H12A-01
Simulation of Solute Flow and Transport in a Geostatistically Generated Fractured Porous System
Fractured aquifer systems have provided important natural resources such as petroleum, gas, water and geothermal energy and have also been recently under investigation for their suitability as storage sites for high-level nuclear waste. The resource exploitation and potential utilization have led to extensive studies aiming of understanding, characterizing and finally predicting the behavior of fractured aquifer systems. By applying a discrete model approach to study flow and transport processes, fractures are determined discretely and the effect of individual fractures can be explicitly investigated. The critical step for the discrete model is the generation of a representative fracture network since the development of flow paths within a fractured system strongly depends on its structure. The geostatistical fracture generation (GFG) developed in this study aims to create a representative fracture network, which combines the spatial structures and connectivity of a fracture network, and the statistical distribution of fracture geometries. The spatial characteristics are characterized from indicator fields, which are evaluated from fracture trace maps. A global optimization, Simulated annealing, is utilized as a generation technique and the spatial characteristics are formulated to its objective function. We apply the GFG to a case study at a Pliezhausen field block, which is a sandstone of a high fracture density. The generated fracture network from the GFG are compared with the statistically generated fracture network in term of structure and hydraulic behavior. As the GFG is based on a stochastic concept, several realizations of the same descriptions can be generated, hence, an overall behavior of the fracture-matrix system have to be investigated from various realizations which leads to a problem of computational demand. In order to overcome this problem, a streamline method for a solute transport in a fracture porous system is presented. The results obtained from the streamline simulation and from solving the transport equation are collated.
H12A-02
Two-phase flow through fractured porous media
The prediction of two-phase flows in fractured porous media is a challenging problem, because of the multiple scales that are involved and of the nonlinearity of the governing equations. The present work is based on a three-dimensional discrete description of the fracture network and of the embedding matrix. Any fracture network geometry, any type of boundary condition, and any distribution of the fracture and matrix properties can be addressed, without simplifying approximations. First, the mathematical framework for two-phase flow in fractured porous media is provided, including the transport and the constitutive equations that are eventually reformulated in dimensionless form. Dimensionless parameters and criteria are also introduced to quantify various physical regimes. In particular, an a priori criterion for the possibility of upscaling generalized Darcy's equation is devised, which is later confirmed by the numerical simulations. Then, the 3D meshing of randomly fractured media is described and the spatial and temporal discretizations of the equations and the solution algorithm are presented. The illustrative simple example of an array of infinite parallel fractures is treated analytically in order to provide a direct check of the numerical codes. A generalization for steady-state two-phase flow of the classical result of Snow for single-phase flow in networks of infinite plane fractures is given. The simulation of the flow in a closed regularly compartmented reservoir is detailed and discussed in comparison with homogenized models. Finally, illustrative cases of complex realistic situations with random fractures embedded in a permeable rock matrix are presented. The temporal evolution of the local saturations is illustrated and discussed. Then, systematic results related to the steady-state macroscale phase relative permeabilities are given as functions of saturation, for typical situations with percolating or nonpercolating fracture networks. The influence of the other parameters is briefly considered. Ref: (1) I. I. Bogdanov, V. V. Mourzenko, J.F. Thovert, P. M. Adler, Phys. Rev.E, 68, 026703-1, 2003.
H12A-03
Large-scale Discrete Fracture Network Multiphase Flow Simulations Using TOUGH2- MP
Application of discrete fracture network (DFN) modeling method has in practice been limited by the computational intensity involved in the method, because an extremely large amount of fractures needs to be investigated to accurately describe a fractured rock aquifer. However, DFN models can portray fractures and their connectivity in much greater detail than continuum models, and thus more fully represent the complex field distributions of fracture-matrix systems. Spatial distribution and connectivity of fractures in a reservoir are in general complex, difficult to simulate numerically, and demand enormous computational effort, especially for the multiphase flow simulations. In this study, we use the TOUGH2-MP code for large-scale DFN simulations. TOUGH2-MP is a parallel version of TOUGH2, which simulates nonisothermal flows of multicomponent, multiphase fluids. The code uses an integral finite difference method and unstructured grids for space discretization, which avoids any reference to a global system of coordinates, and thus offers the advantage of being applicable to regular or irregular shaped elements in 1D, 2D, and 3D systems. These features make it an easy task to discretize and represent any 2D or 3D complex fracture networks in simulation. The excellent scalability of TOUGH2-MP makes possible the simulation of DFN models involving a huge number of fractures, using a multiple CPU computer. This approach has been successfully used in large-scale unsaturated flow simulations involving hundreds of thousands of fractures and matrix blocks.
