The Effects of Fluid Rheology and Drillstring Eccentricity on Drilling Hydraulics

Accurate determination of hydraulic parameters such as pressure losses, equivalent circulation density (ECD), etc. plays profound roles in drilling, cement-ing and other well operations. Hydraulics characterization requires that all factors are considered as the neglect of any could become potential sources of errors that would be detrimental to the overall well operation. Drilling Hydraulics has been extensively treated in the literature. However, these works almost entirely rely on the assumption that the drill string lies perfectly at the center of the annulus—the so-called “concentric annulus”. In reality, concentricity is almost never achieved even when centralizers are used. This is because of high well inclination angles and different string geometries. Thus, eccentricity exists in practical oil and gas wells especially horizontal and extended reach wells (ERWs) and must be accounted for. The prevalence of drillstring (DS) eccentricity in the annulus calls for a re-evaluation of existing hydraulic models. This study evaluates the effect of drilling fluid rheology types and DS eccentricity on the entire drilling hydraulics. Three non-Newtonian fluid models were analyzed, viz: Herschel Bulkley, power law and Bingham plastic models. From the results, it was observed that while power law and Bingham plastic models gave the upper and lower hydraulic values, Herschel Bulkley fluid model gave annular pressure loss (APL) and ECD values that fall between the upper and lower values and provide a better fit to the hydraulic data than power law and Bingham plastic fluids. Furthermore, analysis of annular eccentricity reveals that APLs and ECD decrease with an increase in DS eccentricity. Pressure loss reduction of more than 50% was predicted for the fully eccentric case for Herschel Bulkley fluids. Thus, DS eccentricity must be fully considered during well planning and hydraulics designs.


Introduction
Hydraulics plays vital roles in well operations such as drilling, cementing, completion, and well stimulation. Increasing well depth and complexity in geometry such as horizontal or extended reach wells (ERWs) gives rise to more complicated hydraulics than would be encountered in shallower slightly deviated or vertical wells. Proper study of hydraulics in a well is crucial as it translates to a reduction of risk, improvement in efficiency, decrease in the overall cost of well operation and reduction in non-productive time (NPT). Of all hydraulics encountered in well operations, drilling hydraulics is perhaps the most important because it covers more aspects in downhole operations than the others [1]. The aspects encountered in drilling hydraulics are downhole circulating pressures, surge and swab, equivalent circulation density (ECD), bit optimization, hole cleaning and volumetric displacement. Three basic factors affect drilling hydraulics in oil wells. These are the fluid, the hole and the drillstring (DS). Fluid could be Newtonian or non-Newtonian. Drilling fluids (DFs) are subjected to forces, downhole and they shear differently according to their rheological behaviour.
Fluid rheology affects the character and deformation of DFs. Fluid rheology typically depends on the fluid type (whether Newtonian or non-Newtonian), flow regime (laminar or turbulent) and particle size distribution in the fluid (water-based or oil-based, inhibitive or non-inhibitive) [2] [3]. Fluid velocity, density, viscosity, size and shape of flow channel determine if the flow regime is laminar or turbulent or in a transition zone [4]. Fluid rheology helps to characterize fluid flow which helps in the determination of friction factors and frictional pressure losses (FPLs). Accurate knowledge of the FPL in the well helps to check against formation damage or fluid influx into the well. Oil well DFs are very sensitive to pressure and temperature conditions downhole, this makes fluid rheology an important factor in drilling hydraulics [5] [6]. Hole geometry affects the movement of fluids and rock particles in the well. In vertical wells, the influence of cuttings on drilling hydraulics is not profound. Cuttings deposition increases when wells deviate from vertical and become highest in horizontal wells [7] [8].
Cuttings deposit at the lower side of the well and require additional energy to displace them thereby increasing the friction factors and frictional pressures.
Additionally, the hole condition also affects drilling hydraulics. If the hole is cased, then frictional pressure is not expected to be as high as when the well is open-hole. Open holes are noted with the presence of micro-doglegs and micro-tortuosities which increase the roughness of the hole and its frictional pressure [9]. The orientation of the pipe in the hole affects the flow pattern and hence its hydraulics in the well. Pipes in the hole can be concentric or eccentric depending on their placement in the hole. The pipe is said to be concentric when its axis coincides with the axis of the outer pipe or hole; otherwise, the pipe is said to be eccentric. Concentric pipes are usually modeled with less difficulty, but complexity in modeling increases as eccentricity increases [10]. Eccentric pipes in hole are usually the situation most encountered in reality. Eccentricity [24] went further on the work of [21]. They used seven different definitions of equivalent diameter in the estimation of PL in eccentric annuli. They discovered the hydraulic diameter definition to be most essential in their study. [25] used CFD method to compute FPLs in eccentric annuli. They investigated the effects of flowrate, mud type and mesh size. They solved the problem using finite volume method. [26] analyzed the effect of eccentricity on PL and velocity profile. [27] noted that eccentricity effect in laminar flow is more profound than in tur-bulent flow. [

