Numerical Simulation of Oil Recovery Through Water Flooding in Petroleum Reservoir Using Boundary-Fitted Coordinates

Transcription

1 INTERNATIONAL JOURNAL OF MODELING AND SIMULATION FOR THE ETROLEUM INDUSTRY, VOL. 2, NO. 1, FEBRUARY Numerical Simulation of Oil Recovery Through Water Flooding in etroleum Reservoir Using Boundary-Fitted Coordinates Brauner Gonçalves Coutinho 1, Francisco Marcondes 2, and Antônio Gilson Barbosa de Lima 1 1 Federal University of Campina Grande Center of Sciences and Technology Department of Mechanical Engineering Av: Aprígio Veloso, 82, Bodocongó, O Box 10069, CE , Campina Grande, B, Brasil. 2 Federal University of Ceará Center of Technology Dept. of Mechanical and roduction Campus do ici O Box 12144, CE , Fortaleza, CE, Brasil. braunergc@yahoo.com, marconde@dem.ufc.br, gilson@dem.ufcg.edu.br Abstract Efficient mathematical models can be used to predict the behavior of the fluids and fluid flow inside the petroleum reservoir under several operation conditions. The main goal of this study is to obtain a numerical solution for two-phase problems with complex geometry reservoirs using the finite-volume method and boundary-fitted coordinates. The physical model adopted is the standard black-oil, simplified to an immiscible, two-phase (oil-water flow including water flooding process to increase oil recovery. This model can be applied for studies in reservoirs that contain heavy oils or low-volatility hydrocarbons. The mass conservation equations, written in the mass fractions formulation, are solved using a fully implicit methodology and the Newton s method. In spite of computational time consumption, the advantage of this methodology is the possibility to use larger time steps. The UDS scheme is used to evaluate the phase mobilities in each control volume face. Results of the fluid saturation fields, water cut, oil recovery and pressure inside the reservoir along the time are presented and analyzed. Attention are given to the effect of refinement and orientation of grid in the simulation results. Results are presented in terms of Newton s and solver iterations number, CU time used to build the Jacobian matrix and to solve the linear systems and for the whole simulations. Index Terms Reservoirs 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 underground at depths that prohibit extensive measurement and characterization. etroleum reservoir engineers face the difficult task of using their understanding of reservoir mechanics to design schemes for recovering hydrocarbons efficiently. A typical oil reservoir is a body of underground rock, often sedimentary, 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., In the petroleum exploration and production sector, a priority is placed on gaining accurate knowledge and analysis 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 begins with 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 ongoing 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 reservoirs characteristics be accomplished with the most efficient an accurate use of a wide range of data regarding the reservoir (Anderson et al., 2004.

2 18 INTERNATIONAL JOURNAL OF MODELING AND SIMULATION FOR THE ETROLEUM 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 operators pump water into the reservoir through injection wells with the aim of displacing oil toward equipped production wells (Allen III et al., In the tertiary recovery stage, additional steps are taken to enhance the recovery of hydrocarbons and to 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., 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 and gas. 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 laboratory experiments is a complex task. The confident reproduction of all fluid and rock conditions (temperature, pressure, geometry, composition in the surface is almost impossible, or economically difficult. In this sense, oil reservoir engineering 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 wells to 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., Granet et al. (2001 presents a two-phase flow modeling of a fractured reservoir using a new fissure element method. The method 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 permits 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 experimental 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 simulation 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 physical parameters, such as size, resident fluid and rock composition and properties, which are needed by the mathematical model. Given the physical parameters, the mathematical model describes 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 basically 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 (eaceman, In the reservoir 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 components 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 volume methods for solution of the partial differential equations (Dicks, Some influential factors on the modeling are: number of components and phases, well formulation, grid construction and geometry, and physical phenomena considerations. The

