Ashley Johnson
Schlumberger Cambridge Research Ltd.
Cambridge, England
Jeffrey Tarvin
Schlumberger-Doll Research
Ridgefield, Conn.
A new model includes the affects of mud compressibility, well bore elasticity, and fluid loss for accurate calculation of the pressure increases in a shut-in well bore following a kick.
Incorrect estimates of gas migration can lead to additional problems during well control (Fig. 1).
Direct measurements of gas migration velocities in laboratory experiments and full-scale tests have shown that gas can migrate up a well bore at approximately 6,000 ft/hr.
Typical estimates of the gas migration velocity are 1,000 ft/hr or less. These estimates are derived from the rate of increase of surface pressures during shut in and are in error because they rely on a conventional hydrostatic model that neglects several important factors, such as mud compressibility, well bore elasticity, and fluid loss.
In the new model, typical values for elasticity and fluid loss are derived from formation integrity tests for a real well. With these values, the new model calculates rates of pressure rise that are consistent with those observed in the field, even though the gas migration velocity is as large as that measured in the experiments.
The model shows that conventional field practices that neglect these effects can underestimate the gas migration velocity by a factor of ten or more.
GAS MIGRATION
The published literature contains a vast amount of conflicting information on gas migration velocities during the shut-in period of a gas kick.
A widely accepted "rule of thumb" used in the field is that gas bubbles migrate upward at about 0.085 m/sec (1,000 ft/hr). These velocities are usually derived from conventional correlations for casing pressure increase rates. Blount claimed evidence of gas migration rates of around 0.014 m/sec (160 ft/hr), but the details of how the gas migration was measured were not given. (It is assumed that the conventional method was used.)1
Gas migration rates measured in short flow loops and large test wells suggest a much higher migration velocity.
Hovland and Rommetveit reported large-scale tests in a 1,500-m deep test well that had a maximum deviation of 63. The gas slip velocity, 0.55 m/sec (6,500 ft/hr), was measured using the time of flight between pressure transducers mounted at various depths in the well.
Johnson and White described tests in a 15-m flow loop with realistic fluids and a typical drilling geometry.5 In those experiments, the gas migrated at a velocity of about 0.5 m/sec (5,900 ft/hr). Rader reported similar results for gas migration in a 3.7-m flow loop and in a 1,800-m well.8 The gas velocity in the well was measured using the time-of-flight principle.
KICK SIMULATOR ANALYSIS
An analysis of test well data using the SideKick simulator also showed that high migration rates give results close to the field observations as an influx is circulated out. The test well data come from well-control experiments at Rogalands Research Institute, where gas was injected into a 1,500-m cased well (Fig. 2).
In an experiment, the well was shut in for 90 min after the injection ended. The key data from this experiment are pressure measurements in the annulus. The pressure difference between transducers changes abruptly when gas first enters the interval between them. Thus, plots of pressure difference are direct indicators of gas migration. Table 1 lists the positions of the six transducers. Note that the depth for the choke is negative because it is above the reference point (kelly bushing).
To see the effects of the gas-migration relationships, the experiment was modeled two ways using the SideKick simulator (Fig. 3):
- The first model used the standard gas-slip relationship, which gives a slip velocity of about 0.5 m/sec (6,000 ft/hr). The curves derived from this computation are labeled "SideKick."
- The second model used a typical industry estimate of 0.1 m/sec (1,200 ft/hr) for the slip velocity. The curves derived from this computation are labeled "Slow gas."
In the plots, the injection starts at 35 min and ends at 55.5 min.
The arrival of gas at transducer P3 is clear in the plot of P3-P2 (Fig. 4). SideKick calculates this arrival accurately. The jump in the slow-gas curve at 55 min results from the change in circulation rate at the end of injection.
In the slow-gas calculation, gas arrives at P3 at 78 min, which is about 30 min late. Additionally, the accumulation of gas in the interval is much too slow in the slow-gas model.
The errors in the slow-gas model are more pronounced in the plot of P2-P1 (Fig. 5). SideKick is still accurate, showing the gas arriving about 1.5 min early. The computed pressure difference for the slow-gas, however, is absolutely flat during the 90-min shut in. In the slow-gas model, the gas arrives only after circulation starts again at 146 min. Thus, the gas arrival is nearly 90 min late; the slow-gas model is clearly wrong.
For the first 5 min after shut in, the measured choke pressure increases almost as fast as SideKick indicates (Fig. 6).
Beyond that point, SideKick slightly overestimates choke pressure, and the slow-gas model underestimates choke pressure.
The large-scale test data are based on time of flight measurements; thus, a misinterpretation of the experimental data would be very difficult.
