters requiring a fitting process to find the optimal solution for modelling the fluids viscous behaviours. When fitting the Quemada model to the experimental data, these parameters must be defined by upper and lower boundaries through accurate sample preparation and measurements.

The characteristic time is defined by the characteristic shear rate (𝑡_𝑐 = 𝛾̇_𝑐⁻¹). Hodne et al. (2007) recognised the characteristic time tc, as defined by Quemada (1998) to be the time needed for obtaining a suspension of almost mono-disperse SUs. The characteristic shear rate, compared to the shear rate range applied by the viscometer, describe the rheological behaviour of the selected fluids. Pseudo-plastic behaviour requires a characteristic shear rate within the range of applied shear rates (Quemada, 1978). Furthermore, the characteristic time can be given by 𝑡_𝑐 ≈ 𝜂_𝑒𝑓𝑓𝑎³⁄𝑊_𝐼 (Baldino et al., 2018; Quemada, 1998), where tc depends on the effective viscosity ηeff, particle radius a, and particle interaction energy WI. This equation explains how the characteristic time will decrease with increased temperature, due to the increase in WI and decrease in ηeff (Baldino et al., 2018).

Infinite- and zero-shear viscosity was described from the viscosity curve in Fig. 4 (Chapter 2.1.3). These parameters can be estimated indirectly through curve fitting, directly measured, or be estimated through the volume fraction parameters described by Quemada (1998). However, attempts to directly measure these values are not always possible due to the limitation on the measuring equipment (Baldino et al., 2018). When using the method of determining the infinite- and zero-shear viscosity by Eq. (6) and Eq. (7) in Chapter 2.2.2, limiting the boundaries and estimating values of the parameters in the equations are crucial when fitting the model.

The solid volume fraction 𝜙 is an important factor of the viscous behaviour. It is a measure of the fraction of particles suspended in a fluid compared to the total volume of the

suspension. It is important to define it by the volume, and not the weight, due to the rheology’s dependence on the forces acting on the particle surface and not on the particle’s density. The viscosity is heavily dependent on the 𝜙. Increasing the concentration of particles results in increased flow resistance, due to the particles being in each other’s way, i.e. increased internal friction (Barnes et al., 1989). This volume faction can be estimated from the weight of the materials used in the selected drilling fluids and the known/measured material/suspension densities, yielding equation:

𝜙 = 𝑉𝑆𝑢𝑠𝑝𝑒𝑛𝑑𝑒𝑑 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒𝑠

𝑉_{𝑇𝑜𝑡𝑎𝑙} =𝑉_{𝑇𝑜𝑡𝑎𝑙}− 𝑉𝑆𝑢𝑠𝑝𝑒𝑛𝑑𝑖𝑛𝑔 𝑓𝑙𝑢𝑖𝑑

𝑉_{𝑇𝑜𝑡𝑎𝑙} (18)

The maximum packing fraction 𝜙_𝑚 is defined as the solid volume fraction needed for flow to stop i.e. infinite viscosity. This happens when a solid structure, continuous throughout the suspension, is formed by the dispersed particles (Barnes et al., 1989; De Visscher &

Vanelstraete, 2004). The 𝜙_𝑚 of a suspension depends on several factors. One factor is the particle shape, where spherical particles having better space-filling properties than non-spherical particles. Another is the particle size distribution, where large distribution of particle sizes in a dispersion lead to the pores between the larger particles being filled by smaller particles. Lastly, including the arrangement of packing, where monodisperse spheres range from simple cubic (𝜙_𝑚 = 0.52) to hexagonal close packed (𝜙_𝑚 = 0.74) (Barnes et al., 1989).

3 Previous Work

The structural model proposed by Quemada (1998) is not normally used for modelling drilling and well fluids. The oil industry prefers simple models like Bingham, Power Law, and Herschel-Bulkley. However, there has been conducted some studies on the Quemada model in relation to drilling and well fluids.

Hodne et al. (2007) used this model for cementitious materials to predict the rheological behaviour of the fluid outside the range of measurement. The method utilised by Hodne et al.

