DETAILED NEAR-WELL MODELING AND UPSCALING IN CONDENSATE SYSTEMS

Transcription

1 DETAILED NEAR-WELL MODELING AND UPSCALING IN CONDENSATE SYSTEMS A THESIS SUBMITTED TO THE DEPARTMENT OF ENERGY RESOURCES ENGINEERING OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE By Hyung Ki Kim December 2009

2 I certify that I have read this thesis and that in my opinion it is fully adequate, in scope and in quality, as partial fulfillment of the degree of Master of Science in Petroleum Engineering. Dr. Louis J. Durlofsky (Principal advisor) I certify that I have read this thesis and that in my opinion it is fully adequate, in scope and in quality, as partial fulfillment of the degree of Master of Science in Petroleum Engineering. Dr. Mohammad Karimi-Fard ii

3 Abstract In gas condensate reservoirs, when the pressure falls below the dewpoint, condensate liquid appears. Because pressure is typically the lowest around production wells, this liquid drop out occurs first in the near-well region. This phenomenon often leads to reduced gas production because the condensate causes a decrease in gas relative permeability. This so-called condensate blockage can be difficult to capture in standard reservoir simulation models because well blocks are typically too large to resolve the pressure and saturation variations that lead to this localized effect. In this thesis, unstructured grid models, which include high resolution in the well and near-well region, are constructed and used to simulate condensate systems. The well completion components, such as tubing, gravel pack and perforations, are represented in detail. Simulations involving various completion strategies (open-hole, cased-hole with different types of perforations) are performed to quantify the impact of completion type on well productivity. As would be expected, open-hole completions are shown to provide greater productivity than cased-hole completions, and longer perforations result in higher production rates than shorter perforations. The unstructured grid simulations are, however, very computationally demanding. For this reason, single-phase and two-phase near-well parameter upscaling procedures are applied to coarsen the unstructured fine-scale descriptions to coarse structured models. The single-phase upscaling provides coarse-scale well indices and well-block transmissibilities, while the two-phase upscaling generates coarse-scale well-block relative permeability functions. Both upscaling techniques determine the coarse-scale parameters through use of optimization procedures that minimize the mismatch in flow results between fine and coarse-scale simulations over a near-well region referred to iii

4 as the local well model. Comparison of global coarse-model solutions to the reference fine-scale results demonstrates that the models that use only upscaled single-phase flow parameters provide a reasonable degree of accuracy when only gas is flowing, though less accurate results are observed when both gas and liquid are flowing. The additional use of upscaled well-block relative permeabilities is shown to improve the accuracy of simulation results when two phases are flowing. Thus the overall upscaling procedure provides a means for generating coarse-scale models that capture the detailed interaction between the well completion and the near-well reservoir region. iv

5 Acknowledgements First and foremost I would like to express my sincere thanks to my advisor Professor Louis J. Durlofsky for his support, continuous encouragement and guidance during this work. He has always been deeply involved with my work and has given me great feedback and knowledge. I also would like to thank Dr. Mohammad Karimi-Fard and Dr. Toshinori Nakashima for their contributions to this work. Dr. Karimi-Fard always kindly shared his thoughts and discussed the problems I encountered during this work. Dr. Nakashima provided his upscaling code and shared his experience on this topic. I am grateful to all of the faculty, staff and students in the Department of Energy Resources Engineering at Stanford University. They provided me with a wonderful opportunity to study in an inspiring and friendly environment. I also would like to acknowledge the industrial affiliates of the Stanford University Advanced Wells (SUPRI-HW) Consortium for their financial support, which made this research possible. Finally, I would like to extend the most special gratitude to my parents and brother in South Korea for their unconditional support and belief. v

6 Contents Abstract Acknowledgements Table of Contents List of Tables List of Figures iii v vi viii ix 1 Introduction Overview Previous work Overview of thesis work Simulations with unstructured grids Unstructured grid generation Simulations with simple tubing well completion Simulation of wells with open-hole and cased-hole completions Three-component model and effects of perforations Summary and directions Upscaling of detailed unstructured models Single-phase parameter upscaling Application of near-well upscaling with complex well completions 29 vi

7 3.1.2 Application to condensate systems Two-phase parameter upscaling Simulation results using two-phase upscaling Summary Conclusions and future work Conclusions Future work vii

8 List of Tables 2.1 Condensate composition and component properties (nine-component system) Well component properties (colors correspond to the model shown on the right of Figure 2.13) Condensate composition and component properties (three-component system) Initial and optimized parameters defining fine and upscaled well-block relative permeability functions viii

9 List of Figures 1.1 Phase diagram of a gas condensate system (from Fan et al. [1]) Three regions of flow behavior in a gas condensate reservoir (from Fan et al. [1]) Schematic of realistic well completion (courtesy of Karimi-Fard) Unstructured grid representing the complex wellbore illustrated in Figure Wellbore and near-well models using unstructured grid (red: tubing, green: reservoir) Relative permeabilities of gas and oil phases Bottomhole pressure for simple tubing simulation Oil production rate for simple tubing simulation Productivity index for simple tubing simulation Schematic of the refined structured areal grid around a vertical well (well block shown in red) Comparison of BHP computed using unstructured and refined structured grids for gas condensate system Comparison of oil production rate computed using unstructured and refined structured grids for gas condensate system Simulation results illustrating liquid fallback inside the tubing Schematics of open-hole (left) and cased-hole (right) well completions (courtesy of Karimi-Fard) Unstructured grid for detailed well completion (left) and for well and near-well reservoir regions (right) ix

