iN SITU, 15(1), 87-114 (1991)

NUMERICAL SIMULATION OF TRANSIENT GAS FLOW DURINGUNDERBALANCED DRILLING INTO A GAS SAND.

by K.A.Berg and P.SkalleDepartment of Petroleum Engineering

University of Trondheirn

and

A.L.PodioDepartment of Petroleum Engineering

University of Texas at AustinAustin, Texas 78712

ABSTRACT’

Shallow gas drilling has long been recognized as a serious problem inoffshore operations. Low fracture gradients and shallow casing do not permitshutting-in the well. Computer simulations of gas kicks during drilling requireaccurate description of the gas flow rate from the formation into the welibore. Theproblem is complicated by the fact that during drilling into a gas sand the effectivewelibore area exposed to flow is continually changing until the formation has beencompletely drilled.

This paper describes a numerical model developed to calculate gas flow intothe wellbore while drilling underbalanced into a gas sand. A two-dimensional finitedifference model of transient flow from the reservoir has been coupled with a one-dimensional finite element model of two-phase flow in the welibore.

A 1-D version of transient flow model in the reservoir was implemented forkick simulation purposes to minimize CPU time. The difference between the 1-Dmodel and 2-D model was found to be insignificant for the cases studied.

Accurate transient model representation of gas influx results in significantdifferences in the kick simulation performance, especially for the cumulative effectsover some simulation time. It is recommended that the transient flow behavior ofgas from the formation should be considered in development of gas kicksimulators.

87

88 BERG, SKALLE, AND PODIO

INTRODUCTION

Several well control simulators have been developed in recent years1’2’3.

Computer simulation of the pressure conditions in a well when formation gas is

entering the wellbore is useful, both for personnel training and for analyzing

different aspects of the well control process in more detail. In the latter case, it is

especially important that the model reflect reality as closely as possible. When

developing a mathematical model describing the well control system, one must take

into consideration the gas flow into the welibore from a gas sand during

underbalanced conditions. The results presented in this paper show that the

treatment of gas inflow has a great influence on simulation results. Hence, a model

that gives a correct description of the transient behaviour of the gas reservoir in

response to welibore pressure is desirable. The gas reservoir should he treated as a

separate model coupled to the welibore. The simulators mentioned above do not

have a complete description of the gas inflow, since they include assumptions

which oversimplify the gas inflow model. This makes the calculations of the gas

inflow easier and faster, but does not, however, produce satisfactory results.

A numerical solution of the difitsivity equation reflects the true transient

reservoir behaviour. In the case of kick simulation, an analytical solution to the

diffusivity equation can not be obtained, because it is impossible to establish the

proper boundary condition in the weilbore. The weilbore pressure at the gas sand

level can have any sort of variations during a simulation, depending on practically

every input parameter and the actions of the operator. Hence, a numerical solution

is the only practical way to calculate the gas inflow due to variations in wellbore

pressure.

PHYSICAL DESCRIPTION OF TIlE WELLRORE -RESERVOIR SYSTEM

The weilbore pressure is composed of the hydrostatic mud pressure in the

annulus plus frictional losses caused by flow of mud and gas in the annulus. Flow

from the reservoir is initiated whenever the weilbore pressure is lower than the

reservoir pressure. Underhalance is often experienced when drilling with low mud

density into a shallow gas sand. Drilled gas or minor gas influxes from the sand

may also cause the well to become unstable. This typical situation, as illustrated in

Fig. 1, forms the physical basis for all the subsequent simulation runs that were

used to test the numerical reservoir model.

0?

FIG. 1. Drilling into a gas sand. Typical drilling situation fortesting the numerical simulator.

TI IEORY

The Mathematical Model

Transient gas flow in a porous medium in radial and vertical directions isdescribed by the diffusivity equation:

1 _Pr dr [z drj dz Lz - q -

which is obtained from the three principles of- mass conservation- Darcy’s law- gas equation of state

In Eq. (1) Z represents the compressibility factor, y hydrostatic pressuregradient, porosity and k permeability. The parameters r and z are distance inradial and vertical direction respectively while t is time. When establishing amathematical model describing gas flow from a reservoir towards the wellbore, onemust take into consideration the fact that the welibore pressure at the gas zone istime-dependent. The welibore pressure will depend on the actions of the operator,

90 BERG, SKALL, AND PODIO

which may cause both increases and decreases in weilbore pressure. Thus, the

model must allow any sort of variation in welibore pressure. This means that the

diffusivity equation can not be solved analytically, since no boundary conditions

can be defined in the welibore and a numerical solution technique is required.

