SPEIIADC 25710

Advanced Influx Analysis Gives More Information Following a KickJohn Billingham and Martin Thompson, * Schlumberger Cambridge Research, andD.B. White, * Sedco ForexlSchlumbergerlADe Members'SPE Members

Copyright 1993, SPE/IADC Drilling Conference.

This paper W\lS prepared for presentation at the 1993 SPElIADC Drilling Conference held in Amsterdam 23-25 February 1993.

This paper was selected for presentation by an SPElIADC Program Committee following review of information contained in an abstract submitted by the author(s). Contents of the paper,as presented, have not been reviewed by the International Association of Drilling Contractors or the Society of Petroleum Engineers and are subject to correction by the author(s). Thematerial, as presented, does not necessarily reflect any position of the SPE or IADC, their officers, or members. Papers presented at SPElIADC meetings are subject to publicationreview by Editorial Committees of the SPE and IADC. Permission to copy is restricted to an abstract of not more than 300 words. Illustrations may not be copied. The abstract shouldcontain conspicuous acknowledgment of where and by whom the paper is presented. Write Librarian, SPE, P.O. Box 833836, Richardson, TX 75083-3836, U.S.A. Telex, 183245 SPEUT.

ABSTRACT

We present an efficient, new procedure for characterisinga kick by analysing changes in surface pressures, surfacemud flow rates and pit level. This procedure allows us todetermine, faster and more reliably than is possible withconventional methods: the times when the influx startedand stopped, the formation pressure, the influx type (evenfor a horizontal well), the formation permeability and themaximum casing shoe pressure. The performance of thisprocedure has been tested using data from both a gas kicksimulator and a series of controlled experiments in a 4500 fttest well. We present examples of our results, which showgood agreement between predictions and observations. Thisanalysis has been implemented in a rig based monitoringsystem, [1].

INTRODUCTION

It is well-known that an influx of gas into the wellborefrom a permeable formation (a kick) can be extremely dan-gerous. An uncontrolled kick can develop into a blowout,which threatens the life of rig personnel, destruction of therig and significant environmental damage. It is very im-portant that a kick is detected and the blowout preventors(BOPs) closed as rapidly as possible in order to stop theformation from flowing and secure the well. The influxmust then be circulated out and the well killed. During thekill, bottomhole pressure must be maintained high enoughso that no further influx occurs, whilst the pressure at thecasing shoe must remain below the fracture pressure. Es-timates must be made of the formation pressure, the influxvolume and type (gas/liquid/mixture) and the greatest cas-ing shoe pressure during the kill. The accuracy of theseestimates is vital to the success of the operation.

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Industry standard kick control and kill procedures are verysimple and rely on calculations which can be performed byhand under conditions of extreme stress (see [2]). The den-.sity of the influx, and.hence its type, can ~_ estimated.froma simple hydrostatic calculation using the measured pit gainat shut in and the difference between the stabilised surfacepressures in the drillstring and the annulus. In addition,the formation pressure can be calculated from the stabiliseddrillpipe pressure. Finally, the maximum casing shoe pres-sure during the control/kill procedure can be calculated byconsidering the change in hydrostatic pressure as the influxmoves up the well. These analyses assume that the influxfills the annulus and exists as a single bubble.

There are a number of disadvantages to these calculations:

1. Under severe stress, errors can be. made during thehand calculation of the various important quantities.

2. The determination of the time when the influx ceasesis made by simply looking at a plot of the surfacepressures, or even just a mechanical gauge. Thismethod can be inaccurate, and the driller may evenmiss the end of the influx altogether.

3. Inaccuracies in pressure measurements can give er-roneous, and even negative, estimates of the influxdensity.

4. The determination of the influx density, and henceinflux type, relies on a pressure difference betweendrillpipe and annulus. This will not work for hori-zontal wells.

5. The assumption that the influx exists as a single bub-ble can lead to a significant overprediction of the max-imum casing shoe pressure and an inaccurate estimateof influx density.

