Hussein Hoteit

Compositional Modeling by the Combined Discontinuous Galerkin and Mixed Methods

In this work, we present a numerical procedure that combines the mixed finite-element (MFE) and the discontinuous Galerkin (DG) methods. This numerical scheme is used to solve the highly nonlinear coupled equations that describe the flow processes in homogeneous and heterogeneous media with mass transfer between the phases. The MFE method is used to approximate the phase velocity based on the pressure (more precisely average pressure) at the interface between the nodes. This approach conserves the mass locally at the element level and guarantees the continuity of the total flux across the interfaces. The DG method is used to solve the mass-balance equations, which are generally convectiondominated. The DG method associated with suitable slope limiters can capture sharp gradients in the solution without creating spurious oscillations. We present several numerical examples in homogeneous and heterogeneous media that demonstrate the superiority of our method to the finite-difference (FD) approach. Our proposed MFE-DG method becomes orders of magnitude faster than the FD method for a desired accuracy in 2D.

Compositional Modeling of Discrete-Fractured Media Without Transfer Functions by the Discontinuous Galerkin and Mixed Methods

In a recent work, we introduced a numerical approach that combines the mixed-finite-element (MFE) and the discontinuous Galerkin (DG) methods for compositional modeling in homogeneous and heterogeneous porous media. In this work, we extend our numerical approach to 2D fractured media. We use the discrete-fracture model (crossflow equilibrium) to approximate the two-phase flow with mass transfer in fractured media. The discrete-fracture model is numerically superior to the single-porosity model and overcomes limitations of the dual-porosity model including the use of a shape factor. The MFE method is used to solve the pressure equation where the concept of total velocity is invoked. The DG method associated with a slope limiter is used to approximate the species-balance equations. The cell-based finitevolume schemes that are adapted to a discrete-fracture model have deficiency in computing the fracture/fracture fluxes across three and higher intersecting-fracture branches. In our work, the problem is solved definitively because of the MFE formulation. Several numerical examples in fractured media are presented to demonstrate the superiority of our approach to the classical finitedifference method.

Numerical Modeling of Diffusion in Fractured Media for Gas-Injection and -Recycling Schemes

Diffusion in fractured reservoirs, unlike in unfractured reservoirs, can affect significantly the efficiency of gas injection in oil reservoirs and recycling in gas/condensate reservoirs. The physical diffusion, similar to gravity, results in the change of the path of the injected gas species from the fractures to the matrix, giving rise to late breakthrough. In this work, we present, for the first time, a consistent model to incorporate physical diffusion of multicomponent mixtures for gas-injection schemes in fractured reservoirs. The multicomponent diffusion flux is related to multicomponent diffusion coefficients, which are dependent on temperature, pressure, and composition. These coefficients are calculated from a model based on irreversible thermodynamics. Current simulation models of fractured reservoirs that include diffusion are based on inconsistent models of gas-to-liquid diffusion at the fracture/matrix interface. We avoid this deficiency by assuming that the gas and liquid phases are in equilibrium at the interface. The concept of crossflow equilibrium (i.e., vertical equilibrium) is invoked in our model to avoid the use of transfer functions. In this work, we use the combined discontinuous Galerkin and mixed methods on 2D structured grids to calculate fluxes accurately and to have low numerical dispersion to study physical diffusion. Four examples are presented. In one of the examples, a field-scale study is performed to investigate gas injection in a fractured reservoir away from miscibility pressure and close to miscibility pressure. Results show a significant effect of diffusion on recovery performance away from miscibility pressure. In another example, recycling in a fractured gas/condensate reservoir is presented to demonstrate that diffusion has a significant effect on condensate recovery.

Understanding Shale Gas Production Mechanisms Through Reservoir Simulation

Abstract Shale gas has changed the energy equation around the world, and its impact has been especially profound in the United States. It is now generally agreed that the fabric of shale systems comprise primarily of organic matter, inorganic material and natural fractures. However, the underlying flow mechanisms through these multi-porosity, multi-permeability systems are poorly understood. For instance, debate still exists about the predominant transport mechanism (diffusion, convection and desorption) as well as the flow interactions between organic matter, inorganic matter and fractures. Furthermore balancing the computational burden of precisely modeling the gas transport through the pores versus running full reservoir scale simulation is also contested. To that end, commercial reservoir simulators are developing new shale gas options but some, for expediency, rely on simplification of existing data structures and/or flow mechanisms. We present here the development of a comprehensive multi-mechanistic (desorption, diffusion and convection) multi-porosity (organic materials, inorganic materials and fractures), multi-permeability model that uses experimentally determined shale organic and inorganic material properties to predict shale gas reservoir performance. Our multi-mechanistic model takes into account gas transport due to both pressure-driven convection and concentration-driven diffusion. The model accounts for all the important processes occurring in shale systems, including desorption of multi-component gas from the organics surface, multi-mechanistic organic-inorganic material mass transfer, multi-mechanistic inorganic material-fracture network mass transfer, and production from a hydraulically fractured wellbore. Our results show that Dual-porosity Dual-permeability (DPDP) with Knudsen diffusion is generally adequate to model shale gas reservoir production. By comparing Triple-porosity Dual-permeability (TPDP), DPDP and Single-porosity Single-permeability (SPSP) formulations under similar conditions, we show that Knudsen diffusion is a key mechanism and should not be ignored. We also guide the fractures design by analyzing flow rate limiting steps. This work provides a basis for long-term shale gas production analysis and also helps define value-adding laboratory measurements.

Numerical Simulation of Complex Reservoir Problems and the Need for a Different Line of Attack

17 We began a major research effort on production and recovery mechanisms in fractured petroleum reservoirs in 1990. By 1998, the research efforts gave us a clear understanding of various processes including reinfiltration and the effect of capillary pressure contrast. During the same period, we also made efforts to simulate laboratory flow testing. The senior author was also involved in the study of several fractured reservoirs in different parts of the world. It was gradually recognized that the available numerical-simulation models were not fit for the study of some of the fractured reservoirs, nor can they be used for some laboratory-scale problems. After an extensive literature review in various disciplines, we embarked on reservoir simulation research with the intention of developing new models for simulation of fractured reservoirs. We were convinced that a new approach is needed. In this article, after a brief review of the three approaches in reservoir simulation, results of our recent efforts in the simulation of complex reservoir problems are presented.