The finite difference method, which is well known from the field of reservoir

simulation, is applied4. The gas reservoir is divided into a grid system as shown in

Fig. 2. The number of gridblocks in radial and vertical directions will influence

computer time; therefore it is recommended to use relatively few gridhiocks. In the

simulations presented below, 10 gridblocks were used in both vertical and radial

directions. This requires that one gridblock is penetrated each timestep in order to

avoid partially penetrated gridblocks. Hence, the timestep is adjusted accordingly.

Normally, when drilling underbalanced through a gas sand, the rate of penetration

is relatively high, known as a “drill-break”, so that the timesteps will be small. The

gridhlocks are indexed with I andj in horizontal and vertical directions respectively.

Since central differences are applied, a single gridblock step in the vertical direction

would be termed i,j + 1/2.

Gas Reservoir

FIG. 2. Reservoir and annulus gridhiock systems.

As shown in the Appendix, the diffusivity equation can be written as

Tr,i+ l/2,j(P+ i ,jPj,j)+Tr,i_ l/2,j (Pj_ 1 ,-p,)+Tz,i,j+ l/2(Pjj+1-Pjj-yjj+112)+Tz,i,j- 1/2(P, -1 P jji’ij- l/2)-qg=

±[(Cg+Cr)] (pi,j-p)At i.i (2)

in which Tr and Tz are the transmissibilities in radial and vertical directions, while qis the flow rate. As an example Tz,jj+1/2 is defined as

pk

T— Zi.t i,j+112

z,i,j+l/2 —Az2 (3)

The gas and pore compressibilities are defined by Cg and Cr respectively.

The initial conditions are given by static conditions with the pore pressure asinput to the simulation. The boundary conditions are given as no-flow over thevertical and outer radial reservoir boundaries.

Eq. (2) can be rearranged to form the following linear equation:

Pi-i,j + pi,j—1 + Pi,j + Pi,j+1 + C,j pi+1,j = (4)

known as implicit formulation. The coefficients are all functions of thetransmissibilities, Tr and Tz. The gridblock pressures can now be solved directlyby the solution of NR x NZ linear equations with NR x NZ unknowns. Arrangingthe equations as a matrix, the gridblock equations yield a pentadiagonal matrix.

The two-dimensional reservoir model presented above is intended to be usedon a mainframe computer. On a personal computer the solution of thepentadiagonal matrix will be too time consuming, with up to 20 s for one timestep.In modern work stations time consumption will decrease. This problem is heredealt with by introducing a one-dimensional reservoir model for use on personal

92 BERG, SKALLE, AND PODIO

Partialpenetration

FIG. 3. The one-dimensional reservoir model.

I

computer simulators. In this case, only flow in the radial direction is assumed as

shown in Fig. 3, and the coefficient maix will be reduced to a tridiagonalmatrix.

The effect of partial penetration of the gas zone is taken care of by introducing the

skin-factor, s, given as5:

=- i [ n ( - 2] (5)

where is the partially penetrated part of the reservoir.

Computation of the solution of the tridiagonal matrix is very fast by use of the

“Thomas - algorithm”, so that the computer time added by introduction of a more

complex gas-inflow model is practically negligible.

Implementing the Model in the Existing Pressure Control Simulator

To study the effect of the new transient inflow model on a simulatedgas

inflow case, it was decided to implement the transient inflow model into anexisting

simulator and to use the pit level change calculated by the two differentinflow

models as the basis of comparison.

A pressure control simulator developed by Podio and Yang3 in 1986 wasavailable. The simulator is interactive and makes it possible to simulate all thesequences from before the start of drilling into a gas sand until the gas is circulatedout by conventional methods. The principal characteristic of the existing model4used in this study is that it is capable of describing the correct distribution of fluidsin the welibore whenever a gas kick is experienced and during the subsequent wellcontrol operations. Most of the other published kick control simulators lump thegas into a single section. The model also accounts for multiple kicks, which maydevelop during drilling and/or well control activities. The welibore geometry maytake any realistic shape, for instance the large change in welibore diameter at theeasing/conductor/riser interfaces. The simulator was successfully applied tosimulate the blowout on Haltenbanken off Norway in 19866. The inflow modelapplied by Podio and Yang to calculate the gas influx i:

2 2(Pf

-

h k

qg=clog(R) (6)

where R is the outer radius of the reservoir affected by gas flow. During thecalculation R is increased in proportion to qg, thus attempting to take care of thetransient flow behavior.

