We investigate the discretization of Darcy flow through fractured porous media on general meshes. We consider a hybrid dimensional model, invoking a complex network of planar fractures. The model accounts for matrix-fracture interactions and fractures acting either as drains or as barriers, i.e. we have to deal with pressure discontinuities at matrixfracture interfaces. The numerical analysis is performed in the general framework of gradient discretizations which is extended to the model under consideration. Two families of schemes namely the Vertex Approximate Gradient scheme (VAG) and the Hybrid Finite Volume scheme (HFV) are detailed and shown to satisfy the gradient scheme framework, which yields, in particular, convergence. Numerical tests confirm the theoretical results.

International audienceIn this work, we extend, to two-phase flow, the single-phase Darcy flow model proposed in [26], [12] in which the (d − 1)-dimensional flow in the fractures is coupled with the d-dimensional flow in the matrix. Three types of so called hybrid-dimensional two-phase Darcy flow models are proposed. They all account for fractures acting either as drains or as barriers, since they allow pressure jumps at the matrix-fracture interfaces. The models also permit to treat gravity dominated flow as well as discontinuous capillary pressure at the material interfaces. The three models differ by their transmission conditions at matrix fracture interfaces: while the first model accounts for the nonlinear two-phase Darcy flux conservations, the second and third ones are based on the linear single phase Darcy flux conservations combined with different approximations of the mobilities. We adapt the Vertex Approximate Gradient (VAG) scheme to this problem, in order to account for anisotropy and heterogeneity aspects as well as for applicability on general meshes. Several test cases are presented to compare our hybrid-dimensional models to the generic equi-dimensional model, in which fractures have the same dimension as the matrix, leading to deep insight about the quality of the proposed reduced models

Generally, in classical reservoir studies, the geomechanical behavior of the porous medium is taken into account by the rock compressibility. Inside the reservoir simulator, the rock compressibility is assumed to be constant or to vary with the pressure of the oil phase. It induces some changes in the porosity field.During the depletion phase or the cold-water injection of highpressure/high-temperature (HP/HT) reservoirs, the stress state in and around a reservoir can change dramatically. This process might result in rock movements such as compaction, induced fracturing, and enhancement of natural fractures and/or fault activation, which continuously modify the reservoir properties such as the permeabilities and the fault transmissibilities.Modifications of such parameters strongly affect the flow pattern in the reservoir and ultimately the recovery factor.To capture the link between flow and in-situ stresses, it becomes essential to conduct coupled reservoir-geomechanical simulations.This paper compares the use of five types of approach for the reservoir simulations:• A classical approach with rock compressibility using only a reservoir simulator.• A loose coupled approach between a reservoir simulator (finite volumes) and a geomechanical simulator (finite elements). At given user-defined steps, the hydrocarbon pressures calculated by the reservoir simulator are transmitted to the geomechanical tool, which computes the actual stresses and feeds back iteratively the modifications of the petrophysical properties (porosities and permeabilities) to the reservoir simulator.• A one-way coupling: this approach is a simplification of the loose coupled approach in that the modifications are not fed back to the reservoir simulator.• A simplified approach using permeability and porosity multipliers inside a reservoir simulator. These multipliers are userdefined curves and vary with the pressure of the oil phase. This approach uses only a reservoir simulator.• A coupled approach in which the structural and the flow unknowns (displacement, pressure, and saturations) are solved simultaneously.These approaches are compared for two validation cases and two field cases described in the following.

TX 75083-3836, U.S.A., fax 01-972-952-9435. AbstractA full analysis of the problem of coupling two independent industrial simulators, a reservoir simulator and a steady-state network simulator, has been performed.

