FracRisk Reporting form for milestones

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1 FracRisk Reporting form for milestones Milestone Number: M18 Work package number: 5 Milestone title Review of dual-porosity and MINC approaches and necessary adaptions in the Open-Source simulator DuMu x Lead participant Contributing scientists and other personnel University of Göttingen Dr. Alexandru Tatomir (UGOE) Mashia Foghi (UGOE) Dr. Ulrich Maier (UGOE) Prof. Dr. Martin Sauter (UGOE) Dennis Gläser (USTUTT) Martin Beck (USTUTT) Prof. Dr. Holger Class (USTUTT) Prof. Dr. Rainer Helmig (USTUTT) Schedules delivery date from DOW Actual / forecast delivery data 12 month from the project start, Comments (optional) This milestone reviews the current MINC models available in the free open-source simulator Dumux. It further provides a list of future extensions of the simulator and of the work steps to be conducted. Submitted Reviewed Final submission Till end of the 12 th month....

2 1. DuMu X numerical simulator: Introduction DuMux (DUNE for multi-{phase, component, scale, physics, }) ( (Flemisch et al. 2011) is a free (under GPL version 2 or later) and open-source simulator for flow and transport processes in porous media. The basis of the simulator is the Distributed and Unified Numerics Environment (DUNE) cf. (Bastian et al. 2008) (Figure 1). DUNE ( is a toolbox for solving partial differential equations with grid-based methods. Applying C++ techniques, both Dumux and DUNE have a modular structure. DUNE ensures efficiency in scientific computations and supports high-performance computing applications by offering numerous modules with variety of solvers and grids. The current (May 2016) DuMu X stable release branch is 2.9 based on DUNE version 2.4. There are two standard approaches for the obtaining a solution for flow in porous media: implicit approach (described by strongly coupled equations fully-implicit); and a sequential approach (decoupled semi-implicit). The fully-implicit approach refers to discretizing the differential equations with an implicit method (i.e., backward Euler) for time while for the space discretization an appropriate method (e.g, box method) can be chosen. DuMu x modules are divided into two categories a stable module (dumux) available for download using git tool as indicated by and a development branch (dumux-devel) where unstable code is being continuously developed (available only for the developers). An overview of the modules available in the stable release branch of DuMu x v.2.9 is given in Table 1. Figure 1: Modular design of DuMu x

3 Table 1: The current stable porous media flow models available in DuMu x (v. 2.9) Branch Implicit Sequential Porous media flow 1p, 1p2c, 2p, 2p2c, 2pdfm, 2pminc, 2pnc, 2pncmin, 3p, 3p3c, co2, mpnc, richards 1p, 2p, 2p2c, DuMu x algorithmic cycle used by the fully coupled models is shown in Figure 2 and consists of four steps: 1. Main 2. Time step 3. Newton step 4. Element (Dumux handbook, 2016). Figure 2: Structure of implicit (left) and a sequential (right) scheme in DuMu x (Dumux Handbook, 2016) 2. The MINC Approach in DuMu x Two conceptual modelling approaches for flow in fractured porous media are available within DuMu x : the Discrete Fracture Model (DFM), and the Multiple Interacting Continua (MINC) method. The DFM for two-phase flow in fractured porous is the model 2pdfm, using a lower-dimensional representation of the fractures. The MINC model implementation for two-phase flow in porous media is named 2pminc. The description of 2pminc development is done in (A. Tatomir 2012; A. B. Tatomir et al. 2011). Both models belong to the fully-implicit fully coupled model category. The two numerical simulators (2pdfm, 2pminc) were validated in a number of tests as described in (A. Tatomir 2012).