Since the gas inflow model was contained in one subroutine, it was simple tointerchange the two different inflow models in the Podio and Yang simulator. Thissimulator is based on an iterative solution technique, and the 2-D gas inflow modelwill therefore be solved several times for every tirnestep. A 2-D model with 8 x 9blocks requires tip to 10 seconds for every timestep.

This problem was solved by implementing the l-D reservoir model, describedabove. The 1-D model simulates flow only in the radial direction; only one verticalgrid block exists. Partial penetration into the reservoir is taken care of by includinga skin factor (Eq. 5) in the productivity index. The two models, 1-D and 2-D, wereimplemented in the Podio and Yang model, and a 5-ft reservoir was drilled through.The results are presented in Fig. 4, which shows that the 1-D model is a goodapproximation of the 2-D model, especially after complete penetration at 41 secondsof drilling. But even for partial penetration of the reservoir the two models comparewell.

25.0

(J•)

20.0C-)

Ln

x 15.0 -:34-

_____

C

1 0.0

5.0 -

I I I I I I I I I I I III I I I I I I I I Fj

0.0 20.0 40.0 60.0 80.0100.0

Tid (s)

94 BERG, SKALLE, AND PODIO

Pt = 9Obax

85bar

35.0-: h = 5ft

k = lOOrnd

:30.0/

/

/

/2-D

1-I)

0.0

FIG. 4. Comparison between l-D and 2-D reservoir models.

The model has been tested extensively. It yields realistic results and the

calculated pressures and pit level profiles agree with those of a published blow-out

case6. All calculations were undertaken on a VAX computer system.

SIMULATION RUNS

The objective of the simulation runs was to show what influence the new

gas inflow model has on pit level variation arid compare it with the existing model

in the Podio and Yang simulator. A sensitivity study was performed on the two

inflow models, the Podio and Yang model indicated with OLD and the new gas

inflow model with NEW. The following parameters were varied:

* Pressure differential

* Permeability

* Penetration depth

* Pump rate, after drilling 5 ft into sand

t;m iftpr 1rjBini 5 ft into sand

FIG. 5. The drilling configuration during the shallow gas blowouton Haltenbanken, Norway, in 1986.

Table 1. Base Data Applied For All Simulations. The Subsequent SimulationStarted at 390 Seconds.

Time Depth ROP Flow rate EventMm Sec ft m fl/mm GPM0 0 1657 505 0 0 Connection3 180 1657 505 0 465 Start circulating4 240 1657 505 148 687 Drilling started6.5 390 1662 506.5 148 687 The bit is now

immediatelyabove the gas sand.

DRILL FLOOR

_______________

TO MUD CLEANING

0 (Om)

S’YSTi

DIVERTER PIPES

— ANNULARBOP

SLIP JOINTwrr SEALING

72 (22)

797 (243)

1043 (318)

MwcGAMMA RAY

1529 (466)

1562 (476)

1596 (486.5)

1627 (496)

1660 (506)

1692 (516)

1715 (523) .4— BIT (122)

BERG, SKALLE, AND PODIO96

Influx whenprevious influxat surfaceInflux when first

influx arrives

at surface

First influx

9.5 OLD

9.5 NEW

0 1000 2000 3000 4000 5000

TIME (SECOND)

IOLDI

10.0 PPG

10U)IU

80

z 6

thC, 4

2

0

400

______

-J300UJ-J

I200

100

0

— 400-J

_____

0

-J300

Lii-J

I

200

100

04000 5000

TIME (SECOND)

FIG. 6a. Effect of underbalance. Mud weight is 9.01 PPG. Top graphshows influx of gas at the bottom of the well for a pore pressureof 9.5 PPG.

9.5 PPG

0 1000 2000 3000 4000 5000

TIME ( SECONDS)

Pm9.O1 PPGk = 10 mDh = lofthp = 5 ft

lwl

11.5 PPG

11.0 PPG

10.5 PPG

10.0 PPG

9.5 PPG

0 1000 2000 3000

400

DnIIed gas at surface lIEWI

9.5 PPG

11.5 PPG

pp= PPG

11.5 PPGm = 9.01 PPG

k= lOmDh lofthp = 5 ft

TIME (SECOND)

FIG. 6b. Effect of underbalance on gas influx at bottom, bottomholepressure and gas flow at surface for the NEW model.

