Analysis of the Brinkman equation as a model for flow in porous media

The fundamental solution or Green's function for flow in porous media is determined using Stokesian dynamics, a molecular-dynamics-like simulation method capable of describing the motions and forces of hydrodynamically interacting particles in Stokes flow. By evaluating the velocity disturbance caused by a source particle on field particles located throughout a monodisperse porous medium at a given value of volume fraction of solids ø, and by considering many such realizations of the (random) porous medium, the fundamental solution is determined. Comparison of this fundamental solution with the Green's function of the Brinkman equation shows that the Brinkman equation accurately describes the flow in porous media for volume fractions below 0.05. For larger volume fractions significant differences between the two exist, indicating that the Brinkman equation has lost detailed predictive value, although it still describes qualitatively the behavior in moderately concentrated porous media. At low ø where the Brinkman equation is known to be valid, the agreement between the simulation results and the Brinkman equation demonstrates that the Stokesian dynamics method correctly captures the screening characteristic of porous media. The simulation results for ø ≥ 0.05 may be useful as a basis of comparison for future theoretical work

The spatial stability of a class of similarity solutions

The spatial stability of a class of exact similarity solutions of the Navier–Stokes equations whose longitudinal velocity is of the form xf′(y), where x is the streamwise coordinate and f′(y) is a function of the transverse, cross‐streamwise, coordinate y only, is determined. These similarity solutions correspond to the flow in an infinitely long channel or tube whose surface is either uniformly porous or moves with a velocity linear in x. Small perturbations to the streamwise velocity of the form x^λg′(y) are assumed, resulting in an eigenvalue problem for λ which is solved numerically. For the porous wall problem, it is shown that similarity solutions in which f′(y) is a monotonic function of y are spatially stable, while those that are not monotonic are spatially unstable. For the accelerating‐wall problem, the interpretation of the stability results is not unambiguous and two interpretations are offered. In one interpretation the conclusions are the same as for the porous problem—monotonic solutions are stable; the second interpretation is more restrictive in that some of the monotonic as well as the nonmonotonic solutions are unstable

Many-particle hydrodynamic interactions in parallel-wall geometry: Cartesian-representation method

This paper describes the results of our theoretical and numerical studies of hydrodynamic interactions in a suspension of spherical particles confined between two parallel planar walls, under creeping-flow conditions. We propose a novel algorithm for accurate evaluation of the many-particle friction matrix in this system--no such algorithm has been available so far. Our approach involves expanding the fluid velocity field into spherical and Cartesian fundamental sets of Stokes flows. The interaction of the fluid with the particles is described using the spherical basis fields; the flow scattered with the walls is expressed in terms of the Cartesian fundamental solutions. At the core of our method are transformation relations between the spherical and Cartesian basis sets. These transformations allow us to describe the flow field in a system that involves both the walls and particles. We used our accurate numerical results to test the single-wall superposition approximation for the hydrodynamic friction matrix. The approximation yields fair results for quantities dominated by single particle contributions, but it fails to describe collective phenomena, such as a large transverse resistance coefficient for linear arrays of spheres

Far-field approximation for hydrodynamic interactions in parallel-wall geometry

A complete analysis is presented for the far-field creeping flow produced by a multipolar force distribution in a fluid confined between two parallel planar walls. We show that at distances larger than several wall separations the flow field assumes the Hele-Shaw form, i.e., it is parallel to the walls and varies quadratically in the transverse direction. The associated pressure field is a two-dimensional harmonic function that is characterized by the same multipolar number m as the original force multipole. Using these results we derive asymptotic expressions for the Green's matrix that represents Stokes flow in the wall-bounded fluid in terms of a multipolar spherical basis. This Green's matrix plays a central role in our recently proposed algorithm [Physica A xx, {\bf xxx} (2005)] for evaluating many-body hydrodynamic interactions in a suspension of spherical particles in the parallel-wall geometry. Implementation of our asymptotic expressions in this algorithm increases its efficiency substantially because the numerically expensive evaluation of the exact matrix elements is needed only for the neighboring particles. Our asymptotic analysis will also be useful in developing hydrodynamic algorithms for wall-bounded periodic systems and implementing acceleration methods by using corresponding results for the two-dimensional scalar potential.Comment: 28 pages 5 figure

