A three-dimensional non-hydrostatic coupled model for free surface – Subsurface variable – Density flows

Nonlinear solutions of free surface flow under a sluice gate are studied. Upstream, the fluid is assumed to be infinitely deep, and the gate is at an angle to the horizontal axis. The flow emerges from the gate and produces a uniform flow far downstream. The problem is solved numerically by a boundary element method derived from an integral equation along the free surface. An analytical function is constructed, related to the upstream flow, so that the integral equation is solvable. As a result, a free surface flow with smooth detachment from the edge of the gate is obtained for relatively large upstream Froude numbers. As the Froude number decreases, a free surface with back flow near the edge of the gate occurs, until at a certain Froude number a stagnation point occurs. References D. D. Frangmeier, T. S. Strelkoff, Solution for gravity flow under a sluice gate, ASCE J. Engng. Mech. Div. , 94 , 1968, 153--176. B. E. Lorock, Gravity-affected flow from planer sluice gate, ASCE J. Engng. Mech. Div. , 96 , 1969, 1211--1226. Y. K. Chung, Solution of flow under sluice gates, ASCE J. Engng. Mech. Div. , 98 , 1972, 121--140. J. Asavanant, J.-M. Vanden-Broeck, Nonlinear free surface flows emerging from vessels and flows under a sluice gate, J. Austral. Math. Soc. , 38 , 1996, 63--86. doi:10.1017/S0334270000000473 J.-M. Vanden-Broeck, Numerical calculations of the free-surface flow under a sluice gate, J. Fluid Mech. , 330 , 1996, 339--347. http://journals.cambridge.org/abstract_S0022112096003849 B. J. Binder, J.-M. Vanden-Broeck, Free surface flows past surfboards and sluice gates, Eur. J. Appl. Maths. , 16 , 2005, 601--619 doi:10.1017/S0956792505006443 B. J. Binder, J.-M. Vanden-Broeck, The effect of disturbances on the flows under a sluice gate and past an inclined plate, J. Fluid Mech. , 576 , 2007, 475--490 doi:10.1017/S0022112007004806 J.-M. Vanden-Broeck, E. O. Tuck, Flow near the interaction of a free surface with a vertical wall, SIAM J. Appl. Math. , 54 , 1994, 1--33. doi:10.1137/S0036139992240838 L. H. Wiryanto, J. Widjaja, B. Supriyanto, Free-surface flow under a sluice gate from deep water, Bul. Malay. Math. Soc. , 34 , 2011, 601--609. http://www.emis.de/journals/BMMSS/vol34_3_17.html L. H. Wiryanto, E. O. Tuck, A back-turning jet formed by a uniform shallow stream hitting a vertical wall, Proc. Int. Conf. on Dif. Eqs. , Kluwer Academic Press, 1997. L. H. Wiryanto, E. O. Tuck, An open-channel flow meeting a barrier and forming one or two jets, J. Austral. Math. Soc. , 41 , 2000, 458--472. doi:10.1017/S0334270000011760 L. H. Wiryanto, Small Froude number solutions of flow caused by a line source, J. Indones. Math. Soc. , 6 , 2000, 1--6. G. C. Hocking, L. K. Forbes, A note on the flow induced by a line sink beneath a free surface, J. Austral. Math. Soc. B , 32 , 1991, 251--260. doi:10.1017/S0956792507006924 G. C. Hocking, L. K. Forbes, Subcritical free-surface flow caused by a line source in a fluid of finite depth, J. Eng. Math. , 26 , 1992, 455--466. doi:10.1007/BF00042763 L. H. Wiryanto, Zero gravity of a jet emerging from a slit, J. Indones. Math. Soc. , 12 , 2006, 89--98. L. M. Milne-Thomson, Theoretical hydrodynamics . Dover Publications, Inc., 1968, 274--281.

