oil brazil

8/12/2019 Oil Brazil

1/18

INTERNATIONAL JOURNAL OF MODELING AND SIMULATION FOR THE PETROLEUM INDUSTRY, VOL. 2, NO. 1, FEBRUARY 2008 17

Numerical Simulation of Oil Recovery Through

Water Flooding in Petroleum Reservoir Using

Boundary-Fitted Coordinates

Brauner Goncalves Coutinho1, Francisco Marcondes2, and Antonio Gilson Barbosa de Lima1

1Federal University of Campina Grande

Center of Sciences and Technology

Department of Mechanical Engineering

Av: Aprgio Veloso, 82, Bodocongo, PO Box 10069, CEP 58109-970, Campina Grande, PB, Brasil.

2Federal University of Ceara

Center of Technology

Dept. of Mechanical and ProductionCampus do Pici

PO Box 12144, CEP 60455-760, Fortaleza, CE, Brasil.

[email protected], [email protected], [email protected]

AbstractEfficient mathematical models can be used to predictthe behavior of the fluids and fluid flow inside the petroleumreservoir under several operation conditions. The main goal ofthis study is to obtain a numerical solution for two-phase prob-lems with complex geometry reservoirs using the finite-volume

method and boundary-fitted coordinates. The physical modeladopted is the standard black-oil, simplified to an immiscible,two-phase (oil-water) flow including water flooding process toincrease oil recovery. This model can be applied for studies inreservoirs that contain heavy oils or low-volatility hydrocarbons.The mass conservation equations, written in the mass fractionsformulation, are solved using a fully implicit methodology and theNewtons method. In spite of computational time consumption,the advantage of this methodology is the possibility to use largertime steps. The UDS scheme is used to evaluate the phasemobilities in each control volume face. Results of the fluidsaturation fields, water cut, oil recovery and pressure inside thereservoir along the time are presented and analyzed. Attentionare given to the effect of refinement and orientation of gridin the simulation results. Results are presented in terms of

Newtons and solver iterations number, CPU time used to buildthe Jacobian matrix and to solve the linear systems and for thewhole simulations.

Index TermsReservoirs simulation, finite-volume, black-oil,boundary-fitted coordinates

I. INTRODUCTION

A petroleum reservoir is a complicated mixture of porous

rock, brine, and hydrocarbon fluids, usually residing under-

ground at depths that prohibit extensive measurement and

characterization. Petroleum reservoir engineers face the dif-

ficult task of using their understanding of reservoir mechanics

to design schemes for recovering hydrocarbons efficiently. Atypical oil reservoir is a body of underground rock, often sed-

imentary, in which there exists an interconnected void space

occupying up to 30 percent of the bulk volume depending on

location. This void space harbors oil, brine, water, and possibly

injected fluids and hydrocarbons gas. The structure of the void

space can be quite fine and tortuous, and as a consequence the

resident fluids flow rather slowly - typically less than a meter

per day (Allen III et al., 1988).In the petroleum exploration and production sector, a pri-

ority is placed on gaining accurate knowledge and analy-

sis regarding the characteristics and changes over time of

petroleum reservoirs (for instance, reservoirs of crude oil and

/or natural gas) as oil, gas, and water are being extracted to the

surface. Because petroleum deposits occur underground, often

far below the surface of the Earth, and because the contents

of a petroleum reservoir (for instance, an oil or gas field) may

be dispersed throughout a spatially and geologically extensive

and diverse underground region (the reservoir), the evaluation

over production lifetime of petroleum reservoirs is a complex

and economically essential task.

The goal of evaluating reservoirs are manifold and beginswith the earliest stages of speculative exploration activity

(at a point when it is not necessarily known whether a

geologic region or structure contains accessible petroleum in

commercially marketable quantities), and goes through the

production lifetime of an identified reservoir (when it may be

important, for example, to evaluate and/or vary the best sites

for placing wells to tap the reservoir, or the optimal rate at

which petroleum may be removed from a reservoir during on-

going pumping). Because companies in the petroleum industry

invest very large sums of money in exploration, development,

and exploitation of potential or known petroleum reservoirs, it

is important that the evaluation and assessment of reservoirscharacteristics be accomplished with the most efficient an

accurate use of a wide range of data regarding the reservoir

(Anderson et al., 2004).


2/18

18 INTERNATIONAL JOURNAL OF MODELING AND SIMULATION FOR THE PETROLEUM INDUSTRY, VOL. 2, NO. 1, FEBRUARY 2008

The production of hydrocarbons from a petroleum reservoir

is often characterized as occurring in three stages. Despite

these inhibiting phenomena, a variety of natural sources of

energy actually facilitate the production of oil from reservoir.

In this situation, when these energy sources suffice to allow

production by pumping alone, without the injection of other

fluids, the reservoir is said to be under primary production.

Among the mechanisms promoting primary production we

can cite: dissolved gas drive, gas cap expansion and natural

water drive. Eventually, fluid production exhausts these natural

energy sources, and pumping alone ceases to be economical.

To recover oil beyond primary production, reservoir engineers

usually implement secondary recovery production. Typically,

these consist of water flooding, a process in which field opera-

tors pump water into the reservoir through injection wells with

the aim of displacing oil toward equipped production wells

(Allen III et al., 1988). In the tertiary recovery stage, additional

steps are taken to enhance the recovery of hydrocarbons andto aid the fluid replacement process. These steps may include

the injection of special hydrocarbons solvents as well as other

selected fluids into the formation. Further, in-situ thermal

agitation such as the injection of steam and the ignition of

the hydrocarbons may be employed.

In order to produce the hydrocarbons as efficiently as

possible during each of these stages, it is important to know

the distribution of the fluid in the reservoir at any time during

the production process (Wason et al., 1990). The successful

characterization and management of petroleum fields depends

strongly on the knowledge of the hydrocarbons volumes in

place and the flow conditions of the phases (water, oil andgas). These data are the support for the economic and strategic

decisions, like drilling new wells or the field abandonment.