Influx whenprevious intl

Influx when first at surface9.5 OLDinflux arrives

at surf

NEW

•04000 5000

TIME (SECOND)

I I I ——

0 1000 2000 3000

8

6

4.

2-

0-

800 -

700 -

600

500

U,

I—L.C)U)

z(1)IXC,

U)0

w

U)C’)w

0

0

C,

ID0

Infuxed gas at surface

10000

100

80

60

40

20

0

2000 3000 4000 5000TIME (SECOND)

Drilled gas at surface INEWIInfluxed gas at surface

11.5 PPG

9.5 PPG

I I I — p

0 1000 2000 3000 4000 5000

98 BERG, SKALLE, AND PODIO

Base Data Set

The data applied in the simulation were all taken from the published data6

mentioned above , consisting of the physical properties of the fluids, the geometry

of the weilbore and drillstring systems, the properties of the formation, and the

sequence of events and values of operational parameters as a function of time.

The drilling operation immediately prior to the blowout is shown in Fig. 5. In

Table 1 the operational variables applied as basic for all simulations are presented.

Pressure Differential

Pressure differential is the difference between the pore pressure and the well

pressure at the bottom. h Figs. 6a and 6b the NEW and the OLD models are tested

by simulating drilling underbalanced 5 ft into a 10-ft thick gas reservoir. The

simulation was repeated for several cases by holding the mud density at 9.0

PPG(pounds per gallon) and gradually increasing the formation pressure gradient

(equivalent density). In addition to the difference between the two models, these

first simulation runs also indicate how the new simulator itself works. Drilling

starts at 240 s. At 390 s the gas pocket is reached and at 500 s the bit has drilled 5

ft into the sand and drilling is stopped. The pump is running at 687 GPM

throughout the simulation. For both the OLD and NEW models the gas production

at the bottom of the well and the resulting pit level change are shown. For a pore

pressure gradient of 9.5 GPM, the following observations can be made. At 400-

500 s the first gas, indicated with 1, has entered the bottom of the well. At 2100

and 3300s second and third influxes are seen. Looking at the surface pit level the

expansion of the gas causes a slow but accelerating pit gain. At approximately

2100 s the first gas reaches the surface and the expansion effect is completed. As

the gas reaches the surface the bottomhole pressure decreases, and a second influx

of gas at the bottom is seen. This gas causes a new pit level increase, indicated

with 2, and the same effect is repeated. At a given pressure differential the OLD

model produces at a higher rate than the NEW model, but the OLD model levels off

at a pore pressure gradient of 11.5 PPG. For a pore density of 9.5 PPG,the bottom

pressure variation and the surface gas percentage are shown for the NEW model in

Fig. 6b. None of the two models caused any complete blowout; the reservoir was

probably too thin for that.

lOOmD 5OmD60

ID

_j 50LU>LU—J 40

I—

3

20

10 mD

10

5 mD

0

3000

FIG. 7. Effect of permeability on pit level after having drilled 5 ft intoa 10-ft gas sand.

Permeability

The effect of permeability on the two models is presented in Fig. 7. It isdrilled 5 ft into a 10-ft gas sand, pump still running at same speed (687 GPM). Nosignificant difference between the OLD and NEW model is detected. The reservoiris probably too small, leading to a small production capacity. Comparing the twomodels in a thicker reservoir shows that the OLD model produces a higher initial pitgain than the NEW model. In Fig. 8 a 75-ft gas sand was penetrated 5 ft, and themud pump was shut off. This is not a normal duration (7000 s = 2 h) for a flowcheck procedure, but other operations may often require a 2 h drilling pause. In thefirst 4000 s (ca. 1 hr.) of simulation the OLD model produces twice the gas influxat the bottom than the transient model. As the pressure differential becomes moredominant, the transient model produces more gas influx at the bottom. At one pointthe two models result in the same inflow, which is at the 765 psi pressure point.Both models cause blowout approximately at the same time but the NEW modelcauses a much steeper pit level increase just prior to the blowout. This is inaccordance with observed pit level behaviour at West Vanguard prior to theblowout.

IN EW I

0 1000 2000

TIME (SECOND)

100 BERG, SKALLE, AND PODIO

40OLD

0D = 9.33 PP-Jw 30 P-,=g.opp

k=lOmD

h 75ft-J

1 20

__________

NEW

10

0 2000 4000 6000 8000TIME (SECOND)

3-

C.)

