Recently, a flexible gridding technique known as Voronoi or perpendicular bisection (PEBI) grid has received some attention in perpendicular bisection (PEBI) grid has received some attention in the petroleum literature. The main advantage of this grid is that individual grid points can be specified anywhere inside the domain regardless of the position of any other points. Several important issues related to the practical use of such flexible grids have not been addressed in previous works. This paper describes a practical way of using Voronoi grids for field scale simulation. Grid points are easily generated by using modules of known geometry which can be located, scaled or rotated in any position. Physical properties are specified at points that are independent of the computational grid. A new well points that are independent of the computational grid. A new well model for Voronoi grids is proposed and tested. The results of the well model are in close agreement with analytically calculated pressure drop for different injection patterns and grid geometries. pressure drop for different injection patterns and grid geometries. It is shown that, while the results of hybrid-Cartesian and hybrid-hexagonal grids are less sensitive to grid orientation than pure hexagonal grids or the 9-point scheme for the cases pure hexagonal grids or the 9-point scheme for the cases investigated, they are still dependent on the number of grid blocks. A 5-spot pattern in a heterogeneous reservoir was simulated using different grid geometries. The results art in close agreement with a fine grid simulation performed with a commercial simulator. This shows that, at least for the problems investigated, the proposed gridding technique is able to capture the main features proposed gridding technique is able to capture the main features of reservoir heterogeneities. Introduction Field scale simulation of petroleum reservoirs is performed by dividing the real domain into imaginary grid blocks and locally applying conservation laws for each fluid component in the system. The fluid flow between blocks is calculated by the discretized form of Darcy's law. Additionally, PVT and relative permeability relations are also honored. The outcome of any flow permeability relations are also honored. The outcome of any flow simulation depends on how the reservoir is divided and how the flow equations are formulated. Although several grid geometries have been presented in the literature, the combination of different geometries in a single grid system is restricted to very specific cases like local grid refinement and hybrid-Cartesian grids. Even in these situations, the specification of an individual grid block is dependent on the location of adjacent blocks. The Voronoi grid has the property that individual grid points can be specified anywhere inside the domain regardless of the position of any other points. Because of its flexibility, it has position of any other points. Because of its flexibility, it has been used in several different areas like physics, rock characterization, crystallography, electrical engineering, biology, mathematics, fluid mechanics and petroleum engineering. Although the grid blocks have been referred to by different names like Wigner-Seitz cells or PEBI grid, the vast majority of the literature refers to it as Voronoi polyhedra or grid, in honor of the mathematician who first defined it.

Summary This paper discusses the treatment of wells in a flexible Voronoi grid. A new model problem is proposed to evaluate exact well indices for multiwell configurations and homogeneous reservoirs. A simplified model also is proposed and discussed. New exact and simplified well models for heterogeneous reservoirs are presented. Introduction An exact well index can be derived by comparing the solution of the differential equation (obtained analytically or numerically) for a given problem with the numerical solution of the difference equation (exact well model). Peaceman1 first published this approach, and well models based on it are defined here as Peaceman-type models. In this sense, well models described by Kuniansky and Hillestad,2 Abou-Kassem and Aziz,3 and Babu et al.4 also are considered to be Peaceman-type models because, although these authors used different reference solutions, the well models were based on Peaceman's1 concept. In this paper, we propose new exact Peaceman-type well models for Voronoi grid and multiwell configurations in homogeneous and heterogeneous reservoirs and propose and discuss a simplified model for this grid. The exact well model can be used easily to investigate the effect of different well configurations (location and rates) over the value of the well index. Because the well index is assumed to be constant during the simulation process, a good grid geometry should result in a constant (within a tolerance) value for each well, regardless of the well configuration. Therefore, this procedure is an additional tool that can be used to select the appropriate grid size for the problem of interest. Current Wen Models Although van Poolen et al.5-type models were extensively used in the past, Peaceman1 showed that this approach is incorrect. For this reason, only Peaceman-type models are considered here. The relationship between wellblock pressure and bottomhole flowing pressure (BHFP) is a function of fluid rates, rock and fluid properties, and grid geometry:Equations 1 and 2 where Iw=well index, ?