grid coning horizontal

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2 FLEXIBLE GRIDIDING TECHNIQUES FOR CONING STUDIES IN VERTICAL AND HORIZONTAL WJ ‘iLS SPE 2556s

ahowrr in Fig.1. Coning

athctiag

he producer

(either vertical

or horizontal) ia then studied ia s two-otep pproach. At tlr~t,

heterogeneity ie

aaaumed

to be negligible snd sverage comtsnt

valueaare used for permesbtity sad porosity. Ins second phue

rezervoir heterogeneity ie included and its

fects on coning are

otudied. Thus four cues are coneidcred:

Ifomogeneoue reeervoir & vertical producer;

Homogeneous reservoir 4X

lroriaontal producer;

Heterogeneous

reeervoir

vertical produceri

Heterogeneous reservoir k hotixontal producer.

For each cue s fine grid simulation (45x33x23 = 3415S Blocke)

iz used ae the reference response througho>i the study. Differ-

rmt hybrid geometries on s coarse Carteuian grid zte then in-

veetigsted aa to their abtity to reproduce the fine grid reeults.

For the vert ical producer, the hybrid geometries me either s

cyl indrical module ( l?ig.4) or a finer Carteaian grid (F~.5) .

For the horizontal producer only C*rtesisn

refinement ie con-

sidered (Fig.8). Thie alao includee the caze in which the weIl ie

not aligned with t ie underlying cosrse grid (Fig.10). The use

of

full

thtee-dirnenaionaf hybrid gridz ia not considered in thiz

p-per, M of the grid rctkrements used here ue special cues

of

Voronoi grid. 9

Reservoir Geometry Data

The reeervoit considered iz of simple rectangular geometry

with *flat top, horiaontd lsyera, znd bottom aquifer (Fig.1).

The reeervoir dimenaiorw, and the grid used for the r~ference

aolutionl ue summarized ue

followx

Top Depth:

Areal Dmeneiorm:

Thickneaa:

Diacretisation:

Total Celia:

1420 m

L. = 1350 m, Lv = 990 m

40 m p~y

sone

210 m aquifer

Az=Ay=SOm

Az = 2 m in pay none

As = 10 ,40 ,160 m in aquifer

45z33z23 = 34155.

The well locstion~ ue:

Injector:

a=465m

~=4Q5m

z = 1490 m

Vertical Well:

z=885m

U=4915m

z = 1420-1434 m

Horizontal Well: z=885m

y= 390- 600m

z = 1431 m

Injection/Production data

e:

Ir@ctor:

Producers:

A more detaiIed

Appendix.

-300 Std,ms /rfay water rate;

+300 Std.mJ /dav liquid rate

description of the problem ia given in the

Numerical Simulators

AU

hybrid grid caaea were inveetigater using

numen-

cal aimulstor baaed on a Voronoi grid geometry developed at

StanforJ University. A complete description of the airmrhtor

(SIVOR) maybe

found clzewhere. 1316 Only a

brief dizcua-

aion of the Voronoi gndding technique ie given here. The fine

grid reference results were obtained using a commercial code,

ECLIPSE 100.17

The Voronoi Grid

The Voronoi grid u a generalktion of point-distributed

grid. 10 Grid points msy be dwtributed anywhere in the d~

main irrenpect i~e of the pocit ion of any other points. A Voronoi

block ia then constructed around each point by considering a

line that ie orthogonal to, and centered along, a line connecting

two adjzcent points. 9 The flow equation are written by the

control volume finite differencing (CVFD) method, 11M

Geometries such zz Carteeh, cylindrical, curvilinear,

hexagonal and their combhmtions me to be viewed ae spe-

cial cazee of Voronoi grid. It should be noted though, that

Voronoi blockz edwaye have atrzight boundaries and systems

with curved bounduy blockz can therefore only be approxi-

mated.

It haa however been demonstrated thet for or ail prac-

t ical purpoeee thi i approximation ixexcellent provided volumee

and tranmniaaibilitiee

ue

calculated correctly. 9

Beaidee the tkxibility in generating zny type of regular

grid, an interesting feature of the Voronoi grid ie th~t locallY

refined

grids can be connected to baae couae grids in a very

natural w-y. 11

Finally itahorrldbe pointed out thd the above diacua-

aion ie

completely general with regard

to dimeneionzlity. For

the casee investigated here though, only areal Voronoi grids

are

considered. The thkd dimermion iz simply treatzd az a projec-

tion

of

the

ueal geometry.