NEW

0 2000 4000 6000 8000TIME (SECOND)

0

0II—0

760-I—

740 -

NEW

720- • •0 2000 4000 6000 8000

TiME (SECOND)

FIG. 8. Simulation of pump shut-off after drilling 5 ft into a 75-ft thick gassand and its effect on pit level, gas influx and bottomhole pressure.

-J

-JLU>LU-J

I—0

-J

-JLU>Lii-J

I

20ff loft

IOLI

6 ft

3.6 ft

0 1000 2000

2 ft1 ft

TIME (SECOND)

50

40

30

20

10

0

3000

50

40

30

20

4_6 ft

+ 3.6 ft10 4._lOft2 ftift

0 420f13000

TIME (SECOND)

FIG. 9. Effect of drilling deeper and deeper into a gas sand. Pit levelchange is shown as a function of time. Two different reservoirmodels are applied, the OLD and NEW, where NEW representstransient inflow performance. The effect on pit level is the sameuntil 6 ft penetration, then NEW model produces a smaller pitlevel increment.

k= 10 mDPp = 9.308 PPGPm=9.01 PPG

h = 20 ft

INEW]

0 1000 2000

102 BERG, SKALLE, AND PODIU

Penetration Depth

From Fig. 9 we see that the pit level of the OLD model increases with

reservoir penetration depth. In the OLD simulator program this effect is caused by

increasing the gas reservoir pressure with increasing bottomhole pressure. In the

NEW inflow model the gas sand pressure gradient is assumed vertical, which is a

better assumption. However, the vertical grid division in this implementation was

limited to one. The radial flow from the reservoir is thus flowing against a well

pressure equal to the bottomhole pressure, which is increasing with depth. This is

a peripheral error, solved during simulations by a small increment in pore pressure.

Pump Manipulation After a Drilling Break

The effect of different flow rates after drilling 5 ft into a 10-ft gas sand is

shown in Figs. 10 and 11. No significant difference between the two models was

observed in this thin sand layer. However, in Fig. 10 it is demonstrated that, for

flow rates above 300 GPM, annular friction controls the pore pressure. For flow

rates 300 GPM blowout occurs in all cases.

In Fig. lithe pump is shut down for 2, 6 and 16 mm after a drilling break.

Both gas influx at the bottom and pit gain are shown for this flow check simulation.

As in Fig. 9, downhole influx is also in this case affected by what takes place at the

-J

-JLU>LU-J

I0.

k = 10 mD687 GPM Pp = 9,308 PPG

Pn.,=9,ol PPGh = lofthp = 5 ft

80

60

40

20

0

INEWI Q=300GPM

o = 250 GPMo 200 GPM

Q 687 GPM

0=50 GPM

0 1000 2000 3000 4000 5000 6000TIME (SECOND)

FIG. 10. Effect of different flow rates after drilling 5 ft into a 10 ft gas sand.Pump flow rate was 687 GPM during drilling.

INEV1-JILl> 30-

pump shut down16mm

20

687 GPMI 687GPM

r 2mm14 ft/hr

0•0 1000 2000 3000 4000 5000

TIME (SECOND)

INEWILLC)

- 6 mm. pump stop

1

0 1000 2000 3000 4000 5000TIME (SECOND)

Ci, 3P)=g3(cjpp

u Pri = 9.01 PPG tOLDC) k=lOmD

2h 10 ft 5 mm. pump stop

0 1000 2000 3000 4000 5000TIME (SECOND)

FIG. 11. Simulation of flow check. Pump shut down for a certain period oftime after drilling 5 ft into a gas sand. Pit level and gas influx atbottom are shown as functions of time.

104 BERG, SKALLE, AND PODIO

surface. The influx is repeated successively in the NEW model, while in the OLD

model the influx increase levels off at about 4000 s of simulation.

Other simulation runs show similar trends for the two models; the NEW

model creates a lower and a slower pit gain in the beginning of the simulation, but

increases at a high rate when bottomhole pressure has been reduced. The OLD

model causes the pit increase to level off too early.

CONCLUSIONS

1. A two-dimensional numerical model has been developed to calculate transient

gas flow into the weilbore while drilling underbalanced into a gas sand and

has been coupled with a one-dimensional finite element model of two-phase

flow in the welibore.

