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Abstract Upscaling is often applied to coarsen detailed geological reservoir descriptions to sizes that can be accommodated by flow simulators. Adaptive local-global upscaling is a new and accurate methodology that incorporates global coarse scale flow information into the boundary conditions used to compute upscaled quantities (e.g., coarse scale transmissibilities). The procedure is iterated until a self-consistent solution is obtained. In this work, we extend this approach to three-dimensional systems and introduce and evaluate procedures to decrease the computational demands of the method. This includes the use of purely local upscaling calculations for the initial estimation of coarse scale transmissibilities and the use of reduced border regions during the iterations. This is shown to decrease the computational requirements of the reduced procedure by a factor of about six relative to the full methodology, while impacting the accuracy very little. The performance of the adaptive local-global upscaling technique is evaluated for three different heterogeneous reservoir descriptions. The method is shown to provide a high degree of accuracy for total flow rate, local flux and oil cut. In addition, it is shown to be less computationally demanding and significantly more accurate than some existing extended local upscaling procedures. Introduction Fine scale heterogeneity can have a significant impact on reservoir performance. Because it is usually not feasible to simulate directly on the detailed geocellular model, some type of upscaling is often applied to generate the simulation model from the geological description. Here we focus on the upscaling of single-phase flow parameters, particularly absolute permeability. The algorithms we consider can provide either coarse-scale permeability, designated k*, or coarse-scale transmissibility, designated T*. It is important to emphasize that the accurate upscaling of permeability (which can be studied within the context of single-phase flow) is essential for the development of accurate coarse models of two-phase or multiphase flow. Thus the applicability of the methods developed here is very broad and includes all types of displacement processes. Permeability and transmissibility upscaling algorithms can be classified in terms of the solution domain over which the governing single-phase pressure equation is solved to compute the coarse scale quantities. Purely local methods (e.g., Durlofsky, King and Mansfield) consider only the fine scale cells comprising the target coarse block, while extended local methods include some number of neighboring cells in the local problems. Both of these methods require assumptions regarding the boundary conditions to be imposed, which can lead to inaccuracy in some cases. At the other extreme are global methods, in which the flow solution used to compute the upscaled quantities is performed over the entire domain. White and Horne5 considered a set of global flows in the derivation of coarse scale properties, while Holden and Nielsen applied a specific global flow scenario (e.g., driven by wells) in their calculations. These methods may achieve high degrees of accuracy but have the drawback of requiring global fine scale solutions. With these techniques, in order to avoid spurious values of coarse scale properties, some iteration is also usually required in the calculation of the upscaled parameters. Quasi-global upscaling methods use some type of approximate global flow information in the calculation of k* or T*. These techniques can provide accurate upscaled models if the approximate global flow information is sufficiently representative of the global fine scale solution. We recently developed new quasi-global upscaling techniques referred to as "local-global" methods. The basic idea of these approaches is to perform a global coarse scale simulation in order to determine the boundary conditions to be applied for the local fine scale calculation of k* or T*. Consistency between the specific global flow and the upscaled model is achieved through iterations between the global coarse simulation and local fine scale calculations. We have shown that the local-global method provides high degrees of accuracy for difficult problems involving highly heterogeneous channelized systems and changing flow conditions, although to date we have considered mostly idealized cases in two dimensions.
