Shallow water equations with binary porosity and their application to urban flooding

Modelling urban floods using a finite element staggered scheme with an anisotropic dual porosity model

As data analysis becomes increasingly important in the field of flood disasters, a data-based analytical model that considers hydraulic factors has emerged as an alternative to flood analysis. Data-driven modeling requires complex and extensive data that depend on actual and diverse hypothetical scenarios. Thus, preliminary data processing is required to utilize extensive input data efficiently and obtain accurate results. In this study, a gamma test was applied as a preliminary data processing technique to the efficient prediction of flood disasters in urban areas. Specifically, probability rainfall was calculated for the Daejeo and Dadae districts in Busan, and manhole overflow was estimated by inputting relevant data into a storm water management model (SWMM) with a Huff distribution. A preliminary uncertainty and sensitivity analysis was performed with the gamma test using combined time-series data for probability rainfall and manhole overflow, and then an optimal combination of data was selected for a simulation using correlativity analysis (Gamma, Gradient, Standard error, and V-ratio). The proposed method employs quantitative criteria for efficient selection of optimal input data, thus saving a greater amount of time and effort than when using the traditional method that derives results from repeated execution of models. Keywords: Data Analysis, Rainfall Scenario, Urban Flood, Gamma Test

We present a new multi-OS platform named SW2D-LEMON (SW2D for Shallow Water 2D) developed by the LEMON research team in Montpellier. SW2D-LEMON is a multi-model software focusing on shallow water-based models. It includes an unprecedented collection of upscaled (porosity) models used for shallow water equations and transport-reaction processes. Porosity models are obtained by averaging the two-dimensional shallow water equations over large areas containing both a water and a solid phase. The size of a computational cell can be increased by a factor 10–50 compared to a 2D shallow water model, with CPU times reduced by 2–3 orders of magnitude. Applications include urban flood simulations as well as flows over complex topography. Besides the standard shallow water equations (the default model), several porosity models are included in the platform: (i) Single Porosity, (ii) Dual Integral Porosity, and others are currently under development such as (iii) Depth-dependent Porosity model. Various flow processes (friction, head losses, wind, momentum diffusion, precipitation/infiltration) can be included in a modular way by activating specific execution flags. We recall here the governing equations as well as numerical aspects and present the software features. Several examples are presented to illustrate the potential of SW2D.KeywordsShallow water equationsUrban floodsSoftwareUpscalingFinite volumes

A non-linear two-dimensional vertically stratified cross-sectional model of a constant depth basin without rotation is used to investigate the influence of vertical and horizontal diffusion upon the wind-driven circulation in the basin and the associated temperature field. The influence of horizontal grid resolution, in particular the application of an irregular grid with high resolution in the coastal boundary layer is examined. The calculations show that the initial response to a wind impulse is downwelling at the downwind end of the basin with upwelling and convective mixing at the opposite end. Results from a two-layer analytical model show that the initial response is the excitation of an infinite number of internal seiche modes in order to represent the initial response which is confined to a narrow near coastal region. As time progresses, at the downwind end of the basin a density front propagates away from the boundary, with the intensity of its horizontal gradient and associated vertical velocity determined by both horizontal and vertical viscosity values. Calculations demonstrate the importance of high horizontal grid resolution in resolving this density gradient together with an accurate density advection scheme. The application of an irregular grid in the horizontal with high grid resolution in the nearshore region enables the initial response to be accurately reproduced although physically unrealistic short waves appear as the frontal region propagates onto the coarser grid. Parameterization of horizontal viscosity using a Smagorinsky-type formulation acts as a selective grid size-dependent filter, and removes the short-wave problem although enhanced smoothing can occur if the scaling coefficient in the formulation is too large. Calculations clearly show the advantages of using an irregular grid but also the importance of using a grid size-dependent filter to avoid numerical problems.