10 2.14 Bottomhole pressure of open-hole and cased-hole well completions Productivity index of open-hole and cased-hole well completions Liquid buildup near the wellbore at 2.7 days for open-hole well completion Liquid buildup near the wellbore at 2.4 days for cased-hole well completion Uniform (left) and spiral (right) perforation distributions Comparison of P I between uniform (red) and spiral (blue) distributions Productivity indices for various perforation lengths (l p ) Workflow for the calculation of the coarse-scale well index and transmissibilities Unstructured grids and local well model: reservoir and local well model (red border) (left), coarse well-block region (upper right), and zoom-in of well (lower right) Unstructured grids for perforated well: reservoir (upper left), coarse well-block region (right), and zoom-in of well (lower left) W I as a function of the number of perforations Bottomhole pressure for fine and coarse solutions Liquid production rate (surface conditions) for fine and coarse solutions Productivity indices of gas for fine and coarse solutions Initial (solid) and optimized (dashed) well-block relative permeabilities of gas (red) and oil (blue) phases Liquid production rate (surface condition) for fine-scale simulation, model with single-phase upscaling, and model with two-phase upscaling (Q g = 500 mcf/d) Productivity indices for fine-scale simulation, model with single-phase upscaling, and model with two-phase upscaling (Q g = 500 mcf/d) Liquid production rate (surface condition) for fine-scale simulation, model with single-phase upscaling, and model with two-phase upscaling (Q g = 250 mcf/d) x

11 3.12 Productivity indices for fine-scale simulation, model with single-phase upscaling, and model with two-phase upscaling (Q g = 250 mcf/d).. 39 xi

12 Chapter 1 Introduction 1.1 Overview Gas condensate reservoirs contain only gas at original reservoir conditions. This gas consists mainly of methane and other lighter components but some long-chain hydrocarbons (heavy ends) may also exist. As a result, some amount of oil is typically produced along with the gas. On a pressure-temperature phase diagram, the initial reservoir condition is above and to the right of the critical point, as illustrated in Figure 1.1. Reservoir pressure decreases during production, and once pressure falls below the dewpoint, a liquid phase appears. As the pressure decreases further, the liquid revaporizes; this is referred to as retrograde behavior. This behavior can be understood with reference to Figure 1.1. The numbers on the blue dashed curves indicate the percentage of vapor. For this particular system, the percentage of vapor decreases from 100% to a minimum of about 82%, though with continued pressure decline it subsequently increases to more than 90%. Several regions within gas condensate reservoirs can be identified [1], as depicted in Figure 1.2. The location of the production well is shown in the figure. Far from the well, in region 3, the pressure is above the dewpoint, so in this region only the gas phase exists. Closer to the well, in region 2, the pressure is below the dewpoint and condensate liquid forms. This liquid is, however, immobile because the liquid saturation is below the critical saturation. The decrease in gas mobility in region 2 1

13 2 CHAPTER 1. INTRODUCTION Figure 1.1: Phase diagram of a gas condensate system (from Fan et al. [1]) may be negligible. The situation in region 1, which is nearest the production well, is different. Here the condensed liquid saturation is significant, which means that it is mobile and that it impedes gas flow. In other words, the increase in liquid saturation in this condensate bank leads to a decrease in gas relative permeability. This phenomenon is referred to as condensate blockage, which is a key reason for reduced production in gas condensate reservoirs. Along these lines, productivity losses due to condensate blockage of 50% or more have been reported in field observations in [2] and [3]. It is clearly important to accurately model this effect if we are to predict the behavior of gas condensate reservoirs and mitigate productivity losses. In reservoir simulation models, well performance predictions can be very sensitive to the size of the well block and neighboring blocks. Field-scale models typically use large grid blocks (e.g., 100 m 100 m) for computational efficiency. However, models with blocks of this size may underestimate condensate blockage around the well because they do not accurately capture the condensate bank, which is typically only several meters in extent. Furthermore, the detailed well components (tubing, casing, gravel pack, perforations and fractures) can significantly impact near-well

14 1.2. PREVIOUS WORK 3 Figure 1.2: Three regions of flow behavior in a gas condensate reservoir (from Fan et al. [1]) behavior and productivity [4], and it can be difficult to account for these effects in standard coarse-scale simulations, where the only well parameter is the well index (and associated skin). The detailed aspects of the well completion and the near-well reservoir region can be resolved using fine-scale unstructured (e.g., tetrahedral) grids, and the resulting models can be used to understand the interplay of flow physics and well completion strategies. In this study, we develop such models and simulate a number of cases to assess the impact of various completion types on the productivity of condensate reservoirs. These fine-scale simulations are, however, very time-consuming due to the use of very small grid cells (needed to resolve details of the well). Thus, we implement and test upscaling techniques to accelerate the compositional simulations. 1.2 Previous work Much of the research on gas condensate reservoirs concerns the prediction of productivity loss due to condensate banking. A recent paper by Kamath [5] presents an overview of different steps required to achieve this goal, from laboratory measurements to full-field modeling.

