Three parallel algorithms for classical molecular dynamics are presented. The first assigns each processor a fixed subset of atoms; the second assigns each a fixed subset of inter-atomic forces to compute; the third assigns each a fixed spatial region. The algorithms are suitable for molecular dynamics models which can be difficult to parallelize efficiently—those with short-range forces where the neighbors of each atom change rapidly. They can be implemented on any distributed-memory parallel machine which allows for message-passing of data between independently executing processors. The algorithms are tested on a standard Lennard-Jones benchmark problem for system sizes ranging from 500 to 100,000,000 atoms on several parallel supercomputers--the nCUBE 2, Intel iPSC/860 and Paragon, and Cray T3D. Comparing the results to the fastest reported vectorized Cray Y-MP and C90 algorithm shows that the current generation of parallel machines is competitive with conventional vector supercomputers even for small problems. For large problems, the spatial algorithm achieves parallel efficiencies of 90% and a 1840-node Intel Paragon performs up to 165 faster than a single Cray C9O processor. Trade-offs between the three algorithms and guidelines for adapting them to more complex molecular dynamics simulations are also discussed.

Fast parallel algorithms for short-range molecular dynamics

This is a review of Collected Works of John Tate. Parts I, II, edited by Barry Mazur and Jean-Pierre Serre. American Mathematical Society, Providence, Rhode Island, 2016. For several decades it has been clear to the friends and colleagues of John Tate that a “Collected Works” was merited. The award of the Abel Prize to Tate in 2010 added impetus, and finally, in Tate’s ninety-second year we have these two magnificent volumes, edited by Barry Mazur and Jean-Pierre Serre. Beyond Tate’s published articles, they include five unpublished articles and a selection of his letters, most accompanied by Tate’s comments, and a collection of photographs of Tate. For an overview of Tate’s work, the editors refer the reader to [4]. Before discussing the volumes, I describe some of Tate’s work. 1. Hecke L-series and Tate’s thesis Like many budding number theorists, Tate’s favorite theorem when young was Gauss’s law of quadratic reciprocity. When he arrived at Princeton as a graduate student in 1946, he was fortunate to find there the person, Emil Artin, who had discovered the most general reciprocity law, so solving Hilbert’s ninth problem. By 1920, the German school of algebraic number theorists (Hilbert, Weber, . . .) together with its brilliant student Takagi had succeeded in classifying the abelian extensions of a number field K: to each group I of ideal classes in K, there is attached an extension L of K (the class field of I); the group I determines the arithmetic of the extension L/K, and the Galois group of L/K is isomorphic to I. Artin’s contribution was to prove (in 1927) that there is a natural isomorphism from I to the Galois group of L/K. When the base field contains an appropriate root of 1, Artin’s isomorphism gives a reciprocity law, and all possible reciprocity laws arise this way. In the 1930s, Chevalley reworked abelian class field theory. In particular, he replaced “ideals” with his “idèles” which greatly clarified the relation between the local and global aspects of the theory. For his thesis, Artin suggested that Tate do the same for Hecke L-series. When Hecke proved that the abelian L-functions of number fields (generalizations of Dirichlet’s L-functions) have an analytic continuation throughout the plane with a functional equation of the expected type, he saw that his methods applied even to a new kind of L-function, now named after him. Once Tate had developed his harmonic analysis of local fields and of the idèle group, he was able prove analytic continuation and functional equations for all the relevant L-series without Hecke’s complicated theta-formulas. Received by the editors September 5, 2016. 2010 Mathematics Subject Classification. Primary 01A75, 11-06, 14-06. c ©2017 American Mathematical Society

The collected works

Matrix-related pore networks in mudrocks are composed of nanometer- to micrometer-size pores. In shale-gas systems, these pores, along with natural fractures, form the flow-path (permeability) network that allows flow of gas from the mudrock to induced fractures during production. A pore classification consisting of three major matrix-related pore types is presented that can be used to quantify matrix-related pores and relate them to pore networks. Two pore types are associated with the mineral matrix; the third pore type is associated with organic matter (OM). Fracture pores are not controlled by individual matrix particles and are not part of this classification. Pores associated with mineral particles can be subdivided into interparticle (interP) pores that are found between particles and crystals and intraparticle (intraP) pores that are located within particles. Organic-matter pores are intraP pores located within OM. Interparticle mineral pores have a higher probability of being part of an effective pore network than intraP mineral pores because they are more likely to be interconnected. Although they are intraP, OM pores are also likely to be part of an interconnected network because of the interconnectivity of OM particles. In unlithifed near-surface muds, pores consist of interP and intraP pores, and as the muds are buried, they compact and lithify. During the compaction process, a large number of interP and intraP pores are destroyed, especially in ductile grain-rich muds. Compaction can decrease the pore volume up to 88% by several kilometers of burial. At the onset of hydrocarbon thermal maturation, OM pores are created in kerogen. At depth, dissolution of chemically unstable particles can create additional moldic intraP pores.