=angle open to flow, and req=equivalent wellblock radius. Because only single-phase cases are discussed, the subscript P (phase index) will be dropped from all remaining equations. Peaceman1,6 proposed the following simplified model based on the comparison between numerical and analytical solutions for the total pressure drop in a homogeneous, isotropic, repeated five-spot pattern under steady-state, single-phase flow conditions.Equation 3 The conditions that must be satisfied to apply this model safely may be as important as the model itself and are discussed by Peaceman.7,8 In his first paper, Peaceman1 showed that the block pressures increase logarithmically with the radial distance measured from the gridpoint to the well when a regular Cartesian grid with square blocks (?x=?y) is used to discretize an isolated one-quarter of a five-spot pattern. The extension of this concept led to the development of Kuniansky and Hillestad's2 and Abou-Kassem and Aziz's3 analytical well model. This model does not produce good results for wellblocks with large grid-aspect ratios (Ry=?y/?x) and may not produce good results for wells close to reservoir boundaries.7 However, it does yield good results for isolated wells in blocks with small Ry. This model's main advantage is that it can be applied to any kind of grid geometry and discretization scheme. Kuniansky and Hillestad2 and Peaceman7 proposed exact well models for multiwell configurations. They used different model problems for a reservoir with constant pressure at the external boundaries. In principle, these models can be used to derive well indices for grids of any geometry. However, conventional simulators assume closed boundaries, and the exact representation of constant pressure boundaries in these codes may be very tedious, although it is possible. Babu et al.4 proposed a well model based on the analytical solution presented by Babu and Odeh9 for a single well producing at constant and uniform. flux from a closed, box-shaped (3D) drainage volume under pseudo-steady-state conditions. The solution is valid for a well that partially penetrates the reservoir length. On the basis of a series of numerical experiments, they also presented a simplified model for regular Cartesian grids. Although their analytical solution is valid for 3D configurations, the actual well model was restricted to 2D (areal or cross-sectional) cases. Further research on fully 3D models is needed to investigate such factors as the effect of boundary conditions at the well (uniform flux, uniform pressure, or mixed boundary conditions) and partial penetration. All the authors mentioned above have focused their attention on the treatment of wells in a Cartesian grid, which is a special case of the Voronoi grid10,11,12 discussed in this paper (Fig. 1). For this reason, currently used well models were used as a foundation to develop the well models presented here for the Voronoi grid and heterogeneous reservoirs. Exact Well Model for Homogeneous Reservoirs By definition, the use of an exact well index in numerical simulation yields the same well pressure, pw, as that calculated analytically for a given model problem. Because this approach was used first by Peaceman,1 this condition characterizes the well model as Peaceman-type. A model problem consists of defining the reservoir geometry and its external and internal (well) boundary conditions, The linear, single-phase flow equation then is solved in the proposed domain to obtain the analytical solution for pressure at the well of interest, pw. The same problem is solved numerically to compute the pressure of the block containing the well, po. The equivalent radius, req, is evaluated by arranging Eq. 1 asEquation 4 This approach is also applicable to Voronoi grids because there is no restriction on grid geometry. The results presented in the literature show that different model problems produce similar well indices provided there are "enough" gridblocks between wells (or images). While the best model problem is the one that is closest to the actual field configuration, its choice should be based on ease of use, flexibility to represent flow geometries and boundary effects, and capability of conventional simulators to model the same problem. p. 15–21

