Rainer Helmig

Numerical simulation of non-isothermal multiphase multicomponent processes in porous media. 1. An efficient solution technique

Modeling non-isothermal multiphase multicomponent flow and transport processes in the subsurface requires the consideration of the transfer of mass and energy between the phases in addition to the flow processes such as advection and diffusion. We developed a multidimensional numerical simulator, in which we implemented new efficient solution techniques. The description of the physical and thermodynamical state yields a system of four strongly coupled partial differential equations. The set of phases (phase state) present in the porous medium is variable in space and time. In order to take this into account, we implemented an algorithm that allows an adaptive switching of the primary variables according to each phase state. We apply a Newton algorithm for the linearization of the equations. For the solution of the arising linear equations, we extended a multigrid method and adapted it to the problem of variable phase states. First results of this new solution technique are given in the paper.

Hydraulic fracturing in unconventional gas reservoirs: risks in the geological system part 1

Hydraulic fracturing is a method used for the production of unconventional gas resources. Huge amounts of so-called fracturing fluid (10,000–20,000 m3) are injected into a gas reservoir to create fractures in solid rock formations, upon which mobilised methane fills the pore space and the fracturing fluid is withdrawn. Hydraulic fracturing may pose a threat to groundwater resources if fracturing fluid or brine can migrate through fault zones into shallow aquifers. Diffuse methane emissions from the gas reservoir may not only contaminate shallow groundwater aquifers, but also escape into the atmosphere where methane acts as a greenhouse gas. The working group “Risks in the Geological System” as part of ExxonMobil’s hydrofracking dialogue and information dissemination processes was tasked with the assessment of possible hazards posed by migrating fluids as a result of hydraulic fracturing activities. In this work, several flow paths for fracturing fluid, brine and methane are identified and scenarios are set up to qualitatively estimate under what circumstances these fluids would leak into shallower layers. The parametrisation for potential hydraulic fracturing sites in North Rhine-Westphalia and Lower Saxony (both in Germany) is derived from literature using upper and lower bounds of hydraulic parameters. The results show that a significant fluid migration is only possible if a combination of several conservative assumptions is met by a scenario.

Black-Oil Simulations for Three-Component - Three-Phase Flow in Fractured Porous Media

Discrete-fracture modeling and simulation of two-phase flow in realistic representations of fractured reservoirs can now be used for the design of IOR and EOR strategies. Thus far, however, discrete fracture simulators fail to include a third compressible gaseous phase. This hinders the investigation of the performance of gas-gravity drainage, water alternating gas injection, and blow-down in fractured reservoirs. Here we present a new numerical method that expands the capabilities of existing Black-Oil models for threecomponent – three-phase flow in three ways: (i) It utilizes a finite element - finite volume discretization generalized to unstructured hybrid element meshes. (ii) It employs higher-order accurate representations of the flux terms. (iii) Flash calculations are carried out with an improved equation of state allowing for a more realistic treatment of phase behavior. We illustrate the robustness of this numerical method in several applications. First, quasi-1D simulations are used to demonstrate grid convergence. Then, 2D discrete fracture models are employed to illustrate the impact of mesh quality on predicted production rates in discrete fracture models. Finally, the proposed method is used to simulate three-component – three-phase flow in a realistic 2D model of fractured limestone mapped in the Bristol Channel, U.K. and a 3D stochastically generated discrete fracture model.

Streamline-oriented grid generation for transport modelling in two-dimensional domains including wells

Flownets are useful tools for the visualization of groundwater flow fields. Using orthogonal flownets as grids for transport modeling is an effective way to control numerical dispersion, especially transverse to the direction of flow. Therefore tools for automatic generation of flownets may be seen both as postprocessors for groundwater flow simulations and preprocessors for contaminant transport models. Existing methods to generate streamline-oriented grids suffer from drawbacks such as the inability to include sources in the interior of the grid. In this paper, we introduce a new method for the generation of streamline-oriented grids which handles wells in the grid interior, and which produces orthogonal grids for anisotropic systems. Streamlines are generated from an accurate velocity field obtained from the solution of the mixed-hybrid finite element method for flow, while pseudopotentials, which are orthogonal to the streamlines, are obtained by a standard finite element solution of the pseudopotential equation. A comprehensive methodology for the generation of orthogonal grids, including the location of stagnation points and dividing streamlines, is introduced. The effectiveness of the method is illustrated by means of examples. A related paper presents a compatible formulation of the solution for reactive transport, while a second related paper gives a detailed quantitative assessment of the various forms of modelled mixing and their effect on the accuracy of simulations of the biodegradation of groundwater contaminants.