6. There is much more information available in the sur-face measurements than is currently used.

2 ADVANCED INFLUX ANALYSIS GIVES MORE INFORMATION FOLLOWING A KICK SPE/IADC 25710

We present a new method of analysing surface pressures andmud flowrates which addresses each of these points. By for-mulating simple models for the behaviour of the downholefluids during and after a kick we can determine the expectedbehaviour of surface measurements. Simple curve-fittingthen reveals various important downhole quantities.

The analysis can be used for kicks taken whilst drilling withwater-base mud. The method works in verticaVdeviated/horizontal and slimjnonnal holes. The analysis, which givesan estimate of fonnation pressure and the influx type. is alsovalid in oil-base mud. This procedure has been automatedand include4 in a rig-based monitoring system (see [1]).

SEQUENCE OF EVENTS DURING AKICK

Let us now consider the sequence of events and the qualita-tive behaviour of surface pressures, mud flow rates and pitgain during a kick in water-base mud taken while drilling.

Kick: As the bit penetrates an overpressured fonnation,fluid starts to enter the well. The delta flow and pit gainincrease as the influx fluid enters the annulus.

Flowcheck: Once the kick has been detected, the pumps areshut off and a visual check is made for flow out of the well.During this period, mud may flow down the drillstring andup the annulus, pulling a vacuum at the top of the drillstring.This is known as V-tubing.

Shutin: The BOP is closed and, if V-tubing has occurred,the flow from the fonnation will drive mud up the drillstringuntil it is full again, whilst the annulus pressure increases.Both annulus and drillpipe pressures will then increase alongwith bottomhole pressure until the fonnation stops flowing.The surface pressures are then affected by any migration ofthe influx. If the influx is gas, surface pressures will riseas the gas migrates upwards. If the influx is liquid, surfacepressures will not rise significanLly, since the influx onlymigrates slowly, if at all.

Kill: The well is killed using either the wait and weightor driller's methods. In each case the influx is circulatedout through the choke, which is used to maintain a sufficientback-pressure to keep bottomhole pressure above fonnationpressure.

ANALYSIS OF SURFACE MEASUREMENTS

We now describe the advanced analysis of surface measure-ments during the kick, flowcheck and shut in phases, alongwith the benefits it brings. Full details can be found in the

334

appendix.

Pit gain/delta flow during the kick

By using a simple fonnation model, we can show that thepit gain, G. is given by,

Here, Go is the initial pit volume before the influx startsat time t = t K By fitting this curve to the pit gain, orequivalently the integrated delta flow, we can determine thetime when the kick started, tK, and the constant, CK, fromwhich we can calculate, kf).PdlJ1.in, the product of the for-mation penneability and the drilling underbalance dividedby the influx viscosity. A typical curve is shown in fig-ure 1. This analysis is very similar to that applied to thedraw down and build up of a Drill Stem Test

Analysis of surface pressures during shut in

We use a model for Darcy flow,

qJ = F f).p for PJ > PwO fi (2)qJ = or PJ :$ Pw

which simply states that the fonnation flowrate, qJ' is pro-portional to the underbalance, f).p, whenever the fonnationpressure, PJ' exceeds the bottom hole pressure, Pw. Werefer to the constant F as the fonnation producibility.

We analyse the build up of surface pressures after the wellis shut in to detennine the compressibility of the downholefluids. The drillpipe pressure is given by,

Pd =CS 1 (I - e--'cs,(t-tu )for tu :$ t :$ te

. .(3)Pd =CS 1 [1- e-Cs,(tc-tu) {I- C S2 (t - t e )}]for t > t e

Here, CS 1 and CS2 are constants which are related to prop-erties of the wellbore fluids and the fonnation (see the ap-pendix). A typical curve given by equation 3 is shown infigure 2. Once the fonnation flow has shut off, the pressurerise becomes linear as any free gas migrates. By fitting thiscurve to the drillpipe pressure, Pd, we can determine thetime when the fonnation stopped flowing, t e This is thetime at which to take shut in pressures and start to con-trol the well. It is difficult to determine t e accurately usingthe standard method, which is simply to wait for surfacepressures to stabilise.

