Nonlinear solver for three-phase transport problems based on approximate trust regions

Implicit transport solvers used in reservoir simulation can take longer time steps than explicit solvers, but for long time steps, the commonly used Newton-Raphson’s method will often fail to converge. The convergence issues may manifest themselves as oscillating residuals even though the implicit discretization itself is stable. This behavior occurs because the fractional flow-type flux functions often change between convex and concave during long time steps, resulting in multiple contraction regions for the Newton-Raphson solver. The common strategy to overcome this is to set limits on the saturation changes during the nonlinear iteration, but such a limit has to be determined on a case by case basis, excess iterations may be required, and practical convergence is not guaranteed for a given problem. Previous work on this problem by multiple authors has resulted in solvers based on trust regions, where unconditional convergence can be obtained for incompressible two-phase flow provided a priori analytical knowledge of the flux function exists. The goal of our work is to extend this methodology to a solver where inflection points demarking the different contraction regions do not need to be explicitly known. Instead, these values are estimated during the solution process, giving improved convergence by a local computation for each interface in the simulation model. By systematically reducing updates over regions known to produce convergence issues, it is possible to greatly reduce the computational expense, making the same formulation suitable for an arbitrary number of components. We present a series of numerical results, including arbitrary time-step lengths for two and three-phase gravity segregation, as well as three-dimensional gas and water injection problems with wells and a mixture of both viscous and gravity-dominated flow regimes. The test cases are a systematic validation on a wide variety of both analytical and tabulated relative permeability curves.     

Combining the modified discrete element method with the virtual element method for fracturing of porous media

Simulation of fracturing processes in porous rocks can be divided into two main branches: (i) modeling the rock as a continuum enhanced with special features to account for fractures or (ii) modeling the rock by a discrete (or discontinuous) approach that describes the material directly as a collection of separate blocks or particles, e.g., as in the discrete element method (DEM). In the modified discrete element (MDEM) method, the effective forces between virtual particles are modified so that they reproduce the discretization of a first-order finite element method (FEM) for linear elasticity. This provides an expression of the virtual forces in terms of general Hook’s macro-parameters. Previously, MDEM has been formulated through an analogy with linear elements for FEM. We show the connection between MDEM and the virtual element method (VEM), which is a generalization of FEM to polyhedral grids. Unlike standard FEM, which computes strain-states in a reference space, MDEM and VEM compute stress-states directly in real space. This connection leads us to a new derivation of the MDEM method. Moreover, it enables a direct coupling between (M)DEM and domains modeled by a grid made of polyhedral cells. Thus, this approach makes it possible to combine fine-scale (M)DEM behavior near the fracturing region with linear elasticity on complex reservoir grids in the far-field region without regridding. To demonstrate the simulation of hydraulic fracturing, the coupled (M)DEM-VEM method is implemented using the Matlab Reservoir Simulation Toolbox (MRST) and linked to an industry-standard reservoir simulator. Similar approaches have been presented previously using standard FEM, but due to the similarities in the approaches of VEM and MDEM, our work provides a more uniform approach and extends these previous works to general polyhedral grids for the non-fracturing domain.     

Modelling a real-world buried valley system with vertical non-stationarity using multiple-point statistics

Stationarity has traditionally been a requirement of geostatistical simulations. A common way to deal with non-stationarity is to divide the system into stationary sub-regions and subsequently merge the realizations for each region. Recently, the so-called partition approach that has the flexibility to model non-stationary systems directly was developed for multiple-point statistics simulation (MPS). The objective of this study is to apply the MPS partition method with conventional borehole logs and high-resolution airborne electromagnetic (AEM) data, for simulation of a real-world non-stationary geological system characterized by a network of connected buried valleys that incise deeply into layered Miocene sediments (case study in Denmark). The results show that, based on fragmented information of the formation boundaries, the MPS partition method is able to simulate a non-stationary system including valley structures embedded in a layered Miocene sequence in a single run. Besides, statistical information retrieved from the AEM data improved the simulation of the geology significantly, especially for the deep-seated buried valley sediments where borehole information is sparse.     

Nitrogen Retention in Coastal Marine Sediments—a Field Study of the Relative Importance of Biological and Physical Removal in a Danish Estuary

The aim of this study was to elucidate the relative importance of physical versus biological loss processes for the removal of microphytobenthic (MPB) bound nitrogen in a coastal environment at different times of the year via a dual isotope labeling technique. We used ⁵¹Cr, binding to inorganic sediment particles but not participating in any biological processes, and ¹⁵N–NO₃ ^?, taken up by the MPB and turned over as part of the MPB nitrogen pool. Retention, down-mixing, and export of ¹⁵N were due to both biological and physical processes, so that by comparing retention of the two isotopes, we were able to discern the relative importance of physical and biological processes. The isotope marking was supplemented with measurements of sediment chlorophyll biomass and oxygen fluxes, allowing us to evaluate MPB biomass as well as primary production vs. respiration in the sediment. In spring/early summer, the system was characterized by tight N cycling and high N retention: any remineralized N was immediately taken up and retained in the MPB biomass. In late summer and autumn, the system was still physically stable, but high biological mediated N losses were observed. In early winter, the system was physically dominated due to low MPB biomasses and activity combined with a significant storm event. Our data support the hypothesis that the relative balance between physical and biological processes in determining retention and removal of MPB-bound nitrogen changes seasonally.     