3 COUTINHO et al.: NUMERICAL SIMULATION OF OIL RECOVERY THROUGH 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 Darcy s law is used to represent the flux in porous media. Finally, some complementary equations and the boundary 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 finitevolume 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 is governed 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 [φρm Z p [ λp ] ] =. m p (1 where the superscript p indicates the phase p, φ is the porosity, ρ is the average density of the mixture, Z is the mass fraction, and is the pressure inside the reservoir. In this equation m p and λ p represents the mass flow per unit of volume of the reservoir and phase mobility, respectively, and are defined as follows m p = ρ m q p (2a λ p = ρp kk rp µ p (2b where q p is the volumetric flow rate of the phase p per volume. In Eq. (2, k is the absolute permeability, k rp is the relative permeability, and ρ p and µ p are density and viscosity of phase p, respectively. Writing Eq. (1 for the oil and water phases, there are three unknowns (Z O, Z W, and and two equations. The equation needed for the complete solution comes from global mass conservation as follows: Z w + Z o = 1 (3 More details of the Black-oil formulation in terms of mass fractions can be found in rais 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 this problem. 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 and CU 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 distribution of wells and grid orientation effect (Todd et al., 1972; Aziz and Settari, Non-orthogonal boundary-fitted grids can turn the numerical method flexible to treat reservoirs with more complex geometries (Maliska, 2004; Cunha et al., Numerical solution of two-dimensional displacement problems can be strongly influenced by the orientation of the underlying grid. Under certain situation, vastly different numerical 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, The numerical error of a solution of a set of differential equation 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 discretization 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 terms induced by a non-trivial metric may lead to errors that are independent of the grid spacing (Soleng and Holden, 1998; Fletcher, In this study, the cross terms are used in two

4 20 INTERNATIONAL JOURNAL OF MODELING AND SIMULATION FOR THE ETROLEUM INDUSTRY, VOL. 2, NO. 1, FEBRUARY 2008 ( = N + NE S SE e 4 η (7b To evaluate the phase mobility in each control volume face it was employed the Upwind Differencing Scheme. Using again the east face, λ is given by, λ e = λ if ū e > 0, and λ e = λ E otherwise. (8 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 in boundary-fitted coordinates as follows: 1 J t (φρm Z p + mp J = [ ] D1 ξ ξ + D 2 + ] [ D 2 ξ + D 3 where J is the Jacobian and the coefficients D p i are given by: D1 = λ p ( ξ 2 J x + ξy 2 (5a D2 = λ p J (ξ xη x + ξ y η y (5b D3 = λ p ( η 2 J x + ηy 2 (5c Equations (5 a-c have all grid information (Maliska, 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: V J [(φρm Z p (φρ m Z p o ] + mp V t = [( ( J ] D1 ξ + D 2 D1 e ξ + D 2 η γ [( ( w t + D2 ξ + D 3 D 2 n ξ + ] D3 ξ γ t (6 s where V = ξ η γ is the volume dimensions on generalized 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, ( = E (7a ξ ξ e (4 Flow velocity can be calculated through Darcy s law. Written in generalized coordinates, for the east face of the volume, for example, this field can be determined by: [ ū p e = λ p (Φ p E e G Φp 1e + ξ (Φ p N G + Φp NE Φp S Φp SE ] 2e (9 4 η where C. Fully implicit methodology G i = D i i = w, e, n, s (10 λ w In this methodology the unknowns and Z o are implicitly calculated at the current time step. The equations are linearized by Newton s method. assing to the left side all terms of Eq. (6 the following residual equation is obtained: F p = V J [(φρm Z p (φρ m Z p o ] + mp V t [( ( J ] D1 ξ + D 2 D1 e ξ + D 2 η γ [( ( w t D2 ξ + D 3 D 2 n ξ + ] D3 ξ γ t (11 s Expanding the equation (11 by Taylor s series, we have: (F p k+1 = (F p k + ( p k X = 0 (12 X X where k is the iteration level and X represents the unknowns and Z o. In the Newton s method, the solution in every time step is considered to converge when the residues are smaller than the convergence criterion. Therefore, Eq. (12 in the short form is given by: (F p k = ( p k X (13 X X In the matrix form, Eq. (12 can be written by: A X = F (14 where A is the Jacobian matrix of the residual function F on the k-th iteration. The solution of the linear system, Eq. (14, allows calculating the and Z o values till the mass conservation in each time step is obtain. The Jacobian matrix A is a block matrix, i.e., all its elements are square matrices.