SPE/IADC 25710 J. BILLINGHAM, M. THOMPSON" AND D. WHITE" 3

. From the drillpipe pressure at t = t e we can calculate thefonnation pressure, PI' from simple hydrostatics. It is thiscalculation of t e and PI which remains possible in oil-basemud. The other calculations are strongly affected by thehigh solubility of gas in oil.

Next, by estimating the proportion of the measured frictionalpressure loss during drilling which occurs in the drillstring,we can calculate the bottomhole pressure, and hence thedrilling underbalance, Llpd.

The delta flow, Llq, just before the pumps are turned off isa good estimate of the rate of influx from the fonnalion, ql'From LlPd, Llq and the simple model given by equation 2,we can deduce the fonnation producibility, F.

From our estimates of the constant, CS2, and the fonnationproducibility, F, we can calculate the influx volume com-pressibility, Cin Vin. Since we know the influx volume Vinfrom the pit gain, we can detennine the influx compress-ibility, Cin'

The compressibility of a gas is at least an order of mag-nitude greater than that of a liquid. The calculated valueof Cin is therefore an excellent indicator of the type.orin-flux. In particular, this method still works in horizontalwells where the standard field method for calculating influxdensity fails. This has been found to be much more reliablethan conventional indicators in the examples which we haveanalysed.

We have calculated LlPd and can estimate /lin from ourknowledge of the influx type. Therefore, we can estimatethe fonnation permeability, k, by using the calculation ofkLlpd//lin from the pit gain curve-fit. Whilst this informa-tion is not needed for well control, the permeability, k, canbe considered a "risk factor". If the permeability is high, thewell could flow strongly if any mistakes are made duringwell control.

The above sequence of calculations is illustrated in figure 3.

If a continuous drillpipe pressure measurement is not avail-able (i.e. there is a non-return valve in the drillstring), thiscurve fit must be made to the annulus pressure. A singlemeasurement of drillpipe pressure needs to be taken, as isnonnal practice. This allows the drillpipe pressure at t = t e ,and hence the fonnation pressure, to be estimated.

MAXIMUM CASING SHOE PRESSURE

Before starting the kill circulation after a gas influx it wouldbe extremely useful to have an estimate of the maximumcasing shoe pressure during the kill in order to minimisethe risk of fracturing the fonnation. The standard method,which is not nonnally used on the rig, assumes that the gasexists as a single bubble which fills the annulus at bottom-hole at the start of the kill circulation. The maximum shoe

pressure occurs when the leading edge of this bubble of gasreaches the casing shoe. The expansion of the gas can be al-lowed for by assuming ideal behaviour. These calculationsoverestimate the true pressures. If the shoe is weak, pres-sure may be bled off to prevent breakdown when it wouldnot actually have occured.

In reality, the influx gas does not exist as a single bubblebut as a distributed, bubbly mixture of gas and mud whichoccupies a greater length of the annulus. This means thatthe leading edge of the gas reaches the casing shoe earlierthan is usually assumed. In addition, the gas will haveexpanded less since the centre of mass of the gas cloud islower, so the maximum casing shoe pressure will be smallerthan that calculated with the standard method. Indeed, if thegas has reached the casing shoe before the start of the kill,the maximum casing shoe pressure will occur at the start ofthe kill. This is often the case when the open hole sectionis short.

An example of the effect of gas distribution on casing shoepressure, as calculated by a simple gas kick simulator isshown in figure 4. In each case the void fraction has a

. different, unifonn value whilst the mass of gas remai,.ns thesame. The lower the void fraction, the greater the length ofthe influx. The effect described above can clearly be seen.

Recent studies of the dynamics of a rising cloud of influxgas in drilling mud enable us to estimate the downhole gasdistribution, [3]. This estimate can be used in conjunctionwith the standard assumption of ideal gas behaviour to im-prove the estimate of the maximum casing shoe pressure.This lower estimate may be crucial in deciding whether agiven kill procedure is likely to fracture the fonnation. Thedetails of the calculations are given in an appendix. It isnow possible to use the advanced analysis of the kick de-velopment and shut in phases to estimate the casing shoepressure at any time during the kill.

The analysis for the determination of:

shutin pressures,

influx type, independent of wellbore inclination,

earliest time to start controlling the well,

is valid for all mud types. The other calculation~, for for-mation penneability and shoe pressure, are strongly affectedby the high solubility of gas in oil.

