Numerical simulation for a one well

8/18/2019 Numerical simulation for a one well

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Numerical Simulation of Individual

Field Simulation Model

AL I M. AK B AR

M. D. ARNOLD

A. HERBERT HARVEY

MEMBERS SPE.AIME

I

KUWAIT U,

KUWAIT

I

U. OF MISSOURI –ROLLA

ROLLA, MO.

Wells in a

INTRODUCTION

Pressures and fluid saturations in hydrocarbon

reservoirs may be described at any point by

differential equations involving reservoir rock and

fluid properties. Numerical simulation of field

performance is accomplished by establishing some

type of reference grid, writing the appropriate

equations for each mesh poirrt, then solving the

system of equations by a finite-difference t~chnique,

Since the number of mesh points must be finite,

there is a necessary assumption that each mesh

point is representative of a finite segment of the

reservoir. Actually, however, pressures are not

equal throughout such a segment of a producing

field.

This inequality of pressures within an

element of the reference grid creates problems when

the element contains a production or injection well.

Since the finite-difference technique calculates a

pressure that is representative of the entire element,

this pressure is not the bottom-hole pressure of the

well. This situation will exist even though the well

location may coincide with the grid point used to

represent the element.

Furthermore, this characteri-

stic of the finite-difference approximation is not

unique to the pressure calculation. Fluid saturations

computed for a mesh point actually repesent

saturations of a finite segment of the reservoir.

Fluid produced from the area of the wellbore is

handled mathematically as if it were withdrawn from

the entire area associated with a mesh point. Since

the conventional finite-difference technique does not

adequately describe reservoir conditions near a

well, .sPecial mathematical techniques are required

to handle the problem.

AVAILABLE TECHNIQUES

Several methods have been employed to predict

well bottom-hole pressure for numerical simulation

work.

Attempts to use mesh-point pressure as

bottom-hoJe pressure were generally unsatisfactory

Original manuscript received in Society of

Petroleum Engineers

office Sept. 5, 1972. Revised manuscript received Jan, 10, 1974.

Paper (SPE 4073) was first presented at the SPE-AIME 47th

Annual Fall Meeting, held in San Antonio, Tex., Oct. 8-11, 1972.

@ Copyright 1974 American Institute of Mining, Metallurgical,

and Petroleum Engineers, Inc.

lr?eferences listed et end of paper.

AI’ GI’ST ,1974

for reasons previously discussed. A more useful

technique is to reduce permeability arbitrarily at

mesh points corresponding to producing wells, thus

obtaining mesh-point pressures that correspond to

estimated bottom-hole pressures. It has also been

suggested that it might be possible to represent

pressure distributions by means of piecewise-

polynomial approximations.l The technique involves

the use of high-order polynomials to represent the

immediate vicinity of the wellbore. and lower-order

polynomials to represent points more remote from

the well.

Another procedure that has been used with some

success is to estimate bottom-hole pressure by

extrapolating pressures from grid blocks adjacent

to the block in which the well is located. The

extrapolation is based on Darcy’s law written in

radial form and

integrated

for

steady - state

conditions. The result of this integration may be

written

BHP . pa –

~oBoPo ‘n ‘ r i , j i rw )

2rrk k ,. b

where is the average pressure

,. ...

1)

on the edges of

grid blo;k, and ri, i is ‘the radius of a hypothetical

circle with an area equal to that of the grid block.

Although Eq, 1 is entirely adequate for estimating

bottom-hole pressure in some instances, it can lead

:0 erroneous results under unfavorable conditions.

For example, a well with a large drawdown below

bubble-point pressure

may generate a high gas

saturation in the vicinity of the borehole. This

change in saturation reduces relative permeability

to oil near the wellbore. Since the areal model does

not account for this effect and the extrapolation

technique uses the saturation computed by the areal

model, this approach may predict a bottom-hole

pressure that is too high.

