Hydraulic model

The hydraulic system parameters are determined to for hydraulic optimizations. The main objected is to provide an appropriate nozzle jet impact. However there are other factors which are involved for selection of these parameters. As mentioned earlier critical flow velocity for cutting transport and borehole cleaning are those factors. Due to high variation in hole and tools and pipes dimension the flow regimes are different which also must be taken into consideration.

As mentioned earlier there are both annular frictional pressure and static pressure are factors affecting the downhole pressure. By calculating the ECD one can control the downhole pressure more accurately. ECD is a function of frictional pressure and static pressure. Static pressure is controlled by changing the density of the drilling mud. The frictional pressure is however more complex. The frictional pressure in annulus is affected by following factors:

 Rheological behavior of the drilling fluid

 The flow regime of drilling fluid

 The drilling fluids density

 Drilling string eccentricity

 The flow rate of drilling fluid

In order to overcome the frictional pressure drops in different section of the wellbore, the pump pressure has to deal with following pressure drops:

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 Pressure drops thorough the surface equipment like swivel and pipes ∆𝑃_𝑠

 Pressure drops across the drilling string∆𝑃_𝑑𝑠 and drilling collar ∆𝑃_𝑑𝑐

 Pressure drops across the nozzles of drilling bit∆𝑃_𝑏

 Annular pressure drops outside the drilling collar∆𝑃_𝑎𝑐

 Annular pressure drops outside the drilling string and riser ∆𝑃_𝑎𝑑𝑠

Figure 21 Hydraulic system and pressure drops [modified]

The total pressure drop for the hydraulic system is the sum of the mentioned pressure drops.

∆𝑃_{𝑇𝑜𝑡𝑎𝑙} = ∆𝑃_𝑠+ ∆𝑃_𝑑𝑐+ ∆𝑃_𝑑𝑠+ ∆𝑃_𝑏+ ∆𝑃_𝑎𝑐 + ∆𝑃_𝑎𝑑𝑠 (25)

MSc Thesis, 2015

34 Other pressure drops are not directly related to drilling operation. They are more related to borehole cleaning and pump pressure. Aadnøy (2010) divided these pressures drops into two groups. The pressure drop across the nozzle and other pressure drops known as parasitic pressure drops. The values of parasitic pressure are directly related to the flow regimes. The pressure drop across bit nozzle is not negligible and has an impact on drilling operation.

According to Kendal and Goins, for having maximum jet impact force, the bit pressure drop must be close to 49 % of the pump pressure. Another optimization is to provide the maximum bit hydraulic horsepower. In this case the pressure drop across the bit must be kept as high as 66 % of the pump pressure. The fraction must be kept until the target depth is reached or the flow rate is down to minimum annular velocity for cutting transport. [3] The pressure drop across the bit nozzle (∆𝑃_𝑏) assuming field units, is usually calculated by equation (26):

∆𝑃_𝑏 = 𝜌𝑞²

12034.7𝐴²𝐶_𝑑² (26)

Where 𝑞 is the volumetric flow rate across the bit nozzles (GPM), 𝜌 is the density of the drilling fluid (ppg), 𝐴 is the sum area of the bit nozzles (in²) and 𝐶_𝑑 is the bit discharge coefficient which is normally set equal to 0.95. The parasitic pressure drop in annulus and drill pipe is around 10-20 % of the total pressure drop of the system. For calculating annular pressure drop one has to use a rheological model. The Bingham plastic and power law model are best fitted for the mud behavior for pressure drop prediction. The obtained annular pressure drop obtained is normally less accurate due to simplifying assumption. Another reason is the complex behavior of mud as a Non-Newtonian fluid. Another factor which affects the annular pressure drop is the annular wellbore geometry.[5] It is well known that the pressure drop in concentric annular flow is significantly higher than eccentric annular flow. According to Haciislamoglu and Cartalos the pressure drop in fully eccentric pipe can be as low as 40% less than a fully concentric drilling pipe. Eccentricity is not a parameter which can be easily controlled. It is a function of depth and bore inclination.[6] Variation in cross sectional area in the inclined section due to cutting beds which occupy part the area, prediction of pressure drop becomes more challenging. A flow profile for a non-Newtonian in an eccentric drill pipe is shown in figure 22.

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Figure 22 Velocity profile of non-Newtonian fluid in an eccentric drill pipe

The unified rheology model is given as [41]:

𝜏 = 𝐾𝛾̇^𝑛+ 𝜏_𝑦 (27)

Where, the shear yield (𝜏_𝑦 ), k and n values are calculated directly from Fann rheology data. Table 1 shows unified hydraulic model, which is used to analyze the best fluid system to be formulated in experimental part of this thesis.

Table 1 Unified hydraulic model [41]

Pipe Flow Annular Flow

μ_p = R₆₀₀− R₃₀₀ τ_y = R₃₀₀− μ_p τ₀ = 1.066(2R₃− R₆)

μ_p = cp

np = 3.32 log (^2μ_μ^{p + τy}

p + τ_y) k_p =1.066 (^μp ₅₁₁ ^{+ τy})

np = 3.32 log (^2μ_μ^{p + τy – τy}

p + τ_y−τ_y ) kp =1.066 (^{μp + τ}₅₁₁ ^{y− τo} )

G=(⁽³⁻₍₄₋^Α)N+1

Α)N ) (1 +^Α₂) Α=1 FOR ANNULL

α= 1 for pipe v_p =24.51 q

D_P² v_a = 24.51 q

D₂²− D₁² v=ft/min

MSc Thesis, 2015

Solid particle follow the hooks law and there is proportionality between the shear rate and share stress. The slope of the stress strain is called rigidity modulus G, indicating that the material is rigid.[40]

𝜏 = 𝐺𝛾 (28)

Viscosity is the same constant that relates shear and shear rate for Newtonian fluids. They follow Newton’s law. However some materials have the characteristics of both liquid and solids meaning that are both elastic and viscous. Polymer such as Xanthan is an example of such viscoelastic materials. Polymers that are used in drilling mud give the mud the viscoelastic properties. The definition of viscoelasticity can easily be misunderstood by plasticity. Plastic materials like metals are ductile. Ductility is as same as plasticity.