Current topographic survey technology provides high-resolution (HR) datasets for urban environments. Incorporating this HR information in models aiming to provide flood risk assessment is desirable because the flood wave propagation is depending on the urban topographic features, i.e. buildings, bridges and street networks. Conceptual, numerical and practical challenges arise from the application of shallow water models to HR urban flood modeling. For instance, numerical challenges are occurrence of wet-dry fronts, geometric discontinuities in the urban environment and discontinuous solutions, i.e. shock waves. These challenges can be overcome by using a Godunov-type scheme. However, the computational cost of this type of schemes is high, such that HR two-dimensional shallow water simulations with practical relevance have to be run on supercomputers. The porous shallow water model is an alternative approach that aims to reduce computational cost by using a coarse resolution and accounting for unresolved processes by means of the porosity terms. Usually, a speedup between two and three orders of magnitude in comparison to HR simulations can be obtained. This study reports preliminary results of a practical test case concerning pluvial flooding in a district of the city of Nice, France, caused by the intense rainfall event on October 3rd, 2015. HR topography data set on a 1 m resolution is available for the district, whereby street features of infra-metric dimensions have been included. A reference solution is calculated by a HR shallow water model on a 1 m by 1 m structured computational grid. The porous shallow water model is run on a 10 m by 10 m grid and the influence of the drag source term is studied. The model results show a large deviation, which is caused by the poor meshing strategy of the porous shallow water (AP) model. The study also summarizes practical challenges that arise during the application of the AP and HR models to a large urban catchment. The main difficulty is to obtain a good mesh. In smaller scale investigations, the mesh is currently constructed by hand such that the cell edges align with buildings. This approach is not feasible for large scale urban catchments with a large number of buildings. Future steps that have to be taken, such as a strategy for automatic mesh generation, are reported on.

A practical method to adapt fractures both micro and macro as flow enhancing properties in a single porosity model is introduced to simulate Ujung Pangkah fractured carbonate reservoir. This approach is taken because dual porosity modeling attempt fails to explain the behavior of many wells which experience early water breakthrough and/or excessive water production in Ujung Pangkah field. The enhancement factor term is used to define the degree of permeability enhancement by diffuse or micro fractures. At well location, the enhancement factor can be determined by the ratio of production test to the production of matrix-only model. The enhancement factor 3D distribution is derived from well data and seismic minimum curvature attributes as trend. Fracture corridors correspond to macro fractures in the order of meters extending vertically and/or laterally. As normally scattered spatially, fracture corridors cannot be modeled in a discrete fracture network model which is the integral part of dual porosity model. Some wells show behavior anomalies, such as rapid, early water breakthrough with excessive water production in unlikely location. It is observed that the location of these anomalies coincide with the fracture lineaments derived from the seismic incoherency attribute. As the fractures are well characterized, diffuse fractures as permeability enhancement and fracture corridors as high permeability streaks, the further improvement of history match is then easily achieved by calibrating two other key parameters; relative permeability curves and aquifer strength. The relative permeability is calibrated to the shape of fracture relative permeability. Oil rate match is greatly improved. Water rate match is achieved by placing adequate aquifer strength. Reservoir dynamic of Ujung Pangkah carbonate fractured reservoir can be simulated as a single porosity model with permeability enhancement adapted from two types of fracture distribution. Diffuse fractures enhance the overall permeability and fracture corridors dominantly influence flow dynamic in certain local area. Compared to dual porosity model, adapting fractures as permeability enhancement in single porosity model is more practical, more efficient in simulation run time – computational cost.

Shallow-water equations are widely used to model water flow in rivers, lakes, reservoirs, coastal areas, and other situations in which the water depth is much smaller than the horizontal length scale of motion. The classical shallow-water equations, the Saint-Venant system, were originally proposed about 150 years ago and still are used in a variety of applications. For many practical purposes, it is extremely important to have an accurate, efficient and robust numerical solver for the Saint-Venant system and related models. As their solutions are typically non-smooth and even discontinuous, finite-volume schemes are among the most popular tools. In this paper, we review such schemes and focus on one of the simplest (yet highly accurate and robust) methods: central-upwind schemes. These schemes belong to the family of Godunov-type Riemann-problem-solver-free central schemes, but incorporate some upwinding information about the local speeds of propagation, which helps to reduce an excessive amount of numerical diffusion typically present in classical (staggered) non-oscillatory central schemes. Besides the classical one- and two-dimensional Saint-Venant systems, we will consider the shallow-water equations with friction terms, models with moving bottom topography, the two-layer shallow-water system as well as general non-conservative hyperbolic systems.