For the other side, the study of oil reservoirs using labora-

tory experiments is a complex task. The confident reproduc-

tion of all fluid and rock conditions (temperature, pressure,

geometry, composition) in the surface is almost impossible,

or economically difficult. In this sense, oil reservoir engineer-

ing encompasses the processes of reservoir characterization,

mathematical modeling of the physical processes involved in

reservoir fluid flow, and finally the numerical prediction of a

given fluid flow scenario. The basic problem associated with

oil recovery involves the injection of fluid or combinations of

fluids and/or chemicals into the reservoir via injection wellsto force as much oil as possible towards and hence out of

production wells. Accurate prediction of the performance of a

given reservoir under a particular recovery strategy is essential

for an estimation of the economics, and hence risk, of the

oil recovery project. Therefore a large amount of research,

and money, is directed towards the above processes, by the

oil industry (Wason et al., 1990; Dicks, 1993; Marcondes,

1996; Granet et al., 2001; Giting, 2004; Gharbi, 2004; Hui

and Durlafsky, 2005; Mago, 2006; Matus, 2006; Di Donato et

al., 2007; Lu and Connell, 2007; Escobar et al., 2007).

Granet et al. (2001) presents a two-phase flow modeling of

a fractured reservoir using a new fissure element method. Themethod has been validated by comparison with results from a

black-oil simulator run on a finely gridded Cartesian model.

According to authors the computational code developed per-

mits the accurate description of the phenomena occurring

within the fissure and the matrix blocks, and an understanding

of the production mechanism of fractured reservoirs.

Di Donato et al. (2006) report an analytical and numerical

analysis of oil recovery by gravity drainage. The numerical

model is validated by predicting previously-published exper-

imental measurements. According to authors, when gravity

dominates the process, the oil recovery scales as a power law

with time an exponent that depends on the oil mobility.

Ridha and Gharbi (2004) report a study about reservoir sim-

ulation for optimizing recovery performance by fluid injection.

The following techniques were tested: water-alternating-gas,

simultaneous water-alternating-gas, and gas injection in the

bottom of the reservoir with water injection in the reservoir

top. By comparing among the situations, the most economical

method to oil recovery was gas injection in the bottom of the

reservoir with water injection in the reservoir top.

The reservoir characterization process provides the physicalparameters, such as size, resident fluid and rock composition

and properties, which are needed by the mathematical model.

Given the physical parameters, the mathematical model de-

scribes the fluid flow with a set of partial differential equations,

initial and boundary conditions, and other relations, which are

derived from physical principles.

The processes occurring in petroleum reservoirs are basi-

cally fluid flow and mass transfer. Up to three immiscible

phases (water, oil, and gas) flow simultaneously, while mass

transfer may take place between the phases (chiefly between

gas and oil phases). Gravity, capillary, and viscous forces all

play a role in the fluid flow process (Peaceman, 1977). In thereservoir simulation, a frequent boundary condition is that the

reservoir lies within some closed curve C across which there

is no flow, and fluid injection and production take place at

wells which can be represented by point sources and sinks,

for example.

A number of mathematical models exist for the description

of fluid flow in oil reservoirs. These can be divided into

categories as to whether the fluid flow is considered to be

compressible or incompressible and whether the fluid compo-

nents are immiscible or miscible. The mathematical models

that describe most isothermal flow situations are derived from

four main physical principles. There are: conservation of

mass of the fluid components; conservation of momentum;thermodynamic equilibrium, which determines how the fluid

components combine to form phases, and lastly the condition

that the fluid fills the rock pore volume. Several analytical

models are available, but its application is restricted to small

models, due to the complexity and mathematic effort required

in most of the practical applications. So the solution for

intermediate and large models is the numerical simulation.

Different methods such as finite difference, finite element and

finite volume methods are used in oil reservoir simulation;

although in this work we concentrate solely on finite vol-

ume methods for solution of the partial differential equations

(Dicks, 1993).Some influential factors on the modeling are: number of

components and phases, well formulation, grid construction

and geometry, and physical phenomena considerations. The


3/18

COUTINHO et al.: NUMERICAL SIMULATION OF OIL RECOVERY THROUGH . . . 19

most common model is the Black-Oil, where three phases and

three components are considered. The first basic equation is

the mass conservation through a control volume, the continuity

law. The Darcys law is used to represent the flux in porous

media. Finally, some complementary equations and the bound-

ary and initial conditions are used.

In this work, theoretical development of the Black-oil model

(applied to petroleum reservoir) and numerical solution for the

governing equations are presented and numerical examples

are demonstrated. The purpose is to simulate oil recovery

through water injection in petroleum reservoir with complex

geometry using boundary-fitted Coordinate and the finite-

volume method.

II. MATHEMATICAL MODELLING

The standard black-oil is a mathematical model that can be

used in reservoirs with heavy or low-volatility hydrocarbons. It

is an isothermal model where the behavior among the phases isgoverned by pressure, temperature and volume relationships.

The characteristics of the model are:

There are three components (water, oil and gas) and three

phases (water, oil and gas);

Water and oil phases neither mix nor interphase mass

transfer;

The gas component is dissolved in oil phase;

Water and oil components cannot be found in the gas

phase.

In the present study, a two-phase (oil-water) immiscible flow

was considered. Here, gravitational and capillarity effects are

neglected, therefore, in all phases only one pressure is used.Based in these assumptions, mass conservation equation for a

generic phase p is given by

t[mZp] = .

pPmp (1)where the superscript p indicates the phasep, is the porosity, is the average density of the mixture, Z is the mass fraction,and P is the pressure inside the reservoir. In this equationmpandp represents the mass flow per unit of volume of thereservoir and phase mobility, respectively, and are defined as

follows

mp =mqp (2a)

p = pkkrpp

(2b)

whereqp is the volumetric flow rate of the phasep per volume.In Eq. (2), k is the absolute permeability, krp is the relative

permeability, andp andp are density and viscosity of phasep, respectively.

Writing Eq. (1) for the oil and water phases, there are three

unknowns (ZO,ZW, andP) and two equations. The equationneeded for the complete solution comes from global mass

conservation as follows:

Zw + Zo = 1 (3)

More details of the Black-oil formulation in terms of mass

fractions can be found in Prais and Campagnolo (1991), Cunha

(1996) and Coutinho (2002).

III. NUMERICAL SOLUTION

Due to nonlinearities present in the governing equations,

specially that one in the phase mobility, those equations do not

have known analytical solution. A numerical solution, such

as finite-volume method, can be an alternative to solve thisproblem.

One of the inputs to a numerical reservoir simulator is a

reservoir geometric description to obtain the grid. Gridding

for petroleum simulators has been relatively conservative,

with most commercial simulators being restricted to structured

grid with local grid refinement. However, in the last decade,

unstructured grid was introduced. Reservoir simulation are

normally being performed on rectangular Cartesian grid, radial

grid was developed later to simulate flow near the wellbore. In

principle, if extremely fine grid could be created it would be

possible to represent reservoir easily. However, the number of

control-volume in the grid is limited by computer capacity andCPU time. In order to solve this problem, the concept of local

grid refinement has been introduced. Local grid refinement was

developed to achieve better accuracy in high flow regions.