ANALYSIS VALIDATION

Validation is a key aspect of any safety related innovation.Initially, two distinct methods have been used, which aredescribed below. Further validation is continuing with fielddata.

335

4 ADVANCED INFLUX ANALYSIS GIVES MORE INFORMATION FOLLOWING A KICK SPE/IADC 25710

Data from a commercial gas kick simulator

.Firstly, we consider data from a numerically simulated methanekick. The wellbore geometry and physics are summarisedin table 1. The kick is take by drilling into an overpressuredfonnation with an underbalance of around 300 psi. the in-flux is detected at a pit gain of 10 bbl, the pumps shut offand a hard shutin perfonned. This shut in is rapid enoughthat no V-tubing occurs.

Figure 5 shows the pit gain whilst the kick is taken, alongwith the curve fit of equation 1. This non-linear curve fitwas perfonned using a standard method (see [4]). Thiscalculates the start time of the kick, tk, to be 306 s, inexcellent agreement with the actual time, 305 s.

Figure 6 shows the drillpipe pressure after shut in, alongwith the curve fit of equation 3. The curve fit detenninesthe time when the influx ceases, t e , to be 917 s, within oneminute of the actual time, 865 s. The fonnation pressure isthen calculated to be 8520 psi, in good agreement with theactual value of 8500 psi. From this we estimate the under-balance just before the pumps are turned off to be 380 psi.At this time the delta flow was 3.9 bbl/min, so we calculatethe fonnation producibility, F, defined in equation 2, to be1.0 X 10-2 (bbl/min)/psi. This then allows us to calculatethe influx compressibility from the curve fit shown in fig-ure 6. We find that CS2 = 2.7 X 10-28- 1, and hence thatcin = 1.2 X 10-4 pSi-I; about 25 times that of oil. The ac-tual compressibility of the influx at bottomhole temperatureand pressure is 8.2 x 10-5 psi-I.

The difference between our estimate and the actual valueis due to the simplifications made in the analysis. In spiteof the difference between the estimated and actual influxcompressibilities, it is obvious that the influx is gas ratherthan liquid. This illustrates the robust nature of the method.

It is interesting to compare our new technique with the stan-dard method used to detennine the influx type. This relieson an estimate of the influx density from,

Pa - PdPg =Pm - I 0 (4)fI 9 cos

The difference between the shulin drillpipe and casing pres-sures was 368 psi, and an influx of 36 bbl occupies a ver-tical height 529 fl in the annulus. With these values, theconventional analysis gives a negative influx density. Theerror arises from the assumption that the gas exists as asingle bubble at bottomhole around both the drillstring andthe collars. This lcads to an overestimate of the volume ofgas around the coIlars and hence of the hydrostalic pressurechange across the gas cloud.

We now know the drilIing underbalance and thal the influxis gas. From the curve fil to the pit gain, we have an esti-male of the product of fonnalion penneabilily and drilIingunderbalance divided by influx viscosity. We can lhere-

336

fore estimate the fonnation permeability. Our estimate is60 mdarcy, whilst the actual value is 79 mdarcy. Althoughthis will not replace well testing, we have. a good estimateof the order of magnitude of the risk associated with anymistakes made during the well control operation.

By estimating the maximum easing shoe pressure with themethod described in the appendix, we find that the expan-sion of the gas as it moves from bouomhole to the easingshoe is negligible. Both the standard method and our modi-fied method give a maximum casing shoe pressure ofaround6540 psi. The actual maximum easing shoe pressure afterthe start of the kill is about 6450 psi. The 90 psi differenceis due to the frictional pressure loss across the gas cloud,which we have neglected in our analysis. This means thatthe estimate is conservative. It is worth noting that the eas-ing shoe pressure, illustrated in figure 7, was at its highestduring the shut in period before the start of the kill.

Data from a full-scale experimental kick

In order to test the validity of our analysis we perfonned aseries of experiments ina well at the International Drillingand Downhole Technology Centre (lDDTC) in Aberdeen.This test well has been completed with 14 inch easing, witha second 9 5/8 inch easing hanging freely within it Thereis extensive instrumentation, both on surface and downhole,including all nonnal oilfield sensors.