Numerical methods for reactive transport on rectangular and streamline-oriented grids

Coupling advection-dominated transport to reactive processes leads to additional requirements and limitations for numerical simulation beyond those for non-reactive transport. Particularly, both monotonicity avoiding the occurence of negative concentrations, and high-order accuracy suppressing artificial diffusion, are necessary to study accurately the reactive interactions of compounds transported in groundwater. These requirements are met by non-linear Eulerian methods. Two cell-centered Finite Volume schemes are presented for the simulation of advection-dominated reactive transport. The first scheme is based on rectangular grids, whereas the second scheme requires streamline-oriented grids the generation of which is explained in an accompanying paper. Although excellent results for conservative transport are obtained by the scheme for rectangular grids, some artificial transverse mixing occurs in the case of multi-component transport. This may lead to erroneous reaction rates if the compounds interact. The transport scheme for streamline-oriented grids, on the other hand, avoids artificial transverse mixing. A quantitative comparison is given by two test cases. A conservative tracer simulation for a five-spot configuration in a heterogeneous aquifer shows a high coincidence of the breakthrough curves obtained for the two methods, whereas a test case of two reacting compounds shows significant differences. In this test case, a rate of convergence with respect to the overall reaction rates lower than first-order is calculated for the rectangular grid.

Comparison of Galerkin-type discretization techniques for two-phase flow in heterogeneous porous media

A comparison of Standard Galerkin, Petrov-Galerkin, and Fully-Upwind Galerkin methods for the simulation of two-phase flow in heterogeneous porous media is presented. On the basis of the coupled pressure-saturation equations, a generalized formulation for all three finite element methods is derived and analysed. For flow in homogeneous media, the Petrov-Galerkin method gives excellent results. But this method fails miserably for problems with heterogeneous media. This is because it is not able to capture correctly processes that take place at interfaces when, for instance, the capillary pressure-saturation relationship after Brooks and Corey is assumed. The Fully-Upwind Galerkin method is superior to the Petrov-Galerkin approach because it is able to give correct results for flow in homogeneous and heterogeneous media for the two models of van Genuchten and Brooks-Corey. The widely used dpcdSw grad Sw formulation which is correct for the homogeneous case cannot be used for heterogeneous media. Instead the straightforward approach of grad pc in combination with a chord-slope technique must be utilized.

Macro-Scale Dynamic Effects in Homogeneous and Heterogeneous Porous Media

It is known that the classical capillary pressure-saturation relationship may be deficient under non-equilibrium conditions when large saturation changes may occur. An extended relationship has been proposed in the literature which correlates the rate of change of saturation to the difference between the phase pressures and the equilibrium capillary pressure. This linear relationship contains a damping coefficient, τ, that may be a function of saturation. The extended relationship is examined at the macro-scale through simulations using the two-phase simulator MUFTE-UG. In these simulations, it is assumed that the traditional equilibrium relationship between the water saturation and the difference in fluid pressures holds locally. Steady-state and dynamic “numerical experiments” are performed where a non-wetting phase displaces a wetting phase in homogeneous and heterogeneous domains with varying boundary conditions, domain size, and soil parameters. From these simulations the damping coefficient τ can be identified as a (non-linear) function of the water saturation. It is shown that the value of τ increases with an increased domain size and/or with decreased intrinsic permeability. Also, the value of τ for a domain with a spatially correlated random distribution of intrinsic permeability is compared to a homogeneous domain with equivalent permeability; they are shown to be almost equal.

Node-centered finite volume discretizations for the numerical simulation of multiphase flow in heterogeneous porous media

Three node-centered finite volume discretizations for multiphase porous media flow are presented and compared. By combination of these methods two additional discretization methods are generated. The ability of these schemes to describe flows at textural interfaces of different geologic formations is investigated. It was found that models with nonzero-entry pressures for the capillary pressure-saturation relationship in conjunction with the Box discretization may give rise to spurious oscillations for flows around low permeable lenses. Furthermore, the applicability and sensitivity of the discretization methods with regard to the used computational grids is discussed. The schemes are used for the numerical study of two-phase flow in porous media with zones of different material properties.

Multiphase flow in heterogeneous porous media: A classical finite element method versus an implicit pressure–explicit saturation‐based mixed finite element–finite volume approach

Various discretization methods exist for the numerical simulation of multiphase flow in porous media. In this paper, two methods are introduced and analyzed-a full-upwind Galerkin method which belongs to the classical finite element methods, and a mixed-hybrid finite element method based on an implicit pressure-explicit saturation (IMPES) approach. Both methods are derived from the governing equations of two-phase flow. Their discretization concepts are compared in detail. Their efficiency is discussed using several examples

Multiphase Processes in Porous Media

Models for multiphase flow in porous media are widespread today and can be found in many places in science and engineering. More complex multiphase-multicomponent models that even allow phase changes to occur need sophisticated numerical algorithms. Research in this area has been very successful with a versatile result.