<p>Density-driven flows in porous media are frequently encountered in natural systems and arise from the gravitational instabilities introduced by fluid density gradients. They have significant economic and environmental impacts, and numerical modelling is often used to predict the behaviour of these flows for risk assessment, reservoir characterisation or management. However, modelling density-driven flow in porous media is very challenging due to the nonlinear coupling between flow and transport equations, the large domains of interest and the wide range of time and space scales involved. Solving this type of problem numerically using a fixed mesh can be prohibitively expensive.  Here, we apply a dynamic mesh optimisation (DMO) technique along with a control-volume-finite element method to simulate density-driven flows. DMO allows the mesh resolution and geometry to vary during a simulation to minimize an error metric for one or more solution fields of interest, refining where needed and coarsening elsewhere. We apply DMO to the Elder problem for several Rayleigh numbers. We demonstrate that DMO accurately reproduces the unique two-dimensional (2D) solutions for low Rayleigh number cases at significantly lower computational cost compared to an equivalent fixed mesh, with speedup of order x16. For unstable high Rayleigh number cases, multiple steady-state solutions exist, and we show that they are all captured by our approach with high accuracy and significantly reduced computational cost, with speedup of order x6. The lower computational cost of simulations using DMO allows extension of the high Rayleigh number case to a three-dimensional (3D) configuration and we demonstrate new steady-state solutions that have not been observed previously. Early-time, transient 3D patterns represent combinations of the previously observed, steady-state 2D solutions, but all evolve to a single, steady-state finger in the late time limit.</p>

Comparative performance of free surface and sub-surface flow systems in the phytoremediation of hydrocarbons using Scirpus grossus

Towards non-intrusive reduced order 3D free surface flow modelling

Coupling surface and subsurface flows in a depth averaged flood wave model

Direct numerical simulation of turbulent free-surface high Prandtl number fluid flows in fusion reactors

The problem is a free-surface flow of a fluid, emerging from a semi-infinite container. The fluid is assumed to be inviscid, incompressible and the flow to be two dimensional and irrotational. When surface tension is neglected the free surface leaves the wall of the container tangentially. We show that when surface tension is taken into account, there is, in general, a train of waves on the free surface and a discontinuity in slope where the free surface separates from the wall of the container. These new solutions include, as particular cases, previously obtained solutions for which the free surface is waveless in the far field. Although the calculations are presented for a special flow configuration, the results are general and apply to other potential free surface flows where a free-surface intersects a rigid wall.

A 6M digital twin for modeling and simulation in subsurface reservoirs

Advances in direct numerical simulation for MHD modeling of free surface flows

Accurate modelling of the fate of injected CO 2 is necessary if geological storage is to be used at a large scale. In one form of geological storage, CO2 is injected into an aquifer that has a sealing caprock, forming a CO2 cap beneath the caprock. The diffusion of CO2 into underlying formation waters increases the density of water near the top of the aquifer, bringing the system to a hydro-dynamically unstable state. Instabilities can arise from the combination of an unstable density profile and inherent perturbations within the system, e.g., formation heterogeneity. If created, this instability causes convective mixing and greatly accelerates the dissolution of CO2 into the aquifer. Accurate estimation of the rate of dissolution is important for risk assessments because the timescale for dissolution is the timescale over which the CO2 has a chance to leak through the caprock or any imperfectly sealed wells. A new 2D numerical model which has been developed to study the diffusive and convective mixing in geological storage of CO2 is described. Effects of different formation parameters are investigated in this paper. Results reveal that there are two different timescales involved. The first timescale is the time to onset the instability and the second one is the time to achieve ultimate dissolution. Depending on system Rayleigh number and the formation heterogeneity, convective mixing can greatly accelerate the dissolution of CO2 in an aquifer. Two field scale problems were studied. In the first, based on the Nisku aquifer, more than 60% of the ultimate dissolution was achieved after 800 years, while the computed timescale for dissolution in the same aquifer in the absence of convection was orders of magnitude larger. In the case of the Glauconitic sandstone aquifer, there was no convective instability. Results suggest that the presence and strength of convective instability should play an important role in choosing aquifers for CO2 storage. risk of leakage of CO2 from a storage formation may need to analyze leakage mechanisms and their likelihood of occurrence during the full-time period over which mobile free-phase CO 2 is expected to remain in the reservoir. Once dissolved, risk assessments may well ignore the leakage pathways resulting from the very slow movement of CO2-saturated brines. An accurate assessment of the timescales for dissolution are therefore of the first order of importance. The CO2 injected into a saline reservoir is typically 40 – 60% less dense than the resident brines (4) . Driven by density contrasts, CO 2 will flow horizontally (in a horizontal aquifer) spreading under the caprock, and flow upwards, potentially leaking through any high permeability zones or artificial penetrations, such as abandoned wells. The free-phase CO2 (usually supercritical fluid) slowly dissolves in the brines. The resulting CO2-rich brines are slightly denser than undersaturated brines, making them negatively buoyant, and thus greatly reducing or eliminating the possibility of leakage. The rate of dissolution depends on the rate at which diffusion or convection brings undersaturated brine in contact with CO2. Convective mixing enhances the dissolution rate as compared to diffusion by distributing the CO2 into the aquifer (5) . Therefore, the role of convective mixing in CO2 sequestration and the timescales involved in the process are important. The dissolution time of the injected CO2 into brine is important because during this time the injected CO2 has a chance to leak into the atmosphere through the caprock and wellbores. Accurate modelling of the convective mixing in heterogeneous porous media plays a central role in predicting the fate of CO2 injected into aquifers. In this paper, geological CO2 storage is modelled by solving the convection-diffusion equation while considering the CO2-brine interface as a boundary condition. Geochemical reactions that can reduce the timescale of sequestration of CO2 are not included, since they generally occur on longer timescales (6) . The paper is organized as follows. First, the mathematical model for simulating density-driven flow through porous media is briefly presented. The model is validated with a benchmark problem for density-driven flow in porous media. Then, the geological CO2 sequestrations both in small and field scale are simulated using the model. Two important timescales, the effect of formation properties, as well as sensitivity to temporal and spatial discretisations, are discussed. Finally, the results are summarized and their relevance to geological storage of CO 2 in aquifers is discussed.