H12A-04
Multilevel Multiscale Mimetic Method for Two-Phase Flows in Porous Media
Flow simulations in porous media formations involve a wide range of strongly coupled scales. For example, the length scale of short and narrow channels is on the order of micrometers, while the size of a simulation domain may be several kilometers. In modeling two-phase flow the strongest multiscale influence arises from the heterogeneous structure of the subsurface environment. The permeability of rock formations is highly heterogeneous and may span several orders of magnitude, from nearly impermeable barriers to high-permeable flow channels. For such complex systems fully resolved simulations become computationally intractable. It is also well understood that employing simple averages of the fine-scale parameters in a model of the same form has significant limitations. The goal of multiscale modeling is to developed methods which upscale fine-scale model, not just its parameters. Many different approaches that take into account the multiscale nature of the problem have been proposed, such as the Multiscale Finite Element methods, the Multiscale Finite Volumes method, the Multilevel Upscaling method (MLUPS). All of these methods, except MLUPS, consider a two-level structure: coarse and fine scale partitions. Using a two-level structure most multiscale methods achive a coarsening factor of approximately 10 in each coordinate direction, while the trends in fine-scale realizations of large reservoirs requires a coarsening factor of 100 or more. Multilevel framework was realized in MLUPS method but this approach does not produce conservative velocity fields, which is desirable for two-phase flow simulation. In our work we propose the new Multilevel Multiscale Mimetic method (M3). This approach brings together a novel subgrid modeling algorithm for developing mimetic discretizations on coarse scales with algebraic multigrid for estimating moments of the flux on the edges. Multilevel hierarchy of coarse scale discretizations makes it very flexible and computationally efficient. Due to the algebraic nature of the method it can be naturally adopted to the different types of fine scale discretization, such as: Mixed Finite Element method, Finite Volume method, Mimetic Finite Difference method. Moreover the method can handle full permeability tensor and general types of fine and coarse scale partitions. The idea of the method consists in an algebraic transformation of a fine-scale linear system to a coarse scale. The coarsening procedure is based on approximated values of the flux moments on the edges. To estimate these moments we apply a small number of algebraic multigrid cycles to the global flow problem. The stencil of a coarse-scale discretization as well as the structure of the linear system are similar to the fine scale ones hence the same transformation may be performed recursively. The algebraic transformation is defined in such a way that velocity fields on all levels are conservative. This is the advantage over MLUPS method. The method is applied to two test cases. The first test case consists of two highly heterogeneous quarter five- spot problems in 2D. The permeability fields are generated using GSLIB library with different anisotropy angles. The second test case is 2D version of a 3D upscaling benchmark taken from 10th SPE Comparative Solution project. To preform 2D simulations we consider one of the fluvial layers of 3D reservoir model defined in SPE project which is the challenge test for the most of multiscale methods.
H12A-05
Numerical Upscaling of Relative Permeability in Fractured Porous Media
Fractured reservoir relative permeability, water-breakthrough and recovery cannot be extrapolated from core samples, but computer simulations allow their quantification using discrete fracture models at an intermediate scale. For this purpose, we represent intersecting natural and stochastically generated fractures in massive or layered porous rock with an unstructured hybrid finite element grid. We compute two-phase flow with an implicit finite element - finite volume method (FEFVM) to identify the emergent properties of this system. The results offer many important insights: Total mobility is low over a wide saturation range and very sensitive to small saturation changes. When fractures dominate the flow, but fracture porosity is low (10-4-1\(% \)), grid- block average relative permeabilities, krawg, cross-over during saturation changes of less than one percent. Such upscaled krawg yield a convex, highly dispersive fractional flow function without a shock. Its shape can not be matched with any conventional model and a new formalism based on the fracture-matrix flux ratio will be presented. Counter-current imbibition (CCI) during water flooding occurs only over a small fraction of the total fracture-matrix interface area because water imbibes only a limited number of fractures. Only in some fractures, flow will be sufficiently fast for CCI to significantly enhance recovery. CCI leads to a rate dependence of recovery and water breakthrough which occurs earlier in transient- than in steady state flow. We show how such effects can be included into a new relative permeability model for fractured porous media. http://csmp.ese.imperial.ac.uk/wiki
H12A-06
Upscaling Flow and Transport Parameters for Fracture Network Systems
The scale dependence of flow and transport parameters in fractured rock has been observed at variable scales from column experiments to field tracer tests. To test concepts of parameter scaling from bench-scale fractured- rock column experiments to fracture network characterization to kilometer-scale predictions, we have developed a multi-scale transition probability model to simulate the geometry of fracture network systems (2D at this time) and at the same time to upscale the fracture apertures and fracture block proportions. Flow and reactive solute transport in the upscaled fracture network are simulated with a novel generalized dual-porosity model (GDPM) implemented in Los Alamos National Laboratory's FEHM groundwater flow and transport simulator. The GDPM formulation provides high numerical resolution in secondary matrix nodes near the primary fracture nodes, which enables highly efficient and accurate simulations of diffusive concentration fronts moving between fractures and matrix material. In this presentation, we first describe the geometry simulation of a two-family orthogonal fracture network with an indicator Kriging method based on the multi-scale transition probability model. Then we evaluate upscaling methodologies for flow and transport in the fracture network system that account for heterogeneity in diffusion coefficients, apertures, and radionuclide sorption parameters. The developed methodologies will have broad applications for modeling flow and reactive solute transport in the fractured rocks.