The main advantages of Cartesian grids are the simplicity

of the conservation balances and easy solution of the resulting

linear systems. The disadvantages are: difficulty to model

complex geometries reservoirs, geologic faults, complex dis-

tribution of wells and grid orientation effect (Todd et al., 1972;

Aziz and Settari, 1979). Non-orthogonal boundary-fitted grids

can turn the numerical method flexible to treat reservoirs with

more complex geometries (Maliska, 2004; Cunha et al., 1994).

Numerical solution of two-dimensional displacement prob-lems can be strongly influenced by the orientation of the

underlying grid. Under certain situation, vastly different nu-

merical results are obtained for water flooding, depending

on whether the grid lines are parallel or diagonal to the

line joining an injection-producer well pair. This is called

the grid orientation effect. This effect has been found to be

particularly pronounced in simulation where the displacing

phase is much more mobile than the displaced phase. In water

flooding simulation, both mobility weighting procedure and

discretization scheme affect grid orientation. Therefore both

accurate numerical procedure and correct mobility weighting

are needed to alleviate grid orientation (Abou-Kassem, 1996).

The numerical error of a solution of a set of differentialequation on a grid is caused by the truncation errors due

to the discretization. A non-uniform grid produces additional

terms in the truncation errors. The numerical error and its

propagation depend on the differential equation and discretiza-

tion method. In hyperbolic and parabolic problems, like the

saturation equation, the numerical error propagates easier

between regions. This is not the case in elliptic equations,

like the pressure equation, where the local numerical error is

closely related to the local truncation error. So, independent of

the equation type, it is important to minimize the truncation

error. Non-orthogonality will usually imply that cross terms

should be added in the equations. Neglecting these termsinduced by a non-trivial metric may lead to errors that are

independent of the grid spacing (Soleng and Holden, 1998;

Fletcher, 2003). In this study, the cross terms are used in two


4/18


Fig. 1. Infinitesimal volume in the computational domain.

situations namely five points and nine points scheme. All the

equations are written in boundary-fitted Coordinates.

A. Transformation of the governing equations

Considering only 2D problems Eq. (1) can be written inboundary-fitted coordinates as follows:

1

J

t(mZp) +

mpJ

=

DP1P

+DP2

P

+

DP2P

+DP3

P

(4)

where J is the Jacobian and the coefficients Dpi are givenby:

DP1= pJ 2x+ 2y (5a)DP2=

pJ

(xx+ yy) (5b)

DP3=pJ

2x+

2y

(5c)

Equations (5 a-c) have all grid information (Maliska, 2004).

B. Integration of the governing equations

Integrating Eq. (4) in space and time for the volume shown

on Figure 1, the following equation is obtained:

VJ

[(mZp)P (mZp)oP] +mpJ Vt=DP1P

+DP2

P

e

DP1P

+DP2

P

w

t+

DP2P

+DP3

P

n

DP2P

+

DP3P

s

t (6)

where V = is the volume dimensions on gener-alized coordinates system.

All differential terms in right hand side of Eq. (6) are

approximated by central differencing scheme. The pressure

gradients in the east face, for example, are given by,P

e

=PE PP

(7a)

P

e

=PN+ PNE PS PSE

4 (7b)

To evaluate the phase mobility in each control volume face it

was employed the Upwind Differencing Scheme. Using again

the east face,P is given by,Pe =PP ifuPe >0, andPe =PE otherwise. (8)Flow velocity can be calculated through Darcys law. Writ-

ten in generalized coordinates, for the east face of the volume,

for example, this field can be determined by:

upe = peG1e

(pE pP)

+

G2e(pN+

pNE

pS

pSE)

4

(9)

where

Gi = Diw i= w,e,n, s (10)C. Fully implicit methodology

In this methodology the unknowns P andZo are implicitlycalculated at the current time step. The equations are linearized

by Newtons method. Passing to the left side all terms of Eq.

(6) the following residual equation is obtained:

FpP = V

J [(mZp)P (

mZp)oP] +

mpJ

Vt

DP1P

+DP2P

e

DP1P

+DP2P

w

tDP2P

+DP3

P

n

DP2P

+

DP3P

s

t (11)

Expanding the equation (11) by Taylors series, we have:

(FpP)k+1

= (FpP)k

+X

FpPX

kX= 0 (12)

wherek is the iteration level andXrepresents the unknownsP and Zo.

In the Newtons method, the solution in every time step isconsidered to converge when the residues are smaller than the

convergence criterion. Therefore, Eq. (12) in the short form is

given by:

(FpP)k

=X

FpPX

kX (13)

In the matrix form, Eq. (12) can be written by:

AX= F (14)

where A is the Jacobian matrix of the residual function Fon the k-th iteration.

The solution of the linear system, Eq. (14), allows calcu-lating theP andZo values till the mass conservation in eachtime step is obtain. The Jacobian matrix A is a block matrix,

i.e., all its elements are square matrices.


5/18


1) Nine points scheme: On this scheme, all neighboring

points are considered on the differentiation of the residual

functions. Using this scheme, Eq. (13) will be given by:

FpP

PPPP+FpP

ZoPZoP+FpPPWPW+

FpPZoW

ZoW+

FpPPE

PE+

FpPZoE

ZoE+

FpPPS

PS+

FpPZoS

ZoS+

FpPPN

PN+

FpPZoN

ZoN+

FpPPSW

PSW+

FpPZoSW

ZoSW+

FpPPSE

PSE+

FpPZoSE

ZoSE+

FpPPNW

PNW+

FpPZoNW

ZoNW+

FpPPNE

PNE+

FpPZoNE

ZoNE= F

pP (15)

2) Five points scheme: According to Cunha (1996), to

simplify the linear system, the derivatives of the cross terms

(SW,SE,NW,NE) may be considered only in the residualfunction. This procedure avoids additional terms in the Jaco-

bian matrix when the coordinates lines are non-orthogonal.

Using this scheme, the Eq. (12) can be rewritten as follows:FpPPP

PP+

FpPZoP

ZoP+

FpPPW

PW+

FpPZoW

ZoW+FpPPEPE+ Fp

P

ZoEZoE+

FpPPS

PS+

FpPZoS

ZoS+

FpPPN

PN+

FpPZoN

ZoN= F

pP (16)

This approach simplifies the resultant linear system but it

can either slow down the convergence rate or hamper the

convergence if the mesh is highly non-orthogonal. In the

results section of this work, some comparisons between both

schemes will be shown and analyzed. More details about

whole mathematical formulation can be found in Cunha (1996)and Coutinho (2002).