During the experiments, nitrogen was injected into the wellthrough coiled tubing. This ran from the surface to bot-tomhole between the two concentric easings and into theinner easing via a non-return valve. Great care was takento ensure that the nitrogen injection rate would represent areal fonnation flow. We assumed Darcy flow, so the exper-iments did not test the validity of the fonnation modellingin the analysis.

Many experiments were perfonned for different "fonna-tions" and kick volume. Let us now consider in detail atypical experiment. The wellbore geometry and physics aregiven in table 2. Note that there was no easing shoe -the casing ran from surface to bottomhole. However, sincethere was a pressure tapping at a vertical depth of 2304f1, where a measurement was made, we have assumed anominal casing shoe depth of 2304 ft

Gas was injected at an appropriate rate whilst mud wascirculating at 6.9 bbl/min. Once the pit gain reached 5bbl the pumps were stopped and a hard shut in performed.V-tubing of mud down the driIlstring and up the annulusoccurred before the BOPs were fuIly closed. A total of 7.5bbl of gas was injected at bottomhole.Figure 8 shows the pit gain whilst the kick was taken, alongwith the curve fit of the equation I. The calculated start timeof the kick is 656 s, in good agreement with the actual time,650 s, when the gas injection began.

SPE/IADC 25710 J. BILLINGHAM, M. THOMPSOW AND D. WHITE- 5

Nomenclature

We are able to estimate the formation permeabilityfrom our analysis of pit gain and drillpipe pressuremeasurements.

Recent studies of gas dynamics during a kick enableos to estimate the. distribution of the influx. This al-lows us to estimate the maximum casing shoe pressureduring the kill. This is more accurate than the esti-mate obtained using the single bubble approximation.

These analysis techniques have been implemented ina rig-based monitoring system which can advise thedriller during the course of a kick.

area of a section of drillstring or annuluscross-sectional area of annulus around drillstringcross-sectional area of annulus around collarstotal area of bit nozzlesgas distribution factorconstant in pit gain curve fitconstant in drillpipe pressure curve fitconstant in drillpipe pressure curve fitinflux compressibilitymud compressibilitytotal vertical depthbit nozzle discharge coefficientformation producibilitypit gainpit gain offsetgravitational accelerationvertical extent of influxvertical distance from bit to shoeformation permeabilitylength of a section of drillstring or annulusaxial length of collarsaxial length of influx

- The ratio of gas and liquid compressibilities ismuch greater than the ratio of their densities -compressbility is a more robust measure of in-flux type.

- The standard calculation of influx density is in-accurate since the volume of the influx in theannulus around the collars is overestimated. Inaddition, this calculation is sensitive to small er-rors in the surface measurements. The estimateof the influx compressibility is insensitive to thedetailed distribution of the influx, a small errorin the measured pit gain and to a constant errorin the drillpipe pressure measurement

- The estimate of the influx compressibility worksin horiwntal wells where the standard methodfails.

AAaA coilANCCKCS 1CS2

CONCLUSIONS

A reappraisal of possible analysis methods both dur-ing and after a kick leads us to new techniques.

The measured pit gain during a kick can be analysedto deduce when the kick started.

The drillpipe or annulus pressure during shut in canbe analysed to deduce when the influx stopped, theformation pressure and the influx compressibility.

This estimate of influx compressibility is a better in-dicator of influx type than the standard estimate ofinflux density.

Figure 9 shows the annulus and drillpipe pressures aftershutin, along with the curve fit of the equation 3. The risein the annulus pressure for 1120s < t < 1160s whilst thedrillpipe pressure remains constant clearly indicates that U-tubing occurred and that the drillstring is refilling duringthis period. The time when the influx ceased, teo was cal-culated to be 1246 s (actual value, 1230 s). The apparentformation pressure was then calculated to be 2250 psi, withan apparent drilling underbalance of 220 psi.