15 4 CHAPTER 1. INTRODUCTION To accurately assess well deliverability in gas condensate reservoirs, Fevang and Whitson [6] and Whitson et al. [7] introduced a method to calculate pseudopressure, which is an extension of an earlier method proposed by Evinger and Muskat [8] for solution-gas-drive oil wells. This method requires knowledge of the producing gas-oil ratio in addition to PVT properties and gas-oil relative permeabilities. It is known that key properties that impact productivity loss in gas condensate reservoirs are the relative permeabilities of the gas and liquid phases. Methods for modeling relative permeabilities were introduced by Henderson et al. [9], Pope et al. [10] and Ayyalasomayajula et al. [11]. Controlling the condensate banking and improving the productivity require a more detailed understanding of flow behavior at the completion level. Some work on the effect of perforations on well productivity was carried out as early as 1943 by Muskat [4]. His work was based on analytical calculations for single-phase flow. Extension of this type of study to more complex systems such as gas condensates requires numerical techniques. Research focusing on accurate modeling of the perforation region has been performed by several investigators (e.g., Tariq [12], Behie and Settari [13], Jamiolahmady et al. [14]). These studies considered single-phase flow, multiphase flow and, to some extent, non-darcy effects. Jamiolahmady et al. [15] investigated gas condensate flow in perforated regions using Comsol Multiphysics software [16], which applies a finite element method. With their approach the geometry of the perforations is represented accurately. They specified constant pressure in the wellbore and the perforations. This is justified by the high conductivity of the perforations and their connection to the wellbore. The actual flow through the perforations was not modeled. The present work represents an extension of the work by Karimi-Fard and Durlofsky [17]. They investigated the impact of non-darcy effects on well productivity in single-phase flow. They used highly detailed unstructured grids and the finite volume method to discretize the flow equations. The flow through different parts of the completion was modeled explicitly including the flow through the perforations and the gravel pack around the tubing. In this work we use the same framework to study gas condensate flow and the impact of different types of completions on well productivity.

16 1.3. OVERVIEW OF THESIS WORK 5 The fine-scale simulations performed in this study are very time consuming. Upscaling procedures, which we now discuss, act to coarsen fine-scale models to sizes appropriate for large-scale simulation. Procedures for the determination of upscaled single-phase near-well parameters have been developed by Ding [18], Durlofsky et al. [19], and Muggeridge et al. [20]. These methods provide upscaled well indices in addition to upscaled transmissibilities (for flow between well blocks and non-well blocks). Mascarenhas and Durlofsky [21] and Nakashima and Durlofsky [22] improved the accuracy of these methods by incorporating optimization techniques to minimize the mismatch between flow results over the near-well region (referred to as the local well model or LWM). These techniques were recently generalized to handle unstructured fine-grid models by Nakashima [23], and this approach will be applied here. See [24] for a general overview of single-phase parameter upscaling procedures. Since gas condensate reservoirs can experience two-phase flow in some regions, two-phase upscaling may also be required. Two-phase upscaling approaches, which provide upscaled or pseudo-relative permeability functions, are discussed in detail in Barker and Dupouy [25] and Darman et al. [26]. Two-phase near-well upscaling techniques, which make use of optimization procedures to force agreement in component flow rates over the LWM, have been developed by Hui and Durlofsky [27] and Nakashima and Durlofsky [22]. A simplified variant of these approaches will be used in this work. 1.3 Overview of thesis work In Chapter 2 of this thesis, we develop unstructured simulation models for condensate systems that include high resolution in the near-well region. The well components are represented in detail, which allows us to quantify the effects of various completion strategies on well productivity. The general approach is verified, for an open-hole well (tubing only), through comparison to a refined structured grid model. Detailed simulations for both open-hole and perforated cased-hole completions are presented, and the impact of completion type on productivity is illustrated. In Chapter 3, we evaluate procedures for upscaling the detailed unstructured

17 6 CHAPTER 1. INTRODUCTION models introduced in Chapter 2. Both single-phase and two-phase near-well upscaling techniques are applied. The models that use only upscaled single-phase flow parameters provide reasonable accuracy when the reservoir flow is single-phase gas, though some degradation in accuracy is observed when there is two-phase flow in the near-well region. The use of upscaled well-block relative permeabilities is shown to improve the accuracy of flow predictions in the two-phase flow regime. Finally, in Chapter 4 we present conclusions and some suggestions for future work in this area.