D. Discretized Well model

In reservoir simulation we use an analytical model to

represent flow within a grid as it enter or leaves a well. This

model is called the well model. It is well-known that pressure

of the wellblock is different from the bottomhole well flowing

pressure at the well. This is because the control-volume

dimensions are significantly greater than the wellbore radius.

The flow rate in the well is proportional to the difference

between the block and well pressure (Figure 2). Since thegrid pressure and all other physical properties are assumed to

be centered at the middle of the control-volume, the well is

also assumed to be at the center of the grid cell.

Fig. 2. Radial flux near the well in a generalized grid.

For the generalized grid (Figure 2) the mass conservation

equation is given by:

m= D1eP

e

+D2eP

e

D1wP

w

D2wP

w

+

D2nP

n

+D3nP

n

D2sP

s

D3sP

s

(17)

By using the derivative approximations we have:

m= D1ePE PP

+D2ePN+ PNE PS PSE

4

D1w

PP PW

D2w

PN+ PNW PS PSW

4

+

D2n

PE+ PNE PW PNW

4

+D3n

PN PP

D2s

PE+ PSE PW PSW

4

D3s

PP PS

(18)

where, for example, for the east face,

D1ePE PP = D1e m2khlnrEro (19a)

D2e

PN+ PNE PS PSE

4

= D2e4

m

2kh

ln

rNro

ln

rSro

+ ln

rNEro

ln

rSEro

(19b)

where ro is the equivalent radio and rN, rS, rNE and rSE,are the distance between the center of the volume P and thecenter of the volumes N, S, NE and SE, respectively.

The equivalent radio of the well is given by:

ro =G1e

e2

G1e 1 (20)


6/18


where

=

rG1e+

G2n4 G2s

4E

r

G2e4 G2w

4 +G3n

N

r

G2e4

+G2w

4 +G3s

S

rG1w

G2n4

+G2s

4W

rG1e

G2e4

+G2n

4NE

rG1e

G2e4 G2s

4SE

rG1e

G2w4 G2n

4NW

rG1e

G2w4

+G2s

4SW

(21a)

= G1e+ G1w+ G3n+ G3s (21b)

Gi=Diw i= w,e,n,s (21c)

E. Wells boundary conditions

In numerical simulation problems, in a particular petroleum

reservoir, initial and boundary conditions are required to

initialize the solution of the model. The boundary conditions

used in reservoir simulators can be very complicated as the

differential equations solved by the simulators require that

all boundaries be specified. This includes both internal and

external boundaries. External boundaries are the physical

boundaries of the flow domain, while for internal boundaries,

either well rates or bottomhole pressure can be specified.Initial conditions are initial pressure and saturation distribution

inside the reservoir. Here it is considered that a non-flow

outer boundary exists. So, phase transmissibilities across the

boundary interfaces are set to zero. This implies that there is

an impermeable boundary.

Boundary conditions at the wells are based on the fluids

mobility. By assuming that the flow rate in each phase is

proportional to mobility, we can write:

qw

w

= qo

o

= qT

T

(22)

where superscript Trepresent total (water plus oil)In a injector well, the flow rate of each component that is

being injected is prescribed. All others components have flow

rate equal to zero. For instance, for water injection:

qw =qwinj (23a)

and

qo = 0 (23b)

In the producer well, the total flow rate (water + oil) and

pressure are prescribed as follows:

qT =qTprod (24a)

qp

= p

T qT (24b)and

Pwf =Pi (24c)

F. Physical properties and saturation relationships

Finally, the following relationships were used: a)

1) Formation volume factor

Bp

(P) =

Bpref1 + cp(P Pref) (25)

2) Porosity

= ref[1 + cr(P Pref)] (26)

3) Density of the phases

p =pSTCBp

p= o, w (27)

wherePSTCis the density of the phases in the standardcondition.

4) Saturation of the phases

Sp =

p

pnp

p

p

p= o, w (28)

5) Average density of the mixture

m =nP

pSp p= o, w (29)

All these parameters presented in the equation (21-28) are

labeled in the Tables II and III. The grid generation procedure

can be found in Coutinho (2002) and Maliska (2004).

IV. RESULTS AND DISCUSSIONSManagement of water flooding requires an understanding of

how the injected fluid displaces the oil to the production wells.

This permits to allow the optimization of the oil recovery

and identification of possible allocations of new injection and

production well. However, to predict movements of fluid into

reservoir is not a easy task. The solutions are sensitive to the

grid system because the fluids move between discrete control-

volumes, and the numerical scheme should be carefully cho-

sen.

In water-flooding process in petroleum reservoir, simulation

results are largely influenced by numerical treatment of the

mathematical model, grid refinement and grid orientation. Gridorientation effects arise from an unfavorable mobility ratio

in a displacement process. Grid orientation along with the

level of refinement may produce widely varying quantitative

simulation results. Having demonstrating the importance of the

numerical treatment of the model, orientation and refinement

of the grid, different simulations were run to study their impact

on oil recovery and computational time.

A. Case 1

The example selected to evaluate the behavior of the ap-

proaches just mentioned in the last section was originally

proposed by Hirasaki and Odell (1970) and after studied byHegre et al. (1986) and Czesnat (1998). They considered a

reservoir with two production wells equidistant from an injec-

tion well and the same size of the control-volume, as shown


7/18


Fig. 3. Scheme of the reservoir and wells locations.

TABLE IRELATIVE PERMEABILITIES OF THE PHASES. SOURCE: HEGRE ET AL.

(1986).

Sw krw kro

0.25 0 0.920

0.30 0.020 0.7050.40 0.055 0.420

0.50 0.100 0.240

0.60 0.145 0.110

0.70 0.200 0

in Figure 3. To investigate the grid effects they compared the

volumetric production rate of each producer well. The mesh is

aligned with the line that connects the producer and injector

wells (parallel grid system) and is diagonal to another pair

(diagonal grid system), as shown in Figure 4.