Carmópolis Field in Sergipe/Alagoas Basin in northeastern Brazil is the country's largest onshore oil accumulation at 253 MMm3OOIP and a current total oil production of 2,880 m3/d. Discovered in 1963, it was quickly put into primary production. Water flooding followed in 1971 at the central portion of the field. The combination of adverse fluid mobility ratio, reservoir heterogeneity and the lack of proper selective injection, led to the quick decline of production. Immediately, a major program of selective plugging, stimulation and selective injection was able to stabilize production. Water flooding was then extended to the entire main block of the field. Well pattern was changed from five to nine-spot arrangement, with a corresponding downsizing in well spacing and injection rates. Carmopolis has also experienced several pilot projects for Enhanced Oil Recovery (EOR): polymer flooding, steam flooding and in situ combustion, respectively. This past history of Carmópolis Field and the significance of water flooding to oil production in Brazil with approximately 1,850 MMm3OOIP currently submitted to this method of recovery, led to the selection of Carmópolis as the target for one of the projects in the portfolio of PRAVAP - Petrobras Strategic Improved Oil Recovery (IOR) Program. The scope of this project included the review of the water flooding operation through improved reservoir characterization and flow simulation, as well as the investigation of other IOR methods that might reverse the production declining trend. This paper summarizes the outcome of this project that went from lab research to field testing and led to the approval of operational implementations worth US$ 34 million NPV. Introduction Water flooding has come a long way since its accidental "implementation" in the area around the city of Pithole, Pennsylvania back in 18651. By the mid fifties this improved method of recovery was responsible for more than 10% of the total oil production in the US. By 1986, this share was thought to be in the 50% range2. This scenario is not any different around the world, specially in major producing countries like the former USSR and the Middle East. A good example is the giant field of Ghawar in Saudi Arabia already partially under water flooding. The situation in Brazil is very similar. Close to 2,000 MMm3 of OOIP are currently under the influence of water injection. In the near future water injection rates will reach more than 500,000 m3/d in discovered fields of Campos Basin. In Marlim field alone in the same Basin, water injection is expected to peak around 100,000 m3/d. In fact, the history of water flooding in Brazil dates back to the early fifties and Carmópolis Field onshore Sergipe/Alagoas Basin, as shown in Figure1, is a very representative part of it. Discovered in 1963 and quickly brought on stream, it produces predominantly from the sandstone and conglomerate reservoirs of the Muribeca/Carmópolis formation and secondarily from the deeper Barra de Itiuba formation plus the fractured basement (see Figure 2). Accordingly, oil quality varies considerably throughout the stratigraphic column. General reservoir data is given on Table1 and a good description of the reservoir geology is given by C ndido and Wardlaw3. This paper focuses on the review of the Improved Oil Recovery (IOR) methods tested in Carmópolis field over the last 28 years, the recently concluded R&D project within PRAVAP - Petrobras Strategic IOR Program, and the corresponding field pilot implementations. Past History of IOR Applications in Carmópolis Field. A good review of the past IOR applications in Carmópolis Field is given by Correia4et al., Doria5 and Romeu6et al.

Summary Carmópolis field, in northeastern Brazil's Sergipe/Alagoas basin, is the country's largest onshore oil accumulation at 253×106 m3 original oil in place (OOIP) and a current total oil production of 2880 m3/d. Discovered in 1963, it was quickly put into primary production. Waterflooding followed in 1971 at the central portion of the field. The combination of adverse fluid mobility ratio, reservoir heterogeneity, and the lack of proper selective injection led to the quick decline of production; however, a major program of selective plugging, stimulation, and selective injection was able to stabilize production immediately. Waterflooding was then extended to the entire main block of the field. The well pattern was changed from five- to nine-spot arrangement, with a corresponding downsizing in well spacing and injection rates. Carmópolis also was subjected to an intense improved oil recovery (IOR) campaign with pilot tests on polymer flooding, steamflooding and in-situ combustion. The history of Carmópolis field and the significance of waterflooding to oil production in Brazil, with approximately 2000×106 m3 OOIP currently submitted to this method of recovery, led to the selection of Carmópolis as the target for one of the projects in the PRAVAP (Petrobras Strategic IOR Program) portfolio. The scope of this project included a review of the waterflooding operation through improved reservoir characterization and flow simulation, as well as the investigation of other IOR methods that might reverse the declining production trend. This paper reviews the IOR history of Carmópolis field and summarizes the outcome of the PRAVAP project that led to the approval of field implementations worth U.S. $34 million net present value (NPV). Introduction Waterflooding has come a long way since its accidental implementation in 1865,1 in