Multiple friction tests under low load conditions: application to planar fractures in a shale reservoir
Abstract
Knowledge of the friction coefficient on fracture surfaces within tight reservoirs, geological storage systems, and the seals of conventional reservoirs, is increasingly important in the geomechanical analysis of fracturing treatment and the assessment of seal integration. We measured the static friction coefficient (µ) of 118 samples of bedding fractures and 52 samples of tectonic fractures taken from a borehole core from the depth interval of 3-4 km in the early Paleozoic Baltic Basin shale reservoir (North Poland). In total, 3372 measurements of µ were made. Based on repeated measurements under the same loading and wetting conditions, the repeatability of the data obtained was evaluated. A significant reduction of the µ value with increasing load was determined, due to the wetting of fracture surfaces. The gradient of µ value decrease caused by the increase in load was compared with predictions for realistic ranges of parameters estimated on the basis of Barton’s empirical law. For groups of fractures with similar µ values, the trends are comparable with Barton’s predictions and the gradients systematically decrease with decreasing µ. This allowed a rough extrapolation of values measured under very low-load reservoir conditions.
Revisiting the second-order convergence of the lattice Boltzmann method with reaction-type source terms
Abstract
This study analyses an approach to consistently recover the second-order convergence of the lattice Boltzmann method (LBM), which is frequently degraded by an improper discretisation of the required source terms. The current work focuses on advection-diffusion models, in which the source terms are dependent on the intensity of transported fields. Such terms can be observed in reaction-type equations used in heat and mass transfer problems or multiphase flows. The investigated scheme is applicable to a wide range of formulations within the LBM framework. All considered source terms are interpreted as contributions to the zeroth-moment of the distribution function. These account for sources in a scalar field, such as density, concentration, temperature or a phase field. Further application of this work can be found in the method of manufactured solutions or in the immersed boundary method. This paper is dedicated to three aspects regarding proper inclusion of the source term in LBM schemes. Firstly, it identifies the differences observed between the ways in which source terms are included in the LBM schemes present in the literature. The algebraic manipulations are explicitly presented in this paper to clarify the observed differences, and to identify their origin. Secondly, it analyses in full detail, the implicit relation between the value of the transported macroscopic field, and the sum of the LBM densities. This relation is valid for any source term discretization scheme. It is a crucial ingredient for preserving the second-order convergence in the case of complex source terms. Moreover, three equivalent forms of the second-order accurate collision operator are presented. Finally, closed form solutions of this implicit relation are shown for a variety of common models, including general linear and second order terms; population growth models, such as the Logistic or Gompertz model and the Allen-Cahn equation. The second-order convergence of the proposed LBM schemes is verified on both linear and non-linear source terms. The pitfalls of the commonly used acoustic and diffusive scalings are identified and discussed. Furthermore, for a simplified case, the competing errors are shown visually with isolines of error in the space of spatial and temporal resolutions.
Memory-efficient Lattice Boltzmann Method for low Reynolds number flows
Abstract
The Lattice Boltzmann Method algorithm is simplified by assuming constant numerical viscosity (the relaxation time is fixed at τ=1). This leads to the removal of the distribution function from the computer memory. To test the solver the Poiseuille and Driven Cavity flows are simulated and analyzed. The error of the solution decreases with the grid size L as L−2. Compared to the standard algorithm, the presented formulation is simpler and shorter in implementation. It is less error-prone and needs significantly less working memory in low Reynolds number flows. Our tests showed that the algorithm is less efficient in multiphase flows. To overcome this problem, further extension and the moments-only formulation was derived, inspired by the Multi-Relaxation Time (MRT) approach for single component multiphase flows.
Depth-averaged Lattice Boltzmann and Finite Element methods for single-phase flows in fractures with obstacles
Abstract
We use Lattice Boltzmann Method (LBM) MRT and Cumulant schemes to study the performance and accuracy of single-phase flow modeling for propped fractures. The simulations are run using both the two- and three-dimensional Stokes equations, and a 2.5D Stokes–Brinkman approximate model. The LBM results are validated against Finite Element Method (FEM) simulations and an analytical solution to the Stokes–Brinkman flow around an isolated circular obstacle. Both LBM and FEM 2.5D Stokes–Brinkman models are able to reproduce the analytical solution for an isolated circular obstacle. In the case of 2D Stokes and 2.5D Stokes–Brinkman models, the differences between the extrapolated fracture permeabilities obtained with LBM and FEM simulations for fractures with multiple obstacles are below 1%. The differences between the fracture permeabilities computed using 3D Stokes LBM and FEM simulations are below 8%. The differences between the 3D Stokes and 2.5 Stokes–Brinkman results are less than 7% for FEM study, and 8% for the LBM case. The velocity perturbations that are introduced around the obstacles are not fully captured by the parabolic velocity profile inherent to the 2.5D Stokes–Brinkman model.
Single component multiphase lattice boltzmann method for taylor/bretherton bubble train flow simulations
Abstract
In this study long bubble rising in a narrow channel was investigated using multiphase lattice Boltzmann method. The problem is known as a Bretherton or Taylor bubble flow [2] and is used here to verify the performance of the scheme proposed by [13]. The scheme is modified by incorporation of multiple relaxation time (MRT) collision scheme according to the original suggestion of the author. The purpose is to improve the stability of the method. The numerical simulation results show a good agreement with analytic solution provided by [2]. Moreover the convergence study demonstrates that the method achieves more than the first order of convergence. The paper investigates also the influence of simulation parameters on the interface resolution and shape.