A control volume based finite element method for ... - Springer Link

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$FRQWUROYROXPHEDVHG¿QLWHHOHPHQWPHWKRG for simulating incompressible two-phase ÀRZLQKHWHURJHQHRXVSRURXVPHGLDDQGLWV application to reservoir engineering SADRNEJAD S A1, GHASEMZADEH H1, GHOREISHIAN AMIRI S A1 and MONTAZERI G H2 1 2

)DFXOW\RI&LYLO(QJLQHHULQJ.17RRVL8QLYHUVLW\RI7HFKQRORJ\7HKUDQ,UDQ 5HVHDUFKDQG'HYHORSPHQW'HSDUWPHQW,UDQLDQ&HQWUDO2LO)LHOG&R7HKUDQ,UDQ

‹&KLQD8QLYHUVLW\RI3HWUROHXP%HLMLQJ DQG6SULQJHU9HUODJ%HUOLQ+HLGHOEHUJ

Abstract: $SSO\LQJWKHVWDQGDUG*DOHUNLQ¿QLWHHOHPHQWPHWKRGIRUVROYLQJÀRZSUREOHPVLQSRURXV media encounters some difficulties such as numerical oscillation at the shock front and discontinuity RIWKHYHORFLW\¿HOGRQHOHPHQWIDFHV'LVFRQWLQXLW\RIYHORFLW\¿HOGOHDGVWKLVPHWKRGQRWWRFRQVHUYH mass locally. Moreover, the accuracy and stability of a solution is highly affected by a non-conservative PHWKRG,QWKLVSDSHUDWKUHHGLPHQVLRQDOFRQWUROYROXPH¿QLWHHOHPHQWPHWKRGLVGHYHORSHGIRUWZR SKDVHÀXLGÀRZVLPXODWLRQZKLFKRYHUFRPHVWKHGH¿FLHQF\RIWKHVWDQGDUG¿QLWHHOHPHQWPHWKRGDQG attains high-orders of accuracy at a reasonable computational cost. Moreover, this method is capable of KDQGOLQJKHWHURJHQHLW\LQDYHU\UDWLRQDOZD\$IXOO\LPSOLFLWVFKHPHLVDSSOLHGWRWHPSRUDOGLVFUHWL]DWLRQ RIWKHJRYHUQLQJHTXDWLRQVWRDFKLHYHDQXQFRQGLWLRQDOO\VWDEOHVROXWLRQ7KHDFFXUDF\DQGHI¿FLHQF\RI WKHPHWKRGDUHYHUL¿HGE\VLPXODWLQJVRPHZDWHUÀRRGLQJH[SHULPHQWV6RPHUHSUHVHQWDWLYHH[DPSOHV DUHSUHVHQWHGWRLOOXVWUDWHWKHFDSDELOLW\RIWKHPHWKRGWRVLPXODWHWZRSKDVHÀXLGÀRZLQKHWHURJHQHRXV porous media.

Key words: Finite element method, control volume, two-phase flow, heterogeneity, porous media, ZDWHUÀRRGLQJ

1 Introduction 6LPXODWLRQ RI PDVV WUDQVSRUW DQG IOXLG IORZ LQ SRURXV media is a problem of increasing importance in engineering. Among the practical applications of mass transport in porous media, groundwater flow, oil recovery and enhanced oil recovery processes, contaminant transport in ground water DQGYDGRVH]RQHFRXOGEHPHQWLRQHG 2QHRIWKHPDLQREMHFWLYHVRIVLPXODWLQJPXOWLSKDVHÀRZ in porous media is to properly capture the complex geometry and heterogeneity of rocks. Rock properties, such as porosity DQGSHUPHDELOLW\PD\FKDQJHVLJQL¿FDQWO\IURPRQHUHJLRQWR DQRWKHUDQGVXEVHTXHQWO\ÀXLGYHORFLW\ZLOOYDU\E\VHYHUDO orders of magnitude over a relatively short distance. Due to these facts, multiphase flow simulation in porous media poses a great challenge for most of the available numerical methods. Various numerical methods have been applied to solve the

*Corresponding author. email: [email protected] Received December 2, 2011

governing equations describing subsurface flow. The finite difference (FD) method has been widely used in the solution RIWKHVHHTXDWLRQV7RGGHWDO6HWWDULDQG$]L] Abriola and Pinder, 1985; Chen and Ewing, 1997; Chen et al, 2005; Lu and Wheeler, 2009). However, this method produces excessive numerical dispersion and grid orientation problems (Ewing, 1983) in addition to difficulties in the treatment of FRPSOLFDWHG JHRPHWU\ DQG ERXQGDU\ FRQGLWLRQV ,QWULQVLF mesh flexibility of the finite element method (FEM) could RYHUFRPHWKHVHGH¿FLHQFLHVEXWWKHFODVVLFDO)(0LVQRWLQ general, locally mass conservative (Chen et al, 2006b), and tends to generate numerical solutions with severe nonphysical oscillations (Wang et al, 2003). Recently, mass conservative schemes such as the unstructured finite volume methods (FVM), mixed finiteHOHPHQWPHWKRG0)(0 DQGFRPELQDWLRQRI¿QLWHHOHPHQW and the finite volume methods (FEFVM) have been extensively developed and studied in the literature (Durlofsky, 1994; Verma, 1996; Bergamaschi et al, 1998; Edwards and Rogers, 1998; Edwards, 2002; Korsawe et al, 2003; Rees et DO1D\DJXPHWDO*HLJHUHWDO+RWHLWDQG

