Shear induced breaking of large internal solitary waves

The stability properties of 24 experimentally generated internal solitary waves (ISWs) of extremely large amplitude, all with minimum Richardson number less than 1/4, are investigated. The study is supplemented by fully nonlinear calculations in a three-layer fluid. The waves move along a linearly stratified pycnocline (depth h2) sandwiched between a thin upper layer (depth h1) and a deep lower layer (depth h3), both homogeneous. In particular, the wave-induced velocity profile through the pycnocline is measured by particle image velocimetry (PIV) and obtained in computation. Breaking ISWs were found to have amplitudes (a1) in the range a1>2.24 √h1h2(1+h2/h1), while stable waves were on or below this limit. Breaking ISWs were investigated for 0.27 0.86 and stable waves for Lx/λ < 0.86. The results show a sort of threshold-like behaviour in terms of Lx/λ. The results demonstrate that the breaking threshold of Lx/λ = 0.86 was sharper than one based on a minimum Richardson number and reveal that the Richardson number was found to become almost antisymmetric across relatively thick pycnoclines, with the minimum occurring towards the top part of the pycnoclinePostprintPeer reviewe

Solitary wave interaction in a compact equation for deep-water gravity waves

In this study we compute numerical traveling wave solutions to a compact version of the Zakharov equation for unidirectional deep-water waves recently derived by Dyachenko & Zakharov (2011) Furthermore, by means of an accurate Fourier-type spectral scheme we find that solitary waves appear to collide elastically, suggesting the integrability of the Zakharov equation.Comment: 8 pages, 5 figures, 23 references. Other author's papers can be downloaded at http://www.lama.univ-savoie.fr/~dutykh/ . arXiv admin note: text overlap with arXiv:1204.288

Transmissibility Corrections and Grid Control for Shale Gas Numerical Production Forecasts Corrections de transmissivités et contrôle des maillages pour les simulations numériques de production en faible perméabilité

In the context of shale gas production, the very low effective permeability of the formation leads to flowing conditions that are essentially transient. Even after months or years of production, the pressure drop remains mainly localized around the hydraulic fractures. Using an unstructured grid, finite-volume simulator, we show that the non-linear nature of the pressure field around horizontal wells with multiple hydraulic fractures can have a non-negligible impact on shale gas production forecasts. We first show a very simple synthetic production example with a purely linear PVT (Pressure Volume Temperature). In this case, standard (linear) transmissibility derivations overestimate the forecast after 10 years by 5%, compared to the analytical solution. We propose a new approach for transmissibility derivations, based on numerical integrations of source point solutions. Resulting transmissibility values account for the strong non-linearity of the pressure field in the vicinity of the fractures and for fracture interferences. As a consequence, forecasts are significantly improved. With a real gas PVT, non-linear effects become even more critical in the vicinity of the well. While analytical solutions only partially account for these effects, numerical simulations are more accurate, provided that the grid is fine enough. In order to reduce the computational cost, long-term simulations are usually performed on a coarser grid, with coarse transmissibility corrections obtained from near-well upscaling techniques. We show that even if near-well numerical upscaling is extremely robust for conventional problems, the choice of an optimal simulation grid size becomes essential for shale gas. A recently proposed automatic adjustment of the grid to the considered problem (including permeability and time resolution) is tested. En contexte de roche mère, la perméabilité effective du milieu est si basse que la nature de l’écoulement demeure principalement transitoire, même après des mois ou des années de production. A l’aide d’un simulateur numérique (maillage non-structuré, volumes finis) nous montrons que la non-linéarité du champ de pression au voisinage des puits horizontaux multi fracturés doit être prise en compte lors des prévisions de production. Un premier exemple, basé sur un PVT (Pressure Volume Temperature) linéaire, montre que les transmissivités classiques (linéaires) conduisent à surestimer la production de l’ordre de 5 % par rapport à la réponse analytique. Nous proposons une nouvelle approche de calcul des transmissivités, basée sur des intégrations numériques successives des distributions de points sources. Les résultats prennent en compte la non-linéarité du champ de pression et les interférences entre fractures. La précision des prévisions de production est alors nettement améliorée. Avec un PVT de gaz réel, les effets non-linéaires deviennent encore plus critiques à proximité des puits. Alors que les solutions analytiques sont insuffisantes dans ce cas, les simulations numériques sont plus précises, à condition toutefois que les maillages utilisés soient suffisamment fins. Afin de réduire les temps de calcul, les simulations de production à long terme sont en général effectuées sur des grilles grossières dont les transmissivités sont corrigées par des méthodes éprouvées de mise à l’échelle. Nous montrons néanmoins que la taille du maillage doit être soigneusement adaptée lorsque la perméabilité est faible. Une méthode d’ajustement automatique de la grille est testée qui prend en compte la perméabilité et la résolution temporelle du problème