Comparison of Simulators

Before beginning the study, the new simulator (SIVOR)

nd the commercial simulator (ECLIPSE) were run with iden-

tical data. As shown in Fig,12, the two simulators produce

almost the name responee.

Well Model

For

a complete diacumion on the well model uned by

SIVOR the reader irr referred to 13,16. In the vereion of SIVOR

uoed for thin etudy, only the vertical well model WaII imple-

mented and thus the well indiceu (WI’e) for the horizontal well

had to be calculated separately and given ae input, The r.

for the homogeneous horiaontzI well case WM determined by

a ~emi.an~Yticd method, It The same ro together with the

block permeabilities waz then used to find the V,M for the

heterogeneous caae.

Homogeneous Cases

Vertical WeU

A vertical injector

nd

vertical producer in a homoge-

neous reservoir waz considered fht, But different responses

from both, Grid 1

nd Grid 3, with s vertical diacretisation

given by “a” (F’ig.2), demonstrated early on in the investigation

that even though the refinement

round the producer improves

the shape of the responee it does not improve the predict ion of

398


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SPE 2556? PIE i’RO CONSONNI, MARCO R. THIELE, C*SAR L. PALAGI AND KHALID AZIZ

3

the break-through time (Fig.Is) . Thie ouggeeted the tw-step

pproach of fimt finding

vertical discretixation that would

match the break-through time and subsequently improve the

shape of the reeponee by refining mealfy.

Vertical Layering

The orighml vertical layering for the

fine grid reference caee wed 20 lsyers in the oil sone and 3

luyero in the aquifer. Lewing the l*yers in tht aquifer un-

touched, riudy wu done to

find

the minirmm

number of

layere that would capture the reference break-through time ue-

ing the couee Cai teehn grid, Grid 1 For convenience, thie

part of the study wu undertaken using ECLIPSE.

A number of vertical diecretiration were tried. Vertical

discretisstion “b” (Fig.2), in which the 20 original l~yera were

reduced to 7, two in the completion interval and five in the re-

maining part of the oil Bone, matched the break-through time

best and wae thus used for the rcmtining put of

the

vertical

well caee atudy.

With the vertical dmretisstion fixed d “b”, an areal re-

fi~*ment

study to correct the

shspe O*the response of

the coarse

Cuteeian

grid, Grid 1, wae undertaken. Of the various caem

investigated, only four ue diacueeed here u they capture the

eeaence of all reoultr obtained.

1.

2.

3.

4,

Grid 1- Coaraa Carteehn Grid

Au mentioned earlier, thie grid is not ble to match the

shspe

of

the reference

reeponee

indicating that the detaile

of the the coning phenomenon uound the production wefl

ue not being captured. The reeponee ie shown

Fig.14.

Grid $- Coaraa Carteeiarr Grid with Cylindrical

Module

The cylindrical refinement uound the producer mbstan-

tially clmngec

the

shape of the reaponee u compared to

Grid 1 and, together with the improvement of the break-

through time, now producee a good match of the reference

Ce8e

(Fig.15).

Grid 6- Coerae Carteaian Grid with Carteeiur

Refinement

The reaponae for this caae h shown in Fig. 16 and is com-

puable to thst obtained using Grid 3. This servee to

ithwtrate the fact that for the homogeneous caee the ge-

ometry of the refinement dote not neem to have a major

impact. A fine L

~-tesian

geometry around the producer

u ble to capture the coning phenomenon an well ar a

radial refinement.

Grid 0- Coarae Cartesian Grid with Luger Carte-

sian Refinement

Thie case wea run to verify the dependence

of

the break-

through on the areal extent of the refinement. The re-

cponce u shown in

Fig. 17

with a

break-through that is

slightly shifted.

Horizontal Well

AD for the vertical well, the first step in the analysis of

the horizontal well caae wae to determine

n adequate vertical

diacretisation ao u to capture the water break-through time

uoing Grid 1.

