Drillstring Pressure Losses

Drillstring Pressure Losses

All pressure losses, at first, assume a laminar flow regime. Power Law Model calculations begin with:

where: Plf = Pressure Loss in Laminar Flow (psi)

L = Length of Section (feet)

VP = Velocity in Section of drill string (ft/min)

d = Inside Diameter of drillstring (inches)

k = Consistency Index

n = Power Index

Fluid velocity in the drillstring can be determined by using:

where: Q = Pump flow rate (gpm)

d1 = pipe I.D. (inches)

The equivalent viscosity (m) is then determined, using:

Which, in turn, is used to determine the Reynolds Number.

Flow behavior, with the Power Law Model, will vary depending on the “n” value of the fluid. The critical Reynolds Number (Rec) is found using: 3470 - 1370n (from laminar to transitional)

4270 - 1370n (from transitional to turbulent)

If: Re < Rec flow is laminar

Re is between laminar and turbulent, flow is transitional

Re > Rec flow is turbulent

If the flow is determined to be turbulent, the pressure losses will have to be re-calculated using turbulent flow. This will also require a friction factor (f) to be included:

Drilling Fluids- Pressure Loss in the Drillstring

Pressure Loss in the Drillstring

Once passed the surface equipment, the fluid will flow through the drillstring. In hydraulic calculations, these parts of the circulating system are considered circular pipes. In typical field operations, fluid velocities are in the order of 1000 ft/min (300 m/min). At such velocities, the fluid is in turbulent flow.

The pressure required to circulate fluid in turbulent flow varies by approximately 1.8 power of the flowrate. Doubling the flowrate would increase the pressure drop in the drillstring by approximately 3.5 times. Typically, the pressure losses in the drillstring are about 35 percent of the total pump pressure. 

With this in mind, it will be necessary to know how much pressure will be required to pump the fluid through the drillstring, at a given rate.

Drilling fluid - Surface Pressure losses

Surface Pressure Losses

System pressure loss calculations begin with the determination of the type/ class of surface circulating equipment. These include the standpipe, rotary hose, swivel, and kelly (if present). Though hardly ever consistent, four  types/classes have been recognized by the IADC as the most common.

They are:

When calculating surface pressure losses, choose the class which is closest to the present rig equipment; if necessary, extrapolate. Most modern rigs will have a surface pressure coefficient between 2 and 10. The coefficient is then used in the following formula:

Pls = 10^-5 x ks x MD x Q^1.86

where: Pls = Surface Pressure Loss (psi)

ks = Surface Pressure Coefficient

MD = Mud Density (lb/gal)

Q = Flow Rate (gal/min)

When extrapolating, bear in mind that increased lengths will increase the coefficient, while increased I.D.'s will decrease the coefficient.

Drilling fluid - Hydraulic Calculations

Hydraulic Calculations

In the “Advanced Logging Procedures Workbook” (P/N 80269H), an introduction to hydraulics illustrates the Bingham method for hydraulic optimization. The second, and more commonly used method is the Power Law Model.

This model fits the actual flow properties more closely, although at low shear rates, it will predict slightly low shear stresses. The model describes a fluid in which the shear stress increases as a function of shear rate, raised to some power. As mentioned earlier, the equation for the Power Law model is:

Shear Stress = k x Shear raten

“k” is known as the “consistency index”, and is indicative of the pumpability of the fluid. “n” is the power index, denoting the degree of how “non-Newtonian” the fluid is. 

Both parameters can be determined from the Fann VG meter. “k” is defined as the viscosity of a fluid at a shear rate of 1 sec-1. When “n” equals 1, the fluid is Newtonian. As the fluid becomes more shear thinning, the “n” value decreases.

where: 300rpm = Fann VG meter dial reading at 300 rpm's

600rpm = Fann VG meter dial reading at 600 rpm's

If the Fann VG meter dial readings are not available, both “k” and “n” can be determined using the Plastic Viscosity and Yield Point.

where: PV = Plastic Viscosity (cps)

YP = Yield Point (lb/100ft2)

Once these values have been determined, they are used in calculating the pressure losses throughout the circulating system. This section will describe the pressure losses, using the Power Law Model, in the surface system, the drillstring, and the annulus.

Hydraulics - Deformation of a Fluid by Simple Shear - Power Law Model

Power Law Model

This model is defined by the relationship:

Shear Stress = Consistency Factor x Shear Rate flow behavior index

It describes the thickness or pumpability of the fluid, and is somewhat analogous to the apparent viscosity. The flow behavior index (n) indicates the degree of non-Newtonian characteristics of the fluid. As the fluid becomes more viscous, the consistency factors (k) increases; as a fluid becomes more shear thinning “n” decreases. When “n” is 1 the fluid is Newtonian. If “n” is greater than 1, the fluid is classed as Dilatant (the apparent viscosity increases as the shear rate increases). If “n” is between zero and 1 the fluid is classified as Pseudoplastic, exhibiting shear-thinning; (i.e., the apparent viscosity decreases as the shear rate increases). For drilling fluids, this is a desirable property and most drilling fluids are pseudoplastics.