Pipe Eccentricity in Wellbore
In reality, concentricity is almost never achieved as pipes tend to deviate from its center to one side of the wellbore. Sometimes centralizers are used to achieve a near-concentric annuli, this is the case with casing centralizers that are used to keep casings from contacting the wellbore. But even with the use of centralizers, the casing between centralizers could still deform leading to contact with the wellbore [28]. Centralization describes how off-centered a pipe is within another pipe or an Openhole. It is mainly expressed as a percentage. Thus a 0% eccentricity implies a concentric annulus while a 100% eccentricity implies that the pipes have contact with the outer pipe or wellbore. Centralization is important because it aids in casing wear studies and hole cleaning especially in the low side of the wellbore. In an eccentric annulus fluid preferentially flows through the wider annulus and thus there would be reduced velocity of fluid flow in the narrower annulus leading to build up of cutting beds due to gravity [7] [29]. As cuttings build up in the lower side of the annulus, hole cleaning becomes a problem and high tendency for pipe sticking exists. Research has shown that drill pipe eccentricity is affected by hole inclination angle, weight on bit and the size of the hole. Figure 1 depicts DS eccentricity in annulus. Figure 1. Drillstring eccentricity in annulus [29].
Well geometry and string stiffness plays profound part in annular eccentricity.
In deviated wells, the DP should be fully eccentric over much of the deviated wellbore. In medium inclined sections of the deviated well, such as between 0˚ -30˚, the drill strings tend to lie on the high side of the wellbore. Meanwhile in

A. Kerunwa et al. Open Journal of Yangtze Gas and Oil
high inclined sections or in horizontal wellbores, the DS lies on the lower side of the wellbore. Eccentricity will affect both the flow and the velocity distribution of fluids in the wellbore. It has been shown by research that the frictional pressure drop in an eccentric annulus is known to be less than the frictional pressure drop in a concentric annulus although this varies with fluid rheology type, the difference being much profound in Newtonian fluids than in non-Newtonian fluids [29]. Standoff is usually used to represent eccentricity expressed in per- centage. An eccentricity of 100% implies a standoff of 0% and means that the inner pipe is in contact with the outer pipe or hole at the low side. Meanwhile, an eccentricity of 0% implies a standoff of 100% and means that the inner pipe is perfectly centered in the outer pipe or wall; this is a concentric situation [7].

Equations for Eccentricity in Annulus
Eccentricity in the annulus is given by the equation: where: e = distance between the centers of inner and outer circle, inch ε = eccentricity ratio "the ratio of offset distance to radial clearance"; r o = inner radius of outer pipe, inch; r i = outer radius of inner pipe, inch.
A correlation was developed by Salem and El-Din (2006) to determine the w b = weight on bit, (Ib), E = modulus of elasticity, (psi), θ = hole inclination angle, (degree), I = moment of inertia of the drill pipe (inch), X = drill pipe horizontal projection, 1000 inches, q = axial component of weight of drillpipe per unit length, lb/ft. The pressure drop in eccentric annular flow is given as: where e C represents the correction factor for eccentricity.
( ) where: ε = drill pipe eccentricity, i o r r = welbore parameter o , , n k τ = fluid rheological parameters. Open Journal of Yangtze Gas and Oil

Fluid Rheology
The rheology of the fluid affects its eccentricity values and hydraulics in general. Fluid rheology affects fluid shearing with the walls of the annulus or pipe [28]. Newtonian fluids behave differently from non-Newtonian fluids [30].