Here, ten

layer model, layering “f”, produced

satisfactory match of the break-through time u u shown in

Fig, 10.

To correct the shape of the curve in

Fig,10 various

eal

refinements were tr ied. It nhould be noted

that in

the z-y plane

the only interesting refinements for a horizontal well ue Cute-

aian,

1.

2.

3.

4.

5.

The following grids were chosen to study the problem:

Grid 1- Coarae Carteaien Grid

Thie wax the grid u~ed to determine the vertical layering

to

match the break-through time. As expected, the shape

of the response does not match the reference case (Fig,18)

oince the grid u too coarae to capture the saturation pro-

file around the well.

Grid

10-

Coarae Cartesian Grid wth Carteaian

Refinement

To improve the shape of the reaponae obtained from Grid

1, a local Cartesian refinement was introduced (Grid

10, Fig.8), The resporme (Fig.19) in improved and now

matches the reference caae even though the break-through

time is tightly off.

Grid 11- Coaree Diagonal Grid

To fur .fier investigri .e the effect of gridding in the hori-

sontti well caee, Grid 11 (Fig.9 ), wan cormtructed. The

rezponse ie shown in Fi,~.20 and is very different from the

response of Grid 1.

Grid X4 - Coarse Diagonal Grid With RArremefrt

To

improve the shape

of the reapomre obtained from Grid

11a Cuteeian refinement aligned with the horizontal well

wae introduced ii;to the coarse diagonal grid (Grid 12,

Fig. 10), The fesponae ie good (Fig.21) and almoet in-

dietinguiahabl, from the reeponse obtained from Grid 10.

Thie is particularly interesting considering that the

re

aponser from the two

underlying

coaue grids were very

dfierent.

Grid 14- Coarae Diagonal Grid WUh Diagonal

Refinement

Finally, this case wu meant to point-out the influence

of the grid geometry. Shown in Fig. 11, Grid 14 usee a

diagonal refinement rather than the

parallel

refinement

of Grid 12. The response, using the same well indices ae

for Grid 12, in shown in Fig.22.

Discussion

Vertical Discretization

Before any tempt could be made to cspture the coning phe-

nomenon by refining areaIly around the production well, a new

vertical discretisation had to be found. The vertical layering

had to be such that the break-through time could be matched

ueing the coarse areal grid, Grid 1. Refining around the well

wae then mfficient to capture the coning and a good match wae

obtained,

Thin two-step approach suggest that the shape of the

wuter-cut is a Iocd phenomenon which depends on the abti-

ity of the refinement to accurately model tho distribution of

water around the well The break-through, on the other hand,

depends ntrongly on the vertical reflnemcid, Thin is mirrored

in the

improvement given by Grid 6 (Fig, 16). Even though the

same

layering M the one

u~ed

to

obtain

the

response from Grid

1 (Fig.14) ie used the break-through is not affected suggesting

that the refinement u local enough to not perturb the baee of

the cone, If the area extension i~ increased, ae in the caae of

Grid 8, then the break-through in changed without modifying

the shape of the curve (Fig 17).

This can tio be seen in the horizontal well cue where

Grid 10 dightly

perturbs the break-through

captured by Grid

1 with layering “f” (compare Fig, 18 with Fig.19),

399


4/10

4 FLEXIBuE

GRIDDING TECHNIQUES

FOR CONING

STUDIES IN VERTICAL AND HORIZONTAL WELLS SPE 25663

Geometry of Areal Rellnement

The refinement itself ix more important than its geometry. This

m-y seem countmintuitive first, puticrduly for the vertical

well, since the coning phenomenon in radially symmetric and

thus

one would expect the radial hybrid grid to fsre better than

the locaUy

Cartecian refined

grid.

But

results from Grid 3 and

Grid 6 (Fig-. 15 d 16) chown thct u long m the blocks are

sufficiently small, the phenomenon can be resolved by either

geometries.

The horiaontai well cues are particularly in teresting in

demonstrating the flexibility and power of the Voronoi grid.