the area around the city of Pithole, Pennsylvania. Less than a century later, this improved method of recovery was responsible for more than 10% of the total oil production in the U.S. By 1986, this share was thought to be in the 50% range.2 This scenario is not any different around the world, especially in major producing regions like the former USSR and the Middle East. A good example is the giant field of Ghawar in Saudi Arabia, already partially under waterflooding. The situation in Brazil is very similar. Close to 2000×106 m3 OOIP are currently under the influence of water injection. In the near future, water injection rates will reach more than 500 000 m3/d in the discovered fields of Campos basin. In that basin, water injection in Marlim field alone is expected to peak around 100 000 m3/d. In fact, the history of waterflooding in Brazil dates back to the early 1950's, and Carmópolis field onshore Sergipe/Alagoas basin, as shown in Fig. 1, is a very characteristic part of it. Discovered in 1963 and quickly brought on stream, it produces predominantly from the sandstone and conglomerate reservoirs of the Carmópolis/Muribeca formation and secondarily from the deeper Barra de Itiuba formation and the fractured basement (see Fig. 2). Accordingly, oil quality varies considerably throughout the stratigraphic column. General reservoir data is given in Table 1; for a good description of the reservoir geology, refer to Candido and Wardlaw.3 This paper focuses on the review of IOR applications in Carmópolis field over the past 28 years, the results of the recently concluded project within the PRAVAP portfolio, and the corresponding field pilot implementations. History of IOR Applications in Carmópolis Field Waterflooding. Waterflooding was first implemented in Carmópolis field in 1968 on a 65-ha inverted nine-spot pattern in the southern part of the field. The target was the 2.97×106 m3 OOIP in zones CPS-1, −2 and −3. This project was operated for 3 years at an injection rate of 0.01 PV/yr, with no significant results. Injection was then confined to zones CPS-1 and −2 alone; still, after 2 years of close monitoring, results were inconclusive. Eventually, poor performance led to project abandonment. Meanwhile, in 1971, waterflooding had been initiated at the main block of the field, where substantial reservoir depletion (40 kg/cm2) had led to an average well productivity decline of 30%. The project was designed as nine inverted nine-spot patterns occupying an area of 576 ha. The target oil was 42×106 m3 in zones CPS-1, −2, −3, and −4 of the Carmópolis/Muribeca formation, as well as the reservoirs in the Barra de Itiuba formation. The final estimated recovery factor (FR) was 25.3% for a projected injection rate of 1800 m3/d (200 m3/d/well). During this 8-year project, injection in the Barra de Itiuba formation and zones CPS-3 and −4 of the Carmópolis/Muribeca was suspended owing to the high oil viscosity in the former and the bad quality of the conglomerate reservoirs in the latter. Despite the difficulties in managing selective injection, the project was considered an overall success. Reservoir pressure was restored and well productivity increased. However, the price hike that culminated with the second oil shock of 1979, as well as the fact that the project life expectancy was estimated at 30 years under the original design specifications, led to an effort to upgrade the project design to anticipate production from that area of the field. The result was the conversion of the original nine-spot arrangement of the waterflooding operation to 44 inverted five-spot patterns of 8 ha each, with water injection concentrated in zones CPS-1 and −2 alone. Forty-four new injectors and 28 production wells were drilled and completed only in zones CPS-1 and −2 for a target oil of 15×106 m3. Former injection wells were converted into producers. Injection began in April 1978 at a rate of 100 m3/d/well totaling 4400 m3/d for the entire operation. Because of the downsizing in well patterns and the increase in injection rate, oil production in the area rose from 600 to 800 m3/d within 3 years. Water/oil ratio then began to increase sharply with massive water breakthrough in the producing wells. Originally thought to be simply inherent to the higher injection rates, this breakthrough was later found to be associated to the high mobility ratio of the fluid displacement and reservoir high-permeability streaks. Together with the decrease of the injection rate and the extension of completion to zone CPS-3, an ambitious program of injection well profile modification, through stimulation and selectivity plugging, was implemented. It ensured the maintenance of oil production at acceptable levels up to the mid-1980's. By 1986, however, with the profile modification treatments losing their effectiveness, production went into a sharp decline once again. Fig. 3 gives an overall view of the exploitation history in the main block of Carmópolis field.

TX 75083-3836, U.S.A., fax 01-972-952-9435. AbstractThis paper addresses the design criteria for a Floating,