5 COUTINHO et al.: NUMERICAL SIMULATION OF OIL RECOVERY THROUGH 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: ( p ( p ( p + Z o Z o + W + ( W p ( p ( ZW o ZW o p + E + E ZE o ZE+ o ( p ( p ( p S + S ZS o ZS o + N + ( N p ( p ( ZN o ZN o p + SW + SW ZSW o ZSW o + ( p ( p ( p SE + SE ZSE o ZSE o + NW + ( NW p ( p ZNW o + NE + NE Z o NW ( p Z o NE Z o NE = F p (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 residual function. This procedure avoids additional terms in the Jacobian matrix when the coordinates lines are non-orthogonal. Using this scheme, the Eq. (12 can be rewritten as follows: ( p ( p + ZW o ZW o + ( p S + ( S p ZN o Z o + E + E ZS o + ( p Z o ( p ( p Z o S ( p W ( p Z o E ( p N W + ZE+ o N + Z o N = F p (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 the grid 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: ṁ = D 1e ξ + D 2e e D 1w e ξ D 2w w + w D 2n ξ + D 3n n D 2s n ξ D 3s s (17 s By using the derivative approximations we have: ( E ṁ = D 1e ξ D 1w ( W ξ ( E + NE W NW D 2n 4 ξ D 2s ( E + SE W SW 4 ξ ( N + NE S SE + D 2e 4 η ( N + NW S SW D 2w 4 η where, for example, for the east face, D 1e ( E ξ r o = D 1e ξ + ( N + D 3n η ( S D 3s (18 η ṁ 2πkhρ ln ( re r o (19a ( N + NE S SE D 2e = D 2e ṁ [ 4 η 4 η 2πkhρ ( ( ( ( ] rn rs rne rse ln ln + ln ln (19b r o where r o is the equivalent radio and r N, r S, r NE and r SE, are the distance between the center of the volume and the center of the volumes N, S, NE and SE, respectively. The equivalent radio of the well is given by: ( r o = G 1e r o r o 1 α G 1e β e 2π (20

6 22 INTERNATIONAL JOURNAL OF MODELING AND SIMULATION FOR THE ETROLEUM INDUSTRY, VOL. 2, NO. 1, FEBRUARY 2008 where G 1e + G 2n α = r 4 G 2s 4 r E G 2e N 4 G 2w 4 + G 3n F. hysical properties and saturation relationships Finally, the following relationships were used: a 1 Formation volume factor G 2e r 4 + G 2w + G 3s 4 G 1w G 2n r 4 + G 2s 4 S G 2e G 1e r 4 + G 2n 4 G 1e G 2e r 4 G 2s 4 NE G 2w G 1e r 4 NW G 2n 4 E. Wells boundary conditions SE W G 2w G 1e r 4 SW + G 2s 4 β = G 1e + G 1w + G 3n + G 3s (21a (21b G i = D i i = w, e, n, s (21c λ w 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: q w λ w = qo λ o = qt λ T (22 where superscript T represent 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: and q w = q w inj q o = 0 (23a (23b In the producer well, the total flow rate (water + oil and pressure are prescribed as follows: and q T = q T prod q p = λ p λ T qt wf = i (24a (24b (24c 2 orosity B p ( = 3 Density of the phases B p ref 1 + c p ( ref (25 φ = φ ref [1 + c r ( ref ] (26 ρ p = ρp ST C B p p = o, w (27 where ρ ST C is the density of the phases in the standard condition. 4 Saturation of the phases α p ρ p S p = np αp ρ p 5 Average density of the mixture ρ m = n p = o, w (28 ρ p S p 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 DISCUSSIONS Management 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 controlvolumes, and the numerical scheme should be carefully chosen. In water-flooding process in petroleum reservoir, simulation results are largely influenced by numerical treatment of the mathematical model, grid refinement and grid orientation. Grid orientation 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 approaches just mentioned in the last section was originally proposed by Hirasaki and O dell (1970 and after studied by Hegre et al. (1986 and Czesnat (1998. They considered a reservoir with two production wells equidistant from an injection well and the same size of the control-volume, as shown