To examine the dependency of the simulation results on thenumerical grid, meshes with246control-volumes were used,Figure 4, and refined with 48 20 control-volumes. Table (I)gives the relative permeabilities of the phases and Table (II)

shows characteristic data of fluids and reservoir used in this

work. Figure (fige) presents saturation fields, in the elapsed

time of 500 and 2000 days and compares the results of thetwo grid systems. With these results, we can write that the

production profiles, especially the oil production, are quite

different. Note that the mesh with less number of control-

volumes have generated results with high grid effects and

numerical dispersion. The water injected in the well reaches

the producer well 1 (aligned with the grid) faster than producer

well 2. Considering that each producer well is equidistant tothe injector well, physically, this phenomenon couldnt occur.

We expected to get similar recovery performance from both

grid systems. This is because the coordinate axes resulted

in differing amounts of truncation errors. This undesirable

problem causes errors in water irruption time on producer

wells. Many researchers have been investigating grid effects

on reservoir simulations nowadays (Marcondes, 1996; Czesnat

et al., 1998).

In the study of the grid orientation effect, the goal is to

simulate the same problem using meshes with many levels of

orthogonality. So, several changes in the grid orientation was

proposed. Initially the grid is Cartesian, Figure (6a), i.e., it hasa90 inclination with a horizontal line. Other grids with 80,60, 45, 30 and 20 inclinations were obtained distortingthe original grid. As can be seen on Figures (6b-f), for each

case, the reservoir boundary was changed, but the distances

from injector to producer wells were kept constant.

Figure (7) shows many simulations results. Each plot con-

tains curves generated using five and nine points schemes for

both grids (24 6 and 48 20 control-volumes). On leftcolumn, the maximum time step used was 50 days, while on

the right one, 100 days. In all figures, the variables of vertical

axis were evaluated from beginning untill the end of simulation

(7500 days). The meaning of each of these variables will be

described next.

Figures (7a) and (7b) illustrate the number of time steps

used with the maximum time step of 50 and 100 days, respec-

tively. The time step for solving governing equation depend on

the numerical scheme and the physical parameters considered

in the model. A larger quantity of increments means that the

average time step used was small. From these figures, it can be

noticed that the reduced Jacobian matrix (five-points) presents

a large variation of this parameters when angles equal to 60

or smaller were used. For small mesh angles, great variations

occurred on mass fractions or pressures, that kept the average

time step less than that observed on orthogonal grid (= 90)or when the full Jacobian matrix (nine-points scheme) was

used. It is also worth noticing that the maximum number of

time steps increased when a more refined mesh was used. With

larger time step, the higher saturation of the previous time step

is used for calculating mobility, and this causes errors in the

prediction of the oil production.

The behavior of the number of time steps used was approxi-

mately linear for the full Jacobian matrix, but it was non-linear

when it was used matrix including only direct neighboringvolumes. The increase of the number of time steps with the

number of volumes on the mesh can be explained by the

increase of the mass fractions and pressure variations in each

time step. Finally, it can be mentioned that the number of time

steps wasnt sensible to the grid orthogonality when using

nine-points scheme. Five-points scheme required many time

steps for skewed grids. Some tests couldnt be performed for

determined grid angles, as seen on Figures (7a) and (7b).

Figures (7c) and (7d) show the total number of iterations to

reach convergence in the Newtons method on each time step.

The bigger the iterations number is, the larger will be computer

costs of the simulation. Note that, for both grids, the two

schemes presented the same efficiency only for 90 inclinationof mesh. In all other geometries, as skewness angle increases,

five-points scheme demand more Newtons method iterations.

When the full Jacobian matrix is used the iterations number

has small sensitivity for the increasing mesh inclination and

time step.

Figures (7e) and (7f) present all solver iterations. Analyzing

these figures, it can be seen that the number of solver iterations

was not dependent on the mesh inclination when the full

Jacobian matrix was used . It can be mentioned that, in the

present work, the diagonal block as pre-conditioner matrix

(which doesnt consider the complete structure of the Jacobian

matrix) was used. This pre-conditioner isnt more efficientthan the ILU pre-conditioner, according to Marcondes et al.

(1996) and Maliska et al. (1998), but it has been robust in all

mesh inclinations analyzed. For five-spot configuration, the


8/18


Fig. 4. Grid with 24x6 control-volumes oriented at45.

TABLE IIFLUIDS AND RESERVOIR DATA. SOURCE: HEGRE ET AL. (1986).

Porosity = 0.19 Residual oil saturation Sor = 0.2

Permeability k= 0.049x1012m2 Water and oil density w =o = 1000kg/m3

Height h= 18.3m Water reference volumetric formation factor Bwref

= 1 on Pref

Initial pressure Pi= 27248x103P a Oil reference volumetric formation factor Bo

ref = 0.96 on Pref

Rock compressibility cr = 0P a1 Reference pressure Pref= 27248x103P a

Oil compressibility co = 1.45x109P a1 Water viscosity w = 0.5x103Pa.s

Water compressibility cw = 0.44x109P a1 Oil viscosity o = 2.0x103Pa.s

Well radius rw = 0.122m Water injection rate qinj= 302.1m3/dia

Initial water saturation Swi = 0.2 Total production rate qprod= 159m3/dia

Fig. 5. Water saturation fields for two grids and two times.

iterations number varied with the increase of mesh angle.

Figures (7g) and (7h) show elapsed time for the compo-

sition of the Jacobian matrix and calculation of the residualfunctions. Note that, as larger grid inclination, more time was

needed when five-points scheme was used. As for full Jacobian

matrix, this computational time hasnt changed when varia-

tions occurred on grid inclination. This fact could be explained

because the number of iterations of Newtons method increases

with the grid angle, Figures (6c-d). All simulations were made

using a Silicon Graphics Workstation - model Onyx 2.

Figures (7i-j) exhibit the time consumed by the solver

to solve the linear system. As commented in Figures (7e)

and (7f), the solver iterations number stayed constant when

the full Jacobian matrix was used for all grid angles. Then

the solver time must keep the same behavior. For the five-points scheme, the increase in the iterations number, as a

function of mesh angle, doesnt increase necessarily the CPU

time during simulation. This occurs due to operations done

by BICGSTAB (Bi-Conjugate Gradient Stabilized) method

to solve the linear equation system, proposed by Van Der

Vorst (1992), such as matrix-vector product that require highcomputational cost. The larger the matrix structure is, the more

expensive will be the computational process of the matrix-

vector products. Comparing the number of solver iterations in

Figure (7c) for60 skewed mesh (4820), it can be seen that,using the incomplete Jacobian matrix,5 approximately timesmore iterations were needed. However, time used by solver is

approximately the same for both schemes.

