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# Surge pressure prediction for wellbore flow

An exceptional flow case is the operation of running pipe or casing into the wellbore. Moving pipe into the wellbore displaces fluid, and the flow of this fluid generates pressures called surge pressures.

## Overview

When the pipe is pulled from the well, negative pressures are generated, and these pressures are called swab pressures. In most wells, the magnitude of the pressure surges is not critical because proper casing design and mud programs leave large enough margins between fracture pressures and formation-fluid pressures. Typically, dynamic fluid flow is not a consideration, so a steady-state calculation can be performed. A certain fraction of wells, however, cannot be designed with large surge-pressure margins. In these critical wells, pressure surges must be maintained within narrow limits. In other critical wells, pressure margins may be large, but pressure surges may still be a concern. Some operations are particularly prone to large pressure surges (e.g., running of low-clearance liners in deep wells). The reader is referred to papers on dynamic surge calculations,and the article on dynamic pressure calculation gives a taste of this type of calculation.

## Surge analysis

The surge pressure analysis consists of two analytical regions: the pipe-annulus region and the pipe-to-bottomhole region (Fig. 1). The fluid flow in the pipe-annulus region should be solved using techniques already discussed, but with the following special considerations: frictional pressure drop must be solved for flow in an annulus with a moving pipe, and in deviated wells, the effect of annulus eccentricity should be considered. The analysis of the pipe-to-bottomhole region should consist of a static pressure analysis, with pressure boundary condition determined by the fluid flow at the bit, or pipe end if running casing. The pipe-annulus model and the pipe-to-bottomhole model then are connected through a comprehensive set of force and displacement compatibility relations.

## Boundary conditions

The following conditions describe the flow for a surge or swab operation.

### Surface boundary conditions

There are six variables that can be specified at the surface:

P1 = pipe pressure.

v1 = pipe fluid velocity.

P2 = annulus pressure.

v2 = annulus fluid velocity.

v3 = pipe velocity.

A maximum of three boundary conditions can be specified at the surface. For surge without circulation, the following boundary conditions hold:

P1 = atmospheric pressure.

P2 = atmospheric pressure.

v3 = specified pipe velocity.

For a closed-end pipe, the following boundary conditions hold:

v1 = v3 , and fluid velocity equals pipe velocity.

P2 = atmospheric pressure. v3 = specified pipe velocity.

For circulation with circulation rate Q, the boundary conditions are

v1 = v3 + Q/A1 (i.e., fluid velocity equals pipe velocity plus circulation velocity). P2 = atmospheric pressure.

v3 = specified pipe velocity.

### End of pipe boundary conditions

There are 11 variables that can be specified at the moving pipe end (see Fig. 2):

P1 = pipe pressure.

v1 = pipe velocity.

P2 = pipe annulus pressure.

v2 = pipe annulus velocity.

Pn = pipe nozzle pressure.

vn = pipe nozzle velocity.

Pr = annulus return area pressure.

vr = annulus return area velocity.

P = pipe-to-bottomhole pressure.

v = pipe-to-bottomhole velocity.

v3 = pipe velocity.

A total of seven boundary conditions can be specified at the moving pipe end with bit (see Fig. 3). For the surge model, three mass balance equations and four nozzle pressure relations were used.

Pipe-to-Bottomhole Mass Balance.

Pipe Annulus Mass Balance

Pipe Mass Balance

Pipe Nozzle Pressures

Annulus Return Pressures. The boundary conditions are greatly simplified for a pipe without a bit.

The boundary condition imposed by a float is the requirement that

If the solution of the boundary conditions does not satisfy this condition, the boundary conditions must be solved again with the new requirement:

### Change of cross-sectional area

Changes in the cross-sectional area of the moving pipe generate an additional term in the balance of mass equations because of the fluid displaced by the moving pipe (see Fig. 3).

The following was already inserted:

where

and

The superscript – denotes upsteam properties, and the superscript + denotes downstream properties.

## Surge pressure solution

Because of the complex boundary conditions, the solution of a steady-state surge pressure is most easily solved with a computer program. For closed-pipe and circulating cases, the flow is defined so that pressures can be calculated from the annulus exit to the standpipe, as discussed previously. For open-pipe surges, the problem is finding how the flow splits between the pipe and the annulus, so that the pressures for both the pipe and the annulus match at the bit. One strategy for solving this problem is given next.

1. Calculate all pressures with all flow in the annulus. Then, check pressures at the bit; annulus pressure will be lower because of fluid friction.

2. Calculate all pressures with all flow in the pipe. Then, check pressures at the bit; pipe pressure will be lower because of fluid friction.

3. Calculate a division of flow between the pipe and annulus that will equalize the pressures at the bit.

4. Repeat Step 3 until the two pressures match within an acceptable tolerance.

The efficiency of this calculation will depend on the method chosen for Step 3. With modern computers, this is not a particularly critical problem, so a simple interval halving technique would work. For the ith iteration of Step 3, fi is the fraction of flow in the pipe, and (1 – fi ) is the fraction in the annulus. Previous steps show that fp gives a higher annulus pressure and fm gives a lower annulus pressure. Our new choice for fi is ½(fp + fm). We perform the pressure calculation and find that the annulus pressure is higher, so we assign fp = fi . If the pressure difference is less than our tolerance, which we chose to be 1 psi, then the calculation is complete. Otherwise, we try another step. How do we establish fp and fm? The initial two steps in the solution step should give us fp = 0 and fm = 1, respectively. In some cases, such as small nozzles or restricted flow around the bit, fluid must flow into either the pipe or annulus, or the fluid level must fall. For these cases, f may be negative or greater than one. It may be necessary to repeat Steps 1 and 2 to establish the initial set fm and fp.

## Nomenclature

 A = flow area (see subscripts), m2 c = average concentration of cuttings overall ca = cuttings concentration in annular region co = feed concentration of cuttings cp = cuttings concentration in plug region C = compressibility Cd = discharge coefficients for the flow through an area change, dimensionless CD = drag coefficient, dimensionless ds = particle diameter, m D = characteristic length in Reynolds number, m De = special equivalent diameter for yield power law fluid, m Deq = equivalent diameter, m Dhyd = hydraulic diameter, m Dh = wellbore diameter, m Di = inside diameter, m Do = outside diameter, m Dp = drillpipe outside diameter, m Dplug = plug diameter, m g = acceleration of gravity, m/s2 G = mass flow rate density of mixture, kg/m3–s Gs = mass flow rate density of solids, kg/m3–s h = specific enthalpy, J/kg h = total friction pressure drop, Pa/m ṁ = mass flow rate, kg/s ṁs = mass flow rate of solid, kg/s P = pressure, Pa Q = heat transferred into volume, W R = ideal gas constant, m3 Pa/kg-K t = time, s T = absolute temperature, °K u = radial displacement, m v* = characteristic velocity for turbulent flow calculations, m/s v = average velocity, m/s ΔP = pressure drop, Pa Δt = time increment, s Δv = change in velocity, m/s Δz = length of flow increment, m ρ = fluid density, kg/m3

Subscripts

1 = properties inside pipe, surge calculations

2 = properties inside annulus, surge calculations

3 = properties of moving pipe, surge calculation

c = concentric

e = eccentric

n = properties in bit nozzle, surge calculations

o = upstream, initial, or inlet

r = properties in annulus outside bit, surge calculations

Superscripts

- = upstream properties