18 Chapter 2 Simulations with unstructured grids In this chapter we first describe the unstructured grid models and our well and nearwell representation. Next, a simulation of a condensate system with production from a simple well is presented and the results compared with those from a structured grid model. Finally, the effects of various components of the well completion on productivity are investigated using the detailed near-well models. 2.1 Unstructured grid generation A schematic of a realistic well completion, which includes perforated casing, gravel pack and tubing, is illustrated in Figure 2.1. It would be very difficult to represent these components using a standard structured grid system. Therefore, we use unstructured grids to capture the geometric complexities of realistic wells. All unstructured grids applied in this study are generated using TetGen, a three-dimensional tetrahedral mesh generator developed by Si [28]. The algorithms used in TetGen are of Delaunay type. An unstructured tetrahedral grid that resolves the completion components shown in Figure 2.1 is presented in Figure 2.2. Stanford s General Purpose Research Simulator (GPRS), described in detail by Cao [29] and Jiang [30], is used for all simulations in this thesis. The discretization 7

19 8 CHAPTER 2. SIMULATIONS WITH UNSTRUCTURED GRIDS Figure 2.1: Schematic of realistic well completion (courtesy of Karimi-Fard) scheme introduced by Karimi-Fard et al. [31], which is based on a two-point flux approximation (TPFA), is applied. This approach is strictly applicable only for orthogonal systems (in the isotropic case), and some error may occur for systems that are significantly nonorthogonal. For such cases a multipoint flux approximation could be applied, though this is not considered in this work. The unstructured grid simulations require that we specify cell volumes, cell connectivities (which are prescribed through use of a connection list), and transmissibility between connected cells. This information is generated and provided to GPRS using the discrete feature modeling procedure developed by Karimi-Fard et al. [31].

20 2.2. SIMPLE TUBING WELL COMPLETION 9 Figure 2.2: Unstructured grid representing the complex wellbore illustrated in Figure Simulations with simple tubing well completion We first consider a simple well model that includes only tubing. For such a well, in a structured grid simulation the standard Peaceman well model is appropriate, so we can directly compare simulation results using an unstructured grid to those using a refined structured grid. The well and near-well regions in the unstructured grid model are shown in Figure 2.3. Here the reservoir is shown in green and the tubing in red. We represent the tubing as a porous medium of very high permeability (10 7 md), which results in an essentially constant-pressure well specification. In addition, the porosity of the tubing cells is set to 1. Production is specified to occur from a well completed in a single cell within the tubing. For this cell, the well index is prescribed

21 10 CHAPTER 2. SIMULATIONS WITH UNSTRUCTURED GRIDS Figure 2.3: Wellbore and near-well models using unstructured grid (red: tubing, green: reservoir) to be very high (10 6 md-ft). This well simply acts to collect and bring to the surface the fluid that is produced from the reservoir into the tubing cells. This well treatment is used in all unstructured fine-scale simulations presented in this thesis. The dimensions of the reservoir model are 546 ft 546 ft 50 ft. The radius of the tubing is 0.36 ft. The model contains 5,848 tetrahedral cells. The reservoir porosity (φ) and permeability (k) are 0.1 and 100 md (tubing properties are given above). The well is specified to produce at a constant gas rate of 12,000 mcf/d (the minimum bottomhole pressure is never reached). Initial reservoir pressure is 3900 psi. The nine-component condensate fluid model is modified from the Third SPE Comparative Solution Project [32]. The properties are given in Table 2.1. The dewpoint pressure is 3050 psi. Relative permeabilities (k rg is shown red and k ro in blue) for this model are presented in Figure 2.4. Simulation results for bottomhole pressure and oil production rate are shown in Figures 2.5 and 2.6. Once BHP reaches the dewpoint (3050 psi), the oil production rate (at surface conditions) in Figure 2.6 is seen to decrease. This occurs because more of the heavier fractions, which form oil at surface conditions, are left in the reservoir (in the liquid phase) as pressure decreases.

22 2.2. SIMPLE TUBING WELL COMPLETION 11 Figure 2.4: Relative permeabilities of gas and oil phases Table 2.1: Condensate composition and component properties (nine-component system) Component Mole fraction MW p c (psi) T c ( R) CO N C C C C C C C

23 12 CHAPTER 2. SIMULATIONS WITH UNSTRUCTURED GRIDS Figure 2.5: Bottomhole pressure for simple tubing simulation Figure 2.6: Oil production rate for simple tubing simulation

24 2.2. SIMPLE TUBING WELL COMPLETION 13 Because gas rate is specified, we quantify gas production in terms of the gas productivity index or P I, defined as P I = Q g p res p wf (2.1) where Q g is the gas-phase production rate, p res is the volume-averaged reservoir pressure and p wf is the wellbore flowing pressure. We expect P I to decrease, as a result of condensate banking, after the BHP drops below the dewpoint. This occurs because the liquid that accumulates around the wellbore impedes gas flow, so larger differences between the well and reservoir pressures are required to achieve the specified gas rate. The decrease in P I is evident in Figure 2.7. The slight increase in P I towards the end of the simulation is presumably due to the retrograde behavior of the condensate fluid. Figure 2.7: Productivity index for simple tubing simulation A comparison between the unstructured model and a standard structured grid model is also conducted. The structured grid model is refined in the near-well region, as shown in Figure 2.8. In this model, the fine blocks in the well region are of dimensions 2 ft 2 ft; away from the well grid blocks are of dimensions 50 ft 50 ft.