In particukr, Grid 10 ~nd Grid 12 ret utn the came responses

(Figs. 19 urd 21) even though the corme underlying grid M

different. Grid 14, on the other hand, ilhmtratee the problem

thst can uiae if the horizontal weU is not aligned with the

grid and the only refinement rwailsble is a simple Csrtenian

subdivtilon of the comae grid. Practical problems can arise

with this type of grid aa weU,f ~rexample, in the calcula tion of

the WI’s or in the distortion of the weil. This ia probably the

reaaon why the results in Fig.22 are not as good aa the ones

obtairred from Grid 10 or 12.

Heterogeneous Cases

To sea how heterogeneity affects the coning problem and

the abtity of the hybrid grid method to capture it, the geome-

try of the homogeneou~ eeae waa replaced with a heterogeneous

domain . Becaum the heterogeneity is spec if ied on the fine grid

(34155 nods), an

upscaling procedure of the petrophysical

parameter, porcaity nd baohrte permeabfity, wae necessary

for invatigation of the coaraer grid caaee.

U@kaJirtg of PowAty and

Absolute Permeability

SIVOR decoupla the numerical grid from the grid on which

petrophyaical date are provided, In other words, petrophysical

propert ied may be specified anywhere irr the domain (property-

points)

nd are then automatically averaged to compute block

voluma, block depths and connection transmiaeibditieti. 9J6

The

veraging though, u the construction of the Voronoi grid,

is done only in the Z-V plane and the upscaiing due to the

reduction in layers had to be introduced separa te ly.

Porosity wu upscaled by constructing an arithmetic av-

erage weighted on block volumes. Permeability, on the other

hand, preaerrted severa l possibilities. The horixontai perme-

abtity component , which b isotropic, was averaged ari thmeti-

cally for each column of block~ lumped together into a single

layer, For the vertical permeability component three different

averages

were

tried: geometric, arithmetic and harmonic as is

shown in Fig.23. These runs were aU performed using Grid 3.

Because the phenomenon hu a strong vertical component, the

harmonic average wu used to upecaie the vertical permeabil ity.

Vertical Weil

To do the vertical upecaling, the fine grid syrtem with 20

orighml layers was again reduced

to

7 layers, u in the homoge-

neous caae. But grouping considered that layers with %irnilar’

vertical permeabtity near the production well should be com-

bined. A dightly different verticai discretiaation scheme, caUed

“c”, resulted (Fig.2 ). The eai effect of diacretisat ion was then

invatigated through the foliowing cases.

1, Grid 1- Coarae Carteaimr Grid

Tbia

grid waa not able to

capture

the reoults of

the

ho-

2,

3.

4,

mogenecw cue and thus is not expected to perform any

better here. The caae war run pnmarly ae a reference to

see how refinement would improve reaultc. The water-cut

response is shown in Fig.24.

Grid 9- Coarae Carteaian Grid With Cylindicei

Module

A first attempt to m~tch the response of the heteroge-

neous reservoir us ing Grid 3 was addressed in the preced-

ing diacuaaion on upncaling, While Grid 3 wae able to

capture the homogeneous c-e it warJno longer adequate

for the heterogeneous reservoir an Fig.23 demonstrates.

Grid 7- Coarae Cartesian G :id With Fine Cylin-

dricei Module

In an attempt to match the fine grid response, the areal

refinement of Grid 3 was increased both areaUy as well ae

in the number of points around the well resulting in Grid

7 (Fig.6). The resporme wan good even though itacked

some of the detaila of the reference caee (Fig.25).

Grid 8- Coarse Cartesian Grid With Cartesian

Refinement

This grid (Fig.7) more or less has the same areal extension

as Grid 7 but less blocks overall which ir probably why

the response (Fig.26) is not ar good aa the one returned

by Grid 7.

Horisontai Weil

The same layering echeme waa used here aa for the homo-

geneous part (Layering “f”) . The area l grid geometr ies consid-

ered

1.

2.

3,

4,

5.

were:

Grid 1- Coarse Cartesian Grid

This caae is shown in Fig.27

and

is not expected

to

match

the reference renponse. It ie pretented to show how sub-

sequent local refinement improvee the match.