4 For modelling for large-scale fractured reservoirs it is important to come with efficient practical strategies that could reduce the computational time and cost. The multiple interacting continua (MINC) method is in fact a generalization of the double porosity single-permeability model. The MINC method (Pruess and Narasimhan 1982; A. B. Tatomir et al. 2011) discretize the flow domain inside each coarse MINC block into several nested volume elements associated with an average thermodynamic state. The implemented MINC model works with two types of nested element representations: 1) equidistant nested elements and 2) constant volume nested elements. A third option is the extension of MINC with the MSR upscaling technique (Karimi-Fard, Gong, and Durlofsky 2006; Gong, Karimi-Fard, and Durlofsky 2008). For that the determination of the effective parameters of the nested volume elements from discrete fracture representations is required. This is done in two steps: determination of inner-block transmissibilities and volumetric fractions and the determination of inter-block transmissibilities. The extended MINC model (MSR-MINC) was applied to several field scale problems (e.g., idealized periodic fracture systems, Bristol fractured limestone formation, etc.). The effective parameters of the extended MINC method (MSR upscaling), as well as the reference solution are computed with the DFM. The test investigated the influence of the flow direction with respect to fracture orientation on the saturation distributions and front propagation speed. The published results showed that MINC model was capable to predict the saturation inside the matrix block considering capillary entry pressure effects, and was able to represent both the fluxes and saturation profiles obtained with the accurate discrete 2pDFM simulator. At the same time very good speedup factors were achieved. Using the MSR upscaling technique, both the matrix-fracture transfer and entry pressure effects were correctly estimated while relying on a physically based approach. 3. Future consideration for MINC modeling in DuMu x Simulator Achievements So Far: 2pminc was updated, commented following the general rules of Dumux Handbook and transferred to the stable repository. Now it is available for download in the stable version using the git tool: git clone Further Developments of the Simulator: Extension of the simulator to 3D problems. This step in general addresses only the introduction of generalized 3D-transfer functions and it is not a limitation of the finitevolume discretization technique.

5 Implementation of the hybrid nested volume elements, such as Schwarz-Christoffel mapping Development of a workflow to generate finite element meshes for 3D geometries obtained from the geostatistical fracture generator FRAC3D. This may require available commercial grid generators (e.g., COMSOL) and development of appropriate convertors to DuMu X grid formats. Development of a fully automatic MSR-MINC model. Currently the transmissibilities are calculated by solving successively individual problems. This consists of overlapping a coarse grid, extracting the geometries of the coarse-blocks, meshing, running successively DFMs, and plugging the resulting transmissibilities into the MINC model. All steps in the process require to be done automatically. Incorporation of the physical characteristics of shale gas reservoirs into the transfer functions of the nested volume elements. Several transfer functions were identified in the literature review of MINC simulators. The MINC model should be extended to incorporate processes such as Klinkenberg or gas-slippage effect, sorption of molecules in the matrix, non-darcy flow. Verification of the implementation. Benchmarking of the implemented model with available published results (e.g., Cai et al., 2014) For the investigation of frac-fluid migration through interconnected system of fractures the MINC model should be further extended from current two-phase flow to a one phase flow multi-component transport (1pnc). The first step shall be a one-phase twocomponent 1p2c. 4. References Bastian, P., M. Blatt, A. Dedner, C. Engwer, R. Klöfkorn, R. Kornhuber, M. Ohlberger, and O. Sander A Generic Grid Interface for Parallel and Adaptive Scientific Computing. Part II: Implementation and Tests in DUNE. Computing 82 (2): Cai, Long, Didier-Yu Ding, Cong Wang, and Yu-Shu Wu Accurate and Efficient Simulation of Fracture Matrix Interaction in Shale Gas Reservoirs. Transport in Porous Media 107 (2): doi: /s x. Flemisch, B., M. Darcis, K. Erbertseder, B. Faigle, A. Lauser, K. Mosthaf, S. Müthing, et al DuMux: DUNE for Multi-{phase, Component, Scale, Physics, } Flow and Transport in Porous Media. Advances in Water Resources 34 (9): doi: /j.advwatres Gong, Bin, Mohammad Karimi-Fard, and Louis Durlofsky Upscaling Discrete Fracture Characterizations to Dual-Porosity, Dual-Permeability Models for Efficient Simulation of Flow With Strong Gravitational Effects. SPE Journal 13 (1). doi: / pa. Karimi-Fard, Mohammad, Bin Gong, and J. Louis Durlofsky Generation of Coarse-Scale Continuum Flow Models from Detailed Fracture Characterizations. Water Resources Research 42 (October): W doi: /2006wr Pruess, K., and T. N Narasimhan A Practical Method for Modeling Fluid and Heat Flow in Fractured Porous Media. Society of Petroleum Engineers 25 (February):

6 Tatomir, A.B., A. Szymkiewicz, H. Class, and R. Helmig Modeling Two Phase Flow in Large Scale Fractured Porous Media with an Extended Multiple Interacting Continua Method. CMES - Computer Modeling in Engineering and Sciences 77 (2): Tatomir, Alexandru From Discrete to Continuum Concepts of Flow in Fractured Porous Media. Stuttgart University: University of Stuttgart.