PRESSURE LOSS CALCULATIONS IN DRILLING HYDRAULICS | BINGHAM PLASTIC & POWER LAW

Pumping a drilling fluid requires overcoming frictional drag forces from fluid layers and solids particles. The pump pressure (Pp) can be described as the summation of the frictional forces in the circulation system:

PP : pump pressure, psi

PDS : drill string friction pressure, psi

PB : drilling bit pressure drop, psi

PA : annulus pressure, psi

The pressure drop in the bit results from fluid acceleration and not solely friction forces. As a result, it will be discussed in a separate section.

Equations to determine drill string or annulus pressure loss vary according to the flow regimes, such as laminar and turbulent. In addition, Bingham Plastic and Power Law models differ in form. Since these models are frequently used in drilling applications, they will be presented in the following sections. Newtonian-based equations will not be presented.

Surface Lines Pressure Loss (Friction Pressures).

Calculating the pressure drop in surface equipment such as the standpipe and drilling kelly is normally accomplished by equating it to an equivalent length of drill pipe. The surface equipment is separated into four groups (below Fig.) to determine an equivalent length. For example, if a rig has group three surface equipment and 4Y2-in. drill pipe was used, an additional 479 ft of pipe would be used to calculate pressure losses in the surface equipment.

Pressure Loss ( Friction Pressure) In Annulus And Drill String

A) Bingham Plastic Friction Pressures

The Bingham Plastic model is used primarily to compute friction pressure associated with laminar flow. This restriction is based on its inability to accurately describe shear stresses associated with high shear rates. Laminar and turbulent flow calculations will be presented, however, since they are frequently used in the drilling industry.

V = fluid velocity, ft/sec

Q = flow rate, gal/min

d = pipe diameter, in.

Vc = critical velocity, ft/sec

PV = plastic velocity, cp

YP = yield point, Ib/l 00 fe (check also Yield Point In Drilling Mud Formula)

p = mud weight, Ib/gal

L = section length, ft

In the annulus, the same series of operations is performed but with slightly different equations to account for the hole geometry:
1) The velocity of the fluid in the annulus is described as following:

dH = casing or hole 10, in.

dp = pipe or collar 00, in.

Use the results from Last Example and the following data to calculate friction pressures for flow rates of 100 and 200 gpm. Use the Bingham model.

pipe ID = 3.5 in.; mud weight = 12.9Ib/gal; PV = 29 cp; YP = 6 Ib/100 ft; length = 10,000ft

V = 100 / 2.448 x 3.5^2 = 3.33 ft/sec (at 100 gal/min)

V = 200 / 2.448 x 3.5^2 = 6.66 ft/sec (at 200 gal/min)

Vc = (1.08 x 29 + 1.08 x ( 29^2 + 12.34 x 3.5^2 x 6 x 12.9)^(1/2)) / (12.9 x 3.5 ) = 3.37 ft/s

3. Calculating Friction Pressure For 100 gal/min
For the flow rate of 100 gal/min, the actual velocity (Va) is slightly less than the critical velocity (Vc)of 3.37 ft/sec. Use the laminar flow equation. (Note that the difference between Va and Vc is small. Therefore, it might be advisable in some cases to consider calculating pressure losses

for laminar and turbulent flow and use the larger value.)

Pds = (29 x 10000 x 3.33) / (1500 x 3.5^2) + (6 x 10000) / ( 225 x 3.5) = 52.5+76.1= 128.6 psi

4. Calculating Friction Pressure For 200 gal/min
At a flow rate of 200 gal/min, the actual velocity of 6.66 ft/sec is significantly greater than the critical velocity of 3.37 ft/sec. Therefore, use the turbulent flow equation:

Pds = ( 12.9 ^ 0.75 x 6.66^1.75 x 29^ 0.25 x 10000) / ( 1800 x 3.5 ^ 1.25) = 505.7 psi

The Difference Between Friction Pressure Calculations In Both Laminar & Turbulent Flow

The laminar and turbulence equations can be used to illustrate the basic difference between these two flow systems. In the laminar equations, a value for the yield point (YP) is a significant part of the pressure loss, particularly when it is observed that the PV value is divided by a squared diameter. The

turbulent flow equations do not contain a YP term. The yield point is one of the forces creating the inter-particle attractions, causing the mud to move in laminar. When the shear force exceeds the yield stress, turbulence begins and the yield point is not a factor thereafter.

Power Law calculations follow the same sequence as the Bingham model. Actual and critical velocities are compared to determine the flow regime before calculating the pressure loss.

A word of caution must be given at this point relative to Bingham and Power Law equations. Many forms of these computations exist in the industry with units that differ slightly. Velocity can be expressed in ft/sec or ft/min, which obviously would make a significant error in the calculations, particularly when V is in exponent form. The Power Law model demands additional attention because several methods exist for computing the basic parameters of n and K. This is not the case for the Bingham model because only one accepted method is used for PV and YP calculations. The equations presented in this text are those of Moore et al.

A – Friction Pressure Calculations in Drill String

For computation simplicity, NR = 3,000 is assumed for turbulence criteria. Basic assumptions for friction factor correlations result in the critical velocity equation:

Calculating pressure loss (friction pressures) in the drill string using the Power Law equations for laminar flow: 

Calculating Pressure loss (friction pressures) in the drill string using the Power Law equations for turbulent flow:

B – Friction Pressure Calculations in The Annulus

Note that above equation uses the PV term, which is a Bingham model value. Since it is value to relate viscosity to a turbulent flow, a common practice uses μ related to PV, as shown below:

Example Of Using Power Law To Calculate Friction Pressure

Refer to Above Examples and compute the friction pressures for the system in above Example. Use the Power Law model and a flow rate of 125 gal/min. If Va nearly equal Vc, compute the pressure drop for laminar and turbulent flow and choose the larger value.

2) Determine the actual velocity at 125 gal/min:

3) Use Below Equation to compute the critical velocity, Vc:

Vc = (6.947 x 10^2) ^ 0.884 x (0.474 ^ 0.7699) = 325 x 0.563 = 183 ft / min = 3.05 ft / sec

6) Since 158.6 psi> 93.4 psi, assume the pressure loss is the greater value.

7) Some groups within the industry bypass Step 3 altogether and compute the pressure drops from both laminar and turbulent equations

• Friction Coefficients for Bingham and Power-Law Fluids in Abrupt Contractions and Expansions
• Pressure drop-flow rate profile of some locally formulated drilling fluids using Bingham plastic and power law rheological models
• Prediction of Stand Pipe Pressure Using Conventional Approach