H12A-07
Modeling the Permeability Anisotropy due to Reservoir-Scale Fault Damage Zones Using Dynamic Rupture Propagation: Applications to a Faulted Hydrocarbon Reservoir and the Nojiima Fault
Secondary fractures and faults associated with reservoir scale faults affect both permeability and permeability anisotropy and hence may play an important role in controlling production from a faulted reservoir. It is well known from geologic studies that there is a concentration of secondary fractures and faults in a damage zone adjacent to larger-scale faults. Because there is usually inadequate data to incorporate permeability anisotropy due to these damage zone fractures and faults into reservoir flow models, in this study we utilize the principles of dynamic rupture propagation from earthquake seismology to predict the nature of fractured/damage zones associated with reservoir scale faults. We discuss the concepts of dynamic rupture propagation and propose a workflow to model damage zones on the real field scale faults. The model we propose calculates the extent of the damage zone along the fault plane by estimating the stress perturbation associated with dynamic rupture propagation. To verify this technique we compare the modeling results of damage zone width for a reservoir scale fault with field observations. Also, we model the damage zone width associated with the Nojima Fault for the rupture that occurred in the 1996 Kobe earthquake and compare the results with the measurements on core samples from a scientific borehole drilled through the fault after the earthquake. In both the cases this technique gives a reasonable first order approximation of the damage zone width. Using fine scale simulations we show that the fractures associated with the damage zone effects the permeability distribution in both horizontal and vertical directions and defines the permeability anisotropy of the reservoir.
H12A-08
Numerical Simulation of Geomechanic Fracture Networks With Isotropic Damage and Transition to Fracture Discrete Representation
Fracture networks have a great influence on flow patterns in subsurface hydrological systems. However, these are very difficult to characterize as they are hidden under several layers of subsurface strata. Stochastic methods used to generate fracture datasets rely on sparse data and frequently generate fracture sets that not always obey mechanical laws of fracture initiation and propagation. Alternative geomechanically generated fracture datasets can be developed to generate such fracture networks based on first principle fracture mechanics. However, traditional approaches assume the matrix to be homogeneous, isotropic, and linear elastic. In this work we do not assume the matrix to be linear elastic and implement a finite element based tracking of damage by a sub-grid representation of integrity loss and strain localization. We present an integrated finite-element model which is able to transit from a continuum representation to a discrete representation of damage leading to the formation of new material domains, and the dynamic adaptation of model geometry. The fracture propagation model is based on the growth of a finite set of randomly oriented initial flaws with varying size, capturing the inelastic process zone around fracture tips with a non-linear isotropic damage model. Fracture initiation is determined via a combined Rankine-Mohr-Coulomb failure criterion, and propagation is estimated using a stress intensity-based quasi-static velocity approximation of crack propagation at the fracture tip. Fracture propagation angle is based on the maximum circumferential stress method. Straight and curved fracture geometry is generated on-the-fly by keeping track of fracture-matrix interfaces on the deformed mesh. Fracture apertures are an emergent property of the model. Mesh is adaptively refined around the tips and damaged regions to better capture stress and damage localization patterns. Fracture arrest, closure, coalescence, and intersection are handled geometrically by disallowing displacement over fracture boundaries, and merging polygonal representations of fractures. The finite element method is used to compute stresses, damage of the matrix and propagation of the discrete fractures through the matrix. We use isoparametric quadratic triangles to represent the matrix: the inside of the fractures is not meshed. Material properties are defined at the integration points. Continuous remeshing in the locality of fracture propagation facilitates fracture geometry generation, avoiding costly global remeshing used in classic approaches. This model, implemented in 2D, has been integrated into the mechanical module of the Complex Systems Platform (CSMP++).