- Europe (0.69)
- North America > United States > Texas (0.68)
Abstract Practical production optimization problems typically involve large, highly complex reservoir models, thousands of unknowns and many nonlinear constraints, which makes the numerical calculation of gradients for the optimization process impractical. This work explores a new algorithm for production optimization using optimal control theory. The approach is to use the underlying simulator as the forward model and its adjoint for the calculation of gradients. Direct coding of the adjoint model is, however, complex and time consuming, and the code is dependent on the forward model in the sense that it must be updated whenever the forward model is modified. We investigate an adjoint procedure that avoids these limitations. For a fully implicit forward model and specific forms of the cost function and nonlinear constraints, all information necessary for the adjoint run is calculated and stored during the forward run itself. The adjoint run then requires only the appropriate assembling of this information to calculate the gradients. This makes the adjoint code essentially independent of the forward model and also leads to enhanced efficiency, as no calculations are repeated. Further, we present an efficient approach for handling nonlinear constraints that also allows us to readily apply commercial constrained optimization packages. The forward model used in this work is the General Purpose Research Simulator (GPRS), a highly flexible compositional/black oil research simulator developed at Stanford University. Through two examples, we demonstrate that the linkage proposed here provides a practical strategy for optimal control within a general-purpose reservoir simulator. These examples illustrate production optimization with conventional wells as well as with smart wells, in which well segments can be controlled individually. Introduction Most of the existing major oilfields are already at a mature stage, and the number of new significant discoveries per year is decreasing [1]. In order to satisfy increasing worldwide demand for oil and gas, it is becoming increasingly important to produce existing fields as efficiently as possible, while simultaneously decreasing development and operating costs. Optimal control theory is one possible approach that can be deployed to address these difficult issues. The main benefit of the use of optimal control theory is its efficiency, which makes it suitable for application to real reservoirs simulated using large models, in contrast to many existing techniques. The above-mentioned problem is essentially an optimization problem, wherein the objective is to maximize or minimize some cost function J(u) such as net present value (NPV) of the reservoir, sweep efficiency, cumulative oil production, etc. Here, u is a set of controls including, for example, well rates and bottom hole pressures (BHP), which can be manipulated in order to achieve the optimum. In other words, u is anything that can be controlled. It should be understood that the optimization process results in control of future performance to maximize or minimize J(u), and thus the process of optimization cannot be performed on the real reservoir, but must be carried out on some approximate model. This approximate model is usually the simulation model of the reservoir. This simulation model is a dynamic system that relates the controls u to the cost function J(u). Consider for example the simple schematic of a reservoir shown in Fig. 1, where the cost function is cumulative oil production and the control is the injection rate. Changing the injection rate changes the dynamic states of the system (pressures, saturations), which changes the oil production rate of the producer, which in turn impacts the cost function. Thus, the controls u are related to J(u) through the dynamic system. The dynamic system can also be thought of as a set of constraints that determine the dynamic state of the system given a set of controls. Further, the controls u themselves may be subject to other constraints that dictate the feasible or admissible values of the controls, such as surface facility constraints or fracture pressure limits. It is these additional constraints that in many cases complicate the problem and the solution process.
- Europe (1.00)
- North America > United States > Texas (0.46)
- Europe > United Kingdom > North Sea > Central North Sea > Ness Formation (0.99)
- Europe > Norway > North Sea > Tarbert Formation (0.99)
- Europe > Norway > North Sea > Central North Sea > Ness Formation (0.99)
- Europe > Germany > North Sea > Tarbert Formation (0.99)
Abstract Coupled geomechanical-fluid flow models are needed to account for rock deformation resulting from flow-induced pressure changes in stress sensitive reservoirs. There are, however, issues of numerical stability that must be addressed before these coupled models can be used reliably. Specifically, it is known that standard procedures can lead to pressure oscillations as a result of the violation of the Babuska-Brezzi (B-B) condition, which requires unequal order interpolation of the displacement and pore-pressure variables. In this paper, a number of different types of coupled models are considered. A novel finite element method is developed to circumvent the B-B condition. The method applies a stabilized finite element technique to solve the force balance and pressure equations along with a finite volume method to solve the remaining component mass balance equations. All of the equations are solved in a fully coupled fashion. This method is compared with fully coupled and iteratively coupled models developed using non-stabilized finite elements for the force balance with finite volume methods for all of the component mass balance equations. These comparisons demonstrate that all methods perform reliably on homogeneous reservoirs over long time scales. The stabilized method is shown to provide improved stability at early times and for reservoirs with very low permeability barriers. Introduction