(2007), consisted of viscosity measurements and a thorough sample preparation to measure the particle packing fractions and the solid volume fractions. These measurements were necessary to limit the upper and lower boundaries of the six parameters used in the curve fitting, 𝑝, 𝑡_𝑐, 𝜙, 𝜙_𝑚, 𝜙_𝐴0, 𝜙_𝐴∞.

The exponent p was limited to 0 < p < 1 according to Quemada (1998). The applied range of shear rates were 0.05-511 s^-1. However, the characteristic time (𝑡_𝑐 = 𝛾̇⁻¹) was limited to the shear rate range of 3.1-511 s^-1. Cementitious mixtures often contain air, resulting in air bubbles acting as particles, increasing the solid volume fraction, and affecting the experiment.

Therefore, the limits of the solid volume fraction 𝜙 depended on the measured air content of the samples. For the maximum packing fraction 𝜙_𝑚, Hodne et al. (2007) were able to determine the lower limit packing fractions through a packing experiment. The maximum packing fraction for face centred packing of mono-disperse spheres, at 0.74 (Barnes et al., 1989), were set as the upper limit. The last two parameters, the limiting aggregated volume fractions, were limited to 𝜙_𝐴∞ ≤ 𝜙_𝐴0 ≤ 𝜙 according to Quemada (1998) (Hodne et al., 2007).

Hodne et al. (2007) found two optimal solutions for each of their samples, where one was optimal for high shear rates (HSRs) and the other for low shear rates (LSRs). The two different curves showed two different behaviours, denoted by J- and S-curve. The J-curve only indicates the existence for infinite-shear viscosity and fits the LSR data. While the S-curve indicates both the zero- and infinite-shear viscosity and fits the HSR data. Hodne et al. (2007) concluded that the Quemada model (Quemada, 1998) could be used to predict the rheological behaviour of the cementitious materials. However, care had to be taken when choosing a solution to study a phenomenon at HSRs or LSRs.

Baldino et al. (2018) used a different method to fit the model to describe the behaviour of a synthetic based drilling fluid (SBF). Instead of looking at the volume fractions of the fluid, they tried to measure the infinite-shear viscosity (𝜂_∞) and the zero-shear viscosity (𝜂₀) by using

evaluating the zero-shear viscosity and were expecting a larger value. To correct for this, they introduced the correction parameter α into the model and defined the viscosity by:

𝜂 = 𝜂_∞[ 1 + 𝛤^𝑝 (𝜂_∞⁄α𝜂₀)^0.5+ 𝛤^𝑝]

2

(19) The only parameters requiring fitting were 𝑡_𝑐, α, and 𝑝. The correction parameter 𝛼 had to be greater than 1 to give a larger value of 𝜂₀ (Baldino et al., 2018), and the exponent 𝑝 was treated the same as Quemada (1998) and Hodne et al. (2007). Unlike Hodne et al. (2007), they used the applied range of shear rates, 0.15 𝑠⁻¹≤ 𝛾̇ ≤ 1000 𝑠⁻¹, to define the limits of the characteristic time.

The method applied by Baldino et al. (2018) resulted in a J-curve accurately describing the rheological behaviour at LSRs. However, like Hodne et al. (2007), the resulting J-curves introduced large discrepancies between the model and measured values at HSRs.

This thesis presents another modelling approach compared to Hodne et al. (2007) and Baldino et al. (2018). Instead of trying to limit the limiting viscosities, though the volume fraction like Hodne et al. (2007) or trying to measure the limiting viscosities like Baldino et al.

(2018), the limiting viscosities were treated as empirical curve fitting parameters. The resulting optimal fit and the calculated solid volume fraction were then used to calculate the structural index and the limiting maximum volume fractions. These values were used to verify the rheological behaviour of the selected drilling fluids. Further explanation of this approach to the Quemada model can be found in Chapter 4.3.2.

4 Methodology

Four different drilling fluid recipes, both oil and water based, were prepared in the lab.

Two comparable batches of each recipe were made to ensure the integrity of the experiment and to reduce the uncertainties one fluid sample would present, i.e. a total of eight samples were prepared and measured. Sample preparation, equipment, and method for measuring is presented in chapters 3.1 and 3.2, followed by model implantation methods in chapter 3.3.

4.1 Materials