Another approach to the simulation of performance

near a well has been described by MacDonald and

Coats2 and by Letkeman and Ridings.3 Techniques

described by those authors use radial coordinate

grids to solve the problems of gas coning and water

coning at individual wells. Since these computa-

tional methods have proved to be useful in solving

coning problems,

it seems logical to extend the

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ideas by combining the radial simulation of

individual wells with the conventional rectangular-

coordinate simulation of multiwell reservoirs. This

study presents the results of such a simulation.

DEVELOPMENT OF THE MODEL

The mathematical model developed by the authors

incorporates

a radial coordinate well simulator

within a two-dimensional, three-phase, Cartesian

coordinate reservoir simulation model. This areal

model is essentially a conventional numericai

simulator that provides for variable g,ritl spacing,

and accounts for effects of relative permeability,

reservoir heterogeneity, anisotropy, and structural

dip.

The well simulator is a one-dimensional, three-

phase, radial coordinate

model

emplo; ed

automatically

for those wells that have been

selected for detailed analysis. The radial modei

was based on the equation

However,

since the circular shape of the radial

model corresponds more closely to a square than to

a rectangle, this study was made with Axi = AYj for

grid blocks that contain wells. The use of square

grid blocks near producing wells is consistent with

conventional modeling techniques. We customarily

use rectangular grid spacing only to represent the

reservoir system at points remote from the area of

primary interes t.

We have not endeavored to determine the maximum

deviation from a square grid pattern that can be

successfully employed with this technique. Until

such an investigation has been made, it is suggested

that the method be used only with square or nearly

square) grid systems,

Another criterion selected to assure correspond-

ence of the two models is that the volumetrically

weighted average pressure within the radial model

equal the corresponding block pressure in the areal

model. Since accumulated round-off error could

2 9

eventually cause a discrepancy in pressures

Bo-f3 R )

calculated by the two models, pressures in the

~ “s or f30po

radial model were automatically adjusted to maintain

s

M.

s

113

r

q

c, 4

s ’

c,’, -

—0 — -

2 .-4’

/30 d ’

Bg

dP

so f3 ,

)

J/>

,—. - —.

B“dl ’ ””””’

) I

. . . . .

2)

To minimize discretization error, it was considered

desirable to use uniform increments in the discretized

equation

.4 However, to minimize computation time,

a fine grid spacing should be used only near the

wellbore, where pressure gradients are large. This

problem was resolved by transforming Eq. 2

according to the relationship s = in r. The resulting

equation was discretized with constant ,1s, thus

achieving both uniform increments of the spatial

variable

and a grid

spacing

that

increases

logarithmically with distance from the wellbore.

Ten radial grid segments were used in this study.

Since the radial model represents a rectangular

or square) grid block in the areal model, it was

necessary to establish criteria for equivalence of

the two systems. One criterion is that the pore

volume of the radial model must equa~ the pore

volume of the rectangular block. Since porosity and

bed thickness in the radial model equal the

corresponding terms in the areal model, the radius

of the well simulator was calculated by

—..

ri j : ~’,lxi.Ayjj’rr, . . . . . . . . . . . . 3)

.

The effect of relative sizes of Axi and Ayj on

accuracy of the simulation has not been investigated.

316

the desired equivalency with areal model pressure.

These small adjustments were made without altering

the pressure gradients calculated by the radial

model.

If the radial model is to represent the system

predicted by the areal simulator, then the fluid

fluxes must be the same for both models. This

condition was achieved by summing the fluxes into

the four vertical faces of the grid block in the areal

model, and considering the total flux for each phase

as influx into the closed outer boundary of the

radial model. It should be noted that small errors in

calculating pressures in

the areal model may

invalidate the flux calculation. Thus it is essential

that pressure residuals in the areal model be quite

low. Several approaches to reduction of residuals

were investigated. One method that was found to

converge rapidly even for highly heterogeneous

reservoir conditions

was the strongly implicit

procedure. 5 ‘

Pressures,

saturations,

and ner oil flux were

calculated by the areal modeI on the basis of oil

production rates that were fixed by computations

external to the model. The total oil rates were

computed by Darcy’s law on the basis of the

gradients

calculated by Eq. 2. All relative

permeabilities were baxed on the upstream fluid

saturations. Water and gas fluxes were determined

in the same manner as the oil flux. It was assumed

that all phases would flow toward the producing

weJ1. Although it is possible for counterblow to

occur in

an oil reservoir,

the assumption of

unidirectional flow is usually valid near a producing

well.