2. The one-dimensional model gives a reasonable representation of transient

flow of gas into the weilbore and is preferred because of smaller

computational time requirements.

3. The accurate transient model representation of gas influx results in significant

differences in the kick simulation performance when compared with other

state-of-the-art models.. These differences increase as simulation time

increases, showing a cumulative effect.

4. It is recommended that the transient flow behavior of gas should be accurately

considered in development of gas kick simulators.

NOMENCLATURE

c = compressibility constant, dimensionless

Cr = pore compressibility

h = thickness of gas reservoir [m]

k = permeability [m2j

N = number of reservoir blocks

p = pressure [Pa]

q = volumetric rate [m3/sj

r = radial coordinate [m]

t = time

T = temperature

Tr = radial transmissibility

V volume [m3]

z = vertical coordinate [mJ

Z = compressibility factor

Greek

= hydrostatic pressure gradient [Palm]

p. viscosity [Pa s]

= density of mud [kglm3]

= porosity F-]

Subscripts

b = bulk

c critical

f = formation

g = gas

p = penetrated by bit

r = radial

s = standard

w = well

z = vertical

ACKNOWLEDGEMENTS

The authors wish to express their appreciation to SAGA Petroleum for providingartial financial support for this study.

REFERENCES

1. Ekrann, S., Rommetveit, R.: “A Simulator for Gas Kicks in Oil-BasedDrilling Muds”, paper SPE 14182 presented at the 1985 SPE 60th AnnualTechnical Conference and Exhibition, Las Vegas, September 22-25.

106 BERG, SKALLE, AND PODIO

2. Nickens, H.V.: “A Dynamic Computer Model of a Kicking Well: Part 1-TheTheoretical Model”, paper SPE 14183 presented at the 1985 SPE 60th AnnualTechnical Conference and Exhibition, Las Vegas, September 22-25.

3. Podio, A. L., Yang, A. P.: “Well Control Simulator for IBM PersonalComputer”, paper IADC/SPE 14737 presented at the 1986 IADC/SPEDrilling Conference, February 10-12.

4. Berg, K. A.: “Sirnulering av transient gass-innstrørnning ved boring inn i engrunn gass-sone”, Master thesis at IPT, University of Trondheim, (March1988).

5. Lee, J.: “Well testing”, SPE Textbook Series Vol. 1., New York, 1982.

6. Podio, A. L., Skalle, P. and Yang, A. P.:”Analysis of Events, Leading to anOffshore Shallow Gas Blowout, by History Matching Field Data Using anAdvanced Gas-Kick Simulator”, paper presented at the 1989 InternationalWell Control Symposium, Baton Rouge, November 27-29.

APPENDIX. DISCRETIZATION OF THE DIFFUSIVITY EQUATION

The diffusivity equation can not be analytically solved in our case, since

neither constant well pressure nor constant gas flow from the formation can be used

as boundary conditions. Also it is not possible to obtain an analytical expression

for the boundary conditions in the well. Hence, the diffusivity equation must be

solved numerically by the method of finite differences.

The reservoir is divided into a grid as shown in Fig. A.l with NJ? blocks in

the radial direction and NZ blocks in the vertical direction. Each gridhiock has the

same thickness, Az. The term in Eq. 1 that represents vertical flow is

approximated as follows (see Fig. A.2):

zLz az -

rP!(P i rPi(P iI \jj -Vi. ‘-‘‘—LZp JJ2 LZ1 J 2

_(c)

Az

Inserting

(R) 1P(j÷1) -

z j+ Az

Pj-Pi-

2 Az

NR

3Nz

FIG. A. 1. Definition of gridbloeks for reservoir model.

__________

1L,J

2

i,j+

i,j_ 1

il

i,j 41

FIG. A.2. Vertical indexing of gridblocks.

108 BERG, SKALLE, AND PODIO

givesp.-P.E(PI+l-Pi

- -I 1-1 -

[zt Ji+ [z z(*)

(PiS) (Pz) 1— Zi Zi -

— Az2(P+i - - Az)

-

(P - P1 - Az)

(P!z)i (P!S)i= Zi -‘÷2 Zi

Az2(P+1 - - Az)

+ Az2(P - P + i Az)

The radial term in Eq. 1 is first transformed by using the analogy between linear

and radial flow profile at steady-state flow. The flow profiles can be expressed as

follows (see Fig. A.3):

p=A+Bxp = A + Bln(r)

where A and B are constants. The radial term can be transformed by setting s =ln(r),

p

from whichs

r=eS =r eS

p

2: r

FIG. A.3. Pressure profiles for linear and radial flow.