The delta flow just before the pumps were turned off was1.4 bbVmin, so the apparent formation producibility, F, was6.4 x 10-3 bbVmin/psi. The constant, CS2 , was estimatedto be 4.3 X 10-2 S-I, and hence we calculate the influxcompressibility, Cin' as 2.5 x 10-4 psi-I; about 50 timesthat of oil. Clearly the influx was gas.

The difference between the shutin drillpipe and casing pres-sures was 63 psi, and an influx of 7.5 bbl occupies 259 ftin the annulus. The field calculation gives an influx den-sity of 3.9 ppg, compared to an actual influx density of1.4 ppg. The standard method wrongly suggests a inixedgas/liquid kick. This error can be attributed to small errorsin the measured pit gain and surface pressures, to which thecalculation given by equation 4 is very sensitive.

By calCUlating the position of the influx using the methoddescribed in the appendix, we find that the gas reaches thecasing shoe at about the same time as the kill circulationbegins. We estimate a maximum casing shoe pressure of1390 psi which decreases during the kill as the gas passesthe shoe. The measured casing shoe pressure is shown infigure 10, where it is clear that the maximum casing shoepressure during the kill is about 1400 psi. The standardmethod predicts a maximum casing shoe pressure about50 psi higher.

Validation of the analysis methods presented here continueswith the collection of field data.

Standard kick calculations can be significantly im-proved and currently do not make efficient use of theinformation available in surface measurements.

337

6 ADVANCED INFLUX ANALYSIS GIVES MORE INFORMATION FOLLOWING A KICK SPE/IADC 25710

I..n

PdPd/rp!PlrPhPhePmaxPnozz!rPwpwUqqdrVr

ThThettotetvtvtKto!!tuV;nVmVgV,lX

!::J.p!::J.Pd!::J.qepinpge

PinPmLdrLdr+ann -w

axial distance from bit to casing shoeconstant in pit gain curve fitdrillpipe pressurefrictional pressure loss in drillstringfonnation pressuretotal frictional pressure losspressure at midpoint of influxpressure at midpoint of influx when t = temaximum casing shoe pressurefrictional pressure loss across bit nozzlesbottomhole pressurebottomhole pressure when t =tuvolume flowratepump rate during drillingvolume flowrate from fonnationbit radiustemperature at midpoint of influxtemperature at midpoint of influx when t = t etimefonnation time scaletime when fonnation stops flowingdimensionless fonnation timedimensionless fonnation time since penetrationtime when influx startedtime when pumps are stoppedtime when U-tubing has recoveredinflux volumevolume of mud in wellgas velocitygas slip velocitydistance from top of fonnationfonnation underbalancefonnation underbalance during drillingdelta flowinclination of open-hole sectioninflux viscositygas density when fonnation stops flowinginflux densitymud densitysum over all sections of drillstringsum over all sections of drillstring and annulusfonnation porosityrate of penetration

[4] Press, W.H., Flannery, B.P., Teukolsky, S.A. and Vet-terling, W.T.: Numerical Recipes. Cambridge Univer-sity Press. (1986) 523.

[5] Dake, L.P.: Fundamentals of reservoir engineering,Elsevier (1978) 154.

[6] Govier, G.W. and Aziz, K.: The flow ofcomplex mix-tures in pipes. Krieger (1972) 322.

[7] Collins, R., De Moraes, F.F., Davidson, J.F. and Har-rison, D.: "The motion of a large gas bubble ris-ing through liquid flowing in a tube," J. Fluid Meek.(1978) 89, 3,497.

APPENDIX

Derivation of pit gain curve fit

We assume that the bit enters a large. horiwntally-bedded,over-pressured reservoir, where the flow from the fonnationis radial and axisymmetric. The analysis given in [5] showsthat, to a good approximation, the flow rate per unit heightof producing fonnation is,

21rk Apq -;::, - 1 ( ) ... (A-I)

pin 2' In tv + 0.81

Here, the dimensionless time is tv = (t - tK) Ito, thereservoir timescale is to = JPinCinr2I k and the fonnationstarts to flow at t = tK. This result can be used to showthat the rate at which the flow rate changes with penetrationinto the fonnation is

dq 21rk Ap-d =- 1 (. ) ....... (A-2)

X Pin 2' In tv + 0.81

where, tv = {w (t - tK) - x} Iwto.There is no analytical expression for the integral of thisequation. However, we can show that,

References

[1] Jardine, S.I., White, D. and Billingham, J.:"Computer-aided real-time kick analysis and control,"paper SPE/IADC 25711.