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)LURR]DEDGL&DUYDOKRHWDO  $QRWKHULQWHUHVWLQJDSSURDFKLVWKHFRQWUROYROXPH¿QLWH element method (CVFEM) which has been developed as a fully conservative method. Application of CVFEM to simulate multiphase fluid flow in porous media has been rapidly increased (Fung et al, 1991; Li et al, 2003; Chen et al, 2006b; Mello et al, 2009), since this method combines the mesh flexibility of FEM with the local conservative characteristic of the finite difference method at the control volume level. Another important object in the field of multiphase ÀRZVLPXODWLRQLVWRGHYHORSDVWDEOHHI¿FLHQWUREXVWDQG accurate solution for solving the multiphase flow equations LQ WKH WLPH GRPDLQ 7KUHH W\SHV RI WLPH GLVFUHWL]DWLRQ method are commonly used to solve these equations: the ,03(6 LPSOLFLW SUHVVXUH DQG H[SOLFLW VDWXUDWLRQ  PHWKRG the sequential method; and the fully implicit method (Chen et DOD 7KH,03(6VROXWLRQPHWKRGPDNHVWKHSUHVVXUH variables implicit in time, whereas, the saturation variables are considered explicit in time. This method of solution has a stability limit that is inversely proportional to the fluid velocity, and this limitation is often too restrictive in practice. The sequential solution method attempts to remedy WKHSUREOHP¿UVWO\E\VROYLQJWKHSUHVVXUHHTXDWLRQVEDVHG on the results of the last time step, secondly, by computing WKHQHZÀXLGVYHORFLW\¿HOGVEDVHGRQWKHQHZSUHVVXUHDQG finally solving the transport problem based on the updated YHORFLWLHV,QWKHDEVHQFHRIFDSLOODULW\WKHVWDELOLW\OLPLWDWLRQ RI WKH VHTXHQWLDO PHWKRG LV OHVV WKDQ WKH ,03(6 PHWKRG however, decoupling of equations in this method could generate mass balance errors that are proportional to the length of time steps. Finally, the fully implicit method solves the system of coupled equations simultaneously with an implicit time scheme. This method is unconditionally stable and can take large time steps. The main goal of this paper is to present a fully conservative method for solving two-phase flow equations ZKLFKFRXOGKDQGOHKHWHURJHQHLWLHVLQDUDWLRQDOZD\,QRUGHU to avoid any stability limitation, a fully implicit scheme is DSSOLHGWRWHPSRUDOGLVFUHWL]DWLRQRIWKHJRYHUQLQJHTXDWLRQV The accuracy and efficiency of the proposed method are verified by comparing the model results with some ZDWHUÀRRGLQJH[SHULPHQWV)XUWKHUPRUHLQRUGHUWRHYDOXDWH the model capability for handling discontinuous materials properties, some representative examples are presented.

2 Governing equations The system of concern in this paper is considered as a mixture consisting of a rigid porous media, water and oil which are considering as wetting and non-wetting phases, UHVSHFWLYHO\ ,W LV DVVXPHG WKDW RLO DQG ZDWHU DUH WZR immiscible and incompressible fluids. The basic equations for describing this system are derived based on the classical FRQWLQXXPWKHRU\RIPL[WXUH*RRGPDQDQG*RZLQ ,Q this approach, the constituents of the mixture are considered as overlapping continua. This means that each point in the mixture is simultaneously occupied by materials of all phases. The flow of each phase is described by conservation of

mass

UD

wD  ’.( D UD wD ) w

 D

(1)

DQGJHQHUDOL]HG'DUF\¶VODZ rD K D wD > UD g  ’D @

(2)

PD

where nĮ represents the volume fraction of phase Į[Įdenotes water (w) and oil (o)]; ȡĮ is the mass density of phase Į; wĮ is the relative velocity of phase Įwith respect to the solid skeleton; MĮ is the sources or sinks terms; K is the absolute permeability tensor of the solid skeleton; krĮ and ȝĮ are the relative permeability and the dynamic viscosity of phase Į, respectively. The relative permeability for water and oil phases are calculated by the relations proposed by Lujan (1985)

rw ro

  O  O ew

(3a)

  O  O (1  ew ) (1  ew )

(3b)

nw  nw res

where Sew ( Sew

) denotes the effective degree of nw sat  nw res saturation of water phase and ȜLVD¿WWLQJSDUDPHWHUUHODWHG WRWKHSRUHVL]HGLVWULEXWLRQ ,Q DGGLWLRQ WKHUH LV D FRQVWUDLQW EHWZHHQ IOXLG YROXPH fractions w  o



(4)

where n represents the rock porosity. This relation means that WKHÀXLGSKDVHVMRLQWO\¿OOWKHYRLGV $QRWKHUFRQVWUDLQWHTXDWLRQLVGH¿QHGE\FRQVLGHULQJWKH fact that when two fluid phases flow in a porous medium, IORZRIHDFKSKDVHLVDIIHFWHGE\WKHRWKHUSKDVH,QRUGHU to specify this interacting motion in the model, one requires equations which link each phase pressure to its volume fraction. The most practical method for considering this interacting motion is to use empirical correlations relating the capillary pressure (pc) to the volume fraction of the wetting SKDVH +DVVDQL]DGHK DQG *UD\   ,Q WKLV SDSHU WKH relation proposed by Brooks and Corey (1966) is employed O

ª º w w res  w sat  w res « d » (5) ¬ c ¼ where w sat  LV WKH YDOXH RI ZDWHU YROXPH IUDFWLRQ DW ]HUR suction; w res is the value of water volume fraction at very high suction; pd is the displacement pressure for oil; and Ȝis GH¿QHGLQWKHUHODWLYHSHUPHDELOLW\UHODWLRQV(T  Partially differentiating Eq. (5), one obtains



d w



d w  d o  d w d c

wc  d o  d w

(6)

and from Eq. (4)

d o

 d w

(7)