Grid 10- Coarae Carteaimr Grid with Cartesian

Retlnemcnt

As in the homogeneous caae, thin Cartesian refinement is

sufficient to match the reference response as is shown in

Fig.26. This is probably due to the fact that the coning

is less severe in the horizontal weU

to

begin with and the

heterogeneity is ‘less’ of a factor than in the vertical well

caae.

Grid 12- Coarse Diagonal Grid With Refinement

Also in thin case the rotation of the coarse underly ing grid

does not affect the response which in shown is Fig.29 and

the loca l refinement is sufficient to capture the gradient

of the water saturation.

Grid 14- Coarse Diagonal Grid With Diagonal

Refinement

The response is illustrated in Fig,30 mrd, contrary to the

homogeneous caee, does not reproduce the fine grid result.

T% is probably due to the fact that rotating the refine-

ment now gives rise to a different permeability distribu- ,

tion around the well became of the averaging procedure.

This averaging also affecta

the

weU indite..

Grid 14 (Shifted) - Coarae Diagonei Grid With

Diagonal Refinement

This case waa run to nhow the influence of the averaging

procedure on the response, Here, the refinement of Grid

14 was simply shifted by half a block diagonal in the y-

direction giving rise to another permeabil ity distr ibut ion

400


5/10

SPE 2SS6S

PIETRO CONSONNI, MARCO “R.THIELE, Cl XAR L. PALAGI AND KHALID

AZIZ

5

around the well due

to areal tveraging. The result is

shown in

Fig.31

and is,

in fact, quite different from the

previorm case

(Fig.30).

Discussion

The uet eiTectof the heterogeneity is to extend the coning

phenomenon aredly snd thus force a laryer extension of the

grid refinement. Thu is suggested by Gri i 3 which is no longer

mdlfcierrtto capture the saturation gradie~ttaround the vertical

well, while Grid 7 returns an acceptable mdch (Fig.25). For

the horizontal case both, Grid

10 and Grid 12,

re large enough

to begin with and thus mstch the response as ie shown in Fig-.

28 and 29 which in part is dso due to the fact that coning b

leas severe in horizontal well.

The

two-step approach in

m-tching the reference case by

firstinding

the

correct break-through using the

vertical dis-

cret ination and then refining aredly

to

correct the shape of the

response is applicable but not as robust as in the homogeneous

cases. This is due to the additional difficulty of ‘correctly’ aver-

aging permezbfity for the various grids. This is well ilhmtrated

for

the horizontal well cases where the only grid able to repro-

duce the reference response is Grid 10 which has

a

refinement

thst does not require an upscding in the z - y plaoe Ace itc

cells are aredly equal to the original fine grid. This is shown in

Fig.29. Grid

14, on

the other hand, which requires areal verag-

ing is not sble

to

capture the reference reeponse. Furthermore,

the results change significantly if the rotsted refinement of Grid

14 is transhted by mall amount in the y-duection (Figs. 30

and 31).

Conclusions

b

For vertical wells the refinement itself is more important

than its geomet~y u long as there are enough points

uound the wells. Heterogeneity requires thst a larger

area be refined in order to capture the cone.

For horisontxl wells the Voronoi grid becomes particu-

larly useful as it avoids orientation problems by aligning

the refinement along the well. This also has practical

implicatiorm when it comes to the welf model,

If heterogeneities are considered, then the dominant fea-

ture to resolve remains the upacaling of the petrophyaicai

propertied, irrespective of the type of grid used to model

the phenomenon. The flexible grid must be robust with

respect to rotation of the grid on the underlying petro-

physical distr ibut ion. This is part icularly important when

the refined module is rotated to be aligned with the hor-

isontsl well,

Ackn~wledgments

The

uthors would like to thank AGIP Spa for the support

given

to this work, Irrparticular, we

would like exprees our

gratitude

to Antwrell t Godi snd Davide Ddl’Olio for preparing

the

data set

and reference cues, Pietro Consonni would dao

like to thank all the people of the Petroleum Engineering Dept t

Stanford University for their kind assistance, The work at

Stanford was supported by the Reservoir Simulation Industrial

AtRliates (SUPRI-B) progrtm,

1.

2.

3,

40

6.

6.

7.

6.

9.

10,

11.

12.

13.

14,

15.