The simulator analysis demonstrates that gas migrates at these large velocities during shut in. For the slow-gas model, the field data are normally based on a derivation of the slip velocity from the well bore pressure increase rate during shut in, but this derivation may not be entirely accurate.
CONVENTIONAL CALCULATIONS
Conventional field practice is to calculate the gas migration velocity during shut in from the casing or standpipe pressure increase rates.
The casing pressure increase rate, Pc, is used to calculate the gas migration rate, Vslip using Equation 1.7
This model is based on simple hydrostatics. The model assumes that the well bore is a rigid and sealed vessel filled with incompressible mud. It is important to consider whether these assumptions are appropriate.
NEW MODEL
A shut-in well bore can be thought of as a large elastic vessel made of a permeable material and filled with a compressible fluid. For a better model, then, the stiffness of the vessel, the compressibility of the fluid, and the rate of fluid loss from the vessel must be considered.
For a well bore with mud volume, Vm, and mud compressibility, Xm, the time rate of change of mud volume Vm, caused by pressure increasing at a rate Pc is given by Equation 2.
When the pressure in the well bore changes, the well bore wall starts to deform. For small variations in pressure, the deformation is linearly elastic. For larger changes, when the stress passes the yield stress the well bore wall deforms plastically. This article discusses elastic deformation only.
The rate of change of volume of a well bore deforming elastically is given by Equation 3. Xw is the effective well bore elasticity, and Vw is the well bore volume. The effective well bore elasticity depends on the stiffness of all sections of the well bore. Xw can be calculated from Equation 4.
Xi and Vi are the effective elasticity and volume of each formation of the open hole section together with the cased section of the well bore. The effective elasticity of the cased section of the well bore is discussed in the accompanying box.
For the open hole section, however, a detailed knowledge of each formation is required, and this information is usually unknown while the well is being drilled. This information can be derived from the results of a leak-off test.
FLUID LOSS
Fluid leaking off from a shut-in well bore either reduces the well bore pressure or reduces its rate of increase. The size of the net flow determines the pressure change, so it is important to characterize the rate at which fluid leaks off from the well bore, especially,, if the influx rate is small.
Drilling mud flowing past a porous medium forms a filter cake on the surface, restricting flow rate. For formations with permeability greater than 1 md, a mature filter cake, rather than rock permeability, limits the flux into the formation. When the overpressure across a filter cake increases, the cake is compacted and its permeability is reduced.
The effect of this drop in permeability almost balances the increase in overpressure. Thus, for an environment with constant or increasing pressure, the fluid loss flux is nearly independent of the pressure. For a mature filter cake grown in static fluid, the flux rate (volume of flow rate per unit area) is likely to be in the order of 0.1 mm/sec. For a freshly opened section of permeable formation, the flux can be much greater.
Filter cakes grown during a drilling operation are thinner and allow a larger fluid flux. Most of the time during drilling operations, mud is pumped around the well bore, so filter cake grows under the influence of a cross flow. Fordham and Ladva show that for mature filter cakes, the flux rate lies in the range of 0.1-1.0 mm/sec. As with the static filter cakes, this rate is almost completely independent of overpressure.
These flux rates are into permeable, unfractured formations. To calculate the fluid loss rate from the entire well bore, qe, an estimate of the permeable fraction of the open hole section is needed. Obtaining this estimate during drilling is unlikely; however, it is possible to derive the information from a leak-off test.
PRESSURE INCREASE
For a single large gas bubble rising in a shut-in well, the rate of change of gas pressure, pg, is given by Equation 5. Equation 6 can be used to calculate the growth rate, Vg, of the gas bubble if the gas has a volume Vg and compressibility Xg.
Equation 7 sets the volume of the well bore equal to the volume of its contents. In this equation, qi is a fluid influx into the well, and qe is an efflux.
By substituting Equations 2, 3, 3, and 6 into Equation 7 and rearranging terms, the casing pressure increase can be calculated with Equation 8. Equation 8 simplifies to the basic field model if well bore elasticity, mud compressibility, and fluid influx and efflux are neglected.
CALCULATED PRESSURE RISE
Predicting the elastic and fluid loss characteristics of a well bore is almost impossible. The results of a careful leak-off or formation integrity test, however, can be used to evaluate these characteristics.
The rate of fluid injection, together with the pressure rise and fall rates, must be recorded. It is also important to have details of the mud type to calculate the mud compressibility. The other accompanying box covers the use o( a leak-off test to derive the well bore elasticity and fluid loss rates for a particular well.