The next step i;, the calculation procedure was

to compute pressures, saturations, and oil production

rate from the radial model.

The volumetrical ly

averaged pressures in the two models were then

compared, and adjusted if required in the radial

model to maintain equality. Water and gas production


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rates from the radial model were then determined

from Darcy’s law,

and production rates for this

model were compared with corresponding flax rates

predicted by the areal model. If any of these rates

were found to differ significantly in the two models,

areal model pressure calculations were repeated

using production rates calculated by the radial

model. This iterative process was repeated until

the two models predicted the same fluid production

rates. Convergence was frequently achieved during

the first computation, with no iteration.

This rapid convergence can be attributed to the

fact that production rates seldom changed signifi-

cantly during a time step. The maximum time

increment used during the simulation was 90 days.

Since the production rates changed slowly, accurate

prediction of these rates for the subsequent time

step could usually be achieved. The predicted

production rates for the areal model were computed

on the basis of the values of Elo, PO,

kfo,

and

potential gradients that were calculated from

pressures

and saturations predicted for the new

time step by extrapolation in the radial model,

It should be noted that the method of computation

can lead to an ambiguous situation. The radial

model may predict the production of free gas while

the areal model indicates that the block pressure is

greater than bubble-point pressure. This situation

arises because the areal model fails to provide an

adequate

simulation

of reservoir conditions near the

wellbore. The problem was handled by comparing

free gas production predicted by the two models.

Any excess gas predicted by the radial model was

considered as additional solution gas production in

the areal model. Since the material balance must be

maintained in each model, ??~ in the areal model

was

reduced to account for rhis excess gas

production.

APPLICATION OF THE MODEL.

The model has not yet been tested against actual

field performance .

However, tests that have been

conducted indicate that the technique is successful

in combining individual well simulation with the

numerical simulation of an entire reservoir. Table 1

describes a small hypothetical oil field that has

TABLE 1 —

DATA FOR SIMULATION OF HYPOTHETICAL

OIL FIELD

POrO~ity, percent

20

Permeabi li ty, md

50

Thickness, ft 20

Initial water saturation, Well s 1 ond 2, percent 18

Initial reservoir pressure at -6,130 ft, psia

2,855

Bubblepoint pressure, psio

2,172

Oil viscosity at initial reservoir conditions, CP

1.08

Initial soluticm GOR, cu ft /bbl

.573

Reservoir pore volume, million bbl 19.86

Natural woter inf lux

None

Allowoble production rote, B/D/well 197

Water iniection rote, B/D

300

Well radius all wells), in, 5

Number of grid blocks for areal model 10X 10)

100

Size of grid blocks for oreal model

528 ft

X

528 ft

Critical gas soturotion, percent 8.9

been

studied.

The reservoir is homogeneous,

isotropic, and uniformly thick. As illustrated by

Fig.

1, development consists of two producing

wells, one water injection well, and sufficient dry

holes to define the limits of the field. Water

injection is begun 3 years after first production,

Since the injection rate exceeds the production rate,

a gradual ~ncrease in both reservoir pressure and

p:oducrivity index occurs.

The method used to determine production rrtes

requires some discussion. Calculation of these

rates for a mathematical model requires data that

are not included in the differential equations upon

which the model is based. Pertinent considerations

include production method flowing or artificial

lift), proration regulations, separator pressures,

tubing size, etc. Almost invariably, one or more of

these factors will be altered during the depletion of

the reservoir. The relationship between bottom-hole

f]owing pressure and production rate may be

described by an IPR curve. b Although the curve can

be linear, factors such as reservoir stratification

and changes in relative permeability usually cause

significant deviation from a straight line.

Since the problem of predicting well productivity

pertains only indirectly to this study, a simplified

method for specifying production rates was used.

