# TECHNOLOGY Equations help calculate surge and swab pressures in inclined wells

Sept. 18, 1995
Zhong Bing, Zhou Kaiji, Yuan Qiji Southwest Petroleum Institute Nanchong China A set of equations and some simple charts helps predict the surge and swab pressures during pipe trips in inclined or horizontal wells. Running pipe or pulling pipe in a well bore can cause pressure fluctuations (surge and swab, respectively), which can contribute to lost circulation or kicks. The prediction of surge and swab pressures has always been an essential part of well control. These surge and swab
Zhong Bing, Zhou Kaiji, Yuan Qiji
Southwest Petroleum Institute
Nanchong China
A set of equations and some simple charts helps predict the surge and swab pressures during pipe trips in inclined or horizontal wells. Running pipe or pulling pipe in a well bore can cause pressure fluctuations (surge and swab, respectively), which can contribute to lost circulation or kicks. The prediction of surge and swab pressures has always been an essential part of well control. These surge and swab pressures have been researched extensively since 1934.1 2 The steady-state surge and swab pressure model widely used in the oil industry was first developed by Burkhardt in 1961.3 In 1977, Lubinski, et al., developed the first dynamic model.4 This work was later improved upon by Lal and Mitchell.56 Because of the complexity of the model, however, it is difficult to use. The current steady-state models in the oil field use formulas to calculate pressure loss in a concentric annulus. Because the clinging constant in the models was derived from Newtonian fluids, the models are not correct in theory and field applications. In addition, every model used today assumes the pipe is concentric with the well bore. This assumption is incorrect in directional wells, horizontal wells, and even in vertical wells. To provide accurate data of surge and swab pressures for the design of hole structure and drilling fluid density in directional and horizontal wells, it is necessary to improve the calculation method of the steady-state surge and swab pressures. This article analyzes the surge and swab pressures caused by clinging power during tripping in inclined wells. The calculation method presented here is for steady-state surge and swab pressures for axial flow of a Herschel-Bulkley fluid in an eccentric annulus. The accompanying charts show the application of this method to oil field operations.

### Differential equation

The rheological behavior of the drilling fluid was based on the Herschel-Bulkley model, because it is more accurate than the Bingham model and Power law model and involves the characteristics of these two models. Because an equal velocity core exists for the Herschel-Bulkley fluids structure flow, the flow field in the annulus consists of three regions. If the outer radius of the pipe is Ri, the radius of the well bore is Ro, the distance from the pipe axis to the well bore is R, the inner and outer boundaries of the equal velocity core are r1 and r2, then the three regions are the following:
• Inner velocity gradient region (Ri
• Equal velocity core region (r1
• Outer velocity gradient region (r2
Based on the assumption that the cross-sectional area of the annulus is constant along the length, L, the differential equation of the axial steady-state flow in an eccentric annulus can be written as Equations 1 and 2 (see nomenclature and equation box).(434335 bytes) In Equations 1 and 2,(434335 bytes) DPz/L denotes the frictional pressure loss per unit length, !equation is the shear stress of drilling fluid, rm is the drilling fluid density, g is the gravitational acceleration, c is the hole deviation angle, and r is the distance from the pipe axis to any point in the eccentric annulus.

### Velocity distribution

The surge and swab pressure gradient is defined as Pg = DPz/L !equation cos c.

Combining the Herschel-Bulkley model with Equations 1 and 2 (434335 bytes) and appropriate boundaries yields Equations 3 and 4 (434335 bytes) for the inner velocity gradient region (RiEquations 7-10 (434335 bytes) for the outer velocity gradient region. In these equations, to is the yield point, K is the fluid consistency index, n is the flow-behavior index, u is the velocity of any point in eccentric annulus, and Vp is the tripping velocity of the pipe. The velocity distribution formula in an eccentric annulus can be obtained based on these two groups of equations above (Equations 11 and 12)(434335 bytes).