Figures (7k-l) illustrate average time step used in the

simulation. The smaller this value is, the larger will be the

total time of simulation. The full Jacobian matrix presented

a linear behavior with high values of average time step . For

incomplete matrix this value became lower for higher gridangles. It is verified that these curves have a similar behavior

with those shown on Figures (7a) and (7b). This can be easily

explained: the smaller the time steps are during simulation,


9/18


10/18


Fig. 7. Simulation results obtained with five-points and nine-points scheme.

TABLE IIIWELLS COORDINATES AND FLOW RATES. SOURCE: MARCONDES (1996)

Well Flow rates Coordinates (m)

(m3/day) x y

Injector 1 254.02 906 1343

(water flow rate) 2 174.87 1468 2218

1 79.49 593 1031

2 95.04 406 1281

Producer 3 79.49 1093 1843

(liquid flow rate) 4 47.69 1468 1531

5 63.59 1593 1906

6 63.59 1781 2468

one with 560 volumes (4014) and another more refined with1160 volumes (58 20).

To measure orthogonality levels among the three meshes,

we calculated medium and maximum angles of the coordinate

lines in each control volume of the grids. Values are shown in

the Table 5. For grid where = 11, medium and maximumangles are greater than angles in grids with = 37 and=57, in the meshes4014as well as5820volumes meshes.

Figure (12) shows many results obtained with the simula-

tions. Each plot contains curves generated using five and ninepoints schemes for both grids (4014 and 58 20 volumes).On the left column, the maximum time step used was 50 days,

while on right one, 100 days. In all figures, the variables of


11/18


(a) Producer well 1 (non-aligned) (b) Producer well 2 (aligned)

Fig. 8. Water cut for 90 inclination grid, t= 50 days.

(a) Producer well 1 (non-aligned) (b) Producer well 2 (aligned)

Fig. 9. Water cut for 45 inclination grid, t= 50 days.

Fig. 11. Inclination angle of the grid.


12/18


TABLE IVFLUIDS AND RESERVOIR DATA. SOURCE: MARCONDES (1996).

Height H= 15m

Porosity = 0.30

Absolute permeability k= 0.3x1012m2

Well radius rw = 0.122m

Initial pressure Pi = 20685x103P a

Initial water saturation Swi = 0.2

Residual oil saturation Sor = 0.2

Densities w =o = 1000kg/m3 at Pref

Reference volumetric Bwref

=Boref

= 1

formation factor at the Pref

Reference pressure Pref= 20685x103P a

Compressibility factor cw =co = 7.25163x109P a1

Water viscosity w = 103[1 + 1.45x1012(P

1.38x107)]Pa.s

Oil viscosity o = 1.1632[1 + 1.45x1012(P

1.38x107)]Pa.s

TABLE VFLUIDS AND RESERVOIR DATA. SOURCE: MARCONDES (1996).

Grid () Medium angle () Maximum angle ()

11 71.1 122.9

40x14 37 50.3 97.0

57 36.1 78.1

11 71.5 129.8

58x20 37 50.6 101.3

57 36.3 81.2

vertical axis were evaluated from beginning until the end of

the simulation. The meaning of each of these variables will

be described next.

Figures (12a) and (12b) illustrate the number of time steps

used with the maximum time step of 50 and 100 days,

respectively. Great quantity of increments means that the

average time step used by the code was small. From these

figures, it can be noticed that the reduced Jacobian matrix

(five-points) presents a large variation in the time step for

grid with angle 57. For small mesh angles, large variationsoccurred on mass fractions or pressures, that kept the average

time step smaller than the observed when the full Jacobianmatrix (nine-points scheme) was used. It is also worth noticing

that the maximum number of time steps increased when a more

refined mesh was used.

The behavior of the number of time steps was approximately

linear for the full Jacobian matrix, but it was non-linear when

the matrix including only direct neighboring volumes was

used. The increase of the number of time steps with the

number of volumes in the mesh can be explained by the

increase in the mass fractions and pressure that changes in

each time step. Finally, it can be mentioned that the number

of time steps was not sensitive to the grid orthogonality using

nine-points scheme. Five-points scheme required many timesteps for skewed grids. Some tests could not be carried out

for some grid angles, as seen on Figs. (12a) and (12b).

Figures (12c) and (12d) show the total number of iterations

necessary to Newtons method convergence on each time step.

The bigger the iterations number is, the largest will be the

computer costs to simulate each case. Note that, for both

grids, only for the 11 inclination mesh, the two schemespresented the same efficiency. When distortion increases, five-

points scheme demands more Newtons method iterations. For

full Jacobian matrix, the iterations number has small sensitivity

for the increasing of mesh inclination and time step, for each

skew angle.

Figures (12e) and (12f) present all solver iterations. By

analyzing these figures, it can be noticed that the number

of solver iterations wasnt dependent of mesh inclination

for full Jacobian matrix. In the present work, it was used

the diagonal block as pre-conditioner matrix, which doesnt

consider the complete structure of the Jacobian matrix. This

pre-conditioner isnt extremely efficient such as an ILU pre-

conditioner, according to Marcondes et al. (1996), but it has

been robust in all mesh inclinations analyzed. As for five-spotconfiguration, the iterations number varied by increasing of

mesh distortion.

Figures (13g-h) show the time spent in the composition of

the Jacobian matrix and calculation of the residual functions.

Note that, the larger the grid distortion is, the larger the time

cost when the five-point scheme was used. For full Jacobian

matrix case, the time cost has not changed with variations on

grid inclination. This fact could be explained by the increasing

number of iterations of Newtons method with the grid angle in

the case of five-point scheme(Figures (12c-d)). All simulations

were made using a Silicon Graphics Workstation - model Onyx

2. Figures (11i-j) exhibit the time consumed for the solver to

solve the linear system. As commented in Figures (12e) and

(12f), the solver iterations stayed constant with the complete

Jacobian matrix for all grid angles, thus the solver time showed

the same behavior. As for the five-points scheme, the increase

in the number of iterations as a function of the mesh angle does

not necessarily produce an increase in the CPU time during

simulation. This occurs due to operations done by BICGSTAB

to solve the linear equation system, proposed by Van der

Vorst (1992), such as matrix-vector product that require high

computational cost, as explained before.

The larger the matrix structure is, the more expensive the

computational process to compute the matrix-vector productswill be. Comparing the number of solver iterations on Figure

(11c) for refined 37 inclination mesh, it can be seen that,approximately the same number of iterations were needed

to solve both (complete and incomplete) Jacobian matrices.

However, more time was used to solve the complete matrix.