Summary Computer models of oil reservoirs have become increasingly complex in order to represent geological reality and its impact on fluid flow. Memory and CPU time limitations by finite-difference (FD)/ finite-volume (FV) simulators force a coarser resolution of reservoir models through upscaling. Upscaling can lead to significant difficulties in reservoir studies:while the fine-scale geological model is built from petrophysical, log, and seismic data, its dynamic behavior is never checked. As a result, a coarse-scale reservoir study can be linked to a fine-scale geological model, but the two might be inconsistent in their dynamic behavior.Conversely, the upscaled model cannot be properly tested because the flow and production behavior at the fine-scale level are not available. There is no reference solution for guiding important decisions for building a consistent upscaled model.A large number of sector models are required in designing optimal well patterns. Streamline simulation is now an attractive alternative to overcome some of these drawbacks because it offers substantial computational efficiency while minimizing numerical diffusion and grid-orientation effects. It allows the integration of fine-scale geological models into the reservoir engineering workflow. In this paper, we demonstrate the usefulness and efficiency of a streamline simulator in the reservoir engineering workflow. We evaluate its speed, memory requirements, and scalability using tracer and black-oil-test data sets on an SGI Origin 2000™* (250 MHz MIPS). Our data are based on real fields and range from 200,000 to 7 million cells, with cells as small as 30×30×0.5 m. Streamlines allowed us to check the validity of a large geological model and to optimize well patterns with more than 30 producers and injectors. We demonstrate how streamline-based simulation has matured from a research tool to an industrial application providing real benefits to engineers as a complementary tool to existing conventional simulation technology based on FVs. Introduction Dynamic flow simulation is still a bottleneck in most integrated reservoir studies that attempt to reconcile the geological model with seismic data and well data. Three-dimensional, high resolution (3DHR) seismic data, as well as improved 3D static modeling tools, produce models that are ever more detailed and allow significantly more faults than the previous generation of static models. Today's fine-scale models are commonly in the range of 1 to 10 million cells. On the other hand, flow simulation technology based on FVs or FDs is mature. Any improvements are expected mainly from parallel processing of key modules, such as the simultaneous solution of the linearized flow equations or PVT calculations. As a result, only relatively small dynamic models (100,000 active cells) can be considered in routine engineering studies. Dynamic flow simulation also has suffered from recent cost cutting by reserving large-scale computing power (machines with more than 1,000 processors) for seismic processing while shifting most other simulations to PC clusters with a limited number of processors (8 to 32). Upscaling fine-scale geological models remains a reality for most studies, causing significant deterioration in the geological model. In many cases, the fine-scale and coarse-scale models do not superimpose, with coarse blocks being traversed by fine-scale faults. Under realistic reservoir conditions, rigorous upscaling becomes difficult, forcing the engineer to make dubious approximations (fault location and transmissivity, layer resampling, etc.). The fact that these approximations often cannot be quantified because a fine-scale reference solution is not available makes matters worse. A methodology that allows for solutions to the original geological model is therefore desirable, allowing some quantification of errors caused by upscaling. Streamline-based reservoir flow simulation is one alternative currently available.1,2 Streamline Simulation vs. FV Simulation Streamline-based flow simulation has made significant advances in the past 10 years. Today's simulators are fully 3D1,3 and account for gravity1,4 as well as for complex well controls. Most recent advances also allow for compressible flow and compositional displacements.5,6 A number of recent publications demonstrate how streamline-based simulation is now coming into the mainstream.7–14 FV methods are based on the fundamental concept that fluids are moved from cell to cell. The problem with this methodology is an exponential increase in CPU time, with a linear increase in model size. The reason for this is that larger models dramatically reduce timestep sizes (both in implicit and explicit modes) because of reduced cell volumes and (often) increased heterogeneity. This means that locally higher fluxes have to be pushed through blocks with smaller volumes. Routine solutions of million-cell models with FV or FD technology are, therefore, out of reach for most practical applications. Even with significant simulation power, a single solution can take weeks. Data debugging and sensitivity calculations under these circumstances can become difficult. Streamline-based simulation is an attractive alternative because of the fundamentally different approach in moving fluids. Instead of moving fluids from cell to cell, streamline simulation breaks up the reservoir into 1D systems, or tubes. The transport equations are then solved along the 1D space defined by the streamlines using the concept of time of flight (TOF).15,16 By decoupling the transport problem from the underlying 3D geological model, fluids can be transported much more efficiently. Large timesteps can be taken, numerical diffusion is minimized, and CPU time varies nearly linearly with model size. Description of the Streamline Simulator Modern streamline-based simulation rests on five key principles:tracing streamlines in a velocity field;15writing the mass conservation equations in terms of TOF;16numerical solution of conservation equations along streamlines;17periodic updating of the streamlines;18,2 andoperator splitting to account for gravity.4 Details of the methodology can be found elsewhere;1 we give only a brief overview here.