Bingham Plastic Model
The model for Bingham Plastic is characteristically defined by equation (7)   25.8

Herschel Bulkley Fluid
The equation that defines the behaviour of Herschel Bulkley fluids is given as The yield stress for Herschel-Bulkley fluid is calculated at low shear rate and is given as The pressure loss in the annulus for Herschel Bulkley fluids is given as 25.8

Case Study
Well NDX2 is a well in the Niger Delta field and the well's drilling data are given below. The data was utilized for hydrauics simulation to determine the effect of

Results and Discussions
The results from the Wellplan T & D simulation are given below for Herschel Bulkley, power law and Bingham Plastic fluid rheology models at various pipe eccentricities.

Pressure Losses for Hershel Bulkley Fluid in Concentric Annulus
The overall pressure losses when Herschel Bulkley fluid model was utilized are Open Journal of Yangtze Gas and Oil presented in Figure 2 for a concentric annulus. The system pressure loss comprises all the pressure losses experienced in the system from the surface to the bit. It includes the surface pressure losses if there is any, the string pressure losses (DS i.e. drill pipe, drill collar, etc.), the APLs, and bit pressure losses. The pressure loss estimation helps to know the pump size and horse power requirement for efficient pumping of fluid into the wellbore and controlled hydraulics operations. From Figure 2, the system pressure loss line touches the maximum pump pressure line at 527 gpm pump rate. Thus to overcome the pressure losses prevalent in the system in pumping the fluid, the flowrate must be greater than 527 gpm. Generally, the pressure losses increase with increase in pump flowrate. The APL is the lowest; this is because of the wider annular space as compared to the bit or the DS. However, all the pressure losses in Figure 2 happen at the laminar flow regime.

Minimum Flowrate for Hershel Bulkley Fluid in Concentric Annulus
Presented in Figure 3 is the minimum flowrate as a depth function for Herschel Bulkley fluid in a concentric annulus. This helps to determine the minimum or critical pump flow rate at which cuttings bed will begin to form. To avoid cuttings bed formation, it is required to maintain a flowrate at each particular depth greater than the minimum flowrate. The variation in minimum flowrate is a function of depth and geometry of strings at that depth. Therefore, cuttings will only be avoided if the flowrate is greater than the minimum flowrate at that depth. Because of the variation of minimum flowrate at each depth, the pump rate should be greater than the greatest minimum flowrate for the well. From the Open Journal of Yangtze Gas and Oil figure, the greatest minimum flowrate occurs at the last casing shoe which is at 9280 ft and it corresponds to 398 gpm. Since ECD is the density of the DFs under dynamic conditions, ie when the pump is on and there is circulation. For the Hershel Bulkley fluid for concentric annulus, the ECD at casing shoe is 11.24 ppg and at the bit it is 11.36 ppg. Thus the ECD both at the shoe and at the bit is considerably higher than the static mud weight which is 10.7 ppg. Knowledge of ECD is therefore important in that it helps to avoid fracture of the formation during circulation because the mud appears to have greater weight.

Effect of Pipe Eccentricity on APL for Hershel Bulkley Fluid
Eccentricity affects APLs because of variations in annular flow area called the radial clearance. Higher drill string eccentricity entails higher flow area. Since pressure is indirectly proportional to area, conduits with greater area will have lesser pressure. Thus, it is expected that eccentric annuli will have less annular pressure drop than concentric annuli. This being the case, then APL should decrease with increasing drill string eccentricity and the lowest annular pressure drop should occur when the drill string touches the wellbore (fully eccentric case). Depicted in Figure 4 is the APL for Hershel Bulkley fluid at various pipe eccentricities. From Figure 4, it can be observed that the highest APL occurs when eccentricity is zero, i.e. when the DS is concentric in the annulus. The green line at the top most part of the plot represents the APL profile for concentric annulus. Conversely, the APL begins to decrease with increasing eccentricities from 0.1 to 1.0. At 1.0, the APL becomes lowest. The dark blue line at the bottom of the plot represents the annular pressure profile for Herschel Bulkley fluid for fully eccentric annulus. The result from Figure 4 show that APL de-Open Journal of Yangtze Gas and Oil creases with increasing DS eccentricities. The highest annular pressure drop was witnessed for a concentric annulus (e = 0). The lowest APL was seen at a fully eccentric annulus (e = 1.0). Cuttings bed formation due to gravity forces pulled cuttings to the lower sides of the hole. These cuttings bed proves difficult to clean and may result to severe well problem. [31] averred that these drilled cuttings and gelled DF are best removed through pipe rotation in narrow annulus. This is due to the fact that, in eccentric annulus, if there is no pipe rotation the DF will preferentially flow through wider annulus. [32] posited that pipe rotation creates rotational effect with an induced turbulence and this forces the drilled cuttings and gelled DF out of the narrow annulus. Figure 5 shows the percentage decrease to DS eccentricity in the annulus. It can be seen from the figure that at 200 gpm, there is up to 54% percentage decrease in APL due to full eccentricity from the initial concentric annulus. The lowest percentage decrease is seen at 800 gpm which is 19.2% decrease. It can be observed from Figure 5 also that decrease in APL due to increasing eccentricity gets smaller as flowrate increases. Thus, Eccentricity is important in estimation of the APL because more than 50% reduction is achievable to eccentricity of the annulus. Figure 5 shows that more than 50% reduction was realized for a change to fully eccentric annulus. The reason for the reduction in APL is because of the increase in flow area given by the increased radial clearance in the annulus. APL is inversely proportional to the radial clearance. The higher the annular radial clearance, the less the annular pressure drops.