Geomechanics can play an important role in oil recovery and oil field operations. In particular, geomechanics is crucial in such problems as production-induced compaction and subsidence, borehole stability, and hydraulic fracturing. In these areas, a key mechanism is the rock deformation due to changing in situ stress. Several investigations have shown that models that do not couple flow and geomechanics may give inaccurate predictions. For this reason, recent attention has been focused on modeling of the coupled system. Here we present an accurate and stable numerical technique for coupled problems. In order to build robust and accurate numerical models, it is important to first consider the accuracy and stability properties of existing methods. Vermeer and Verruijt presented a stability condition for a simple consolidation problem. This condition required that time steps be larger than a certain value, otherwise spatial oscillations would occur. Such a requirement may seem counterintuitive, as we generally expect to observe improved accuracy as the time step is decreased. Within the context of coupled soil mechanics - fluid flow problems, Zienkiewicz et al. showed that the problem could be understood in terms of the Babuska-Brezzi stability condition (B-B condition), which requires nonuniform (i.e., different order) interpolation of the displacement and pore-pressure variables to ensure stability. Murad and Loula showed that an incompressible Stokes flow system at early times led to spatial instability if the B-B condition was not satisfied. Hughes et al. developed stabilized methods to circumvent the B-B restrictions for incompressible Stokes flow problems. These methods are within the category of weighted residual methods. Weighted residual methods modify the bilinear form associated with the problem so that improved numerical stability is achieved without compromising consistency. Zienkiewicz andWu showed similar stabilization by modifying the form of the time stepping in transient problems. In this work, we adopt the general approach of stabilized methods and develop stabilized finite element schemes to control the numerical instability arising in consolidation problems without losing consistency. Using this approach, any combination for interpolations of pressure and displacement can be used. We demonstrate both the instability that can be observed using standard methods as well as the improved performance of our stabilized method.
- Reservoir Description and Dynamics > Reservoir Simulation (1.00)
- Reservoir Description and Dynamics > Reservoir Fluid Dynamics > Integration of geomechanics in models (1.00)
- Reservoir Description and Dynamics > Reservoir Fluid Dynamics > Flow in porous media (1.00)
- Reservoir Description and Dynamics > Reservoir Characterization > Reservoir geomechanics (1.00)
New Developments in Multiblock Reservoir Simulation: Black Oil Modeling, Nonmatching Subdomains and Near-Well Upscaling
Lee, S.H. (ChevronTexaco EPTC) | Wolfsteiner, C. (ChevronTexaco EPTC) | Durlofsky, L.J. (Stanford University) | Jenny, P. (ChevronTexaco EPTC) | Tchelepi, H.A. (ChevronTexaco EPTC)
Abstract In multiblock reservoir simulation techniques, the reservoir model is divided into a number of sub domains or blocks. In our current implementation, these blocks are locally structured but globally unstructured. This representation enables the modeling of geometrically complex reservoir features such as fault surfaces and nonconventional wells while avoiding many of the complications of fully unstructured formulations. In this paper, we present several important developments within this framework. These include the extension of a previous two phase flow finite volume formulation to the general black oil case, the implementation of techniques for treating systems in which grid lines do not match between adjacent sub domains, and the application of a near-well radial upscaling technique to the multiblock paradigm. Simulation results illustrating the accuracy and efficiency of these new capabilities are presented. These include a black oil simulation for a local well study, flow through a realistic reservoir with several wells and faults, flow through a fault surface represented by non matching grid lines, and a two phase flow simulation demonstrating the applicability of the near-well upscaling procedure to multiblock models. With the new developments presented in this paper, the finite volume based multiblock simulator can be applied to a variety of problems of practical interest. Introduction Reservoir models commonly contain complex geological features, general fault surfaces, and nonconventional (i.e., inclined or multilateral) wells. As the level of geometric complexity of reservoir models increases, it becomes more and more difficult to represent them using conventional gridding techniques. The most general types of grids are fully unstructured; e.g., triangular grids in two dimensions or tetrahedral grids in three dimensions. The use of such grids for reservoir simulation is still limited, however, because a number of important prerequisites are not yet in place. Specifically, before fully unstructured techniques can be used in practice, rigorous discretization schemes and enhanced gridding and linear solution algorithms are required. In addition, there is a need for improved numerical accuracy and convergence for large aspect ratio cells. Multiblock grids (MBGs) represent an alternative to fully unstructured grids. In the most general sense, multiblock grids describe complex geometries by allowing different types of grids (structured or unstructured) in each subdomain or block, with nonmatching grid lines at block boundaries. The MBG approach considered here is less general as it entails hexahedral cells that are locally structured but globally unstructured. The second characteristic (global unstructuredness) allows for the resolution of complex reservoir geometries while the first yields (locally) structured matrices that can be solved more efficiently than fully unstructured systems.