### Volume rate of flow

The concept of the equivalent gap of an eccentric annulus is used to obtain the flow rate formula for the Herschel-Bulkley fluids structural flow (Equation 13).(434335 bytes)7 The equivalent outer radius of the eccentric radius is Roe (Equation 14)(434335 bytes), the equivalent ratio of the outer radius of the eccentric annulus is koe (Equation 15)(434335 bytes), and the equivalent ratio of inner radius to the outer radius of the eccentric annulus is kie (Equation 16).(434335 bytes) In Equations 13-16,(434335 bytes) k 0.3, and e denotes eccentricity. The flow rate formula for a Herschel-Bulkley fluid in an eccentric annulus can then be obtained (Equation 17).(434335 bytes)

### Surge and swab

If the pipe is treated as a closed pipe, the flow rate is given by Equation 18.(434335 bytes) This treatment conforms to the actual situation for a casing string and does not yield too large of an error. This assumption also tends to be safe for the drillstring because the nozzle equivalent diameter is small. Combining Equation 17 (434335 bytes) with Equation 18 (434335 bytes) yields Equation 19.(434335 bytes) Under the conditions listed in Equation 20,(434335 bytes) one can obtain Equation 21.(434335 bytes) The values of l1 and l2 are defined by combining Equations 19 and 21(434335 bytes). Then, the calculation formula of surge and swab pressures is: Equation 22,(434335 bytes) in which dl = l2 l1. Ps/w is negative when the pipe is pulled and positive when it is run in a well. If the well bore consists of annuluses with different eccentricities and ratios of inner-to-outer diameters, the formula can be written as Equation

23.(434335 bytes)Figs. 1(81832 bytes)-(85857 bytes)3(80564 bytes) give the curves of dl vs. the flow behavior index for different values of e, fp, and k for oil field use. The value of dl can be obtained from the figures if the values of e, fp, k, and n are known. Then the surge and swab pressures can be calculated simply from Equation 22 or 23.(434335 bytes)

### Example

With the mud properties and hole geometry given in Table 1(9597 bytes), the equations are used to determine the surge pressure at a pipe velocity of 2 fps at a depth of 10,000 ft. The eccentricity, e, is first calculated e = e/(Ro-Ri) = 0.6. The dimensionless drill pipe outer radius is calculated as k = Ri/Ro = 0.5. The dimensionless pipe velocity is fp = k/to)1/n Vp/Ro = 0.2. From the charts then, at n = 0.8, k = 0.5, and fp = 0.2, the dimensionless value dl is found as 0.177. From Equation 22,(434335 bytes) the surge pressure per foot of depth is calculated as 0.00395 lbf/sq in./ft. At the depth of 10,000 ft, the surge pressure is 39.5 lbf/sq in.

### Results

From Figs. 1-3, one can obtain the following conclusions:
• The ratio of inner radius to outer radius significantly affects the surge and swab pressures. The larger the value of k, the larger are the surge and swab pressures.
• The surge and swab pressures will increase with increases in the value of fp and the flow behavior index. As the value of fp becomes larger, the flow behavior index has a more significant effect on the pressures.
• The surge and swab pressures are less in an eccentric annulus than they are in a concentric annulus. The larger the eccentricity is, the larger the relative difference between the two annuluses. The maximum relative difference may be nearly 100%. Hence, the effect of eccentricity must be considered while computing surge and swab pressures.

### References

1. Cannon, G.E., Changes in Hydrostatic Pressure Due to Withdrawing Drill Pipe from the Hole, Drilling and Production Practices, American Petroleum Institute, 1934.

2. Goins, W.C., et al., Down-The-Hole Pressure Surges and Their Effect on Loss of Circulation, Drilling and Production Practices, API, 1951.

3. Burkhardt, J.A., Wellbore Pressure Surges Produced by Pipe Movement, Journal of Petroleum Technology, June 1961, pp. 595-605.

4. Lubinski, A., et al., Transient Pressure Surges Due to Pipe Movement in an Oil Well, Revue de Institut Francais du Petrole, May-June 1977, pp. 307-42.

5. Lal, M., Surge and Swab Modeling for Dynamic Pressures and Safe Trip Velocities, Society of Petroleum Engineers paper 11412 presented at the SPE/IADC Drilling Technology Conference, New Orleans, Feb. 20-23., 1983.

6. Mitchell, R.F., Dynamic Surge/Swab Pressure Predictions, SPE Drilling Engineering, September 1988, pp. 325-33.

7. Zhou Fengshi, The Basic Characteristic of the Flow Regularity in Eccentric Annulus and its Effect on Drilling, Oil Drilling & Production Technology, Vol. 3, 1983.

### The Authors

Zhong Bing is an oil and gas exploitation lecturer in the offshore oil department of the Southwest Petroleum Institute in Nanchong, Sichuan, China. He has authored more than 30 technical papers and reports on drilling engineering. Zhou Kaiji is oil and gas exploitation vice-professor at the Southwest Petroleum Institute in Nanchong, Sichuan, China. Yuan Qiji is an oil and gas exploitation lecturer in the petroleum engineering department of the Southwest Petroleum Institute in Nanchong, Sichuan, China.Copyright 1995 Oil & Gas Journal. All Rights Reserved.