6XEVWLWXWLQJ(TV  DQG LQWR(T WKHILQDO

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form of mass balance equations for water and oil phases are obtained:

D

ww w   wc U w o w w ’.^ U w K w > U w g  ’w @`   w

 wc U w

wo w   wc Uo w w w ’.^ Uo K o > Uo g  ’o @`   o

krD K

PD

0

D0

T

(9)

0

) is the mobility of phase Į,WLVZRUWK

in : and on *

(11)

^UD KD > UD g  ’pD @`

noting that viscosities of water and oil phases are assumed to be pressure independent. Eqs. (8) and (9) represent a system of two highly nonlinear partial differential equations. Relative permeability and phase volume fractions are the major nonlinearity sources in these equations. ,QWKLVSDSHUSKDVHSUHVVXUHVDUHVHOHFWHGDVWKHSULPDU\ unknowns. Choice of the primary variables is a crucial step in HI¿FLHQF\RIPXOWLSKDVHÀRZPRGHOVLQSRURXVPHGLD:XD and Forsyth, 2001). Ataie-Ashtiani and Raeesi-Ardekani (2010) showed that selection of the phase pressures as the SULPDU\ YDULDEOHV WHQGV WR PLQLPL]H FRPSXWDWLRQDO FRVW However, Wua and Forsyth (2001) did not agree with them. On the other hand, this work is the first step towards the development of more general numerical algorithm for the FRXSOHGK\GURPHFKDQLFDOVLPXODWLRQRIPXOWLSKDVHÀRZLQ porous media. Therefore, based on some of the recent hydromechanical models (Lewis et al, 1998; Pao et al, 2001; Li et al, 2005), in this paper, the phase pressures are chosen as the primary variables. The equations in this system are strongly coupled. Capillary pressure plays a key role in this type of formulation, because coupling of Eqs. (8) and (9) is due to existence of FDSLOODULW\0RUHRYHUÀXLGVDWXUDWLRQLVFDOFXODWHGEDVHGRQ the existence of capillary pressure function. Theoretically, this system of equations collapses at the absence of an oil phase, because two pressures are no longer independent, and at the same time two mass balance equations are still needed to be solved. While applying this situation in the model, if pc LVGH¿QHGZLWKDFHUWDLQYDOXHHTXDOLQJWRpd in the capillary pressure relation (Eq. (5)), water and oil pressures will remain independent and the problem will be solved automatically. ,Q RUGHU WR FRPSOHWH WKH GHVFULSWLRQ RI JRYHUQLQJ equations, it is necessary to define appropriate initial and boundary conditions. The initial conditions specify the full ¿HOGRIZDWHUDQGRLOSKDVHSUHVVXUHVDWWLPHt=0

D

on *p

D

(8)

 wc Uo

where K D ( KD

ERXQGDU\FRQGLWLRQLQZKLFKWKHYDOXHVRISKDVHÀX[HVDWWKH boundaries ( *q) are imposed (where * *   * )

˜n

qD

on * q

(12)

where n denotes the outward unit normal vector on the boundary and q is the imposed mass flux normal to the boundary.

3 Numerical solution ,QWKHSUHVHQWZRUND&9)(0LVDGRSWHGIRUQXPHULFDO solution of Eqs. (8) and (9) in spatial domain. As common in FEM, the discrete form of differential equations is presented in a transformed space with local coordinate system ([ , K, ] ) GHILQHGIRUHDFKHOHPHQW)LJ ,QWKHSURSRVHGPHWKRG WKHSK\VLFDOGRPDLQGLVFUHWL]DWLRQLVGRQHXVLQJKH[DKHGUDO elements. And further subdivision of elements into control volumes is performed in the transformed space, as shown in Fig. 2. The coordinate transformation at the element level is performed by

 ([ , K , ] )

N ([ , K , ] ) xˆ 

 1, 2, 3

(13)

where xi is the value of the ith coordinate in the physical space; N is the vector of standard finite element shape functions and xˆ  is the nodal value vector of the ith z

(a)

x y

(b) Ș

ȗ

ȟ

Fig. 1 Element representation (a) in physical domain and (b) in the transformed space

(10)

ZKHUHȍLVWKHGRPDLQRILQWHUHVWDQG * is its boundary. The boundary condition can be of two types or a combination of these: the Dirichlet boundary condition, in which the phase pressures on the boundaries ( *p DUHNQRZQDQGWKH1HXPDQQ

coordinate. 7KHFRQWUROYROXPH¿QLWHHOHPHQWGLVFUHWL]DWLRQRI(TV  and (9) is expressed in terms of the nodal phase pressures, i.e. pˆ D , which are selected as the primary variables. The values of phase pressures (pĮ) at any point within an element are

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of the adjacent element; and subsequently mass does not conserve at the element level (Fig. 4(a)). However, owing to FRQWLQXLW\RIWKHYHORFLW\¿HOGDWWKHFRQWUROYROXPHIDFHVLI the mass flux is integrated on each control volume surface, conservation of mass will be obtained on the control volumes (Durlofsky, 1994) (Fig. 4(b)). 1RZ E\ LQWHJUDWLQJ (TV   DQG   RYHU D FRQWURO volume and applying Gauss theorem, the weak form of balance equations are derived

(a)

wp ³  nc U wt

wpo d: wt T ^Uw K w > Uw g  ’pw @` ˜ n d *  ³ M W d : w

:C.V.