[2] Moore, L.P.: Drilling Practices Manual, PennWell,Tulsa (1986) 497.

[3] Johnson, A.B. and White, D.: "Gas rise velocities dur-ing kicks," paper SPE 20431.

338

k!::J.p n-lq ex: - (t - tK) (A-3)Pin

By integrating this expression with respect to time we obtainthe pit gain as,

G -;::, CK (t - tKt (A-4)

This equation suggests that we can fit a curve of the fonngiven by equation I to the measured pit gain data. In equa-tion 1 the constant Go allows for a possible constant offsetin the pit gain data.

SPE/IADC 25710 J. BILLINGHAM, M. THOMPSON" AND D. WHITE- 7

Derivation of drillpipe pressure curve fit aftershut in

After the well has been shut in and V-tubing has recovered,the well is sealed and full of a mixture of drilling mudand influx fluid. The fonnation continues to flow until thebottomhole pressure is higher than the fonnation pressure.The rise in wellborepressures is driven by two mechanisms:

1. The continuing influx "pumps up" the closed systemof wellbore fluids.

2. The migration of the influx fluids carries the bottom-hole pressure upwards.

This can be expressed mathematically as

dpw/dt = dpd/dt = . (A-5)(cmVm + Cin \l;n)-1 (qJ + Cin \l;nPmgv.d

Note that, when qJ = 0 (no flow from the fonnation), equa-tion A-5 simply gives the rate at which the drillpipe pressurerises because of the migration of the influx. If we then ne-glect the compressibility of the drilling mud (cm = 0) weobtain the standard equation which is used to deduce theinflux migration rate, V.I, from the rate of rise of drillpipepressure, [2].Equation A-5 shows that mud compressibility reduces theestimate of the rate of rise of drillpipe pressure for a giveninflux migration velocity. In the two examples described inthis paper the reduction is about 50% for the numericallysimulated kick and 75% for the experimental kick. In ad-dition, gas solubility in drilling mud (even for water-basemud), the dependence of gas density on wellbore temper-ature and other effects reduce the rate of rise of drillpipepressure for a given migration velocity.

If we now use the simple formation model given by equa-tion 2 to give the formation flowrate, qJ in equation A-5and solve the resulting equation, we obtain

Pw =pwu + CS 1 (1 - e-Cso(l-I" J)for tu ~ t ~ tc

(A-6)

From simple hydrostatics Pd = Pw - PmgD and we arriveat equation 3.

Note that the constant CS2, from which we derive the com-pressibility of the influx after curve-fitting, is independentof the rate of gas migration. The gas rise velocity onlyappears in the constant C S1 . The constant CS2 remainsindependent of V,I when the effects of gas solubility andwenbore temperature gradients are included in the analysis.We can still estimate Cin Vin from the constant CS2

Estimation of the frictional pressure loss in thedrillstring

The pump pressure during drilling gives an estimate of thetotal frictional pressure loss in the system, PJr. We canestimate the pressure loss across the bit nozzles using thecorrelation,

PJrnozz = .!.Pm (~)2 (A-8)2 DNAN

The discharge coefficient, DN , is approximately 0.95. Wenow assume that the rest of the frictional pressure loss isdistributed between each section of the drillstring and theannulus in proportion to the length of the section and theinverse of the area squared. The frictional pressure loss inthe drillstring is then given by

This allows us to calculate the bottomhole pressure at anytime before the pumps are turned off.