Immediately after World War II, economic and human losses due to water disasters were enormous in Japan. In the years of 1947 and 1953, for example, economic loss reach 10% of personal income, and the number of lives lost in the 1959 Isewan Typhoon exceeded 5,000. The Japanese government then implemented successive 5-year flood control plans that dramatically reduced such disasters. Economic loss by flooding now is on the order of 0.2% of personal income, and fewer than 100 lives are lost per year. The situation has begun changing in the last decade, however, ostensibly due to global warming and local climatic change such as the heat island phenomenon. The most typical change is the increase in heavy precipitation. Meteorologists sometimes call 1997 the turning point in climate change. The year 1998 was one of extreme heavy rain in Japan, with downpours exceeding 100 mm/h occurring 10 times and those of 50 mm/h almost 440 times. The record for 50 mm/h was broken in 2004, when some 470 such downpours occurred. Another marked change has been the increase in the fluctuation of precipitation, suggesting that drought may follow floods as a typical pattern portending major water disasters in the future. Lifestyle changes are another factor inducing water disaster. Increased urban populations inherently induce concentrated land use, paving of land surfaces, hazardous living conditions, etc. Frequent urban flooding and high underground use in Japan increases the danger for inundations. Wind disasters are also increasing. In September 2006, a tornado in Nobeoka, Kyushu, killed 3 people. In November 2006, another tornado struck Saroma, Hokkaido, killing 9 workers when a construction company's meeting room was destroyed. Polluted material transported long distances by wind is a big problem in Asia. Smoke from forest fires and chemical pollutants increasingly endanger people outside of the countries of origin, spreading throughout the continent and to islands beyond. This issue reviews recent meteorological and hydrographic disasters in urban areas that threaten to become major problems in the 21st Century.

Multi-fractured horizontal wells (MFHWs) completed in low-permeability (‘tight’) oil reservoirs recover only a small fraction of the oil in place using the primary recovery scheme. The main objective of this study is to investigate various continuous injection techniques for enhanced oil recovery (EOR), including water and CO2 flooding schemes, as well as water-alternating-gas (WAG) injection, to increase production from a MFHW completed in a tight oil reservoir in Alberta, Canada. This study employs numerical compositional simulation using two model types: a single porosity (SP) model with an enhanced fracture region (EFR), and a local dual-permeability, dual-porosity (L-DP-DK) model with a limited enhanced fracture region (LEFR). The SP and L-DP-DK models incorporate laboratory-derived rock and fluid data and are utilized to history-match production data from a MFHW operated under primary depletion. Multiple history-matched models are obtained to account for variability in the core- measured matrix permeability (ranging from 30 to 300 μd). The calibrated models are then utilized to predict incremental oil recovery using continuous injection and WAG EOR schemes over a 30-year period. Finally, a sensitivity analysis is conducted to study the impact of matrix permeability, fracture half-length, and fracture geometry (i.e., tip-to-tip and parallel overlapping fractures) on the simulated incremental oil recovery. The results demonstrate that the L-DP-DK model predicts a greater oil recovery (79% higher on average) than the SP model. This increase is attributed to an improved mixing and extraction process predicted by the L-DP-DK model due to more effective communication between the fracture network and the matrix. Simulation results using the L-DP-DK reveal that the CO2-gas flooding scheme has a higher recovery than WAG and water flooding techniques. CO2-gas flooding provides the greatest incremental recovery factor (i.e., 53%) when the history-matched model includes the largest permeability (300 μd) and the smallest fracture half-length (150 ft). In addition, the L-DP-DK model predicts 24% incremental recovery using the matched model with the smallest matrix permeability (30 μd) and the largest fracture half-length (500 ft). Gas flooding provides a better recovery because the oil viscosity and surface tension are two and ten times lower, respectively, than those calculated from the WAG. Moreover, the water cycle of the WAG scheme incurs blockage, leading to a quicker gas breakthrough for up to four months, and lower sweep efficiency. Finally, the sensitivity analysis reveals that the selected completion method and assumed fracture pattern significantly influence the recovery factor, with the staggered zipper completion (parallel, overlapping fractures) providing 140% greater oil recovery compared to a tip-to-tip geometry. This study provides practical insights into the impact of model uncertainty and completion methods on the design and performance of various continuous injection and WAG EOR schemes in tight oil reservoirs. Wherever possible, customized laboratory data, such as relative permeability collected for low-permeability rock samples, have been utilized.