³

*C.V.

Control volume (b)

w

w

d:  ³

:C.V.

 nwc U w

:C.V.

(15)

0

Pressure

Velocity

e=1

e=2

e=4

e=3

Sub-control volume (c)

Control volumes

Control volumes

Fig. 3 'LVFRQWLQXLW\RIWKHYHORFLW\¿HOGDFURVVWKHHOHPHQWIDFHVDQG FRQWLQXLW\RIWKHYHORFLW\¿HOGDFURVVWKHFRQWUROYROXPHIDFHVIRUWKH RQHGLPHQVLRQDO¿QLWHHOHPHQWPHVK

(a) Element out-flux Element 1 Element 2 Fig. 2 D  6\VWHP RI ILQLWH HOHPHQW PHVK LQ WKH WUDQVIRUPHG VSDFH E  control volume representation around a node and (c) illustrating a sub-control volume belongs to the eliminated element

Element in-flux

approximated by the following expressions (b)

w ([ , K , ] )

N ([ , K , ] ) pˆ w

(14a)

o ([ , K , ] )

N ([ , K , ] ) pˆ o

(14b)

,Q WKH FODVVLFDO )(0 WKH ILUVW RUGHU SRO\QRPLDOV DUH commonly used as the shape functions. This lead the pressure field to be continued in the whole physical space; EXWWKHYHORFLW\¿HOGZLOOEHGLVFRQWLQXRXVEHWZHHQDGMDFHQW elements. However, because the control volume faces are ORFDWHGLQVLGHWKHHOHPHQWVWKHYHORFLW\¿HOGVDUHFRQWLQXHV EHWZHHQ DGMDFHQW FRQWURO YROXPHV )LJ   ,W PHDQV WKDW the mass out-flux through a surface of an element is not QHFHVVDULO\HTXDOWRWKHPDVVLQÀX[WKURXJKWKHVDPHVXUIDFH

Control volume out-flux

Control volume 1 Control volume in-flux Control volume 2 Fig. 40DVVLQÀX[DQGRXWÀX[WKURXJKWKHHOHPHQWVDQGFRQWUROYROXPHV faces (a) mass out-flux from element 1 is not necessarily equal to mass LQÀX[LQWRHOHPHQWDQGE PDVVRXWÀX[IURPFRQWUROYROQPHLVDOZD\V HTXDOWRPDVVLQÀX[LQWRFRQWUROYROXPH

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: C.V.

³

489

wpo wp d :  ³  nwc U o w d : : C.V. wt wt T ^Uo K o > Uo g  ’po @` ˜ n d *  ³ M o d :

 nwc Uo

* C.V.

: C.V.

(16)

0

ZKHUHȍC.V. indicates the domain of a control volume bounded by * C.V.. 6XEVWLWXWLQJ WKH LQWHUSRODWRU\ UHSUHVHQWDWLRQ RI SKDVH pressures (Eq. (14)) into Eqs. (15) and (16), one obtains

wpˆ wpˆ ª nc U N d : º w  ª ³  nwc U w N d : º o «¬ ³:C.V.  w w »¼ wt «¬ :C.V. »¼ wt T  ª ³ ^ U w K w ’N ` ˜ n d * º pˆ w ¬« *C.V. ¼»

³

:C.V.

M w d :  ³

* C.V.

U

2 w

 Kw g

(17) T

˜ nd*

wpˆ wpˆ ª nc U N d : º o  ª ³  nwc Uo N d : º w «¬ ³:C.V.  w o »¼ wt «¬ :C.V. »¼ wt T  ª ³ ^Uo K o ’N ` ˜ n d * º pˆ o ¬« *C.V. ¼»

³

:C.V.

M o d :  ³

* C.V.

U

2 o

 Ko g

(18) T

˜ nd*

For the boundary nodes, at least one face of the associated control volume lies on the physical boundary. The boundary conditions must be implemented to the model at these nodes. Dirichlet boundary conditions (Eq. (11)) are enforced by direct substitution of prescribed nodal values, as in the FODVVLFDO)(0)RU1HZPDQERXQGDU\FRQGLWLRQV(T  the prescribed fluxes through the physical boundaries are added to the right hand side of the equations. Consequently, Eqs. (17) and (18) will have the form

wpˆ wpˆ ª  nc U N d : º w  ª ³  nwc U w N d : º o «¬ ³:C.V.  w w »¼ wt «¬ :C.V. »¼ wt T  «ª ³ U w K w ’N ` ˜ n d * »º pˆ w (19) ^ * * ¬ C.V. q ¼ T U w2  K w g ˜ n d * ³ M w d :  ³ qw d *  ³ :C.V.

*q

PDWHULDOSURSHUWLHVLQWKHSK\VLFDOGRPDLQVHH6HFWLRQ  To aim this goal, each control volume around a node is divided to some sub-control volumes belonging to different HOHPHQWVDVVRFLDWHGZLWKWKHQRGH)LJ 6LPLODUO\HDFK face of a control volume is divided to some sub-control volume faces. Hence, integrations over a control volume or a control volume face could be calculated by summing up the individual integrations over the sub-control volumes and the sub-control volume faces. By applying this methodology, GLVFUHWL]HGIRUPXODWLRQFRXOGEHUHSUHVHQWHGDWWKHHOHPHQW level and the complete balance equations could be obtained by assembling the element-wise equations, as in the classical FEM. The element-wise representation of Eqs. (19) and (20) is