Creaming and the Likelihood of Discovering Additional Giant Petroleum Fields

The applied context of this paper is the exploration for petroleum resources, like petroleum accumulations in different deposits, where the most promising deposits are likely to be drilled first, based on some size indicators, so-called creaming. The paper explores creaming models in the context of sampling with probabilities in proportion to size, for which a lognormal size distribution has nice analytical features. It departs from the traditional paradigm in petroleum resource assessment. Instead of conceiving a finite population being depleted over time in a decaying fashion with respect to size, the situation is studied within the framework of independent observations (infinite population) and an exploration maturity-dependent creaming factor. The theoretical and practical consequences for inference on the parent population and the probabilities and expectations linked to future discoveries are studied. The theory applies to the issue of remaining sizes of petroleum resources to be found within different future discovery horizons on the mature part of the Norwegian Continental Shelf. The aim is to obtain reasonable and useful predictions, and not to provide the best possible explanation of the exploratory behavior itself.     

Vertical Structure Of Turbulence In Offshore Flow During Rasex

The adjustment of the boundary layer immediately downstream froma coastline is examined based on two levels of eddy correlation data collected on a mast at the shore and six levels of eddy correlation data and profiles of mean variables collected from a mast 2 km offshore during the Risø Air-Sea Experiment. The characteristics of offshore flow are studied in terms of case studies and inter-variable relationships for the entire one-month data set. A turbulent kinetic energy budget is constructed for each case study.The buoyancy generation of turbulence is small compared to shear generation and dissipation. However, weakly stable and weakly unstable cases exhibit completely different vertical structure. With flow of warm air from land over cooler water, modest buoyancy destruction of turbulence and reduced shear generation of turbulence over the less rough sea surface cause the turbulence to rapidly weaken downstream from the coast. The reduction of downward mixing of momentum by the stratification leads to smaller roughness lengths compared to the unstable case. Shear generation at higher levels and advection of stronger turbulence from land often lead to an increase of stress and turbulence energy with height and downward transport of turbulence energy toward the surface.With flow of cool air over a warmer sea surface, a convective internal boundary layer develops downstream from the coast. An overlying relatively thick layer of downward buoyancy flux (virtual temperature flux) is sometimes maintained by shear generation in the accelerating offshore flow.     

Wind shear and hyperbolic distributions

A data set on wind shear, consisting of 24 frequency distributions of changes in headwind speed experienced by aircraft during landing phase, is modelled parsimoniously by means of the hyperbolic distribution.     

Aspects Of The Atmospheric Surface Layers On Mars And Earth

The structures of mean flow and turbulence in the atmospheric surface boundary layer have been extensively studied on Earth, and to a far less extent on Mars, where only the Viking missions and the Pathfinder mission have delivered in-situ data. Largely the behaviour of surface-layer turbulence and mean flow on Mars is found to obey the same scaling laws as on Earth. The largest micrometeorological differences between the two atmospheres are associated with the low air density of the Martian atmosphere. Together with the virtual absence of water vapour, it reduces the importance of the atmospheric heat flux in the surface energy budget. This increases the temperature variation of the surface forcing the near-surface temperature gradient and thereby the diabatic heat flux to higher values than are typical on the Earth, resulting in turn in a deeper daytime boundary layer. As wind speed is much like that of the Earth, this larger diabatic heat flux is carried mostly by larger maximal values of T_*, the surface scale temperature. The higher kinematic viscosity yields a Kolmogorov scale of the order of ten times larger than on Earth, influencing the transition between rough and smooth flow for the same surface features.The scaling laws have been validated analysing the Martian surface-layer data for the relations between the power spectra of wind and temperature turbulence and the corresponding mean values of wind speed and temperature. Usual spectral formulations were used based on the scaling laws ruling the Earth atmospheric surface layer, whereby the Earth's atmosphere is used as a standard for the Martian atmosphere.     

Intra‐event scale bar–bank interactions and their role in channel widening

Channel bars and banks strongly affect the morphology of both braided and meandering rivers. Accordingly, bar formation and bank erosion processes have been greatly explored. There is, however, a lack of investigations addressing the interactions between bed and bank morphodynamics, especially over short timescales. One major implication of this gap is that the processes leading to the repeated accretion of mid‐channel bars and associated widenings remain unsolved. In a restored section of the Drau River, a gravel‐bed river in Austria, mid‐channel bars have developed in a widening channel. During mean flow conditions, the bars divert the flow towards the banks. One channel section exhibited both an actively retreating bank and an expanding mid‐channel bar, and was selected to investigate the morphodynamic processes involved in bar accretion and channel widening at the intra‐event timescale. We repeatedly surveyed riverbed and riverbank topography, monitored riverbank hydrology and mounted a time‐lapse camera for continuous observation of riverbank erosion processes during four flow events. The mid‐channel bar was shown to accrete when it was submerged during flood events, which at the subsequent flow diversion during lower discharges narrowed the branch along the bank and increased the water surface elevation upstream from the riffle, which constituted the inlet into the branch. These changes of bed topography accelerated the flow along the bank and triggered bank failures up to 20 days after the flood events. Four analysed flow events exhibited a total bar expansion from initially 126 m² to 295 m², while bank retreat was 6 m at the apex of the branch. The results revealed the forcing role of bar accretion in channel widening and highlighted the importance of intra‐event scale bed morphodynamics for bank erosion, which were summarized in a conceptual model of the observed bar–bank interactions. Copyright © 2015 John Wiley & Sons, Ltd.