The exact solution to the Shallow water Equations Riemann problem at width jumps in rectangular channels

Countercurrent imbibition is an important oil recovery mechanism in fractured reservoirs. The imbibition process can be described by a nonlinear diffusion equation that has self-similar solutions. In this paper, we use the diffusion equation to conduct numerical experiments of countercurrent imbibition in a single matrix block to identify flow characteristics. The result demonstrates that the widely-used Aronofsky et al.1 exponential relationship is only valid for cases where diffusion coefficients are constant. The nonlinear diffusion equations in one and two-dimensions are analytically solved using a relative flow rate concept to simplify the analysis. The result is a new understanding regarding the effects of various parameters such as capillary pressure, relative permeability, mobility ratio, and fractional flow on countercurrent imbibition. The solutions can be directly used to modify an existing single porosity model to a dual porosity model for simulating fractured reservoirs. Results show that our approach provides fairly good accuracy compared to a full-scale dual porosity model with little increase in computation time over single porosity modeling.

Much of the hydrocarbon reserves are stored in fractured reservoirs with high heterogeneity worldwide. In such reservoirs there likely exists more than two porosity types, and transient flow likely occurs in the matrix. Dual porosity models can characterize only two porosity types and are difficult to accurately capture the transient flow. This paper discusses an advanced model that allows simulating an arbitrary number of porosity types which accurately captures the transient flow in the matrix. The proposed multiple porosity model honors any number of porosity types with different properties, such as permeability, porosity and wettability. Arbitrary connection for intra-porosity and inter-porosity flow are designed in an unstructured scheme, and convenient transformation between multiple porosity and multiple permeability formulations is allowed. A flexible porosity subdivision works in a fashion of hierarchical refinement and allows us to characterize the transient flow in the matrix and multilevel fracture sets with different geometry as well. The formulation is designed to allow the proposed scheme to be generalized. To quantify the improvement available with the new formulation, multiple porosity models are compared with single porosity models and analysis of different shape factors are also conducted. Our results show that they can be made to be equivalent under single phase conditions. Moreover, the application of the new model in shale reservoirs is discussed in two phase flow cases considering mixed wettability in shale matrix, and the results show that through a complete inter-porosity connection mode, the flow pathway of gas and water in shale is effectively identified.

<p>With the possible climate change and increased pace of urbanization in the century, urban flooding has caused more and more attentions nowadays. Shallow water equations are widely used to reproduce the flow hydrodynamics of flooding around the urban areas, which have been proved a powerful tool for flood risk assessment and evacuation management, like river flow or flowing at drainage networks with irregular cross-sections at 1D scale. Over the last two decades, Godunov-type schemes have became popular for its robustness treating complex flow phenomenons. When tacking complex topography in the framework of Godunov-type scheme, sourer term needs to be treated property to preserve steady state, that flux gradient and sourer term are balanced. Capart et al. (2003) reconstructed the momentum flux by considering the balance of hydrostatic pressure with the approximated water surface level, which has the ability to tackle the irregular and non-prismatic channel flow with complex topography. This approximation is exact for two cases: 1) rectangular and prismatic channel; 2) water surface is horizontal. However, for other cases, approximation is employed to achieve the hydrostatic equilibrium, which has reduced the accuracy of the numerical solution and increased the complexity for the model implementation. </p><p>In this work, we present a new well-balanced numerical scheme for simulating 1D frictional shallow water flow with irregular cross-sections over complex topography involving wetting and drying. The proposed scheme solves, in a finite volume Godunov-type framework, a set of pre-balanced shallow water equations derived by considering pressure balancing (Liang and Marche, 2009). HLL approximated Riemann solver is adopted for the flux calculation at the cell interface. Non-negative reconstruction of Riemann state (Audusse et al., 2004) and local bed modification (Liang, 2010) produce stable and well-balanced solutions to shallow water flow hydrodynamics. Bed slope source term can be approximated using central difference and no special treatment is needed for wet and dry bed. The friction source term is discretized using a splitting implicit scheme and limiting value of friction force is used to ensure stability for the dry bottom (Liang and Marche, 2009). The new numerical scheme is validated against two theoretical benchmark tests and then compared with the validated shallow water model with circular and trapezoid cross-sections over complex topography involving wetting and drying. This method is also possible to reproduce the mixed flow in the conduit or for the flow with non-prismatic channel like river flow in the near future.</p><p>References</p><p>Audusse, E., Bouchut, F., Bristeau, M. O., Klein, R., & Perthame, B. T. (2004). A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM Journal on Scientific Computing, 25(6), 2050-2065.</p><p>Capart, H, Eldho, TI, Huang, SY, Young, DL, and Zech, Yves, "Treatment of natural geometry in finite volume river flow computations", Journal of Hydraulic Engineering 129, 5 (2003), pp. 385--393.</p><p>Liang, Qiuhua and Marche, Fabien, "Numerical resolution of well-balanced shallow water equations with complex source terms", Advances in water resources 32, 6 (2009), pp. 873--884.</p><p>Liang, Qiuhua, "Flood simulation using a well-balanced shallow flow model", Journal of hydraulic engineering 136, 9 (2010), pp. 669--675.</p>