25 14 CHAPTER 2. SIMULATIONS WITH UNSTRUCTURED GRIDS Figure 2.8: Schematic of the refined structured areal grid around a vertical well (well block shown in red) Simulation results for BHP (Figure 2.9) and oil rate (Figure 2.10) using the structured model are very close to those for the unstructured model. Additional resolution in the structured grid resulted in only very little change in the simulation results, so they can be considered to be essentially converged. This provides a degree of verification of our unstructured grid model. The detailed unstructured model is able to simulate additional phenomena such as liquid fallback inside the tubing due to gravitational effects. The liquid saturations in the tubing and near-well region are illustrated in Figure This liquid fallback may also contribute to reduced productivity. We note, however, that our results involving multiphase flow in the tubing should be viewed as qualitative at best, since we represent the tubing as a porous medium in these simulations (i.e., we are not using any sort of wellbore flow treatment such as a drift-flux model).

26 2.2. SIMPLE TUBING WELL COMPLETION 15 Figure 2.9: Comparison of BHP computed using unstructured and refined structured grids for gas condensate system Figure 2.10: Comparison of oil production rate computed using unstructured and refined structured grids for gas condensate system

27 16 CHAPTER 2. SIMULATIONS WITH UNSTRUCTURED GRIDS Figure 2.11: Simulation results illustrating liquid fallback inside the tubing 2.3 Simulation of wells with open-hole and casedhole completions We consider open-hole and cased-hole completions to demonstrate the effects of casing and perforations on condensate production. Completion schematics are shown in Figure 2.12 and the detailed unstructured near-well model is shown in Figure The model contains only eight perforations to keep the number of cells to a manageable number. Different values for porosity and absolute permeability are prescribed for the different wellbore components (all model components are treated as porous media), as indicated in Table 2.2. We use the unstructured grid shown in Figure 2.13 for all simulations in this section, though cell properties are varied to represent different completion types. For example, to simulate production with an open-hole completion, cells that represent casing and perforations in cased-hole simulations are specified to have the permeability (100 md) and porosity (0.1) of the reservoir. This acts to convert the casing cells to reservoir cells but avoids the need to re-grid the

28 2.3. COMPLEX WELL COMPLETION 17 Figure 2.12: Schematics of open-hole (left) and cased-hole (right) well completions (courtesy of Karimi-Fard) system. The reservoir dimensions, fluid properties and initial reservoir pressure are the same as in the previous simple tubing example. The reservoir porosity and permeability are again homogeneous. The radius of the tubing is also the same (r w = 0.36 ft), though here we also have gravel pack, which extends out to a radius r g. Here we set r g = 1.54 ft. In addition, the length (l p ) and radii (r p ) of the perforations are set to 1.5 ft and 0.1 ft, respectively, in the cased-hole simulation. The number of cells in this model is 16,350, which is more than in the previous case. The additional cells are required to resolve the complexity of the well and near-well region. For both the open-hole and cased-hole simulations, the well is specified to produce at a constant gas rate of 12,500 mcf/d. Simulation results for these cases are shown in Figures 2.14 and The results for the open-hole completion are represented as red curves and those for the cased-hole completion as blue curves. We again focus on BHP and P I since wells are controlled with a constant gas production rate. It is evident that the BHP for the cased-hole completion is consistently lower than that for the open-hole completion.

29 18 CHAPTER 2. SIMULATIONS WITH UNSTRUCTURED GRIDS Figure 2.13: Unstructured grid for detailed well completion (left) and for well and near-well reservoir regions (right) Table 2.2: Well component properties (colors correspond to the model shown on the right of Figure 2.13) color region k (md) / φ tubing 10 7 / 1.0 gravel pack 10 4 / 0.7 casing 0.01 / perforation 10 4 / 0.7 reservoir 100 / 0.1

30 2.3. COMPLEX WELL COMPLETION 19 Figure 2.14: Bottomhole pressure of open-hole and cased-hole well completions Figure 2.15: Productivity index of open-hole and cased-hole well completions

31 20 CHAPTER 2. SIMULATIONS WITH UNSTRUCTURED GRIDS Figure 2.16: Liquid buildup near the wellbore at 2.7 days for open-hole well completion Figure 2.17: Liquid buildup near the wellbore at 2.4 days for cased-hole well completion

32 2.4. EFFECTS OF PERFORATIONS 21 This occurs since the cased-hole completion only communicates with the reservoir over a limited region (i.e., through perforations), while the open-hole completion communicates over a much larger area. As expected, the cased-hole completion reaches the dewpoint (pink dotted line) earlier than the open-hole completion. This results in earlier productivity loss as illustrated in Figure Maps of liquid (oil) saturation for the two cases are shown in Figures 2.16 and For the open-hole well completion, most of the liquid drop out occurs around the gravel pack region. For the cased-hole well, more liquid appears, and this liquid accumulation is also evident around the perforations. We note that this sort of detailed behavior would be very difficult to capture with standard structured grid models. 2.4 Three-component model and effects of perforations The cased-hole model simulated in the previous section required about seven days of computation time on an Intel 2.66GHz (2GB RAM) platform. As our intent now is to consider cased-hole completions with different arrangements of perforations, we switch to a three-component condensate system rather than the nine-component system used in the previous cases. The resulting models require about 1/3 the simulation time of the nine-component model. The three-component system is defined in Table 2.3. The dewpoint occurs at psi. Table 2.3: Condensate composition and component properties (three-component system) Component Mole fraction MW p c (psi) T c ( R) C C C

