Random walks with negative particles for discontinuous diffusion and porosity

This study develops a new Lagrangian particle method for modeling flow and transport phenomena in complex porous media with discontinuities. For instance, diffusion processes can be modeled by Lagrangian Random Walk algorithms. However, discontinuities and heterogeneities are difficult to treat, particularly discontinuous diffusion or porosity. In the literature on particle Random Walks, previous methods used to handle this discontinuity problem can be characterized into two main classes as follows: “Interpolation techniques”, and “Partial reflection methods”. One of the main drawbacks of these methods is the small time step required in order to converge to the expected solution, particularly in the presence of many interfaces. These restrictions on the time step, lead to inefficient algorithms. The Random Walk Particle Tracking (RWPT) algorithm proposed here is, like others in the literature, discrete in time and continuous in space (gridless). We propose a novel approach to partial reflection schemes without restrictions on time step size. The new RWPT algorithm is based on an adaptive “Stop&Go” time-stepping, combined with partial reflection/refraction schemes, and extended with a new concept of negative mass particles. To test the new RWPT scheme, we develop analytical and semi-analytical solutions for diffusion in the presence of multiple interfaces (discontinuous multi-layered medium). The results show that the proposed Stop&Go RWPT scheme (with adaptive negative mass particles) fits extremely well the semi-analytical solutions, even for very high contrasts and in the neighborhood of interfaces. The scheme provides a correct diffusive solution in only a few macro-time steps, with a precision that does not depend on their size

Upscaling Fractured Heterogeneous Media: Permeability and Mass Exchange Coefficient

In order to optimize oil recuperation, to secure waste storage, CO2 sequestration and describe more precisely many environmental problems in the underground, we need to improve some homogenization methods that calculate petrophysical parameters. In this paper, we discuss the upscaling of fluid transport equations in fractured heterogeneous media consisting of the fractures themselves and a heterogeneous porous matrix. Our goal is to estimate precisely the fluid flow parameters like permeability and fracture/matrix exchange coefficient at large scale. Two approaches are possible. The first approach consists in calculating the large-scale equivalent properties in one upscaling step, starting with a single continuum flow model at the local scale. The second approach is to perform upscaling in two sequential steps: first, calculate the equivalent properties at an intermediate scale called the ”unit scale,” and, second, average the flow equations up to the large scale. We have implemented the two approaches and applied them to randomly distributed fractured systems. The results allowed us to obtain valuable information in terms of sizes of representative elementary volume associated to a given fracture distribution

An advanced general dominant eigenvalue method of accelerating successive substitution during flash calculation for compositional reservoir model

The efficiency and accuracy of phase equilibrium calculations are essential in compositional reservoir models. Usually, a significant part of the computational effort in compositional reservoir simulations is spent on phase equilibrium calculations. The nonlinear nature of phase equilibrium calculations requires an iterative solution procedure. Although the successive substitution method (SSM) is robust and simple to implement, it suffers from slow convergence, especially near the critical point of the mixture. The general dominant eigenvalue method (GDEM) has been widely used to accelerate SSM, but its stability and efficiency deteriorate as the temperature and pressure approach the critical point. This paper proposes a modified form of GDEM to improve its performance in the near-critical region. The modifications have two aspects. First, the liquid phase fraction in the mixture is added as a variable when performing GDEM acceleration, improving both stability and efficiency. The second modification is a post-calibration step imposed to replace the conventional criterion, which is applied before triggering GDEM. With the help of the post-calibration step, the stability of the modified GDEM is ensured, and more importantly, the calculation efficiency can be improved. Numerical tests of three hydrocarbon mixtures, including different numbers of components, show that the stability of the modified GDEM is almost the same as SSM and that its calculation efficiency is much higher than SSM and the conventional GDEM.Cited as: Wang, X., Wei, D., Wang, X., Zhao, X., Li, J., Noetinger, B. An advanced general dominant eigenvalue method of accelerating successive substitution during flash calculation for compositional reservoir model. Advances in Geo-Energy Research, 2022, 6(3): 241-251. https://doi.org/10.46690/ager.2022.03.0

Equivalent Hydraulic Conductivity, Connectivity and Percolation in 2D and 3D Random Binary Media

International audienceThe equivalent hydraulic conductivity () relates the spatial averages of flux and head gradient in a block of heterogeneous media. In this article, we study the influence of connectivity on of media samples composed of a high conductivity () and a low conductivity () facies. The facies is characterized by a proportion , and also by two connectivity parameters: a connectivity structure type (no, low, intermediate, high), and a correlation integral scale .The probability distribution of , and the critical value of at which percolation occurs (), are studied as a function of these connectivity parameters. The distribution of is Gaussian in all cases, so the results are presented in terms of the geometric mean () and the variance ().Both quantities show a data collapse if expressed as a function of (for the variance , notably, even if 2D and 3D data are plotted together). In 3D, when a connectivity structure exists, is always greater than when no structure exists, and increases (while decreases) as increases. The same is observed in 2D, except for the low connectivity structure type (i.e. when the facies is disconnected), that shows an unprecedented behaviour: is greater in the absence of structure, and decreases ( increases) as increases. Our results show that any influence of connectivity on is well accounted for simply by a shift in the percolation threshold , and then, suggest that is controlled mainly by the proximity to percolation

An efficient finite volume discretization to simulate flows on 3D discrete fracture network for transient flow analysis and equivalent permeability upscaling

The organization of natural fracture networks induces flow paths that control fluid flows in reservoirs. Taking into account all heterogeneities is computationally very costly, therefore, equivalent multi-porosity and multi-permeability models have to be used. We present an innovating discretization procedure allowing to simulate flow on 3D Discrete Fracture Networks involving over 100.000 fractures. We then demonstrate how to improve the computation of an equivalent permeability tensor by combining analytical and clever-meshed numerical solutions