Spectrum of pore types and networks in mudrocks and a descriptive classification for matrix-related mudrock pores

The validity of the cubic law for laminar flow of fluids through open fractures consisting of parallel planar plates has been established by others over a wide range of conditions with apertures ranging down to a minimum of 0.2 µm. The law may be given in simplified form by Q/Δh = C(2b)3, where Q is the flow rate, Δh is the difference in hydraulic head, C is a constant that depends on the flow geometry and fluid properties, and 2b is the fracture aperture. The validity of this law for flow in a closed fracture where the surfaces are in contact and the aperture is being decreased under stress has been investigated at room temperature by using homogeneous samples of granite, basalt, and marble. Tension fractures were artificially induced, and the laboratory setup used radial as well as straight flow geometries. Apertures ranged from 250 down to 4µm, which was the minimum size that could be attained under a normal stress of 20 MPa. The cubic law was found to be valid whether the fracture surfaces were held open or were being closed under stress, and the results are not dependent on rock type. Permeability was uniquely defined by fracture aperture and was independent of the stress history used in these investigations. The effects of deviations from the ideal parallel plate concept only cause an apparent reduction in flow and may be incorporated into the cubic law by replacing C by C/ƒ. The factor ƒ varied from 1.04 to 1.65 in these investigations. The model of a fracture that is being closed under normal stress is visualized as being controlled by the strength of the asperities that are in contact. These contact areas are able to withstand significant stresses while maintaining space for fluids to continue to flow as the fracture aperture decreases. The controlling factor is the magnitude of the aperture, and since flow depends on (2b)3, a slight change in aperture evidently can easily dominate any other change in the geometry of the flow field. Thus one does not see any noticeable shift in the correlations of our experimental results in passing from a condition where the fracture surfaces were held open to one where the surfaces were being closed under stress.

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VALIDITY OF CUBIC LAW FOR FLUID FLOW IN A DEFORMABLE ROCK FRACTURE - eScholarship

The nanometer-scaled pore systems of gas shale reservoirs were investigated from the Barnett, Marcellus, Woodford, and Haynesville gas shales in the United States and the Doig Formation of northeastern British Columbia, Canada. The purpose of this article is to provide awareness of the nature and variability in pore structures within gas shales and not to provide a representative evaluation on the previously mentioned North American reservoirs. To understand the pore system of these rocks, the total porosity, pore-size distribution, surface area, organic geochemistry, mineralogy, and image analyses by electron microscopy were performed. Total porosity from helium pycnometry ranges between 2.5 and 6.6%. Total organic carbon content ranges between 0.7 and 6.8 wt. %, and vitrinite reflectance measured between 1.45 and 2.37%. The gas shales in the United States are clay and quartz rich, with the Doig Formation samples being quartz and carbonate rich and clay poor. Higher porosity samples have higher values because of a greater abundance of mesopores compared with lower porosity samples. With decreasing total porosity, micropore volumes relatively increase whereas the sum of mesopores and macropore volumes decrease. Focused ion beam milling, field emission scanning electron microscopy, and transmission electron microscopy provide high-resolution (∼5 nm) images of pore distribution and geometries. Image analysis provides a visual appreciation of pore systems in gas shale reservoirs but is not a statistically valid method to evaluate gas shale reservoirs. Macropores and mesopores are observed as either intergranular porosity or are confined to kerogen-rich aggregates and show no preferred orientation or align parallel with the laminae of the shale. Networks of mesopores are observed to connect with the larger macropores within the kerogen-rich aggregates.

Characterization of gas shale pore systems by porosimetry, pycnometry, surface area, and field emission scanning electron microscopy/transmission electron microscopy image analyses: Examples from the Barnett, Woodford, Haynesville, Marcellus, and Doig units

Gas-producing mudrock systems are playing an important role in the volatile energy industry in North America and will soon play an equally important role in Europe. Mudrocks are composed of very fine grained particles, and their pores are very small, at the scale of nanometers. Gas production from these strata is much greater than what is anticipated given their very low Darcy permeability. In this paper, images of nanopores obtained by Atomic Force Microscopy (AFM) are presented for the first time. Gas flow in nanopores cannot be described simply by the Darcy equation. Processes such as Knudsen diffusion and slip flow at the solid matrix separate gas flow behaviour from Darcy-type flow. We present a formulation for gas flow in the nanopores of mudrocks based on Knudsen diffusion and slip flow. By comparing this new gas flow formulation and Darcy flow for compressible gas, we introduce an apparent permeability term that includes the complexity of flow in nanopores, and it takes the form of the Darcy equation so that it can easily be implemented in reservoir simulators. Results show that the ratio of apparent permeability to Darcy permeability increases sharply as pore sizes reduce to smaller than 100 nm. Also, Knudsen diffusion's contributions to flow increase as pores become smaller. Unlike Darcy permeability, which is a characteristic of the rock only, permeation of gas in nanopores of mudrocks depends on rock, gas type and operating conditions.

