sukirman_consolid_mod_using_fin_elem_meth-_1994[1].pdf

8/10/2019 Sukirman_Consolid_Mod_using_fin_elem_meth-_1994[1].pdf

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ocw of PerroleumEngineers

SPE 28755

Three-Dimensional Fully Coupled Flow: Consolidation ModellingUsing Finite Element MethodY.B. Sukirman, U. Technology Malaysia,and R.W. Lewis,U. College of Swansea

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2 .: Three Dlmensiorral Fully Coupled Flow- Conioiidaiioii Modelling SPE 28755using Ftite Elemet Method

Wilmington .Oil Field, Long Beach, California;Ingleyood fie ~, Los Angeles, Califomi&. theNigata g~ fiel~; the Po river delta gas producingarea, Itafy; the oil fields along the Bolivar Coast ofLake Maracaibo, VenezrrelZ the Groningen gas field,

the Netherlands; and the Ekofisk Field in North Sea.In many case:, the amount of subsidence rangesfrom a few centimeters tip to several meters. Fromfield experiences, a considerable degree of reservoircompaction may damage the subsurface productioncasing, tubing and liners whilst seabed subsidencemay cause instability to the production platform.Alternatively, reservoir compaction can act as anactive drive mechanism and therefore infhrence theultimate recovery of a reservoir. Recent simulationstudies otithe Valhall oil field in the North Sea,indicate that a cornpxction diive mechanism accountsfor more than 80 millions STB of oil, whichapproximately 25 percent of the total recovery.J

Irr this paper, Biots self consistent tfreo~is used todevelop the governing &quations which couple theequilibrium and continuity eqirations for a saturatedoil reservoir. Art elastoplastic soil model has beendeveloped for three-dimensional problems which isbased on a Mohr-Coulomb yield surface. The a non-linear terms accotirrt for the effects of reservoirheterogeneity, relative permeability cmttrasrs, rockand fluid compressibility factors, capillary pr~sureand viscosity variations. The developed model wasusd to investigate c.ornpactioii-subsidenc: problemsof a saturated oi~reserv.tiir where free gas existsinitially.

G V NINs

The derivation of four fully-coupled partialdifferential equations, including tie three inrrniscibleand compwssible fluid flow equations along with theequilibrium ~qrmtion, extends the one-phaseconsolidation model developed by Lewis andSchrefler4. Only the summary of the derivation ispresented here.

Capillary and relative permeability

The fl&dpie&ures at ariy point in the reservoir arerelated by their capillary pressure relationships.Therefore,for water wet oil reservoirs, the folfowingexpressions are used for oil-water system.

Pm= P. Pw O

for oil-gas sys~rn:

p =pp ~.. o _. .. w

Slmifarly, three-phase relative pelmeabiMy functionscan be approximated from two-phase data whichconsists of a set of oil-water and oil-gas relativepermeability data. In the present paper, the three-phase oil relative permeability calculations were

based on a statistical probability model shown below;

Km= KrOw+ K,w K,vg . .X +Lg) (3)

where

= oif-water relative pelmeabifity~: = oil-gas relative perrneabihtyKm =-water relative permeabilityKn? = gas relativeperrrrealifiy

In practice, capillary and refative permeability dataare obtained by experiments. It is also assumed thatthe pore volume is completely ffled by a combinationof the fluids present i.e.

So+sw+-sg=l.o (4)

Equifibrirrm equation

A general equilibrium equation requires both theeffective stress relationship and the corrstitutive lawrelating effective stress .IO the strains of the solidskeleton. The effective stress relationship is givenby

[IS]= [0]- [m]P (5) -

where [m] =-[1, 1,1,O,O,O]T.or agenefilnonlinear rnateriaf, eqrr. (5) is expressed in a tirigentialform thus allowing for plasticity, creep fid otherfactors irrffuencing strains to be included. This can.be written as followsd[d] = [DT (dE- d - d&p- d&o)l .(6a)

. ..:.where [&] represents the total strain of the skeleton,

d[Q = [c]dt is tiecreep strain (6.b). .and

d[sp] = - [m] dP/3Ks (6c)

represents the overafl volumetric wrains caused byuniform compression of the particles (as opposed tothe skeleton) by the pressure of the pore fluid.Fkrally, the telm [s.] represents all other strains notdirectly associated with stress changes (swelling,thermal, chemicaJ etc.) i.e. the mtogeneous strains(see Lewis and Schrefler 4). The tangential stiffness

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SPE 2875.5 .,.. Y/4HYA B. SUKIRJ/fAN and.ROL~ W. LEWIS 3

matrix (DT) and the creep [c] are dependent on the

level of effective stress= [cr] and also, if straineffects are considered, on the total strain of theskeleton [&].Based on the principle of virtual work,

an incremental form of the equilibrium equation canbe written as, .