7 COUTINHO et al.: NUMERICAL SIMULATION OF OIL RECOVERY THROUGH Fig. 3. Scheme of the reservoir and wells locations. TABLE I RELATIVE ERMEABILITIES OF THE HASES. SOURCE: HEGRE ET AL. (1986. S w k rw k ro 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 the numerical grid, meshes with 24 6 control-volumes were used, Figure 4, and refined with 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 the two 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 controlvolumes 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 to the injector well, physically, this phenomenon couldn t 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., 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 has a 90 inclination with a horizontal line. Other grids with 80, 60, 45, 30 and 20 inclinations were obtained distorting the 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 contains curves generated using five and nine points schemes for both grids (24 6 and control-volumes. On left column, 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, respectively. 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 (θ = 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 approximately linear for the full Jacobian matrix, but it was non-linear when it was used matrix including only direct neighboring volumes. 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 wasn t sensible to the grid orthogonality when using nine-points scheme. Five-points scheme required many time steps for skewed grids. Some tests couldn t 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 Newton s 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 inclination of mesh. In all other geometries, as skewness angle increases, five-points scheme demand more Newton s 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 doesn t consider the complete structure of the Jacobian matrix was used. This pre-conditioner isn t more efficient than 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 24 INTERNATIONAL JOURNAL OF MODELING AND SIMULATION FOR THE ETROLEUM INDUSTRY, VOL. 2, NO. 1, FEBRUARY 2008 Fig. 4. Grid with 24x6 control-volumes oriented at 45. TABLE II FLUIDS AND RESERVOIR DATA. SOURCE: HEGRE ET AL. (1986. orosity φ = 0.19 Residual oil saturation S o r = 0.2 ermeability k = 0.049x10 12 m 2 Water and oil density ρ w = ρ o = 1000kg/m 3 Height h = 18.3m Water reference volumetric formation factor B w ref = 1 on ref Initial pressure i = 27248x10 3 a Oil reference volumetric formation factor Bref o = 0.96 on ref Rock compressibility c r = 0 a 1 Reference pressure ref = 27248x10 3 a Oil compressibility c o = 1.45x10 9 a 1 Water viscosity µ w = 0.5x10 3 a.s Water compressibility c w = 0.44x10 9 a 1 Oil viscosity µ o = 2.0x10 3 a.s Well radius r w = 0.122m Water injection rate q inj = 302.1m 3 /dia Initial water saturation S w i = 0.2 Total production rate q prod = 159m 3 /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 composition of the Jacobian matrix and calculation of the residual functions. Note that, as larger grid inclination, more time was needed when five-points scheme was used. As for full Jacobian matrix, this computational time hasn t changed when variations occurred on grid inclination. This fact could be explained because the number of iterations of Newton s 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 fivepoints scheme, the increase in the iterations number, as a function of mesh angle, doesn t increase necessarily the CU 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 high computational cost. The larger the matrix structure is, the more expensive will be the computational process of the matrixvector products. Comparing the number of solver iterations in Figure (7c for 60 skewed mesh (48 20, it can be seen that, using the incomplete Jacobian matrix,5 approximately times more 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 grid angles. 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 COUTINHO et al.: NUMERICAL SIMULATION OF OIL RECOVERY THROUGH Fig. 6. Meshes with many levels of orthogonality between the lines. the larger the number of time steps will be needed to get the total time. To compare the results obtained with the five-points and nine-points schemes, Figures (8 and (9 show water cut curves on producer wells 1 and 2 for grid with 24 6 and control-volumes and 90 and 45 inclination using time step of 50 days. It is observed the effect of water flooding to improve oil recovery. The production profile has two periods. The first period is dominated by the initial viscosity and concentration of the oil inside the reservoir. The water front displaces both water and oil with low mobility toward the producer wells, but only oil is extracted. In the second period a occurs breakthrough of water. In this stage oil and water appear in the producer wells, and the oil production decrease along the time. The breakthrough of the water at the producer well occurred after 2000 days. The water cut is approximately zero until 2000 days, and then increases significantly in the next periods consistent with decrease in the oil production. From this figures, it is possible to notice that results achieved with five-points and nine-points schemes are very similar, with small differences, only when it was used 45 inclination mesh. When water cut starts to increase it is possible to observe the grid orientation effects by analyzing and comparing the process time. The water cut is defined by the relationships between water volumetric flow rate and total liquid (oil+water volumetric flow rate. It can be seen that the non-rotated grid shows the same trend as the rotated grid as long as the number of control-volume increases. Furthermore, the difference between orientations vanishes as the number of control-volumes between the injector well and the producer well increases. B. Case 2 In this selected case, a reservoir with irregular geometry has eight wells (two injectors and six producers wells. They are distributed as shown in Figure (10. Table 3 shows position coordinates and water and liquid (water+oil flow rates of the injector and producer wells. Table 4 presents all fluids and reservoir data. Fig. 10. Reservoir with irregular shape. Relative permeabilities for oil and water are given by k rw = (S w 0.2[ 250(S w S w 55]/27 k ro = 1 k rw (30a (30b To compare behavior of both schemes (five and nine points in meshes with others degrees of orthogonality, the same problem was studied using two others situations obtained distorting the original geometry (37 as shown in Figure (11. Two grids were generated for each one of these geometries,