Nanopores and Apparent Permeability of Gas Flow in Mudrocks (Shales and Siltstone)

Production of gas out of low permeability shale packages is very recent in the Western Canadian Sedimentary Basin (WCSB). The process of gas release and production from shale gas sediments is not well understood. Because of adsorptive capacity of certain shale constituents, including organic carbon content, coalbed methane models are sometimes being applied to model and simulate tight shale gas production behaviour. Alternatively, conventional Darcy flow models are sometimes applied to tight shale gas. However, neither of these approaches takes into account the differences in transport mechanisms in shale due to additional nanopore networks. Hence, the application of existing models for shale results in erroneous evaluation and predictions. Our analysis shows that a combination of a nanopore network connected to a micrometre pore network controls the gas flow in shale. Mathematical modelling of gas flow in nanopores is difficult since the standard assumption of no-slip boundary conditions in the Navier-Stokes equation breaks down at the nanometre scale, while the computational times of applicable molecular-dynamics (MD) codes become exorbitant. We found that the gas flow in nanopores of the shale can be modelled with a diffusive transport regime with a constant diffusion coefficient and negligible viscous effects. The obtained diffusion coefficient is consistent with the Knudsen diffusivity which supports the slip boundary condition at the nanopore surfaces. This model can be used for shale gas evaluation and production optimization.

Nanoscale Gas Flow in Shale Gas Sediments

We study the gas flow processes in ultra-tight porous media in which the matrix pore network is composed of nanometre- to micrometre-size pores. We formulate a pressure-dependent permeability function, referred to as the apparent permeability function (APF), assuming that Knudsen diffusion and slip flow (the Klinkenberg effect) are the main contributors to the overall flow in porous media. The APF predicts that in nanometre-size pores, gas permeability values are as much as 10 times greater than results obtained by continuum hydrodynamics predictions, and with increasing pore size (i.e. of the order of the micrometre), gas permeability converges to continuum hydrodynamics values. In addition, the APF predicts that an increase in the fractal dimension of the pore surface leads to a decrease in Knudsen diffusion. Using the homogenization method, a rigorous analysis is performed to examine whether the APF is preserved throughout the process of upscaling from local scale to large scale. We use the well-known pulse-decay experiment to estimate the main parameter of the APF, which is Darcy permeability. Our newly derived late-transient analytical solution and the late-transient numerical solution consistently match the pressure decay data and yield approximately the same estimated value for Darcy permeability at the typical core-sample initial pressure range and pressure difference. Other parameters of the APF may be determined from independent laboratory experiments; however, a pulse-decay experiment can be used to estimate the unknown parameters of the APF if multiple tests are performed and/or the parameters are strictly constrained by upper and lower bounds.

Gas flow in ultra-tight shale strata

Molecular dynamics simulations of oil transport through inorganic nanopores in shale

We present a pore network model to determine the permeability of shale gas matrix. Contrary to the conventional reservoirs, where permeability is only a function of topology and morphology of the pores, the permeability in shale depends on pressure as well. In addition to traditional viscous flow of Hagen–Poiseuille or Darcy type, we included slip flow and Knudsen diffusion in our network model to simulate gas flow in shale systems that contain pores on both micrometer and nanometer scales. This is the first network model in 3D that combines pores with nanometer and micrometer sizes with different flow physics mechanisms on both scales. Our results showed that estimated apparent permeability is significantly higher when the additional physical phenomena are considered, especially at lower pressures and in networks where nanopores dominate. We performed sensitivity analyses on three different network models with equal porosity; constant cross-section model (CCM), enlarged cross-section model (ECM) and shrunk length model (SLM). For the porous systems with variable pore sizes, the apparent permeability is highly dependent on the fraction of nanopores and the pores’ connectivity. The overall permeability in each model decreased as the fraction of nanopores increased.

Farzam Javadpour

Papers

Nanopores and Apparent Permeability of Gas Flow in Mudrocks (Shales and Siltstone)

Nanoscale Gas Flow in Shale Gas Sediments

Gas flow in ultra-tight shale strata

Molecular dynamics simulations of oil transport through inorganic nanopores in shale

Multiscale, Multiphysics Network Modeling of Shale Matrix Gas Flows