- North America > United States > Texas (0.93)
- Europe (0.68)
Abstract A simplified discrete fracture model suitable for use with general purpose reservoir simulators is presented. The model handles both two and three-dimensional systems and includes fracture-fracture, matrix-fracture and matrix-matrix connections. The formulation applies an unstructured control volume finite difference technique with a two-point flux approximation. The implementation is generally compatible with any simulator that represents grid connections via a connectivity list. A specialized treatment based on a "star-delta" transformation is introduced to eliminate control volumes at fracture intersections. These control volumes would otherwise act to reduce numerical stability and time step size. The performance of the method is demonstrated for several example cases including a simple two dimensional system, a more complex three-dimensional fracture network, and a model of a strike-slip fault zone. The discrete fracture model is shown to provide results in close agreement with those of a reference finite difference simulator in cases where direct comparisons are possible. Introduction Flow through fractured porous media is typically simulated using dual-porosity models. This approach, although very efficient, suffers from some important limitations. One drawback is that the method cannot be applied to disconnected fractured media. In addition, dual-porosity models are not well suited for the modeling of a small number of large-scale fractures, which may dominate the flow. Another shortcoming is the difficulty in accurately evaluating the transfer function between the matrix and the fractures. For these reasons, discrete fracture models, in which the fractures are represented individually, are beginning to find application in reservoir simulation. These models can be used both as stand-alone tools as well as for the evaluation of transfer functions for dual-porosity models. Such models can also be used in combination with the dual-porosity approach. To accurately capture the complexity of a fractured porous media it is usually necessary to use an unstructured discretization scheme. There are, however, some effective procedures based on structured discretization approaches. For example, Lee et al. presented a hierarchical modeling of flow in fractured formations. In this approach the small fractures were represented by their effective properties and the large-scale fractures were modeled explicitly. In the case of unstructured discretizations there are two main approaches - finite element and finite volume (or control volume finite difference) methods. Baca et al. were among the early authors to propose a two-dimensional finite element model for single phase flow with heat and solute transport. In a more recent paper Juanes et al. presented a general finite element formulation for two- and three-dimensional single-phase flow in fractured porous media. There has been some work on the extension of the finite element method to handle multiphase flow. For example, Kim and Deo and Karimi-Fard and Firoozabadi presented extensions of the work of Baca et al. for two-phase flow. They modeled the fractures and the matrix in a two-dimensional configuration with the effects of capillary pressure included. The two media (matrix and fractures) were coupled using a superposition approach. This entails discretizing the matrix and fractures separately and then adding their contributions to obtain the overall flow equations.