Transmissibility Corrections and Grid Control for Shale Gas Numerical Production Forecasts

In the context of shale gas production, the very low effective permeability of the formation leads to flowing conditions that are essentially transient. Even after months or years of production, the pressure drop remains mainly localized around the hydraulic fractures. Using an unstructured grid, finite-volume simulator, we show that the non-linear nature of the pressure field around horizontal wells with multiple hydraulic fractures can have a non-negligible impact on shale gas production forecasts. We first show a very simple synthetic production example with a purely linear PVT (Pressure Volume Temperature). In this case, standard (linear) transmissibility derivations overestimate the forecast after 10 years by 5%, compared to the analytical solution. We propose a new approach for transmissibility derivations, based on numerical integrations of source point solutions. Resulting transmissibility values account for the strong non-linearity of the pressure field in the vicinity of the fractures and for fracture interferences. As a consequence, forecasts are significantly improved. With a real gas PVT, non-linear effects become even more critical in the vicinity of the well. While analytical solutions only partially account for these effects, numerical simulations are more accurate, provided that the grid is fine enough. In order to reduce the computational cost, long-term simulations are usually performed on a coarser grid, with coarse transmissibility corrections obtained from near-well upscaling techniques. We show that even if near-well numerical upscaling is extremely robust for conventional problems, the choice of an optimal simulation grid size becomes essential for shale gas. A recently proposed automatic adjustment of the grid to the considered problem (including permeability and time resolution) is tested

Convectively induced shear instability in large amplitude internal solitary waves

Partially funded by grant no. GR/S27368/01 from EPSRCLaboratory study has been carried out to investigate the instability of an internal solitary wave of depression in a shallow stratified fluid system. The experimental campaign has been supported by theoretical computations and has focused on a two layered stratification consisting of a homogeneous dense layer below a linearly stratified top layer. The initial background stratification has been varied and it is found that the onset and intensity of breaking are affected dramatically by changes in the background stratification. Manifestations of a combination of shear and convective instability are seen on the leading face of the wave. It is shown that there is an interplay between the two instability types and convective instability induces shear by enhancing isopycnal compression. Variation in the upper boundary condition is also found to have an effect on stability. In particular, the implications for convective instability are shown to be profound and a dramatic increase in wave amplitude is seen for a fixed (as opposed to free) upper boundary condition.PostprintPeer reviewe

Dynamics of crescent water wave patterns

The nonlinear dynamics of three-dimensional instabilities of uniform gravity-wave trains evolving to crescent wave patterns is investigated numerically. A new mechanism of generation of oscillating horseshoe patterns is proposed and a detailed discussion on their occurrence in a water wave tank is given. It is suggested that these patterns are more likely to be observed naturally in water of finite depth. A critical wave steepness for the onset of three-dimensional wave breaking due to the nonlinear evolution of quintet resonant interactions corresponding to the phase-locked crescent-shaped structures (class II instability) is provided when the quartet resonant interaction (class I instability) is absent. The nonlinear coupling between quartet resonant interactions (class I instability) and quintet resonant interactions (class II instability) leading to three-dimensional breaking waves, as shown experimentally by Su & Green (1984, 1985), is numerically investigated