33 22 CHAPTER 2. SIMULATIONS WITH UNSTRUCTURED GRIDS Figure 2.18: Uniform (left) and spiral (right) perforation distributions It is reasonable to expect that, for many cased-hole completions, well productivity will be sensitive to the characteristics of perforations such as a shot density, angular phasing and perforation length. In this study, the impact of perforation distribution and length on production is considered. The reservoir dimensions and the properties and dimensions of the completion components are the same as those used in the earlier simulations except where otherwise indicated. We consider uniform and spiral perforation distributions. The grids used to simulate these two perforation distributions are shown in Figure Both cases have the same angular phasing (angle between adjacent perforations, which is 90 in this case) and the same perforation density and perforation length. Simulation results for P I for the two models are presented in Figure It is evident from these results that there is very little difference between the two perforation distributions. We next assess the impact of different perforation lengths (l p = 0.75, 1.0, and 1.25 ft) on P I. These results are presented in Figure As perforation length increases, the area into which reservoir fluid can flow increases. Therefore, a constant production rate can be achieved with a smaller pressure drawdown, which means that P I should increase with increasing perforation length, as observed in Figure 2.20.

34 2.4. EFFECTS OF PERFORATIONS 23 Figure 2.19: Comparison of P I between uniform (red) and spiral (blue) distributions Figure 2.20: Productivity indices for various perforation lengths (l p )

35 24 CHAPTER 2. SIMULATIONS WITH UNSTRUCTURED GRIDS 2.5 Summary and directions In this chapter, we developed unstructured grid models for the simulation of flow in gas condensate reservoirs. Detailed well representations, including resolution of completion components, were incorporated in these simulations. We verified the unstructured simulation model through comparison to a refined structured grid model for the case of a simple well in a gas condensate reservoir with liquid drop out. The effect of casing (versus open-hole completion) and perforation characteristics on productivity was then assessed. It was observed that, as expected, the open-hole completion displayed higher P I than the cased-hole completion and that productivity increased with increasing perforation length. As a result of the very small cells required to resolve the well and near-well region, the unstructured grid simulations required an excessive amount of computation time. For the cased-hole completion with the nine-component system, time steps of less than 10 4 days (8.64 seconds) were often required. These simulations consumed more than seven days of computation time on the available platform, which is comparable to the length of time simulated. Attempts to simplify the grid generated by TetGen, while retaining key completion features, were unsuccessful. It will be useful to pursue the development of improved gridding strategies in future work. In the next chapter, we introduce upscaling procedures which provide us with coarse-scale models that can be simulated very quickly.

36 Chapter 3 Upscaling of detailed unstructured models In this chapter, we compute upscaled single-phase and two-phase well block parameters and use these in coarse-scale simulations of gas condensate systems. The singlephase upscaling entails the computation of upscaled well index (W I ) and well block transmissibilities (Tw) while the two-phase upscaling provides upscaled relative permeabilities. The use of these upscaled parameters is shown to provide coarse-scale models that are of reasonable accuracy relative to the reference fine-scale unstructured grid simulations. 3.1 Single-phase parameter upscaling To capture the interaction of the near-well reservoir region and the well completion components in coarse-grid models, we apply a near-well single-phase parameter upscaling procedure. The basic approach is discussed in, e.g., Ding [18], Durlofsky et al. [19] and Nakashima and Durlofsky [22], though these implementations were for structured grid models. The generalization to upscale unstructured fine models is presented in Nakashima [23], and this is the method (and the code) used here. The upscaled parameters are computed by solving the single-phase incompressible pressure equation over a near-well region referred to as the local well model (LWM). In 25

37 26 CHAPTER 3. UPSCALING OF DETAILED UNSTRUCTURED MODELS the computations applied here, this region is defined by the coarse well block and all adjacent coarse blocks, with an extra layer of fine cells added on the boundary. The LWM is shown in Figure 3.2 for an open-hole completion and in Figure 3.3 for a cased-hole completion. We proceed by solving (k p) = q (3.1) where p is pressure, k is the isotropic permeability and q is the source term. We specify a constant well pressure p wf and a (different) constant pressure at the boundary of the LWM. For the fine-scale LWM solution, the boundary pressure is enforced by introducing constant-pressure wells into the fine-scale boundary cells (red layer in Figures 3.2 and 3.3), which are prescribed to have very high permeabilities. Once the fine-scale LWM is simulated, it provides pressures at the cell centers and flow rates through cell faces. The coarse-block pressures can then be approximated by bulk-volume averaging the pressures of fine cells over the region corresponding to the coarse block. In addition, the flow rates through each face of the coarse well block are estimated by summing the corresponding fine-scale flow rates. We denote the averaged pressure of the coarse well block as p wb and that of its adjacent neighbor as p adj, the sum of the fine-scale flow rates through coarse well block face k as q f k, and the well flow rate as qf w. Then, the coarse-scale well index (W I ) and transmissibilities between the well block and its neighboring blocks (Tw,k) can be estimated as follows: T w,k = q f k p adj p wb (3.2) W I = q f w p wb p wf (3.3) An additional optimization step, which forces agreement between the flow rates computed on a coarse-scale LWM and the summed fine-scale rates, is described in Mascarenhas and Durlofsky [21] and Nakashima and Durlofsky [22]. Here we use the approach of Nakashima and Durlofsky [22] to determine optimium W I and T w,k.