0 º ª pˆ w º ª Pww »« »  « Poo ¼ ¬ pˆ o ¼ ¬C ow

ª Pww « ¬ 0

C wo º d ª pˆ w º Poo »¼ d  «¬ pˆ o »¼

Pww

³

*6&9 *T

Poo

³

Pww

³

*6&9 *T

:e

Poo

³

C wo C ow

fw

WT

W

^ U

K w ’N ˜ n T

o

W T   wc U w N d :

*q

* C.V. *q

(TV   DQG   SUHVHQW WKH GLVFUHWL]HG IRUP RI WKH balance equations at the control volume level. On the other hand, representation of the equations at the element level could help the model to handle probable discontinuous

(22a)

d*

(22b)

(22c)

(22d)

³

W T  wc U w N d :

(22e)

³

W T  wc U o N d :

(22f)

:e

:e

:e

 wW T d :  ³ wW T d * *q

(22g)

T

ª U 2  K w g ˜ nº W T d * *6&9 * T ¬ w ¼

³ fo

³

:e

 oW T d :  ³ oW T d * *q

ª U 2  K o g T ˜ n º W T d * *6&9 * T ¬ o ¼

:C.V.

d*

T

^ U K ’N ˜ n` o

T

`

T

w

W T   wc U o N d :

:e

³

³

T (20)  «ª ³ ^Uo K o ’N ` ˜ n d * »º pˆ o ¬ *C.V. *q ¼ T Uo2  K o g ˜ n d * ³ M o d :  ³ qo d *  ³

(21)

ZKHUHWKHFRHI¿FLHQWVDUHGHVFULEHGDV

* C.V. * q

wpˆ wpˆ ª  nwc U o N d : º o  ª ³  nwc Uo N d : º w  ³ «¬ :C.V. »¼ wt «¬ :C.V. »¼ wt

ª fw º «f » ¬ o¼

(22h)

ZKHUHȍe and * 6&9 indicate the domain of an element and the area of a sub-control volume faces, respectively, and W is the vector of weighting functions. The weighting functions in this model are chosen such that the ith weighting function of an element takes a constant value of unity over the subcontrol volume belonging to node iDQG]HURHOVHZKHUHLQWKH element, i.e.



­1 in the sub-control volume belongs to node  ® (23) ¯0 otherwise

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7KHWLPHGLVFUHWL]DWLRQRI(T LVSHUIRUPHGE\WKH IXOO\LPSOLFLW¿UVWRUGHUDFFXUDWH¿QLWHGLIIHUHQFHVFKHPH

ª Pww  'Pww « C ow ¬

º ª 'pˆ w º C wo » « » Poo  'Pww ¼  1 ¬ 'pˆ o ¼  1

ªP ªf º ' « w »  ' « ww ¬ f o ¼  1 ¬ 0

(24)

0 º ª pˆ w º » « » Poo ¼  1 ¬ pˆ o ¼ 

where ' is the time step length and 'pˆ  1

pˆ  1  pˆ  .

4 Heterogeneous porous media ,Q KHWHURJHQHRXV SRURXV PHGLD LW LV DVVXPHG WKDW WKH change of rock properties is only acceptable on the element edges. However, smooth variation of properties is acceptable inside the elements, because it could be considered with the ¿QLWHHOHPHQWVLQWHUSRODWHGIXQFWLRQV ,Q JHQHUDO WKH HOHPHQWZLVH GLVFUHWL]DWLRQ RI WKH HTXDWLRQVDQGWKHDERYHPHQWLRQHGDVVXPSWLRQDUHVXI¿FLHQW WRVLPXODWHWKHÀXLGÀRZLQKHWHURJHQHRXVSRURXVPHGLD7R achieve the mass balance equations for each control volume in the element-wise format, coefficients of Eq. (24) are integrated separately in each sub-control volume and subcontrol volume faces (Fig. 5) where the rock properties are constant or changed smoothly. Then, the complete balance （a）

A

B

equations are obtained by assembling the coefficients, as usual in the classical FEM. Consequently, heterogeneity of medium could be considered easily and without any PRGL¿FDWLRQRIWKHSURSRVHGIRUPXODWLRQ

0RGHOYHUL¿FDWLRQDQGH[DPSOHV The accuracy and efficiency of the method are verified in this section through simulation of some waterflooding experiments which are reported by Hadia et al (2007; 2008). Furthermore, some representative examples are shown to illustrate the potential of the proposed method to simulate WZRSKDVHÀXLGÀRZLQKHWHURJHQHRXVSRURXVPHGLD

2QHGLPHQVLRQDOZDWHUÀRRGLQJ One-dimensional waterflooding experiments, reported by Hadia et al (2007), are selected as the first example to test the validity of the model. The selected experiments were performed at room temperature of 24 °C and atmospheric pressure. The tests were done on rectangular Berea sandstone core samples (2.5×2.4×54 cm3) saturated with heavy liquid paraffin oil. To simulate the recovery process with waterflooding, brine (water with 1% KCl by volume) ZDV LQMHFWHG LQWR WKH VDPSOH DW D UDWH RI  DQG  P/ h. The measured viscosity of the heavy paraffin oil and brine were 130 and 0.97 cP, respectively. The measured porosity and absolute permeability of the core sample used in the experiments were 38.5% and 1584 mD, respectively. The average value of irreducible water saturation for the experiments was about 32.5%. The schematic of 1D ZDWHUÀRRGLQJH[SHULPHQWLVVKRZQLQ)LJ

1 E

D

Inlet

Outlet

2.4 cm

54 cm

（b）

2.5 cm

Fig. 66FKHPDWLFRI%HUHDVDQGVWRQHFRUHVDPSOHIRU'ZDWHUIORRGLQJ experiments (Hadia et al, 2007) A