160

References

Chappelear, J.A.

and Hirasaki,

G.J, :

“A Model of Oil-

Water Coning for Two Dirneneiond, Areal Reservoir Sim-

ulation, SPEJ Apr. 1976, pp. 6S-74,

Woods,

A.G.

and Khursna, A,K. : “Pseudofunctions for

Water Coning in s Three Dimensional Resevoir Simub

tion”,

SPEJ, Aug.

1977, pp. 253-262.

King, G.R, et al, :

‘A Case Study

of

the Full-Field Simu-

lation of a Reservoir Containing Bottomwater”, paper

SPE 21203 presented at the 1lth SPE Symposium on

Reservoir Simulation, Anaheim, California, Feb. 17-20,

1991.

Chaperon, L :

“Theoret ical Study ofConing Toward Hor-

izontal and Vertical Wells in Anisotropic Formations :

Subcritical and Critical Rates”, paper SPE 15377 pre

sentcd at the 61st Annuxl Technical Conference & Exbi

bition, New Orleans, LA, Oct. S-8, 1986.

Quandalle, Ph. and Bernet Ph. : “The Use of Flexible

Gridding for Improved Reservoir Modell irrg”, paper SPE

12239 presented at the SPE Symposium on Reservoir

Simulation, San Frsncisco, California, Nov. 15-18, 198S.

Wasserman, M.L. : “Local Grid Refinement for Thr=

Dirnensiond Shrmhtors”, paper SPE 16013 presented ~t

the ninth SPE Symposium on Reservoir Simulation, San

Antonio, Texas, Feb. 1-4, 1987.

Ewing, R.E. et al. : “Efficient Uoe of Locally Refined

Grid~ for

MuItiphase Reservoir Simulation”,

paper SPE

18413 ~resented at the SPE Symposium on Reservoir

Simulation, Houston, Texas, Feb. 6-8, 1989.

Pedrost,

O.A.

and Asis, K.: “Use of Hybrid Grid in

Reservoir Simulation”, S R November 1986, 611-621.

Palagi, C.L. and Asis, K.: “Use of Voronoi Grid in

Reservoir Simulation”, paper SPE 22J89 presented at the

SPE Annual Technical Conference and Exhibition, Dal-

1ss, Oct . 6-9, 1991.

Asis, K, and Settari, A. : Petroleum Re:ervoir Simula-

tion,

Applied Science Publishers Ltd, London (1979).

Forsyth, P,A. : “A Control Volume Finite Element

Method for Locxl Mesh Refinement,” paper SPE 18416

presented at the 10th SPE Symposium on Reservoir Sim-

ulation, Houston, TX, Feb., 1989.

Heinemann, Z,E., Brandt, C., Munkx, M., and Chen,

Y,M. : “Modeling Reservoir Geometry with

Irregular

Grids,” SPE 16412 presented at the loth SPE Sympc-

sium on Reservoir Simulation, Hormtonl TX, Feb., 1989.

Palagi, C.L. : Vomnoi

Grid: It~ Genemtion ond

Ap

plicat ion to Reservoir Simulation, Phi) Thesis, Stanford

University, 1992.

Collins, D.A. et al. : “Field Scale Simulation of Horizon-

tal Wells with Hybrid Grids”, paper SPE 21216 presented

at the Ilth SPE Symposium on Reservoir Simulation,

Anaheim, California, Feb, 17-20, 1991,

Economies, M.J, et al. :

“Comprehensive Simulation of

Horisontaf- Well Performance”,

SPIIFE

December 1991,

418,426.

Pdagi , C,L, nd Asis, K.: “The Modelling

of

Vertical and

Horizontal Wells with Voronoi Grid”, paper SPE 24072

presented at the SPE Western Regional Meeting, Bak-

ersf ield, California, Mar. 30-Apr. 1, 1992,

401


6/10

6 FLEXIBLE GRIDDING TECHNIQUES FOR CONING STUDIES

IN VERTICAL AND HORIZONTAL WELLS

S P E 25563

17. ECLIPSE Reference Manuel, Version 88/09, Explo-

ration Consult snta Limited, Henley-on-Thunea, UK

(Sep.1986).

Appendix - Data

Rock Properties.