Monoslope and Multislope MUSCL Methods for unstructured meshes

International audienceWe present new MUSCL techniques associated with cell-centered Finite Volume method on triangular meshes. The first reconstruction consists in calculating a one vectorial slope per control volume by a minimization procedure with respect to a prescribed stability condition. The second technique we propose is based on the computation of three scalar slopes per triangle (one per edges) still respecting some stability condition. The resulting algorithm provides a very simple scheme which is extensible to higher dimensional problems. Numerical approximations have been performed to obtain the convergence order for the advection scalar problem whereas we treat a nonlinear vectorial example, namely the Euler system, to show the capacity of the new MUSCL technique to deal with more complexe situations

Transport in rough self-affine fractures

Transport properties of three-dimensional self-affine rough fractures are studied by means of an effective-medium analysis and numerical simulations using the Lattice-Boltzmann method. The numerical results show that the effective-medium approximation predicts the right scaling behavior of the permeability and of the velocity fluctuations, in terms of the aperture of the fracture, the roughness exponent and the characteristic length of the fracture surfaces, in the limit of small separation between surfaces. The permeability of the fractures is also investigated as a function of the normal and lateral relative displacements between surfaces, and is shown that it can be bounded by the permeability of two-dimensional fractures. The development of channel-like structures in the velocity field is also numerically investigated for different relative displacements between surfaces. Finally, the dispersion of tracer particles in the velocity field of the fractures is investigated by analytic and numerical methods. The asymptotic dominant role of the geometric dispersion, due to velocity fluctuations and their spatial correlations, is shown in the limit of very small separation between fracture surfaces.Comment: submitted to PR

Influence of Hydrodynamic Interactions on Mechanical Unfolding of Proteins

We incorporate hydrodynamic interactions in a structure-based model of ubiquitin and demonstrate that the hydrodynamic coupling may reduce the peak force when stretching the protein at constant speed, especially at larger speeds. Hydrodynamic interactions are also shown to facilitate unfolding at constant force and inhibit stretching by fluid flows.Comment: to be published in Journal of Physics: Condensed Matte

Graph Network Surrogate Model for Subsurface Flow Optimization

The optimization of well locations and controls is an important step in the design of subsurface flow operations such as oil production or geological CO2 storage. These optimization problems can be computationally expensive, however, as many potential candidate solutions must be evaluated. In this study, we propose a graph network surrogate model (GNSM) for optimizing well placement and controls. The GNSM transforms the flow model into a computational graph that involves an encoding-processing-decoding architecture. Separate networks are constructed to provide global predictions for the pressure and saturation state variables. Model performance is enhanced through the inclusion of the single-phase steady-state pressure solution as a feature. A multistage multistep strategy is used for training. The trained GNSM is applied to predict flow responses in a 2D unstructured model of a channelized reservoir. Results are presented for a large set of test cases, in which five injection wells and five production wells are placed randomly throughout the model, with a random control variable (bottom-hole pressure) assigned to each well. Median relative error in pressure and saturation for 300 such test cases is 1-2%. The ability of the trained GNSM to provide accurate predictions for a new (geologically similar) permeability realization is demonstrated. Finally, the trained GNSM is used to optimize well locations and controls with a differential evolution algorithm. GNSM-based optimization results are comparable to those from simulation-based optimization, with a runtime speedup of a factor of 36. Much larger speedups are expected if the method is used for robust optimization, in which each candidate solution is evaluated on multiple geological models

Deep Learning Framework for History Matching CO2 Storage with 4D Seismic and Monitoring Well Data

Geological carbon storage entails the injection of megatonnes of supercritical CO2 into subsurface formations. The properties of these formations are usually highly uncertain, which makes design and optimization of large-scale storage operations challenging. In this paper we introduce a history matching strategy that enables the calibration of formation properties based on early-time observations. Early-time assessments are essential to assure the operation is performing as planned. Our framework involves two fit-for-purpose deep learning surrogate models that provide predictions for in-situ monitoring well data and interpreted time-lapse (4D) seismic saturation data. These two types of data are at very different scales of resolution, so it is appropriate to construct separate, specialized deep learning networks for their prediction. This approach results in a workflow that is more straightforward to design and more efficient to train than a single surrogate that provides global high-fidelity predictions. The deep learning models are integrated into a hierarchical Markov chain Monte Carlo (MCMC) history matching procedure. History matching is performed on a synthetic case with and without 4D seismic data, which allows us to quantify the impact of 4D seismic on uncertainty reduction. The use of both data types is shown to provide substantial uncertainty reduction in key geomodel parameters and to enable accurate predictions of CO2 plume dynamics. The overall history matching framework developed in this study represents an efficient way to integrate multiple data types and to assess the impact of each on uncertainty reduction and performance predictions.Comment: 43 pages, 18 figure