10 26 INTERNATIONAL JOURNAL OF MODELING AND SIMULATION FOR THE ETROLEUM INDUSTRY, VOL. 2, NO. 1, FEBRUARY 2008 Fig. 7. Simulation results obtained with five-points and nine-points scheme. TABLE III WELLS COORDINATES AND FLOW RATES. SOURCE: MARCONDES (1996 Well Flow rates Coordinates (m (m 3 /day x y Injector (water flow rate roducer (liquid flow rate one with 560 volumes (40 14 and another more refined with 1160 volumes ( 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 θ = 11, medium and maximum angles are greater than angles in grids with θ = 37 and θ = 57, in the meshes as well as volumes meshes. Figure (12 shows many results obtained with the simulations. Each plot contains curves generated using five and nine points schemes for both grids (40 14 and 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 COUTINHO et al.: NUMERICAL SIMULATION OF OIL RECOVERY THROUGH (a roducer well 1 (non-aligned (b roducer well 2 (aligned Fig. 8. Water cut for 90 inclination grid, t = 50 days. (a roducer well 1 (non-aligned (b roducer well 2 (aligned Fig. 9. Water cut for 45 inclination grid, t = 50 days. Fig. 11. Inclination angle of the grid.

12 28 INTERNATIONAL JOURNAL OF MODELING AND SIMULATION FOR THE ETROLEUM INDUSTRY, VOL. 2, NO. 1, FEBRUARY 2008 TABLE IV FLUIDS AND RESERVOIR DATA. SOURCE: MARCONDES (1996. Height H = 15m orosity φ = 0.30 Absolute permeability k = 0.3x10 12 m 2 Well radius r w = 0.122m Initial pressure Initial water saturation S w i = 0.2 Residual oil saturation S o r = 0.2 i = 20685x10 3 a Densities ρ w = ρ o = 1000kg/m 3 at ref Reference volumetric Bref w = Bo ref = 1 formation factor Reference pressure Compressibility factor Water viscosity Oil viscosity at the ref ref = 20685x10 3 a c w = c o = x10 9 a 1 µ w = 10 3 [ x10 12 ( 1.38x10 7 ] a.s µ o = [ x10 12 ( 1.38x10 7 ] a.s TABLE V FLUIDS AND RESERVOIR DATA. SOURCE: MARCONDES (1996. Grid θ ( Medium angle ( Maximum angle ( x x 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 variations occurred on mass fractions or pressures, that kept the average time step smaller than the observed 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. 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 time steps 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 Newton s 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 schemes presented the same efficiency. When distortion increases, fivepoints scheme demands more Newton s 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 wasn t dependent of mesh inclination for full Jacobian matrix. In the present work, it was used the diagonal block as pre-conditioner matrix, which doesn t consider the complete structure of the Jacobian matrix. This pre-conditioner isn t extremely efficient such as an ILU preconditioner, according to Marcondes et al. (1996, but it has been robust in all mesh inclinations analyzed. As for five-spot configuration, 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 Newton s 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 CU 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 products will 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 value became 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 easily explained: the smaller the time steps during simulation are, the higher the number of time steps will be needed to get total time.

13 COUTINHO et al.: NUMERICAL SIMULATION OF OIL RECOVERY THROUGH 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 with and control-volumes distorted 11 and 57 with time step of 50 and 100 days. From this figures, it is possible to notice that results achieved with five-points and nine-points schemes are very similar, with small differences when it was used the 57 inclination, mainly using t = 100 days. This effect could be minimized with a mesh refinement study. Figure (17 presents water saturation distribution fields in three V I values. Notice that both grids (560 and 1160 volumes have generated very similar saturation fields. In these figures, V I is given by: V I = q w t φv R (1 S w i S o r (31 where V R is the reservoir volume and Si W and Sr 0 represent initial 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 distorted 11, t = 50 days.