No use of peeudo functions hns been made in the study.

Rock curvee

re reported in table 1, Rock compressibility was

6.110 X 10-4

bar-i

d the reference preenrrreof 173,s8 bar.

Fluid Propertied.

Table 2 reports oil snd gas PVT functiontr, Both water

ud oil densitim re in reference to the tweet water density

sod given by:

s’J’D= ~.g

P.

P:TD

=

1.03

Form~tion

volume factors and viscosities sre cormideredas lin-

eer functions of preuure nd given by:

B. =

1,2215 – 1.412

X

10-4

X p

p.

=

1.2215-1.176 X 10-’ X p

Bw =

1,00- &s 66 x 10-s x

p

P.

= 0.52 – 3.000 X 10-6 X p

where all the prereurea ue in bar.

Equilibration Data.

No free gas war present in the reservoir at any time dur-

ing simulation. Initiel intion ie computed under t hydroetatic

equilibrium hypothesis, using the following datw

GOC

shove the top depth

woc:

1460 m

Initial

Preuure:

177 bar

Capillsry

Preuure: Obar

WOC

Original Fluid in Place.

The original fluid in place ue reported for both the horn-

geneous and heterogeneous

reservoirs

and are m computed io

the reference casew

homogene~us heterogeneous

O.OJ,P. 6.516639 X

10’

Std . rnt

6.664606 x

10° St&m$

O.W.1.P. 63. 374596 x i Oa Std.ms 63,486694 X 10° Std.rns.

Reeervoir Description

The heterogeneous,

field uced in this study was derived

from a real field under production by AGIP, The procedure to

construct the atktic model considered the following ctepe:

1

a.

3,

An unconditional simulation of a normsl score population

using multigsueehn model, A sphericsi variogrsm with

s 0.2 nugget effect, 0.S dil,

1000m rsnge in the z -

plane mtd 6m range in s

Back-transformation of the eirmktion on the buis of one

redietic population of poraeity,

Extrutiorr of the vxlues d

the well

hxtioas.

s.

0


7/10

SPE 2SS63

PIETRO CONSONNI, MARCO il. THIELE,

CEJAR L, PALAGI AND

KHALID AZIZ

~m

2S0m

I

~1

u

?

40

,1 CM zone m

Ii

\l

LI

aquifer

210

m

crom-sodon x-z)

Figure 1: Problem Geometry

Ita m

Figure 2: Vertical Layering

Figure

3:

GRID 1

I@re 4: GRID 3

1 1

L

1

1

t

I

I 1

I I I I I

I 1 1

H+H+tH+H+Hl

twmm

I--4

I

II

I

I

.I

II

II

I i I

I

I I I I

14+H—H+ttt+ttH

i

I I

I

I

1

1

1 1

, ,

Figure 6: GRID

6

F igu re 6: GRID 7

t I I I I I I I I I 1 I I I 1

ltt—i—tHttnTmn

Figure

7:

GRID S

Figure 8: GRID 10

403


8/10

8 FLEXIBLE CXUDDING TECHNIQUES FOR CONING STUDIES IN VERTICAL AND HORIZONTAL WELLS SPE 2556S

Figure 9: GRID 11

?

j

Figure 10: GRID 12

Figure 11: GRID 14

0.2.5

— WpM

*

Skllfad

020

v

0,1s

0.10

0.05

0 ‘m

400 w

600

T&e(dayo)

FMure 12: Siiuhtot Cosnpmison - Grid

1

0.25

0,20

--- S ha fa d . w dl

0.15

0.10 :

0,05 :

000

0

200

400

600

Soo

Time ( dA y8)

Figure 13: Layering %“ - Grid 1 cnd Grid ~

“’or———————————

0,25

[

4

- — I%iipn(rm Wid)

*

shnrmd

0.20 -

0,1s :

0.10 L

O, m

o m 4C Q

600 E lm

Time (days)

Figure 14: Layering “b” - Grid 1

0.25

LA

didi m

pilrJ

020

0.1s

/

,10

13+1k

0,05

0,00

0

200

400

600

IKx)

Time dA) ’S)


0’30

–—--’—””--------’ “ “ ‘]