Maximum casing shoe pressure

The experimental work described in [3] has shown that slugflow (see, for example, [6]) occurs during a gas kick. Inslug flow, gas rises as a series of large bubbles, each ofwhich spans the whole of the annulus. In such flows, gasvelocity can be detennined from the correlation,

Here, Pwu is the bottomhole pressure at time tu when V-tubing recovers, and

Pw =Pwu + CS 1 [1 - e-Cso(lc-lu) {l- C S2 (t - ten]for t > Ie

CS1 = PJ - PwU + Ci" ~i"PmgV.I/ICS2 = F/ (c;n \.~" + Cm \'~n) ... (A-7)

339

Vg = Cq/A + V.I (A-IO)

Here, q is the total fluid flowrate, A is the annulus cross-sectional area, C is a distribution factor and V.I is a bubbleslip velocity. The correlation is discussed in [7]. This al-lows us to estimate the length of the cloud of influx gasafter the formation has stopped flowing as,

8 ADVANCED INFLUX ANALYSIS GIVES MORE INFORMATION FOLWWING A KICK SPE/IADC 25710

GEOMElRY DATA

Ig = V_I (tc - to!!) + (ACqdr + V_I) (toff - tK)co"

for Ig < Icoll(A-ll)

If Ig ~ 1_. the maximum casing shoe pressure occurs at thestart of the kill. Next. we:

neglect friction in the annulus.

OD 5"Drillpipe ID 4.28"

Length 14708 ftOD 5"

Hcaviweighl-

SPE/IADC 25710 J. BILLINGHAM, M. THOMPSON AND D. WHITE 9

fonnalion SlOpSproducing

I = Ic

V-Iubing recoversI=IU

slart of kick1= IKt

0=00 t::===================------------

+-'.....

0..

time, t time, t

Figure 1: The curve used to fit the pit gain during the kick. Figure 2: The curve used to fit the drillpipe pressure aftershut in.

PIT GAINCURVE FIT

,

CK

~,1------------11

Figure 3: The sequence of calculations. Estimates of im-portant quantities are shown in circles.

void fraction = 0.15

void fraction = 0.25

void fraction = 0.5

void fraction = 0.75

void fraction = 1_.-._-_.-

341

Figure 4: The casing shoe pressure during a simulated killfor various uniform void fractions with the same mass ofgas.

10 ADVANCED INFLUX ANALYSIS GIVES MORE INFORMATION FOLLOWING A KICK SPE/IADC 25710

25data

20 curve fit.......

r-.........

.0

.0 t5'-"

.S .......'C':j

.'

.'

bf) 10 .'....

.......

0.. start of kick5

~ ........0 500 600 7000 100 200 300 400time (s)

Figure 5: The curve fit to the pit gain in the numericallysimulated kick.

110010501000850 900 950time (5)

800

r-..1000 estimate

.- data ten0..'-"

curve fit~ 750

, r;:jenen

~ 5000..Q)0...- 250 formatIOn stops0......... producing.......

'1::'"0

Figure 6: The curve fit to the drillpipe pressure in the nu-merically simulated kick.

1oo8000

......._ _.._ predicted maximum...- maximum shoe pressure

after start of kill

1start of

kill

Q6600en0..'-"

6400~;:jen 6200en

~0.. 6000~.8 5800

SPE/IADC 25710 J. BILLINGHAM, M. THOMPSOW AND D. WHITE 11

...

11001000900700 800

time (s)

stall of kick

t600

5data

4 curve fit,.-..........

,.0,.0 3.........,

.5ro 2OJ)......

.-0..

0................-

400 500

Figure 8: The curve fit to the pit gain in the experimentallysimulated kick.

1400I

1350

drillpipe pressure

curve fit

annulus pressure-----------------

I r I

1200 1250 1300

time (s)I

1150

---------------V-tubing ---------------------------recovers /- estimate// I

,/ 'VBOPs shutV/' .// / '0="'00 "0"!: producing

300

250,.-...

.-en2000..

.........,

(1)l-< ISO::lenen 100~0..

SO

01100

Figure 9: The curve fit to the drillpipe pressure in the ex-perimentally simulated kick. The annulus pressure is also'shown to demonstrate that V-tubing has occurred.

predicted maximumstandard method - - .

new method

;,:;- 1500en0..

.........,

~ 1400::lenen~ 13000..~ 1200

..cenOJ) 1100l::.-en

~ 1000 '-.__~_---ll__~____L!--~__-I!L-_~____.JIo 1000 2000 3000 4000

time (s)

Figure 10: The casing shoe pressure in the experimentallysimulated kick.

343

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