14 30 INTERNATIONAL JOURNAL OF MODELING AND SIMULATION FOR THE ETROLEUM 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 distorted 57, t = 50 days.

15 COUTINHO et al.: NUMERICAL SIMULATION OF OIL RECOVERY THROUGH (a 560 volumes (b 1160 volumes Fig. 18. Water saturation distribution for three VI. 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, well 1 (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 OR (porous volume 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: V OR = t 0 q o (tdt φv R (1 S w i S o r (32 where V R is the reservoir volume and Si w and Sr 0 represent initial water saturation and residual oil saturation, respectively. In this study Si w and Sr o 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 inverse process is important for reducing uncertainties in reservoir characterization, which is crucial for evaluating options of field development and predicting future reservoir performance

16 32 INTERNATIONAL JOURNAL OF MODELING AND SIMULATION FOR THE ETROLEUM INDUSTRY, VOL. 2, NO. 1, FEBRUARY 2008 (Arihara, Fig. 22. Oil recovery in the producer well 4 using a grid distorted 37. Fig. 19. ressure in the producer well 1 using a grid distorted 37. Fig. 23. ressure in the producer well 5 using a grid distorted 37. Fig. 20. Oil recovery in the producer well 1 using a grid distorted 37. Fig. 24. Oil recovery in the producer well 5 using a grid distorted 37. Fig. 21. ressure in the producer well 4 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 COUTINHO et al.: NUMERICAL SIMULATION OF OIL RECOVERY THROUGH 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 recovery 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. erformances 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. Although this fact may contribute to increase CU time (in each solver iteration, there is a substantial reduction on the number of iterations in the Newton s 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 CNq, FINE, ETROBRAS, AN/UFCG-RH-25 and CT-ETRO, 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 in porous media. Springer-Verlag, London, 308p. [2] Anderson, R. N.; Boulanger, A.; He, W.; Winston, J.; Xu, L.; Mello, U.; Wiggins, W., 2004, etroleum reservoir simulation and characterization system and method. US. atent No B1, 13p., USA. [3] Arihara, A., 2005, Reservoir simulation technology by streamline-based methods. Journal of the Japan etroleum Institute. V. 48, n. 6, pp [4] Abou-Kassem, J. H., 1996, ractical considerations in developing numerical simulators for thermal recovery. Journal of etroleum science & Engineering. V. 15, pp [5] Aziz, K.; Settari, A., etroleum reservoir simulation. Applied Science ublishers, London, 476p. [6] Coutinho, B. G., 2002, Numerical Solution for roblems of etroleum Reservoirs Using Generalized Coordinates, Master Dissertation, Federal University of Campina Grande, Brazil. (In ortuguese [7] Cunha, A. R., 1996, A Metodology to Three Dimensional Numerical Simulation of etroleum Reservoirs Using Black-Oil Model and Mass Fraction Formulation, Master Dissertation, Federal University of Santa Catarina, Brazil. (In ortuguese [8] Cunha, A. R., Maliska, C. R., Silva, A. F. C., Livramento, M. A., 1994, Two-Dimensional Two-hase etroleum Reservoir Simulation Using Boundary-Fitted Grids, Journal of the Brazilian Society of Mechanical Sciences, Vol. XVI, no. 4, pp [9] Czesnat, A. O., Maliska, C. R., Silva, A. F. C., Lucianetti, R. M., 1998, Grid Effects on petroleum reservoir simulation using boundaryfitted generalized coordinates, VII ENCIT, Rio de Janeiro, Brazil. (In ortuguese [10] Dicks, E. M., 1993, Higher order Godunov black-oil simulations for compressible flow in porous media. hd. Thesis, University of Reading, 1993, UK, 180p. [11] Di Donato, G.; Tavassoli, Z.; Blunt, M. J., 2006, Analytical and numerical analysis of oil recovery by gravity drainage. Journal of etroleum Scinece & Engineering. V. 54, pp [12] Escobar, F. H.; Ibagn, O. E.; Montealegre-M., M., 2007, Average reservoir pressure determination for homogeneous and naturally fractured formations from multi-rate testing with the TDS technique. Journal of etroleum Scinece & Engineering. V. 59, pp [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 optimizing recovery performance. Journal of etroleum Science & Engineering. V. 42, pp [15] Granet, S.; Fabrie,.; Lemonnier,.; Quintard, M., 2001, A two-phase simulation of a fractured reservoir using a new fissure element method. Journal of etroleum Science & Engineering. V. 32, pp [16] Giting, V. E., 2004, Computational upscaled modeling of heterogeneous porous media flow utilizing finite element method. hd. Thesis, Texas A & M University, USA. 148p. [17] Hegre, T. M., Dalen V. And Henriquez, A., 1986, Generalized Transmissibilities For Distorted Grids in Reservoir Simulation, SE 15622, October. [18] Hui, M.; Durlofsky, L. J., 2005, Accurate coarse modeling of weeldriven, high-mobility-ratio displacements in heterogeneous reservoirs. Journal of etroleum Science & Engineering. V. 49, pp [19] Hirasaki, G. J., O Dell,. M., 1970, Representation of Reservoir Geometry for Numerical Simulation, Society etroleum Engineers Journal, V. 10, n. 4, pp SEJ. [20] Lu, M.; Connell, L. D., 2007, A model for the flow of gas mixtures in adsorption dominated dual porosity reservoirs incorporating multicomponent matrix diffusion. art I. Theoretical development. Journal of etroleum Scinece & Engineering. V. 59, pp [21] Marcondes, F., 1996, Numerical Simulation Using Implicit-Adaptative Mehotds and Voronoi Meshs in roblems of etroleum Reservoirs. hd. Thesis, Departament of Mechanical Engineering, Federal University of Santa Catarina, Brazil. (In ortuguese [22] Marcondes, F., Maliska, C. R. & Zambaldi, M. C.,1996, Soluo de roblemas de Reservatrios de etrleo: Comparao entre as Metodologias TI, IMES e AIM, VI Brazilian Congress of Engineering and Thermal Sciences and Latin American Congress of Heat and Mass Transfer(ENCIT - LATCYM, Florianpolis, SC. v.3. pp [23] Maliska, C. R., 2004, Computational Heat Transfer and Fluid Mechanics, LTC, Rio de Janeiro, Brazil. (In ortuguese [24] Maliska, C. R., Silva, A. F. C., Juc,. C., Cunha, A. R., Livramento, M. A., 1993, Desenvolvimento de um Simulador 3D Blackoil em Coordenadas Curvilíneas Generalizadas - ARTE I, Relatrio CENES/ETROBRÁS, SINMEC/EMC/UFSC, Relatrio RT-93-1, Florianpolis, SC, Brasil. [25] Mago, A. L., 2006, Adequate description of heavy oil viscosities and a method to assess optimal steam cyclic periods for thermal reservoir simulation. Master Thesis, Texas A & M University, USA. 79p. [26] Matus, E. R., 2006, A top-injection bottom-production cyclic steam stimulation method for enhanced heavy oil recovery. Master Thesis, Texas A & M University, USA. 81p. [27] rais, F., Campagnolo, E. A., 1991, Multiphase Flow Modeling in Reservoir Simulation. roceedings of the XI Brazilian Congress of Mechanical Engineering (COBEM, So aulo, S. [28] eaceman, D. W., 1977, Fundamentals of numerical reservoir simulation. Elsevier Scientific ublishing Company, New York, 176p. [29] Soleng, H. H.; Holden, L., Gridding for petroleum reservoir simulation. roceedings of the 6th International Conference on Numerical Grid Generation in Computational Field Simulations. pp [30] Todd, M. R., O Dell,. M., Hirasaki, G. J., 1972, Methods for Increasing Accuracy in Numerical Reservoir Simulators, Society of etroleum Engineers Journal, V. 12, n. 6, pp [31] Van Der Vorst, H. A., 1992, BI-CGSTAB: A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems. SIAM Journal on Scientific & Statistical Computing, V. 13, n.2, pp

18 34 INTERNATIONAL JOURNAL OF MODELING AND SIMULATION FOR THE ETROLEUM INDUSTRY, VOL. 2, NO. 1, FEBRUARY 2008 [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 a petroleum reservoir. US atent No , 14p., USA.