38 3.1. SINGLE-PHASE UPSCALING 27 They introduced an objective function L which quantifies the mismatch between fine and coarse-scale flow rates: n c wb [ L = (q c w ) i ( n ] qw) c f f 2 [ + q c k q f ] 2 k (3.4) i=1 Here (qw) c i is the flow rate into the well in coarse well block i, ( ) qw f is the sum of i the fine-scale well flow rates over coarse-block i, and n c wb is the number of coarse well blocks. In addition, qk c is the flow rate across coarse face k, q f k is the sum of the finescale flow rates across fine faces corresponding to coarse face k, and n c f is the total number of coarse-block faces that link well blocks to non-well blocks. The quantities ( q f w ) i and qf k are computed from the fine-scale LWM solution, while (qc w) i and q c k are computed from the coarse-scale simulation. Nakashima and Durlofsky [22] proposed an adjoint procedure to calculate the gradient of the cost function with respect to the optimization parameters (W I and T w,k for all well blocks). This gradient is then used for the minimization. A flow chart of the single-phase upscaling procedure is shown in Figure 3.1. See [22] for full details. i i=1 Figure 3.1: Workflow for the calculation of the coarse-scale well index and transmissibilities

39 28 CHAPTER 3. UPSCALING OF DETAILED UNSTRUCTURED MODELS Figure 3.2: Unstructured grids and local well model: reservoir and local well model (red border) (left), coarse well-block region (upper right), and zoom-in of well (lower right) Before computing the upscaled well index (W I ) and transmissibilities (Tw,k) for complex well completions, it is useful to demonstrate that the single-phase upscaling method provides a result in reasonable agreement with the standard Peaceman well index for a case involving a simple well (tubing only) and a homogeneous reservoir model (see Figure 3.2). The Peaceman expression is given by W I = 2πk z ln (r o /r w ) + s (3.5) r o = 0.14 x 2 + y 2 (3.6) where x, y and z are the well-block dimensions, r w is the wellbore radius and s is skin. Applying the near-well upscaling procedure for this case, we compute a W I of 2523 md-ft. The well index from the Peaceman expression (with s = 0) above is 2674 md-ft. These values are in reasonably close agreement the slight difference may be due to the fact that the wellbore cross section in our fine-scale model is not

40 3.1. SINGLE-PHASE UPSCALING 29 circular (using a circular well would require more cells) or it may be due to numerical error in our procedure. In any event, the fact that these two results are within 6% provides a degree of verification for our upscaling procedure Application of near-well upscaling with complex well completions We now apply the single-phase near-well upscaling procedure to models of well completions containing casing and perforations as well as tubing. One such model is shown in Figure 3.3. In contrast to the models considered in Chapter 2, this model has only four perforations, which reduces the number of fine-grid cells. For the model shown in Figure 3.3, we compute a W I of 441 md-ft. The Peaceman index for this case, using s = 0, is 3343 md-ft. A zero skin factor for this model is clearly inappropriate, however, and methods to estimate s based on perforation characteristics are available. Here we apply the the procedures provided by Karakas and Tariq [33] and Bellarby [34] for this estimation. This approach is as follows: s p = s h + s wb + s v (3.7) ln ( r w s h = ln ( ) 4r w lp α(r w+l p) ) for phasing angles other than 0 for the case of phasing angle 0 (3.8) s wb = C 1 exp (C 2 r wd ) (3.9) where r wd = r w r w + l p s v = 10 a h b 1 D r b pd (3.10) where a = a 1 log 10 r pd + a 2 b = b 1 r pd + b 2 h D = h l p kh kv r pd = rp 2h ( 1 + kv k h )

41 30 CHAPTER 3. UPSCALING OF DETAILED UNSTRUCTURED MODELS Here s p, the perforation skin, is the sum of s h, the horizontal skin, s wb, the wellbore skin, and s v the vertical skin. Other variables appearing in these expressions are the perforation radius r p, the perforation length l p, the spacing between perforations h, and the constants α, C 1, C 2, a 1, a 2, b 1 and b 2, which depend on the phasing angle of the perforations. Values for these parameters are given in [34]. The dimensional variables in Eqs. 3.8 to 3.10 are in units of inches. The variables h D, r pd and r wd are dimensionless. The ratio of horizontal (k h ) to vertical (k v ) permeability appears in the definitions of h D and r pd ; for all cases considered here, k h /k v = 1. Figure 3.3: Unstructured grids for perforated well: reservoir (upper left), coarse wellblock region (right), and zoom-in of well (lower left) For the well model containing four perforations, shown in Figure 3.3, r w = 0.36 ft, l p = 1.25 ft, r p = 0.1 ft, and the phasing angle is 90. The perforation skin for this case is computed to be Using this value for s in Eq. 3.5 gives W I = 1310 md-ft, which is closer to the upscaled well index of 441 md-ft than the Peaceman well index without skin, though it is still quite different. A more detailed investigation of the underlying assumptions in the estimate for s p will be required to isolate the cause of this difference. It is also possible that our result for W I may be impacted by our representation of the well components as porous media; i.e., our upscaled values

