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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).
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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
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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
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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.
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COUTINHO et al.: NUMERICAL SIMULATION OF OIL RECOVERY THROUGH . . . 21
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)
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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
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COUTINHO et al.: NUMERICAL SIMULATION OF OIL RECOVERY THROUGH . . . 23
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
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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,
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26 INTERNATIONAL JOURNAL OF MODELING AND SIMULATION FOR THE PETROLEUM INDUSTRY, VOL. 2, NO. 1, FEBRUARY 2008
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
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COUTINHO et al.: NUMERICAL SIMULATION OF OIL RECOVERY THROUGH . . . 27
(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.
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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.
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COUTINHO et al.: NUMERICAL SIMULATION OF OIL RECOVERY THROUGH . . . 29
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.
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30 INTERNATIONAL JOURNAL OF MODELING AND SIMULATION FOR THE PETROLEUM INDUSTRY, VOL. 2, NO. 1, FEBRUARY 2008
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.
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COUTINHO et al.: NUMERICAL SIMULATION OF OIL RECOVERY THROUGH . . . 31
(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
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32 INTERNATIONAL JOURNAL OF MODELING AND SIMULATION FOR THE PETROLEUM INDUSTRY, VOL. 2, NO. 1, FEBRUARY 2008
(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. 22. Oil recovery in the producer well 4 using a grid distorted37.
Fig. 23. Pressure in the producer well 5 using a grid distorted 37.
Fig. 24. Oil recovery in the producer well 5 using a grid distorted37.
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
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COUTINHO et al.: NUMERICAL SIMULATION OF OIL RECOVERY THROUGH . . . 33
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.
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