jW~[eTldsdW -JwJ[ulTdib]W

-@ii]Td[t]G = O 7

brco~porating equations (5)and (6).inio ~uition (7),the following equation is obtained for three-phaseflow

JQ~[EITIDTI~dQ ~[c]T[m]~;~

wOdQ_w=o-jQ~[ilTIDTl~ dt

(8)where,

M_ ci[U]T ~dS2+jrJ[u]T~drdt -Jn d, - .:-

9

In this paper, the effective average pore pressure iscalculated from

F =.SOPO+SWPW+sxPg - (m

Moh.r Coulomb yield surface

An elastoplastic soil. . .mo.del, based on a. nonassociated Mohr-Coulomb yield surface, was used.The numelical model requires the formulation of ayield surface and the potential surface, Q as givenbelow

For the failure surfice:

F= (Wcost30 sine. sin )q 3psk ~3Ecos(Ila)

For the potential surface

=.Q = (J3 Coseo -sin 90sinr+f)q-3psinv -3ECOSV

(llb)

where parameters p, q, and are fie mean s.tr~,deviator amd angulw stress brvariants respectwely.Strain hardeningkoftening maybe taken into accountby mdhg tbe cohesion c a function of volumetricplastic strain by using

(11.

where x is an empirical constant, which depends onthe void ratio e. The yield surface can therefore bedefined as

F (d, sp)= o 1 Id)Plastic behaviour occurs when the stress. statereaches the yield surface, while for F < 0 thebehaviour is elastic.

Three phase flow equations

The equation governing the behaviour of threeimmiscible and compressible fluids flowing in adeforming porous medium can be obtained .bycombining Darcys linear flow law with massconservatiori balance for each of the flowing phases.

On taking into acZount several factors whichcontribute to the rate of fluid accumulation (seeLewis and Schrefler 4), the general form of thecontinuity equation for each flowing phase may beexpressed as follows (e.g oil and water phase):

[ 1 [1 SAAklkri ~(P, +Pigh) +%-+ I at El_VT _ M

[ ] 1+_..-..--~[m]T[DT][m] ~ =0

K5 (3Ks)

(12)

forl=o, w,g

The mobility terms in equations (12) are stronglydependent on the unknowns, for example the relative

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4 Three Dimensional Fully Coupled - Consolidation Modelfing SPE 28755using Fkdte Element Method

permeabllities depend on the fluid saturations. Forinitial conditions, we assume that the reservoir wasin capillary pressure equilibrium with both initialwater and gas saturations above their critical valuesi.e. all phases were-mobile. Equations (12) aresubject to various boundaly conditions namely, openand closed boundaries.

~INIT13 ELEMENT MET-HOD

The finite element discretization of the equilibriumand flow equations may now be expressed in term ofthe nodal displacements, [iii and nodal fluidpressures i.e [F.], [~WIarid [Pg by using theGalerkin method. The unknowns are related to theitnodal values by the following expressions

P= [Nl[Fk i =[LWiil [u] = [N][ti] (13)

where [NJand [Bj are theshape function tid Ubearoperator respectively.

Upon substitution of equations (13), we obtain thediscretized form of 6quations (8) and (12) as followsFor the equilibrium equation

[K]~+[Lw]~+[Lo]~+

W]._.[Lg]~.Ic]-=_

(14)

For water:

d~;l [wo,~[wpl[~wl -[ivwl~

~[=j .._

[wgl~ [wul= [Fw; =o

15

For oil:-. .=., _,,. ..+

d[Pwl+[Hol~+[Hpl[~ol [HWIT dt

[pd+[Hu]~+[~o]=o-[Hgl= 16

and for the gas flow equation:

152

.- @wl+[Gold+I+ ___ .

[Gpl[~gl [ l~

d[ii] ~ _

[Ggl~+[Gulx+[ J-O 17

The detailed explanation of the above cc@i ientacanbe found in Reference 5. The time dicretizationmethod osed in this text is based on a Kantorovichtype approach which has been described irr detailedby Lewis and Schrefler 4. Therefore, the timeintegration ofeqrratioirs(14- 17) gives the followingresult

..1.HLw Lo Lg AE

wuw~wowg Fw

HuHWH~Hg ~. =

GUGWGOGP ~ g. tn+At

--[._W H; HgGw Go Gp[

LW Lo Lg

[ ..

FwWP Wo Wg P. +

Fg tn

-Fu

Fw

1

AtF.