E

B

1

1

1

1

D

Fig. 5 2D control volume split by four different rock types

7KHILUVWWHVWFDVHP/K LVVHOHFWHGIRUFDOLEUDWLQJ of the numerical model parameters; and the latter test case is adopted to validate the model accuracy. The relative permeability and capillary curves, which are obtained from the calibration process, are shown in Figs. 7 and 8. These curves are also used for the test case with an injection rate of P/KWRHYDOXDWHWKHPRGHODFFXUDF\ Fig. 9 shows the comparative curves of numerical and experimental results for recovery performance of original RLO LQ SODFH 22,3  ,W LV REVHUYHG WKDW QXPHULFDO UHVXOWV are in agreement with experimental data. Fig. 10 shows the

3HW6FL 

491 60

distribution of water saturation inside the Berea sandstone FRUHVDPSOHGXULQJWKHZDWHUÀRRGLQJWHVWZLWKWKHLQMHFWLRQ UDWHRIP/K Recovery, OOIP %

1.0

0.8

Relative permeability

(a) 50

40

30

20

10 0.6

Data (Hadia et al, 2007) Model

krw

0

kro

0.4

0

0.5

1.0

1.5

2.0

2.5

Injection volume, PV 60

0.2

(b) 50 0

0.2

0.4

0.6

0.8

1.0

Effective water saturation Fig. 7 Relative permeability versus effective water saturation IRU'ZDWHUÀRRGLQJSUREOHP

40

30

20

10

6

Data (Hadia et al, 2007) Model 0

5

Capillary pressure, kPa

Recovery, OOIP %

0

0

0.5

1.0

1.5

2.0

2.5

Injection volume, PV 4

Fig. 9 1XPHULFDODQGH[SHULPHQWDOUHVXOWVIRUUHFRYHU\SHUIRUPDQFHRI22,3 D IRUDQLQMHFWLRQUDWHRIP/KDQGE IRUDQLQMHFWLRQUDWHRIP/K

3

2

Beginning of the test

After 0.2 PV of injection

1

0 0

0.05

0.10

0.15

0.20

0.25

Water volume fraction Fig. 8 Capillary pressure versus water volume fraction IRU'ZDWHUÀRRGLQJSUREOHP After 1.0 PV of injection

After 2.0 PV of injection

7KUHHGLPHQVLRQDOZDWHUÀRRGLQJ Hadia et al (2008) performed some three-dimensional ZDWHUIORRGLQJ H[SHULPHQWV RQ D VDQGVWRQH RXWFURS RI VL]H 30×30×10 cm 3 ZLWK KRUL]RQWDO DQG YHUWLFDO ZHOOV RI  cm diameter drilled in the outcrop at different locations. The waterflooding tests were performed with different well configurations (vertical production-vertical injection (VP9, ZHOOVDQGKRUL]RQWDOSURGXFWLRQYHUWLFDOLQMHFWLRQ+3 9,  ZHOOV  DW DQ LQMHFWLRQ UDWH RI  P/K7KH VHOHFWHG experiments were done at room temperature of 25±1 °C and atmosphere pressure. Water and paraffin oil were used as displacing and displaced fluids, respectively. The measured viscosity of oil and water were 130 and 0.97 cP. The measured porosity and absolute permeability of the outcrop, used in

0

0.1

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0.3

0.4

0.5

0.6

0.7

Fig. 10 Distribution of water saturation in the core sample GXULQJWKHWHVWDWDQLQMHFWLRQUDWHRIP/K

the experiments, were 20.8% and 1,500 mD, respectively. The average value of irreducible water saturation for the experiments was about 22.1%. The schematic of experiments is shown in Fig. 11.

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3 cm

(b)

3 cm

(a)

Production well 3 cm

30 cm

28 cm

3 cm

3 cm

6.5 cm

1.8 cm 1.5 cm

10 cm

9 cm

10 cm

9 cm

30 cm

30 cm

30 cm

30 cm

Fig. 116FKHPDWLFRIRXWFURSVDQGVWRQHFRUHVDPSOHVD 939,ZHOOVDQGE +39,ZHOOV+DGLDHWDO

1.0

0.8

Relative permeability

7KHWHVWZLWK+39,ZHOOVLVVHOHFWHGDVWKHEDVHGWHVWIRU FDOLEUDWLQJWKHPRGHOSDUDPHWHUVDQGWKHWHVWZLWK939,LV used to check the model accuracy. Figs. 12 and 13 represent the relative permeability and capillary curves which are obtained from the calibration process. 7KHUHFRYHU\SHUIRUPDQFHRIRULJLQDORLOLQSODFH22,3  and the percent of water in the produced fluid (water cut), resulted from the tests and numerical analyses, are compared in Figs. 14 and 15. The distribution of water saturation inside WKH VDQG VWRQH RXWFURS VDPSOH GXULQJ WKH WHVW ZLWK +39, wells is presented in Fig. 16.