For this example it was assumed that both Wells 1

and 2 would produce at the allowable rate until

TABLE 2 —

RELATIVE PERMEABILITY

so

k

ro

0.18

0

0.20

0.00003

0.25

0.00058

0.30

0.00237

0.35 0.00640

0.40

0.01404

0.45

0.02711

0.50

0.04788

0.55

0.0792

0.60

0.12450

0,65

0.18785

0.70

0.27405

0.75

0.38866

0.80

0.53806

0.82

0.60916

Water saturation = 18 pe, cer,t.

k

rd

0.22126

0.19803

0.14719

0.10590

0.07315

0.04797

0.02935

0.01630

0.00784

0.00297

0.00070

0.00002

0

0

0

Pressure

psio)

200

400

600

800

1,000

1,200

1,400

1,600

1,800

2,000

2,172 BP)

2,400

2,600

2,855

TABLE 3 — FLUID PROPERTIES

Formation

Viscosi ty cp)

Volume Factor

Solution Gas

Oil

Go S

‘Oil

GO;

scf/STB)

—. —

2.0989 0.010

1.0465 0.073S

74.6

1.9649 0.011

1.0675 0.0368

125.2

1.8371

0.012

1.0931 0.0245

175.8

1.7155 0.013

1.1216 0.0184

226.4

1.6000 0,014

1.1580 0.0144

277.0

1.4910 0,015

1.1973 0,0119

327,6

1.3875 0.016

1.2411

0.0096

378.2

1.2905 0.017

1.2895 0.04)83

428.8

1.1997 0,018

1.3425 0.0073

479.4

1.1150 0.019 1.4000 0.0065

530.0

1,0471 0.0199 1.4531 0.0054

573.5

1.0585 0.021

1.4481 0.0053

573.5

1.0685 0.022 1.4438 0.0049

573.5

1,0813 0.0233 1.4383 0.0045

573.5

41’ G1’ST. 1974

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/

\

\

\

\

\

\

\

‘

00

Contouredonlop

of productive

formotnon

. otl well

0 vfoter,nject,on well

0

500

FIG. 1 —STRUCTURAL MAP OF HYPOTHETICAL OIL

FIELD.

30

20

t,. 0.?04

yeot

___ Ateol Model

tp = 04136

yOOf

_ Combmmd Areal and

RodmiNadele

/3 : 062

yoof

t4= )52

years

t 5 : 301

years

16:50

ywrs 2y80r s a ft er ft rs t m ject tm )

“o

100

200

300

400 500

DISTANCE, FEET

FIG.

2 — CALCULATED DISTRIBUTION OF GAS

SATURATION.

?400

1

\

moo

I,

:

\

\

: \

s

,600

\

:

\

\

--- A.,. ..4,3

\

:

— co b . . ...8 ..d . , 6. . 4 . 0, *

? (200

\

\

i

\\

s

\

~ \

800

\

g

L.

~.

;,,.,?., .0,,,,., fl,w

-.”

~ 4

_ _

J__.__J ‘

1

2 3

4

5

,,”f. “fans

FIG. 3 — CALCULATED PRODUCING BOT’IY)M-HOLE

PRESSURE, WELL 1.

318

bottom-hole pressure was reduced to 200 psia.

Thereafter, bottom-hole pressure remained constant,

and production rates declined accordingly. To

provide a basis for comparison, reservoir perform-

ance was calculated both with and without radial

simulation of the two producing wells. Use of the

radial simulator increased computation time by

approximately 15 percent. The areal model used for

comparison included an extrapolation technique for

estimating bottom-hole pressure.

Although the same criteria were i.sed to determine

reservoir behavior for the two metho.~s of simulation,

significant differences

in predicted reservoir

performance

were

observed. As illustrated by Fig.

2, the combined radial and areal models predict an

early buildup in gas saturation near the production

wells, whereas the areal model without the radial

simulator does not anticipate this effect. Figs. 3

and 4 depict rhe producing bottom-hole pressure and

productivity index, respectively, as calculated by

the two types of simulation models. Fig. 5 indicates

the calculated pressure distribution in the vicinity

of Well 1. The increase in reservoir pressure that

occurs during the fourth and fifth years is a result

of water injection in Well 3. Oil production rates

and calculated GOR are illustrated by Figs, 6 and

7, respectively.