Consider a kick in the well described in this box. Assume the influx is 13 bbl of has with a compressibility of xg = 1/pg. The gas pressure is equal to the hydrostatic head at the bottom of the well. The gas migration rate, Vslip, is 0.5 m/sec. The fluid influx has ceased, and the fluid loss into permeable zones is evenly distributed over 70% of the open hole section of the well, with a flux rate of 0.1 mm/sec. The mud volume is 15% less than the well bore volume; the difference accounts for the volume of the drillstring.
Substituting these parameters into Equation 8 yields a surface pressure increase rate of 710 Pa/sec (375 psi/hr). Substituting this value into Equation 1 calculates the slip velocity that would have been derived using the conventional model.
While the gas is actually migrating at 0.5 m/sec (5,960 ft/hr), the conventional model would indicate the gas is moving at 0.058 m/sec (685 ft/hr). This estimated rate of rise is consistent with the rule of thumb for gas migration used in the field. Even though the gas is actually migrating at almost ten times this speed.
This calculation is for the well as completed when the leak-off test was performed. Even with a shorter open hole section, the effects of fluid loss and well bore elasticity would be significant.
Consider the same well completed to a depth of 6,000 ft. Xw is kept the same, and qe is reduced in proportion with the length of the open hole. The pressure increase rate would be 2,100 Pa/sec (1,100 psi/hr). The gas migration rate derived from the field model would be 0.17 m/sec (2,000 ft/hr), which is still much less than the actual 0.5 m/sec (5,900 ft/hr).
To characterize the effect of the fluid loss and elasticity separately, the size of each parameter in Equation 8 must be considered for the well at its full depth.
The source rate due to gas migration is:
XgVgPmGVslip = 360 x 10-6 cu m/sec
The fluid loss rate is qe = 140 x 10-6 cu m/sec. Fluid loss reduces the surface pressure increase rate by 38%. For filter cakes grown in a drilling environment, the flux rate could be more than five times that considered here. In such a case, the efflux rate would be larger than the rate of gas expansion, and the well bore pressure would remain constant despite the gas migration.
The bulk gas compressibility is XgVg = 58 x 10-9 cu m/Pa. The bulk mud compressibility is XmVm = 67 X 10-9 cu m/Pa. The bulk well bore elasticity is XwVw = 187 x 10-9 cu m/Pa. The system compressibility is dominated by the well bore elasticity. While the mud compressibility can reduce the pressure increase by one half, the combination of that with the well bore elasticity reduces the pressure increase rate by more than five times.
ACKNOWLEDGMENT
The authors thank Francois Baret and Gerard Daccord of Dowell Schlumberger and John Haberman of Texaco Inc. for their help in obtaining leak-off test data and for permission to publish the results. The authors also thank Andrew Hamilton for performing the simulations on SideKick. They also thank the U.K. Health & Safety Executive for its support of the R-model gas kick simulator, from which SideKick was developed.
REFERENCES
- Blount, E., editorial comment, SPE Drilling Engineering, 1991, p. 236.
- Fordham, E.J., and Ladva, H.K.J., "Crossflow filtration in bentonite suspensions," Physio-Chemical Hydrodynamics, Vol. 11, No. 4, 1989, p. 411.
- Haberman, J.P., Delestatius, M., Hines, D.G., Daccord, G., and Baret, J-F, "Downhole fluid-loss measurements from drilling fluid and cement slurries," SPE22552 presented at the SPE Annual Technical Conference and Exhibition in 1991.
- Hovland, F., and Rommetveit, R., "Analysis of gas-rise velocities from full scale kick experiments," SPE paper 24580 presented at the SPE Annual Technical Conference and Exhibition, Washington D.C., 1992.
- Johnson, A.B., and White, D. B., "Gas rise velocity during gas kicks," SPE Drilling Engineering, 1991, p. 257.
- McMordie, W.C., Bland, R.G., and Hauser, J.M., "Effect of temperature and pressure on the density of drilling fluids," SPE11114 presented at the SPE Annual Technical Conference and Exhibition in 1982.
- Moore, P., Drilling Practices Manual, PennWell Publishing Co., Tulsa, 1986.
- Rader, D.W., Bourgoyne, A.T., and Ward, R.H., "Factors affecting bubble-rise velocity of gas kicks," Journal of Petroleum Technology, May 1975, pp. 571-585.
- Sherwood, J.D., Meeten, G., Farrow, C.A., and Alderman, N.J., "Concentration profile within nonuniform mudcakes," Journal of the Chemical Society, Faraday Transcripts, Vol. 87, No. 4, 1991, pp. 611-618.
- Timoshenko, S.P., and Goodier, J.N., Theory of Elasticity, McGraw Hill, 1970.
- White, D.B., and Walton, I.C., "A computer model for kicks in water and oil-based muds," SPE paper 19975, Society of Petroleum Engineers, 1996.
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