0.4 krw kro 0.2

5.3 Solution of the elliptic pressure equation in heterogeneous domains

0 0

0.2

0.4

0.6

0.8

1.0

Effective water saturation Fig. 12 Relative permeability versus effective water saturation IRU'ZDWHUÀRRGLQJSUREOHP

35 30

Capillary pressure, kPa

,Q RUGHU WR VKRZ WKDW WKH SUHVHQW PRGHO LV FDSDEOH RI handling discontinuous material properties even with very coarse meshes, the simplest elliptic pressure equation ǻKǻp=0), where K=kI and k is a scalar discontinuous FRHI¿FLHQW LVVROYHGLQWKLVVHFWLRQ7KLVH[DPSOHZDVDOVR adopted by Rees (2004) and Carvalho et al (2007) to show the ability of their model in the simulation of discontinuous materials. However, the model proposed by Rees (2004) produced poor results for coarse meshes, and the method proposed by Carvalho et al (2007) seems complicated for implementation. ,Q WKLV H[DPSOH D îî FXELF GRPDLQ LV FRQVLGHUHG and two cases of material discontinuities are studied by the SURSRVHGQXPHULFDOPRGHO,QWKH¿UVWFDVHWKHFXEHLVVSOLW KRUL]RQWDOO\LQWZRSDUWVSDUW$DQGSDUW% DVVKRZQLQ)LJ 17(a), and in the second case, the cube is divided vertically into two parts (Fig. 17(b)). The non-dimensional diffusive parameters (i.e. permeability) are kA=10 and kB=50 for part A and B, respectively. The pressure at the top and bottom of the domain are set to pT=0 and pB UHVSHFWLYHO\,PSHUPHDEOH boundary conditions are set to the lateral edges of the domain. Fig. 18 shows the extremely coarse mesh adopted for both cases of this example and the pressure contours in the cubes.

0.6

25 20 15 10 5 0 0

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Water volume fraction Fig. 13 Capillary pressure versus water volume fraction IRU'ZDWHUÀRRGLQJSUREOHP

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Recovery, OOIP %

Recovery, OOIP %

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30

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10 Data (Hadia et al, 2008) Model

Data (Hadia et al, 2008) Model 0

0 0

0.5

1.0

1.5

2.0

2.5

0

3.0

0.5

1.0

1.5

2.0

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3

Injection volume, PV

Injection volume, PV

Fig. 141XPHULFDODQGH[SHULPHQWDOUHVXOWVIRUUHFRYHU\SHUIRUPDQFHRI22,3 D +39,ZHOOVDQGE 939,ZHOOV

120

120

(b) 100

80

80

Water cut， %

Water cut, %

(a) 100

60

40

60

40

20

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Data (Haida et al, 2008) Model

Data (Hadia et al, 2008) Model 0

0 0

0.5

1.0

1.5

2.0

2.5

3.0

Injection volume, PV

0

0.5

1.0

1.5

2.0

2.5

3.0

Injection volume, PV

Fig. 15 Comparison of measured and predicted results for water cut curves D +39,ZHOOVDQGE 939,ZHOOV

$QDO\WLFDO VROXWLRQ RI WKH FDVH ZLWK KRUL]RQWDO discontinuity leads to p=0.167 at the central node of the domain and w]=16.667 (fluid velocity) for both parts A and B. For the case with vertical discontinuity, exact solution for the nodal pressure at the central node of the domain is p=0.5, ZKLOHWKHH[DFWÀXLGYHORFLWLHVIRUSDUWV$DQG%DUHwz=10 and 50, respectively. These results are exactly produced by the proposed numerical method for both cases, even with the extremely coarse meshes shown in Fig. 18.

5.4 Confined flow between two perpendicular barriers ,Q WKLV H[DPSOH D îî P 3 domain with two perpendicular barriers, consisting of extremely low SHUPHDELOLW\]RQHVLVFRQVLGHUHG)LJ 7KHJHRPHWU\RI this example is chosen to be similar to that of Carvalho et al (2007). The domain is saturated with 28% water and 72% oil, DWLWVLQLWLDOFRQGLWLRQ,QLWLDO]HURSUHVVXUHLVDVVXPHGLQVLGH WKHGRPDLQ,QRUGHUWRVZHHSWKHRLOIURPWKHGRPDLQZDWHU

is injected at the right-hand side of the domain with a rate of 8 m3GD\ZKLOHWKHPL[WXUHRIRLODQGZDWHULVSURGXFHGDW the left-hand side. The absolute permeability is considered 1.5 D inside the barriers and 15,000 D in the rest of the domain. ,QVXFKDZD\SHUPHDELOLW\LQVLGHWKHEDUULHUVFKDQJHVIRXU orders of magnitude compared with the rest of the reservoir. ,Q IDFW WKHVH EDUULHUV SURGXFH D FKDQQHO DQG IOXLGV PXVW preferentially flow inside the channel. The porosity, fluids properties, irreducible water saturation, capillary curve and relative permeability curves are assumed same as example 2 IRUERWKRIWKH]RQHV Fig. 20 shows the pressure and velocity fields in the domain at t=10 and t=50 days. Fig. 21 shows the water saturation contours at t=10 and t=50 days.

5.5 Five-spot waterflooding problem with a central high permeability zone ,Q WKLV H[DPSOH D TXDUWHU RI ILYHVSRW ZDWHUIORRGLQJ SUREOHPZLWKDFHQWUDOKLJKSHUPHDELOLW\]RQHLVVLPXODWHG

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Beginning of the test

After 0.3 PV of injection

After 1.3 PV of injection

After 2.5 PV of injection

0

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

Fig. 16'LVWULEXWLRQRIZDWHUVDWXUDWLRQLQWKHFRUHVDPSOHGXULQJWKHWHVWIRU+39,ZHOOV

(a)

(a)

(b)

(b)

A A

B

B

Fig. 17 Cubic domain of example 3 D KRUL]RQWDOPDWHULDOGLVFRQWLQXLW\DQGE YHUWLFDOPDWHULDOGLVFRQWLQXLW\