Effect of Pipe Eccentricity on ECD for Hershel Bulkley Fluid
Depicted in Table 1 is the ECD for various eccentricities. From Table 1, it can be observed that the highest ECD both at the shoe and at the bit occurs when the eccentricity is zero (concentric annulus) and the lowest ECD both at the shoe and at the bit occurs when the eccentricity is 1.0 (Fully eccentric annulus). A closer look at the table reveals that ECD decreases with increase in DS eccentricity in the annulus.  ECD at shoe, ppg 11.24 11.2 11.18 11.16 11.13 11.11 11.09 11.07 11.05 11.02 11 ECD at bit, ppg 11.36 11.32 11.3 11.28 11.25 11.23 11.21 11.19 11.17 11.14 11.12

APLs for Various Fluid Rheology Types
Depicted in Figure

Minimum Flowrate for Various Rheology Types
Depicted in Figure 7 is the minimum flowrate as a depth function for concentric Open Journal of Yangtze Gas and Oil annulus for Herschel Bulkley, power law and Bingham Plastic fluids rheology types. From Figure 7, it can be observed that the Herschel Bulkley model gives more minimum annular flowrate than Bingham plastic and power law fluids. Thus, higher flowrate is required to prevent cuttings bed build up in Herschel Bulkley fluids (for pump rates of 250 gpm and above) than in Bingham plastic and power law fluids. Figure 7 shows the minimum flowrate. The minimum flowrate represent the lowest flowrate needed to prevent the buildup of cuttings bed in the wellbore. Analyses of Figure 7 reveal that higher flowrate is required for Hershel Bulkley fluids to avoid the formation of cuttings bed. Cuttings deposition increases the frictional pressure drop in the annulus and may lead to severe well problem such as stuck pipe which translates to huge financial involvement to remedy and requires pipie rotation to clean up as pointed out by [31]. When circulating Herschel Bulkley fluids, care must be taken to circulate at higher pump rates in order to avoid the formation of cuttings bed.

Conclusions
The following conclusion is drawn from this study: 1) Drill pipe eccentricity leads to reduction of APLs; 2) ECD decreases with increasing eccentricity; 3) Bingham plastic and power law fluids give the lowest and highest ECD values respectively; 4) Drill string eccentricity has a considerable effect on APLs while circulating Herschel Bulkley fluids. Pressure loss reduction of more than 50% was predicted for the fully eccentric case. Thus, DS eccentricity must be fully considered during drilling hydraulics planning and designs. The case is more severe in hori-  FPLs-Frictional pressure losses τ = shear stress, lb/ft 2 τ 0 = Yield stress in lb/ft 2 μ p = Plastic viscosity, cp n = flow behavior index τ y = Yield stress in lb/ft 2 γ = Shear rate K = Consistency index ∆P a = Pressure loss for the annulus interval, psi L a = Length of the annulus interval, ft f a = friction factor in the annulus (D 2 − D 1 ) = anuular space, in D 2 = hole diameter or casing internal diameter D 1 = Pipe or drill collar inside diameter, in