Figures (13k-l) illustrate the average time step used during

simulation. The smaller this value is, the larger the total time of

simulation will be. Observe that full Jacobian matrix presented

a linear behavior, with high values of average t. This valuebecame smaller for more distorted grids, when the incomplete

matrix was used. Note that these curves have a similar behavior

as those shown on Figures (12a) and (12b). This can be easilyexplained: the smaller the time steps during simulation are, the

higher the number of time steps will be needed to get total

time.


13/18


Fig. 12. Comparative performance between five and nine points schemes.

To compare the results obtained with both schemes, Figures

(13) to (16) show water cut curves on producer well 1 using

grids with40 14 and 58 20 control-volumes distorted 11

and57 with time step of 50 and 100 days. From this figures, it

is possible to notice that results achieved with five-points andnine-points schemes are very similar, with small differences

when it was used the 57 inclination, mainly usingt= 100days. This effect could be minimized with a mesh refinement

study.

Figure (17) presents water saturation distribution fields in

three V PI values. Notice that both grids (560 and 1160volumes) have generated very similar saturation fields. In these

figures,V PI is given by:

V PI= qwt

VR(1 Swi Sor )

(31)

where VR is the reservoir volume and SW

i and S0

r representinitial water saturation and residual oil saturation, respectively.

Figures (18) to (23) show pressure and recovery curves in

three producer wells (1, 4 and 5). In spite of observing a quite

Fig. 14. Water cut in the producer well 1 using a grid distorted11,t= 50

days.


14/18


Fig. 13. Comparative perfomance between five and nine points schemes.

Fig. 15. Water cut in the producer well 1 using a grid distorted 11

, t=100 days. Fig. 16. Water cut in the producer well 1 using a grid distorted57

,t= 50days.


15/18


(a) 560 volumes

(b) 1160 volumes

Fig. 18. Water saturation distribution for three VPI. 37 inclination grid.

Fig. 17. Water cut in the producer well 1 using a grid distorted 57, t=100 days.

similar behavior among the results, it is noticed that certain

variations exist. Observe that, in some cases, for instance, well1 (Cunha, 1996) and well 5 (present work), the curves obtained

with the meshes do not approximate those obtained with the

hexagonal-hybrid mesh. It is pointed out that, in the work of

Marcondes (1996), two meshes of the type hexagonal-hybrid

were used, with 672 and 1026 volumes, and the obtained

results were practically identical. This discrepancy can be

explained by the variation of the location of the well in the

meshes employed, like mentioned previously.

In these figures, dimensionless parameter V POR (porousvolume of oil recovery) represent the relationships between

the oil volume produced by reservoir with injection process

and the total volume of oil possible to be extracted of the

reservoir. This parameter is given by:

VPOR=

t0

qo(t)dt

VR(1 Swi Sor )

(32)

where VR is the reservoir volume and Swi and S

0r represent

initial water saturation and residual oil saturation, respectively.

In this study Swi and Sor were neglected.

As final comment, this paper can be used to help researchers

in the application of the history matching process. History

matching is an inverse process in which the properties of the

geological model, porosity and permeability, in particular, are

tuned in such a way that the simulation results reproduce

the measured pressure and production data. This inverseprocess is important for reducing uncertainties in reservoir

characterization, which is crucial for evaluating options of

field development and predicting future reservoir performance


16/18


(Arihara, 2005).

Fig. 19. Pressure in the producer well 1 using a grid distorted37.

Fig. 20. Oil recovery in the producer well 1 using a grid distorted37.

Fig. 21. Pressure in the producer well 4 using a grid distorted37.


Fig. 23. Pressure in the producer well 5 using a grid distorted 37.


V. CONCLUSIONS

In any reservoir prediction, a realistic description of the

reservoir behavior under any depletion scheme is probably

the most important factor. In real scenario natural porous


17/18


media are heterogeneous and multi-phase flow is the result

of equilibrium between viscous, capillary and gravity forces.

This equilibrium changes with physical location and time. In

this paper the reservoir was considered heterogeneous and

the sequential solution method for oil-recovery simulation is

similar in several ways from the methods reported in the

literature, but details of the solution method of mass transport

is presented and discussed here.

The sensitivity of the model and numerical treatment

adopted were presented. The water-flooding to improve re-

covery in oil reservoir was analyzed, and the results were

compared with results reported in the literature. An excellent

agreement was obtained for the oil recovery performance.

Performances of the five-points and nine-points schemes

were tested in meshes with many levels of their skewness.

It was observed that it is necessary take into account the

cross terms during the construction of the Jacobian matrix

for meshes with high levels of non-orthogonality. Althoughthis fact may contribute to increase CPU time (in each solver

iteration), there is a substantial reduction on the number of

iterations in the Newtons method and number of time steps

employed. The use of full Jacobian matrix allows the use of

large time steps in all simulations.

Nine points scheme (full Jacobian matrix) keeps a linear

behavior for grids in all cases meanwhile five points scheme

turns more expensive the computational process when used

in distorted grids. So, to study distorted meshes, it is needed

to include all neighboring volumes during composition of the

Jacobian matrix. The grid orientation effect disappears as the

number of grid cells increases. So, grid refinement can help

to reduce the grid orientation effect, however more research

about this theme is recommended.

ACKNOWLEDGMENT

The authors thank to CNPq, FINEP, PETROBRAS,

ANP/UFCG-PRH-25 and CT-PETRO, for the granted financial

support and to Mr. Enivaldo Santos Barbosa for the preparation

of this manuscript.

REFERENCES

[1] Allen III, M. B.; Behie, G. A.; Trangenstein, 1988, Multiphase flow inporous media. Springer-Verlag, London, 308p.

[2] Anderson, R. N.; Boulanger, A.; He, W.; Winston, J.; Xu, L.; Mello, U.;Wiggins, W., 2004, Petroleum reservoir simulation and characterizationsystem and method. US. Patent No. 6826483 B1, 13p., USA.

[3] Arihara, A., 2005, Reservoir simulation technology by streamline-basedmethods. Journal of the Japan Petroleum Institute. V. 48, n. 6, pp. 325-335.

[4] Abou-Kassem, J. H., 1996, Practical considerations in developing nu-merical simulators for thermal recovery. Journal of Petroleum science &Engineering. V. 15, pp. 281-290.

[5] Aziz, K.; Settari, A., 1979. Petroleum reservoir simulation. AppliedScience Publishers, London, 476p.