CONCLUSIONS

Individual well simulation can be included in the

mathematical model of a hydrocarbon reservoir. The

increase in computer time required for running such

a model is not excessive,

provided that a one-

03

G

n

:

<

0

lx

w

a

-J 02

m

o

, fr ons en t per f od

b= m

I

I

I

___ Areol Model

I

_ Comb,ned Areal ond ROdml Mode

\

\

\

IStorl of Wofer

I

I

I

I

o

I

2

3

4

5

TIME, YEARS

FIG. 4 — CALCULATED PRODUCTIVITY

INDEX,

WELL 1.

sOCIEIY

OF PET ROI.};I” M EX(;13EKI{> ]01’RSAL


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dimensional well simulator is used. It is suggested

that this type of model be employed to study

reservoirs where pressure drawdown at producing

wells is large, and bottom-hole pressure is less

than bubble-point pressure.

BHP =

B=

‘- =

g.

h=

i=

j=

k=

k, =

pa .

pg .

p. .

Pfu =

q.

9gf =

T

.

R. =

s.

5.

t=

x=

Y=

z

P=

2800

2400

2000

~

f

Iuoo

2

;

: 1200

n

800

NOMENCLATURE

bottom-hole pressure at producing well

formation volume factor

compressibility

acceleration of gravity

thickness

subscript used to denote position on x-

coordintite axis

subscript used to denote position on y-

coordinate axis

permeability

relat ive permeabili ty

average pressure at edges of grid block

pressure in gas phase

pressure in oil phase

pressure in water phase

production rate

free gas production rate

radial dis tance

solution gas/oil ratio

in r

saturation

time

space coordinate for areal model

space coordinate for areal model

depth

viscosity

,

i

yeor s smw mttl al

productmn

w , l l No I

J

ml

200 300 400

300 m

DISTANCE, FEET

FIG. 5 — CALCUJ-ATED PRJ==URE DISTRIBUTION

FROM COMBINED AREAL AND RADIAL MODELS.

AI’(; I”s T,197\

p =

density

@ = porosity

@g = gas

potentiaI, pg - pggz

@o = oil potential, p. - pogz)

@w = water potential, pw -

pwgz

SUBSCRIPTS

1.

2.

3.

g =

gas

o = oil

w = water

REFERENCES

Cavendish, J. C., Price, H. S., and Varga, R, S.:

“Galerkin Methods for the Numerical Solution of

Boundary Value Problem s,” Sot. Pet. .Eng. J. (June

1969) 204-220; Trans., AIME, Vol. 253.

MacDonald, R. C,, and Coats, K. H.: “Methods for

Numerical Simulation of Water and Gas Coning, “ Sot.

Pet. f?ng. J. (Dec. 1970) 425-436; Trarzs., AIME, Vol.

249.

Le keman, J. P., and Ridings, R, L.: “A Numerical

Coning Model, ” Sot. Pej. .@n j. (Dec. 1970) 418-424;

Trans., AIME, Vol. 249.

200

0

a

o

‘. 150

w

s

a

0

Well No. I

—.— Well No. 2

I ---——- -

stort of woh?r reject/on I

I

o

I

2

3

4

5

TIME, YEARS

FIG. 6 — OIL PRODUCTION RATES.

start of wofer injecflon

I

--e+y

riginol

I

solution

GOR

(S74)

-/

200

0

I

2

3

4 5

TIME , YEARS

FIG, 7 —

CALCULATED GOR, WELL 1.


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4. Settari, A., and Aziz, Khalid: “Use of Irregular Grid

in Reservoir Simulation, ” Sot.

Pet. Eng. ].

(April

1972) 103-114.

6.

5, Stone, Herbert L.:

“Iterative Solution of Implicit

Approximations of Multidimensional Partial Difference

S20

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Numerical simulation for a one well

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