Modeling density-driven flow in porous media by physics-informed neural networks for CO2 sequestration

Multiphase flow in porous media is involved in various natural and industrial applications, including water infiltration into soils, carbon geosequestration, and underground hydrogen storage. Understanding the invasion morphology at the pore scale is critical for better prediction of flow properties at the continuum scale in partially saturated permeable media. The deep learning method, as a promising technique to estimate the flow transport processes in porous media, has gained significant attention. However, existing works have mainly focused on single-phase flow, whereas the capability of data-driven techniques has yet to be applied to the pore-scale modeling of fluid–fluid displacement in porous media. Here, the conditional generative adversarial network is applied for pore-scale modeling of multiphase flow in two-dimensional porous media. The network is trained based on a data set of porous media generated using a particle-deposition method, with the corresponding invasion morphologies after the displacement processes calculated using a recently developed interface tracking algorithm. The results demonstrate the capability of data-driven techniques in predicting both fluid saturation and spatial distribution. It is also shown that the method can be generalized to estimate fluid distribution under different wetting conditions and particle shapes. This work represents the first effort at the application of the deep learning method for pore-scale modeling of immiscible fluid displacement and highlights the strength of data-driven techniques for surrogate modeling of multiphase flow in porous media.

Flow prediction of heterogeneous nanoporous media based on physical information neural network

In this paper we will present an algorithm to perform free surface flow simulations with the lattice Boltzmann method on adaptive grids. This reduces the required computational time by more than a factor of three for simulations with large volumes of fluid. To achieve this, the simulation of large fluid regions is performed with coarser grid resolutions. We have developed a set of rules to dynamically adapt the coarse regions to the movement of the free surface, while ensuring the consistency of all grids. Furthermore, the free surface treatment is combined with a Smagorinsky turbulence model and a technique for adaptive time steps to ensure stable simulations. The method is validated by comparing the position of the free surface with an uncoarsened simulation. It yields speedup factors of up to 3.85 for a simulation with a resolution of 4803 cells and three coarser grid levels, and thus enables efficient and stable simulations of free surface flows, e.g. for highly detailed physically based animations of fluids.

New finite elements have been developed to simulate steady and unsteady two‐dimensional free surface flows. The depth‐averaged velocity components with the free surface elevation have been used as independent variables in the model. The differences between the various elements presented lie in the choice of velocity approximation. The Newton–Raphson method has been used to solve the non‐linear system of equations. Emphasis is put on bench‐mark examples to assess the accuracy and efficiency of the elements. A simple stable new element tested herein shows promising advantages for industrial finite element codes.