While the Power Law Model is more accurate then the Bingham Model at low shear rates, it does not include a yield stress. This results in poor results at extremely low shear rates.

A modification to the Power Law Model, the OXY Model, was proposed for use in oil-based muds. The major difference is the viscometer readings used to determine the “k” and “n” values. Power Law uses the 300 and 600 rpm readings, the OXY Model uses the 6 and 100 rpm readings. In addition, other models have been proposed that tend to exhibit behavior between the Bingham and Power Law models at low shear rates.

Non-Newtonian fluids may show a degree of time-dependent behavior. (For example, the apparent viscosity for a fixed shear rate does not remain constant, but varies to some maximum or minimum with the duration of shear.) If the apparent viscosity decreases with flow time, the fluid is termed “Thixotropic”. Once flow has ceased, a thixotropic fluid will show an increase in apparent viscosity. When apparent viscosity increases with flow time, the fluid is “Rheopectic”. The shear stress developed in most drilling fluids is dependent upon the duration of shear. A time lag exists between an adjustment of shear rate and the stabilization of shear stress. This is due to the breaking up of clay

particles at high shear rates and the aggregation of clay particles when shear rate is decreased, both occurrences take a noticeable length of time.

“Gel strength” is used to measure this time dependent behavior. This gel strength measures the attractive forces of a fluid while under static conditions. If the gel strength increases steadily with time, the gel strength is classed strong or progressive. If it increases slowly with time, it is classed as weak or fragile.

When strong gels occur, excessive pressures may be required to break circulation.

Hydraulics - Deformation of a Fluid by Simple Shear - Bingham Plastic Model

Bingham Plastic Model

The Bingham model is defined by the relationship;

 Shear Stress = Yield Stress + (Plastic Viscosity x Shear Rate)

The major difference between this and Newtonian fluids is the presence of a Yield Stress or “Yield Point” (which is a measure of the electronic attractive forces in the fluid under flowing conditions). No bulk movement of the fluid occurs until this yield stress is overcome. Once the yield stress is exceeded, equal increments of shear stress produce equal increments of shear rate.

Flow Curve for a Bingham Plastic Fluid

Note that the apparent viscosity decreases with increased shear rate. This phenomenon is known as “shear thinning”. As shear rates approach infinity, the apparent viscosity reaches a limit known as the Plastic Viscosity. This viscosity is the slope of the Bingham plastic line. The commonly used Fann V-G meter was specifically designed to measure viscosities for this model. As can be seen in the above illustration, this model does not accurately represent drilling fluids at low shear rates.

Hydraulics - Deformation of a Fluid by Simple Shear

Deformation of a Fluid by Simple Shear

The magnitude of shear between the layers is represented by the shear-rate, which is defined as the difference in the velocities between the layers, divided by the distance of separation. It is this relationship between the shear-stress and shear-rate that defines the behavior of the fluid.

For some fluids the relationship is linear (i.e., if the shear-stress is doubled then the shear-rate will also double, or if the circulation rate is doubled then the pressure required to pump the fluid will double). Fluids such as this are known as “Newtonian fluids”. Examples of Newtonian fluids are water, glycerine and diesel. The Newtonian fluid model is defined by the following relationship:

Shear-Stress = Absolute Viscosity x Shear-Rate

The slope of the flow curve in the diagram is given by the absolute viscosity, this is the shear stress divided by the shear rate. A typical flow profile for a Newtonian fluid in a cylindrical pipe is a parabola, with a maximum shear-rate at the wall and a minimum (0) at the center.

Drilling fluids are generally Non-Newtonian in behavior, and are defined by more complex relationships between shear-stress and shear-rate. When fluids contains colloidal particles (or clays), these particles tend to increase the shear-stress or force necessary to maintain a given flow rate. This is due to electrical attraction between particles and to them physically “bumping” into each other. Long particles, randomly oriented in a flow stream, will display high interparticle interference. However, as shear-rate is increased, the particles will tend to develop an orderly orientation and this interaction will decrease.

In the center of a pipe, the shear-rate will be low and hence particle interaction high, giving it a flattened flow profile. This profile has an improved sweep efficiency and an increased carrying capacity for larger particles.

As can be seen from the previous examples, the ratio of shear-stress to shear-rate is not constant but will vary with each shear-rate. Various “oilfield” models have been proposed to describe this non-

Newtonian shear-rate/shear-stress curve. In order to arrived at “standard” variables, these models require the measurement of shear-stress at two or more shear-rates to define the curve.

The two most common models used at the wellsite are the Bingham Plastic Model and the Power Law Model.