(Fig. 22). The domain is saturated with 28% water and 72% RLO DW LWV LQLWLDO FRQGLWLRQ ,QLWLDO ]HUR SUHVVXUH LV DVVXPHG inside the domain. Water is injected to the domain at a rate of 1,600 liters per day and the model is run for 500 days. The DEVROXWHSHUPHDELOLW\LVFRQVLGHUHG'LQVLGHWKH]RQH with high permeability and 1.5 D in the rest of the domain. 2WKHUSURSHUWLHVRIWKHÀXLGVDQGURFNVDUHFRQVLGHUHGVDPH as example 2. Fig. 23 shows the water saturation contours after 200 days

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Fig. 18 Pressure distribution inside the cubic domain with extremely FRDUVHPHVKD IRUKRUL]RQWDOPDWHULDOGLVFRQWLQXLW\DQGE IRUYHUWLFDO material discontinuity

of injection. At this time, the water front does not reach the KLJKSHUPHDELOLW\]RQHDQGVXEVHTXHQWO\LWLVQRWDIIHFWHG E\WKH]RQH7KHUHIRUHLWLVH[SHFWHGWKDWEHIRUHGD\VRI water injection, the water front should be similar to that in a homogeneous media. As mentioned by Geiger et al (2004) a good numerical scheme should produce quarter circle shaped

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12 m

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40 m

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Injection of water

y

20 m

60 m

Dirichlet boundary condition with p=0.

20 m

(a)

x

1m

z x

0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50 0.52

Fig. 21 Water saturation contours of example 4 (a) t=10 days and (b) t=50 days

Fig. 19*HRPHWULFFRQ¿JXUDWLRQRIH[DPSOH

300 m

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150 m

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100 m

100 m

100 m

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1

2

3

4

5

6

7

8

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10

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Fig. 203UHVVXUHDQGYHORFLW\¿HOGVRIH[DPSOH (a) t=10 days and (b) t=50 days

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VDWXUDWLRQIURQWVRIWKHZDWHUSKDVHIRUD¿YHVSRWSUREOHPLQ a homogeneous medium. This quarter circle shaped of water front is also produced with the proposed method (Fig. 23). Fig. 24 shows the water saturation contours after 500 days of injection. At this time, the water front passed the boundary RIWKH]RQHRIKLJKSHUPHDELOLW\PDWHULDOVDQGFRQVHTXHQWO\ WKHZDWHUÀRZLVFRQFHQWUDWHGDWWKLV]RQH

6 Conclusions A control volume finite element method is proposed for the solution of governing differential equations of immiscible WZRSKDVHIORZLQSRURXVPHGLD,QWKLVPHWKRGWKHPHVK

flexibility of finite element method is combined with the local conservative characteristic of finite difference method at the control volume level. The formulation is presented DWWKHHOHPHQWOHYHO7KHHOHPHQWZLVHGLVFUHWL]DWLRQRIWKH governing equations leads the proposed method to handle the heterogeneity and discontinuity of rock properties in an elegant and accurate manner. The numerical diffusion and dispersion errors at shock fronts are minimal in the proposed PHWKRG6XEVHTXHQWO\WKHSURSRVHGVROXWLRQFRXOGDFFXUDWHO\ preserve the shock fronts. The accuracy and ability of the proposed model was verified with some waterflooding experiments and some representative model examples.

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Fig. 23 Water saturation contours of example 5 at 200 days

0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.90

Fig. 24 Water saturation contours of example 5 at 500 days

Acknowledgements 7KH DXWKRUV ZRXOG OLNH WR WKDQN ,UDQLDQ 2IIVKRUH 2LO &RPSDQ\,22& IRU¿QDQFLDOVXSSRUWRIWKLVZRUN

References Abriola L and Pinder G. A multiphase approach to the modeling of SRURXVPHGLD&RQWDPLQDWLRQE\RUJDQLFFRPSRXQGV1XPHULFDO simulation. Water Resource Research. 1985. 21(1): 19-26 Ataie-Ashtiani B and Raeesi-Ardekani D. Comparison of numerical IRUPXODWLRQVIRUWZRSKDVHÀRZLQSRURXVPHGLD*HRWHFKQLFDODQG Geological Engineering. 2010. 28: 373-389 %HUJDPDVFKL/0DQWLFD6DQG0DQ]LQL*$PL[HG¿QLWHHOHPHQW¿QLWH YROXPH IRUPXODWLRQ RI WKH EODFNRLO PRGHO -RXUQDO RI 6FLHQWLILF Computing. 1998. 20(3): 970-997 %URRNV5DQG&RUH\$3URSHUWLHVRISRURXVPHGLDDIIHFWLQJÀXLGÀRZ -RXUQDORIWKH,UULJDWLRQDQG'UDLQDJH'LYLVLRQ,5  &DUYDOKR':LOOPHUVGRUI5DQG/\UD3$QRGHFHQWUHG¿QLWHYROXPH

IRUPXODWLRQIRUWKHVROXWLRQRIWZRSKDVHÀRZVLQQRQKRPRJHQHRXV SRURXVPHGLD,QWHUQDWLRQDO-RXUQDORI1XPHULFDO0HWKRGVLQ)OXLGV 2007. 53: 1197-1219 Chen Z and Ewing R. Comparison of various formulations of threeSKDVHÀRZLQSRURXVPHGLD-RXUQDORI&RPSXWLRQDO3K\VLFV 132: 362-373 &KHQ=+XDQ*DQG:DQJ+6LPXODWLRQRIDFRPSRVLWLRQDOPRGHOIRU PXOWLSKDVHIORZLQSRURXVPHGLD1XPHULFDO0HWKRGVIRU3DUWLDO Differential Equations. 2005. 21: 726-741 &KHQ = +XDQ * DQG 0D