Abstract Complex well configurations coupled with intelligent completions offer great potential for the efficient production of oil reservoirs. One application of this technology is the use of surface adjustable downhole chokes. This allows wells to be divided into separate inflow zones, with each zone produced under a different pressure drawdown, but with production commingled. Chokes can be set to provide a more uniform inflow profile, which acts to delay the breakthrough of water and gas. The practical benefits of this technology have been demonstrated in several North Sea installations. Despite the large technical and economic risks associated with these complex wells, little work has been presented on the detailed modeling of their performance. While it is possible to apply existing finite difference reservoir simulators, such models can be time consuming to build and the accuracy of the results depends on the grid and the well model. Here, we apply semi-analytical solution methods (based on Green's functions), appropriate for modeling the performance of non-conventional wells operating under single phase flow conditions, to these complex well configurations. The approach entails a fully coupled nonlinear formulation that accounts for pressure drops in the annulus, tubing and downhole chokes. Numerical results for a variety of cases are presented. The high degree of accuracy of the semi-analytical approach is first demonstrated through comparison to results from established methods for model problems. Then, using a geostatistical description of a highly heterogeneous fluvial reservoir, we demonstrate how the method can be used to model the performance of wells with intelligent completions. We show how the technique can be applied to determine downhole choke settings that provide optimal inflow profiles. Introduction The use of horizontal and deviated wells has had a dramatic impact on the industry over the last decade. These wells greatly increase reservoir exposure and can increase the production from a single well significantly. However, horizontal well performance can be impacted significantly by reservoir heterogeneity. Fluvial reservoirs, for example, typically display very large differences in permeability depending on whether the rock type is a channel sand or not, with the result that most of the inflow into a horizontal well might originate from a small proportion of the perforated length. In layered or faulted reservoirs additional complications can arise; e.g., the formation permeability and pressure may vary so much that a single well completion might not be feasible. These complications can be addressed through the use of complex well configurations coupled with so-called "intelligent completions." A key aspect of this technology is the use of surface adjustable downhole chokes, which allow for the isolation of individual reservoirs or zones within a single reservoir. The downhole system makes use of packers that are set between the liner and tubing. It is then possible to control the inflow from different zones along the wellbore through use of downhole chokes, across which a pressure drop can be imposed. This type of completion has been successfully installed and operated in several field applications. Most intelligent completions are very expensive, and it is important to be able to model and predict the performance of these complex systems. Finite difference reservoir simulators can be used to provide an in-depth analysis of the problem, though there are several complications with this type of simulation. Specifically, finite difference models have large data requirements, especially when introducing complex well configurations. As a result, the tool might not be appropriate for quick screening studies. Further, because the reservoir is discretized, well connection factors (or well indexes), linking the well to the discrete model, are required. Default connection factors are typically based on the traditional Peaceman expression, developed under the assumption of two dimensional flow in a homogeneous reservoir. These well indexes may not be valid for complex wells with intelligent completions operating in heterogeneous reservoirs.
- North America > United States > Texas (0.28)
- Europe > United Kingdom > North Sea (0.24)
- Europe > Norway > North Sea (0.24)
- (2 more...)
- Geology > Rock Type (0.48)
- Geology > Sedimentary Geology > Depositional Environment > Continental Environment > Fluvial Environment (0.45)
- Geology > Geological Subdiscipline (0.34)
Application of a New Two-Phase Upscaling Technique to Realistic Reservoir Cross Sections
Wallstrom, T.C. (Los Alamos National Laboratory) | Hou, S. (Los Alamos National Laboratory) | Christie, M.A. (BP Exploration) | Durlofsky, L.J. (Chevron Petroleum Technology Company) | Sharp, D.H. (Los Alamos National Laboratory)
This paper was prepared for presentation at the 1999 SPE Reservoir Simulation Symposium held in Houston, Texas, 14-17 February 1999.
Geologic Modeling, Upscaling and Simulation of Faulted Reservoirs Using Faulted Stratigraphic Grids
Chambers, K.T. (Chevron Petroleum Technology Company) | DeBaun, D.R. (Chevron Petroleum Technology Company) | Durlofsky, L.J. (Chevron Petroleum Technology Company) | Taggart, I.J. (Chevron Petroleum Technology Company) | Bernath, A. (Chevron Petroleum Technology Company) | Shen, A.Y. (Chevron Petroleum Technology Company) | Legarre, H.A. (Chevron Overseas Petroleum Inc.) | Goggin, D.J. (Chevron Overseas Petroleum Inc.)