Fg(18)

where

w; .(WW +Wfolt)

W;= (WW+WP(I - rz)At)

,H~= (HO+HPcrAt)

H;= (HO+HP(l rx)At)

.~P =(Gg +GpizAtj

G;= (Gs +Gp(l rx)At)

Equations (18) represent a fully coupled, and highlynon linear system, for three-phase flow in adeformins uorous media. Since all the coefficientsare depen~r t on the unknowns, iterative proceduresare performed within each time step to obtain thefinal solutions. A convergence criterion is set whichR based on the maximum fluid preswws change sincethe last iteration, i.e.

where &ois the convergence fiiiL


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,-> :,,, -, ,+ .>-.*, ~:=,-,

SPE 28755 ~ YAHYA B.SURIRMAN aiidROL~ W. LEWIS 5

For the stability and accuracy of the solutions,material bafance checks are performed so that themodel conserves mass at all times. A generalmaterial bafance equation, MBE for each of the

flowing pbaws can be given as

[ 1T Sv(px +pzgh)= RHSPzBn (19)where R H S term represents the rate of fluidaccumulation and net flow of the reservoir system.For MBE checks,

ps-LHs~< .m

where em is MBE limit.

This checking is Perfolmed after each iteration withina time step when the unknowns have been calculated.

~ PR E . . . ...

A series of simulations were conducted to indicatethe utility of the developed finite element code forpredicting the surface subsidence above a compactingsaturated oil reservoir. To illustrate tbk application,a hypothetical modeI which c.on.sists of a closedreservoir system, cap rock and overburden layerswas used as shown in Figure 1. The fluid data forthe oil reservoir was obtained from Lewis andSrrkirtnarr(1993) whilst the niitrxial properties usedis given in Table 1.

Table 1: Material Property for Subsidence Analysis

Jrr all simulations, &rosS:s~ctionsOP-Wd MN caneither be flow or no-flow boundary conditions whilstthe remaining secjiogs _@thin the system wefe

assumed to be no-flow boundaries. It was assumedthat the vertical displacement at the base is zero i.e. arigid base rock. The simulations of reservoircompaction and the consequent srrrfac? subsidencefor different types of reselioir aJe presented heie~

Case 1 uses three different values of Youngsmodulus in order to investigate the influence ofdifferent degrees of compaction on the predictedsubsidence at the surface. It can be seen from Figrrie2 that the subsidence at point S increases with thelower value of E and vice versa. TIrk is an expected

result became a higher value of E impliesa morerigid formation rock and therefore lending toincompressible behavioor i.e. deformation can beneglected.

Case 2 uses the same reservoir Win the previouscase, except however that it has a water injection welflocated at point Y. The water was injected at aconstant rate of 1000 STB/D after t = 3 years andwas terminated at t = 16 yearn. Figures 3 and 4show the profile of reservoir compaction andsubsidence at the surface with time. The resultsindicate that the water injection process can beperformed in order to minimize compaction andsubsidence problems. Irr practice, the injected fluidreduces the pressure in the reservoir system andtherefore results irr a minimum subsidence at thesurface.

~:

A numerical simulation using a FEM has beendeveloped for simulating the surface subsidenceabove a compacting oil reservoir. The derivation ofthe four governing equations considered theCquiliblium equation and the continuity of the fluidflow. These balance equations have taken intoaccount the effec~of carrillarity imd the variation ofrelative permeability. A Mohr-Coulomb yieldsurface was used for the elastoplastic soil model.The case studies presented in this paper indicate theutility of the developed model in simulating thesurface subsidence above a compacting oil reservoir.

REFERENC~

1. Ter2aghi, K. Theoretical Soil Mechanics,Wiley, New York. 1943

2.

3.

4.

5.

Biot, M.A, General Theory of threedimensional consolidation, 1. appl. Physics12155-164,1.943

York, S.D, Peng, C.P. and Joslin, T. H.;Numerical Management of the Val.hallField,Norway, JPT, (August, 1992)

Lewis R.W. and Schrefler, B.A. , The finiteelement method in the deformation andconsolidation of orous media, Chichestar,

fohn ,Wiley., 198 .

Lewis, R.W, and Y.. Sukimran, Finiteelement modelling of three uhase flow indeforming satura~d oil rese~oirs, Intl. J.Numerical and Anafy. Methods in Geom, Vol17,577-598 (1993).

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,.

spE 87 o

oo

-200

-300

400

-500I I I I I I

Io 2000 4000 -- 6ooa

Production time (days)

J.

A HYPOTHETICAL MODEL FOR: SUBSIDENCE @L LYSIS.

FIGURE 1

Fkbr- * WfkP-.

EFFECTOF YOUNOS MODULUS.WWJFS

FIGURE 2

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sukirman_consolid_mod_using_fin_elem_meth-_1994[1].pdf

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