Then we achieve the following:

ia1p iaespkpasasr dr

- es s s

—espk 1 1

— es s zji s e es

= e2S i_ (Pi P) (**)ds

zp.

By use of central differencing, we get

2e2S(-[). 11-f--

(**) 2 (P÷ - P)siQ\si+1+Asi)

2e25(). 1+ 2

(P1 - P)Asi(ASi-l+ASi)

From Fig. A.4 we see that

(r+1in (rL) - in(r) = In

(1Xs÷1 + zs) = 2As = 21n (1)

Each gridblock’s inner and outer radius is automatically calculated as afunction of the number of radial gridblocks and the outer and inner radius of thewhole reservoir. The gridblocks near the well should be smaller than those farfrom the well, because the pressure drop is biggest near the well, as shown in Fig.A.3. The reservoir’s outer radius is usually unknown, but can be given a constant

110 BERG, SKALLE, AND PODIO

i+ ,j i—

FIG. A.4. Radial indexing of gridblocks.

‘big” value in the calculations due to transient flow. The center gridblock’s outer

radius is equal to the bit radius and the inner radius is given a “small” value

different from zero, in order to avoid dividing by zero.

= konst.

(NR - l)L\s = Souter - Sinner

(NR- 1)ln

1’IU

(r. ‘\I = in (router)

tri.,)rinner

/2 2’r. 1 - r. 11+ 1-

r21n (_t2)

r1

rj}- 12 (router)

errj

from which rjnner = rl+ and router = rN ‘R+

For the average radius, r, we use the formula

12

2 2r. I - r I

‘221

_______

In— =2r2) 1

The radial term in Eq. 1 can now be written as

7(Pkr) i— z1

. z12 2 (P÷1 - P) + 2 2 (P1 - Pj

in (‘J) (r. 1 - r. i) in (-n--) (r. 1- r. I)r i+- i-i- rj.1 i+- ‘-

1 1 dZBy use of the following relations: Cg = - ()

1Cr

—

we see that the right side term in Eq. 1 can be written as

-

- Zdt

spat

(P

R 1 ldZpl 1Z)pçg

J3 - z

112

1Forward difference gives (*) — [-- (Cg

BERG, SKALLE, AND PODIO

We now have that the diffusivity equation, Eq. 1, can be approximated as

Tr,i4, j (P+i, P1,) + Tr,jj (Pi, -

+ Tz,i,j4 (P,÷i -- i,j4) + (P,i - + - q

= (Ch5)j(PJ- Pi,j)

where ‘r,i+-, j —

r,i-, j —

2 ‘( )+—zIin (+1) 2 1

2 1—i:;-— (r+_- r. —)2 ‘2

2() 1Zt

ln(”) 21 2i:;-i (r+_- r —)2 ‘2

(P!).. 1= zI_L1J+-

Tz,i,j4Az2

(P) 1— zJIij-

Az2

(Chs)j,j +

pp Øp(--) = (--) (cg +Cr)(*)at

+ Cr)] i,j (Pi,j

-, 4-

FIG. A.5. Radial and vertical flow from gridblocks in the vicinity of the wellbore.

The transmissibilities must be replaced by productivity indexes in the case offlow between a reservoir block and a well block. The productivity indexes arederived by assuming steady-state flow during one times tep. We have chosen to usethe “pressure-squared” equation, which gives the following expression for flowbetween the reservoir gridblock (2,j) and the well gridblock (l,j):

(] 7tAzT 2 2Cisc — ‘. ‘ r’ (P2 -z 2 TresPscln(1+—

2tr1÷iT

Vertical flow: cisc = (-h-)2

Zp 11j TrespscAz

The term, q, in Eq. 1, is given as q = jPsc

Hence, the radial and vertical flow from the gridblocks near the weilbore can bewritten as

114 BERG, SKALLE, AND PODIO

radial flow = (PIr)2,j(p2,j- Pi,j)

vertical flow = (PI)i,(pi,- Pi,j-i)

kr 7tAz(p2,1+ Pi,j)where (PIr)2,j =/ rVbZ1 ,j ln( —

27t I (Pi,j + Pi,j-i)

k

________________

(PI7)1= AZ

This is illustrated in Fig. A.5.