42 3.1. SINGLE-PHASE UPSCALING 31 Figure 3.4: W I as a function of the number of perforations may depend on the specific well-component permeabilities used. This too will require further investigation. The upscaled well index in this case is much less than that for the tubing-only well. This is due to the limited area of communication between the reservoir and the well. Figure 3.4 demonstrates how the number of perforations impacts W I. As expected, W I (blue curve) increases as the number of perforations increases. The other curve (red curve) in Figure 3.4 provides the well index computed using s p determined from Eq The perforation skin decreases as the number of perforations increases, which leads to an increase in W I. The mismatch between the two sets of computations persists, however Application to condensate systems We now use the upscaled parameters in coarse-scale models of gas condensate systems. The reservoir is homogeneous and isotropic (k = 100 md and φ = 0.1) and of dimensions 546 ft 546 ft 50 ft. A vertical well is completed in the center of reservoir. The model is shown in Figure 3.3. The fine grid contains 11,448 cells and the coarse

43 32 CHAPTER 3. UPSCALING OF DETAILED UNSTRUCTURED MODELS grid contains (147) blocks. We use the nine-component condensate system introduced in Chapter 2. The well flow rate is specified to be 500 mcf/d. The simulation results for the fine and coarse models are shown in Figures 3.5 through 3.7. During the single-phase flow stage of the simulation (before the dewpoint is reached), the upscaled structured grid model provides accurate results for all quantities. Once two-phase flow begins, however, there are some differences between the fine and coarse-scale models. This is evident in Figure 3.6, which presents results for liquid production rate, and in Figure 3.7 where we show results for P I. The reason that the agreement deteriorates when two-phase flow occurs is that we have not upscaled the relative permeabilities (k rj, where j = gas, oil), which appear in the multiphase flow equations. These impact both total system mobility and the fractional flow of oil and gas. We will introduce a method to upscale these functions in the next section. Figure 3.5: Bottomhole pressure for fine and coarse solutions The use of near-well single-phase upscaling leads to a very large reduction in computation time. The fine-scale unstructured model requires about 21 hours of simulation time. The coarse, structured model needs only 42 seconds for the simulation plus 2 seconds for the single-phase upscaling computations. The upscaling is very

44 3.1. SINGLE-PHASE UPSCALING 33 Figure 3.6: Liquid production rate (surface conditions) for fine and coarse solutions Figure 3.7: Productivity indices of gas for fine and coarse solutions

45 34 CHAPTER 3. UPSCALING OF DETAILED UNSTRUCTURED MODELS fast in this case because we are only solving the steady-state single-phase pressure equation. 3.2 Two-phase parameter upscaling Although the near-well single-phase upscaling applied above is effective, we still observe some mismatch in production quantities when two-phase flow exists. In an attempt to further improve these results, we now consider two-phase parameter upscaling. The method introduced in Nakashima and Durlofsky [22] was developed for black-oil models and is therefore not directly applicable to our case. We now describe an approach that is somewhat simpler than that method but which still uses an optimization procedure to minimize the mismatch between fine and coarse-scale results. In our two-phase upscaling technique, after W I and Tw are computed in singlephase upscaling, we determine the upscaled well-block relative permeability functions kro and krg. These functions are prescribed to be of the following forms: k ro = A (S o S or ) B (3.11) krg = C (S g ) D (3.12) where S o and S g are the oil and gas saturation and S or is the irreducible oil saturation (in this study, S or = 0.05). The parameters A, B, C and D are determined by the upscaling procedure. In the upscaling procedure, we first perform a fine-scale compositional simulation over the LWM. For these simulations, we specify a gas rate of 500 mcf/d and no-flow boundary conditions at the edges of the LWM. Then we determine the parameters A, B, C and D (the vector of parameters is designated u) such that the following objective function L is minimized: L (u) = ω ( P I f g P I c g) 2 + (1 ω) ( Q f o Q c o) 2 (3.13)

46 3.2. TWO-PHASE UPSCALING 35 where P I f g and P I c g are the fine and coarse-scale gas productivity indices and Q f o and Q c o are the oil production rates computed from the fine and coarse-scale models. The parameter ω defines the relative weights of the two terms; in the calculations below we take ω = This value is chosen because it makes the two terms equal using the initial estimates for the parameters. We employ the general pattern search (GPS) optimization algorithm in MATLAB [35] to find the set of parameters u opt which minimizes the objective function L. The fine-scale relative permeabilities are used to define the initial guess for the parameters (see Table 3.1 for the specific values). Note that the fine-scale relative permeabilities are given in tabular form, so we first need to determine the parameters A-D that provide the best fit. The time required for the GPS optimization is very small compared to the time required for the fine-scale global simulation even if many coarse-scale simulations are performed during the minimization. Table 3.1: Initial and optimized parameters defining fine and upscaled well-block relative permeability functions A B C D L Initial Optimized The initial and final (optimized) sets of optimization parameters as well as the associated values of the objective function L are presented in Table 3.1. The finescale and optimized coarse-scale well-block relative permeability curves are shown in Figure 3.8. The maximum and minimum of the krj functions are enforced to be 1 and 0, which leads to shifts in the endpoint S g values. The differences between the two sets of curves are not that large.