0.25

0,20 L

0 15 :

0 10 :

— PkJlpO(flm@d)

sMIlhd

/“’”

41:1

0.05

0,00

0

21xl

400

600 S oo

Time(d9@

Figure 16: Lmyering “b” - Grid 6

404


9/10

“’

4+

SPE

25663

PIETRO CONSONNI, MARCO R. THIELE, C$SAR

L. PALAGI AND KHALID AZIZ

:7

\

02sL

—

iy-JI u@l

O. m :

0.15

0.10

ml

0.05

f

ON ‘

til

400

600

Soo

Tiara ( tiyS)


0.20

0.1s

I

d

/1

*

*”

.

3__44f&....~

200

T kna (& yS)

Figure 18: Layering “f” - Grid 1

?

0.15

0. 0 :

0 .0s

“ al

o

21m

400

lw o

Soo

Time ( ti yX)

Figure 21: Layring “f” - Grid 12

0.20

0.25 :

020 :

0. :5 :

0.10:

0.05

owl

~

o

200

m

600

800

Tlm (&Ys)

F@re 22: Layering “f” - Grid 14

0. 25

t

—

Scrifu(lilnwia

FIEIEI

0.20 -~

~

0.1s

B

0, 10

0.05

00 0

0 m

400

ml

Soo

T~~ (&y )

Figure 19: Lsyering “P -

Grid 10

025E

= %R%’-X %J

_ .?

,.’

,,..,.,.,,..

$-~, ,~.)

,,..”

. . . .

...0.

~~(”~,m)

,.

..*’ ....

,..’”

,,...’

,..

f

.,,

*

0.10

m

.”’’’..:’”::-’”

.,” .’” ,0

0,05:

..*’

,,$.,”

“

,.$+’

,,,

,,,..:,,,, < ,

, , , ,

O XI

.’

o

2rm

—--’%0

00

Soo

Tlma ( ti yl )

Figure 2s: K-Averuging with L*yming “c” - Grid 3

0,1s

0.10

0.0s

Om

o

m

m

600

800

405

0,25

\

— l dlpu nnt8i4

SMrmd

M

02 0

i

0.1s

0,10

0.0s

0.00

0

200

400

6(X2

I

Tknr (w)

Figure

24: Layering ‘c” - Gtid 1

Q

I

10

T- (days)

Figure 20: Lsyerinc “f” -

Grid 11


10/10

10 FLEXIELE GRIDDING TECHNIQUES FOR CONING STUDIES IN VERTICAL AND HORIZONTAL WELLS SPE 2666S

o

0

0

0

0

0

0

‘OF————’—--------7

.1s

. 10

5

1.00

AA~

’200

Wlr

eoo

Time (ti)’S)

Figure 25: Layering “c” - Grid 7

0.23

1—

dlpm(rhlo Srid)

e

SWdd

3+

---

020 i

?

1

.15

1

0 .1 0 :

0 . 03 :

0.00

1

0

2WJ

400

600

Soo

‘rime (dw)

i?igure 29: Layering “f” - Grid 12

).2s

1

—lcrps(llm 8M)

Smnrad

Id

)20

d

).1s

).10

1.05

No

o

“ a

400

Wo

800

Th m d sy s)

Figure 26: Lsyering “c” - Grid 8

I

25 .

I

—rl~ubndd)

Sdd

I a]

-.

02 0

0.15

0.10

I

0, 05 :

0.00

0

200

400

Mm

800

T i me d sy l )

Figure 27: Layering “f” - Grid 1

0 30

—v—-

0.25:,

— W& rm @)

M?

020 :

0.15:

>1

/

O,to :

0,0s

0 00

A—

0

200

400

6(KI

800

Thw(dSY@

Figure 28: Layer~a6 “f’ -

Grid 10

,

?

0 .1 s :

0 ,1 0 :

*

,.-J*

.**

0.0s

*

, * *

,,.0

,*.*

0>00

d]

200 400

600

- 800

Time dsys)

Figure 30: Lmyering “f” - Grid 14

0,25

I

— I}krl pw ilnnuld

e

smnrd

%4J

,.

0,20 .

?

k

~

0 15 .

grid coning horizontal

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