This paper was prepared for presentation at the 1999 SPE Reservoir Simulation Symposium held in Houston, Texas, 14-17 February 1999.
- Africa (1.00)
- North America > United States > Texas > Harris County > Houston (0.54)
- Geology > Structural Geology > Fault (1.00)
- Geology > Geological Subdiscipline > Stratigraphy (0.66)
- Geophysics > Seismic Surveying (0.68)
- Geophysics > Borehole Geophysics (0.47)
This paper was prepared for presentation at the 1999 SPE Reservoir Simulation Symposium held in Houston, Texas, 14-17 February 1999.
Finite Difference Simulation of Geologically Complex Reservoirs With Tensor Permeabilities
Lee, S.H. (Chevron Petroleum Technology Company) | Durlofsky, L.J. (Chevron Petroleum Technology Company) | Lough, M.F. (Chevron Petroleum Technology Company) | Chen, W.H. (Chevron Petroleum Technology Company)
Abstract The grid block permeabilities used in reservoir simulation are commonly determined via the upscaling of a fine scale geostatistical reservoir description. Though it is well established that permeabilities computed in this manner are in general full tensor quantities, most finite difference reservoir simulators still treat permeability as a diagonal tensor. In this paper, we implement a capability to handle full tensor permeabilities in a general purpose finite difference simulator and apply this capability to the modeling of several complex geological systems. We formulate a flux continuous approach for the pressure equation using a method analogous to that of previous researchers (Edwards and Rogers; Aavatsmark et al.), consider methods for upwinding in multiphase flow problems, and additionally discuss some relevant implementation and reservoir characterization issues. The accuracy of the finite difference formulation, assessed through comparisons to an accurate finite element approach, is shown to be generally good, particularly for immiscible displacements in heterogeneous systems. The formulation is then applied to the simulation of upscaled descriptions of several geologically complex reservoirs involving crossbedding and extensive fracturing. The method performs quite well for these systems and is shown to accurately capture the effects of the underlying geology. Finally, the significant errors which can be incurred through inaccurate representation of the full permeability tensor are demonstrated for several cases. Introduction Recent advances in reservoir characterization permit the construction of realistic, highly detailed, heterogeneous reservoir descriptions. Such models typically contain far too many grid blocks to simulate directly and therefore require some type of upscaling before they can be used for reservoir simulation. The most important of the upscaled rock properties, for purposes of flow simulation, is the absolute permeability. Accurate procedures for the scale up of permeability generate full tensor permeabilities on the coarse scale, even in cases where the underlying fine scale permeability description is isotropic. Therefore, simulation models generated through scale up of complex reservoir descriptions will in general be characterized by full tensor permeabilities. For many models, however, the off-diagonal components of the effective (or equivalent grid block) permeability tensors can be expected to be small relative to the diagonal components and can generally be ignored. This will typically be the case, for example, when the fine scale permeability is correlated along the coordinate directions (e.g., strictly layered systems) or is correlated nearly isotropically. However, for other types of systems the cross terms of the permeability tensor can be expected to be quite significant. These include formations containing complex crossbedding, dipping layers not aligned with the coordinate system, or extensive fracturing. In these cases the upscaled simulation model will contain full tensor permeabilities with significant off-diagonal terms, which must be accommodated by the simulator if reservoir performance is to be predicted accurately. Our intent here is to develop an approach for the modeling of complex geological systems using a general purpose reservoir simulator. Toward that end, we first develop and implement a formulation for full tensor permeability models, applicable for curvilinear grids, into a general purpose finite difference reservoir simulator. Second, we apply this formulation to the simulation of flow through a variety of upscaled geologic descriptions. P. 253^