Hybrid steam and in-situ combustion recovery processes have shown advantages over pure steam injection for recovery of oil sands resources, particularly with respect to reducing costs and lowering requirement for water and natural gas. However, it has been very challenging to predict field performance of hybrid steam and combustion processes with a reasonable degree of confidence. Usually, a combustion front has a thickness of only a few inches and high resolution grids are required to capture the steep temperature, saturation and fluid composition gradients in the vicinity of the combustion front. Using high resolution, fine grids to improve accuracy of simulation requires excessive computation time and, therefore, may be impractical for field scale modelling. It is important to have a robust simulation tool to accurately predict reservoir performance without compromising the computational efficiency. In this work, numerical modeling of a hybrid steam and combustion recovery process was performed in a typical Athabasca oil sands reservoir. A comprehensive new reaction kinetics model derived from laboratory results was incorporated to represent the complex chemical reactions in the combustion process. The hybrid recovery process utilized oxygen enriched air co-injection after several years of SAGD operation. In the numerical model, safe limits were set on producing well temperature and oxygen content of produced fluids. The initial grid size in the numerical model was at the centimeter scale resulting in large run time, and thus, in order to improve the computational efficiency, a dynamic gridding feature was applied. Parameters for controlling the creation of a dynamic grid and subsequently reverting back to the coarse grid have been examined in order to properly trigger the dynamic gridding feature in the model. Once the optimized dynamic gridding parameters were determined, several different well configurations were investigated. Comparisons were made between SAGD and hybrid steam/combustion processes in terms of cumulative water (steam) injection, cumulative oil production, and a steam-oil ratio. By comparing the simulation results from the fine grid model and the dynamic gridding model, it has been found that the temperature gradient is the best criterion to use for controlling dynamic gridding compared to fluid saturation and/or composition criteria. The threshold value for the temperature criterion was determined to be 35°C. The model locates the fine grids in close proximity to the combustion front where the temperature and fluid saturation gradients are the steepest and it places the coarse grid blocks elsewhere in the model. Comparisons are made between the computation time and the accuracy of the simulation and these demonstrate that dynamic grid amalgamation reduces the computation time significantly while maintaining reasonable computation accuracy of simulation. Compared with SAGD, the hybrid steam/in-situ combustion process reduced cumulative water usage (steam injection) by 20% to 27%, while the cumulative oil production remained the same. This paper provides a workflow for modelling of hybrid steam and combustion processes. Also, it is expected that this work will provide insights for field design of these hybrid thermal recovery processes.

Urban areas are experiencing considerable damage from urban flooding due to climate change. Vulnerable groups (e.g., older people and infants) are susceptible to such damage, as they generally exhibit higher social vulnerability. Moreover, damage from urban floods differs in terms of type and scale based on the distribution features of local adaptation resources and characteristics of the urban environment. Therefore, the purpose of this study is to identify the major vulnerability factors along with their spatial clustering considering the social vulnerability of socio-economically vulnerable groups, as well as the urban and residential environments under increasing urban flooding. Specifically, the regional distribution features of these vulnerable groups, conditions of the urban and residential environments, and provision status of adaptation resources should be comprehensively studied. This research identified the major vulnerability factors through correlations, principal components, and cluster analysis of the considered vulnerable elements. In particular, our results show that the elderly to are the most vulnerable group. Furthermore, elderly people living in poverty require the most active adaptation measures. Improvements in land cover distribution (e.g., increases in green areas and protected areas) considering regional spatial plans and provision in adaptation resources are also required to reduce vulnerability. The results of this study can be used to assess the status of social vulnerability and improve previous flood vulnerability assessments due to climate change. We expect that consideration of social vulnerability may contribute to successful adaptation policies for urban floods.