[6] Coutinho, B. G., 2002, Numerical Solution for Problems of PetroleumReservoirs Using Generalized Coordinates, Master Dissertation, FederalUniversity of Campina Grande, Brazil. (In Portuguese)

[7] Cunha, A. R., 1996, A Metodology to Three Dimensional NumericalSimulation of Petroleum Reservoirs Using Black-Oil Model and MassFraction Formulation, Master Dissertation, Federal University of Santa

Catarina, Brazil. (In Portuguese)[8] Cunha, A. R., Maliska, C. R., Silva, A. F. C., Livramento, M. A.,

1994, Two-Dimensional Two-Phase Petroleum Reservoir Simulation Us-ing Boundary-Fitted Grids, Journal of the Brazilian Society of MechanicalSciences, Vol. XVI, no. 4, pp. 423-429.

[9] Czesnat, A. O., Maliska, C. R., Silva, A. F. C., Lucianetti, R. M.,1998, Grid Effects on petroleum reservoir simulation using boundary-fitted generalized coordinates, VII ENCIT, Rio de Janeiro, Brazil. (InPortuguese)

[10] Dicks, E. M., 1993, Higher order Godunov black-oil simulations forcompressible flow in porous media. PhD. Thesis, University of Reading,

1993, UK, 180p.[11] Di Donato, G.; Tavassoli, Z.; Blunt, M. J., 2006, Analytical and numer-

ical analysis of oil recovery by gravity drainage. Journal of PetroleumScinece & Engineering. V. 54, pp. 55-69.

[12] Escobar, F. H.; Ibagn, O. E.; Montealegre-M., M., 2007, Averagereservoir pressure determination for homogeneous and naturally fracturedformations from multi-rate testing with the TDS technique. Journal ofPetroleum Scinece & Engineering. V. 59, pp. 204-212.

[13] Fletcher, C. A. J., 2003, Computational techniques for fluid dynamics.V.2, Springer, Berlin, Germany, 2.ed.

[14] Gharbi, R. B. C., 2004, Use of reservoir simulation for optimizingrecovery performance. Journal of Petroleum Science & Engineering. V.42, pp. 183-194.

[15] Granet, S.; Fabrie, P.; Lemonnier, P.; Quintard, M., 2001, A two-phasesimulation of a fractured reservoir using a new fissure element method.Journal of Petroleum Science & Engineering. V. 32, pp. 35-52.

[16] Giting, V. E., 2004, Computational upscaled modeling of heterogeneousporous media flow utilizing finite element method. PhD. Thesis, Texas A& M University, USA. 148p.

[17] Hegre, T. M., Dalen V. And Henriquez, A., 1986, Generalized Trans-missibilities For Distorted Grids in Reservoir Simulation, SPE 15622,October.

[18] Hui, M.; Durlofsky, L. J., 2005, Accurate coarse modeling of weel-driven, high-mobility-ratio displacements in heterogeneous reservoirs.Journal of Petroleum Science & Engineering. V. 49, pp. 37-56.

[19] Hirasaki, G. J., ODell, P. M., 1970, Representation of Reservoir Ge-ometry for Numerical Simulation, Society Petroleum Engineers Journal,V. 10, n. 4, pp. 393-404.SPEJ.

[20] Lu, M.; Connell, L. D., 2007, A model for the flow of gas mixturesin adsorption dominated dual porosity reservoirs incorporating multi-component matrix diffusion. Part I. Theoretical development. Journal ofPetroleum Scinece & Engineering. V. 59, pp. 17-26.

[21] Marcondes, F., 1996, Numerical Simulation Using Implicit-AdaptativeMehotds and Voronoi Meshs in Problems of Petroleum Reservoirs. PhD.Thesis, Departament of Mechanical Engineering, Federal University ofSanta Catarina, Brazil. (In Portuguese)

[22] Marcondes, F., Maliska, C. R. & Zambaldi, M. C.,1996, Soluo deProblemas de Reservatrios de Petrleo: Comparao entre as MetodologiasTI, IMPES e AIM, VI Brazilian Congress of Engineering and ThermalSciences and Latin American Congress of Heat and Mass Transfer(ENCIT- LATCYM), Florianpolis, SC. v.3. pp.1477-1482.

[23] Maliska, C. R., 2004, Computational Heat Transfer and Fluid Mechanics,LTC, Rio de Janeiro, Brazil. (In Portuguese)

[24] Maliska, C. R., Silva, A. F. C., Juc, P. C., Cunha, A. R., Livra-mento, M. A., 1993, Desenvolvimento de um Simulador 3D Black-oil em Coordenadas Curvilneas Generalizadas - PARTE I, RelatrioCENPES/PETROBRAS, SINMEC/EMC/UFSC, Relatrio RT-93-1, Flori-anpolis, SC, Brasil.

[25] Mago, A. L., 2006, Adequate description of heavy oil viscosities anda method to assess optimal steam cyclic periods for thermal reservoirsimulation. Master Thesis, Texas A & M University, USA. 79p.

[26] Matus, E. R., 2006, A top-injection bottom-production cyclic steamstimulation method for enhanced heavy oil recovery. Master Thesis, TexasA & M University, USA. 81p.

[27] Prais, F., Campagnolo, E. A., 1991, Multiphase Flow Modeling inReservoir Simulation. Proceedings of the XI Brazilian Congress ofMechanical Engineering (COBEM), So Paulo, SP.

[28] Peaceman, D. W., 1977, Fundamentals of numerical reservoir simulation.Elsevier Scientific Publishing Company, New York, 176p.

[29] Soleng, H. H.; Holden, L., 1998. Gridding for petroleum reservoirsimulation. Proceedings of the 6th International Conference on NumericalGrid Generation in Computational Field Simulations. pp. 989-998.

[30] Todd, M. R., ODell, P. M., Hirasaki, G. J., 1972, Methods for Increas-ing Accuracy in Numerical Reservoir Simulators, Society of Petroleum

Engineers Journal, V. 12, n. 6, pp. 515-530.[31] Van Der Vorst, H. A., 1992, BI-CGSTAB: A Fast and Smoothly

Converging Variant of Bi-CG for the Solution of Nonsymmetric LinearSystems. SIAM Journal on Scientific & Statistical Computing, V. 13, n.2,pp. 631-644.


18/18


[32] Wason, C. B.; King, G. A.; Shuck, E. L.; Breitenbach, E. A.; McFarlane,R. C., 1990, System for monitoring the changes in fluid content of apetroleum reservoir. US Patent No. 4969130, 14p., USA.

oil brazil

Documents