Hybrid discretization of multi-phase flow in porous media in the presence of viscous,gravitational, and capillary forces

Multi-phase flow in porous media in the presence of viscous, gravitational, and capillary forces is described by advection diffusion equations with nonlinear parameters of relative permeability and capillary pressures. The conventional numerical method employs a fully implicit finite volume formulation. The phase-potential-based upwind direction is commonly used in computing the transport terms between two adjacent cells. The numerical method, however, often experiences non-convergence in a nonlinear iterative solution due to the discontinuity of transmissibilities, especially in transition between co-current and counter-current flows. Recently, Lee et al. (Adv. Wat. Res. 82, 27–38, 2015) proposed a hybrid upwinding method for the two-phase transport equation that comprises viscous and gravitational fluxes. The viscous part is a co-current flow with a one-point upwinding based on the total velocity and the buoyancy part is modeled by a counter-current flow with zero total velocity. The hybrid scheme yields C¹-continuous discretization for the transport equation and improves numerical convergence in the Newton nonlinear solver. Lee and Efendiev (Adv. Wat. Res. 96, 209–224, 2016) extended the hybrid upwind method to three-phase flow in the presence of gravity. In this paper, we present the hybrid-upwind formula in a generalized form that describes two- and three-phase flows with viscous, gravity, and capillary forces. In the derivation of the hybrid scheme for capillarity, we note that there is a strong similarity in mathematical formulation between gravity and capillarity. We thus greatly utilize the previous derivation of the hybrid upwind scheme for gravitational force in deriving that for capillary force. Furthermore, we also discuss some mathematical issues related to heterogeneous capillary domains and propose a simple discretization model by adapting multi-valued capillary pressures at the end points of capillary pressure curves. We demonstrate this new model always admits a consistent solution that is within the discretization error. This new generalized hybrid scheme yields a discretization method that improves numerical stability in reservoir simulation.     

Numerical solution and convergence analysis of steam injection in heavy oil reservoirs

In this paper, the numerical methods for solving the problem of steam injection in the heavy oil reservoirs are presented. We consider a 3-dimensional model of 3-phase flow, oil, water, and steam, with the effect of 3-phase relative permeability. Interphase mass transfer of water and steam is considered; oil is assumed nonvolatile. We apply the simultaneous solution approach to solve the corresponding nonlinear discretized partial differential equation in the fully implicit form. The convergence of finite difference scheme is proved by the Rosinger theorem. The heuristic Jacobian-Free-Newton-Krylov (HJFNK) method is proposed for solving the system of algebraic equations. The result of this proposed numerical method is well compared with some experimental results. Our numerical results show that the first iteration of the full approximation scheme (FAS) provides a good initial guess for the Newton method. Therefore, we propose a new hybrid-FAS-HJFNK method while there is no steam in the reservoir. The numerical results show that the hybrid-FAS-HJFNK method converges faster than the HJFNK method.     

Sensitivity of the Gravity Recovery and Climate Experiment (GRACE) to the complexity of aquifer systems for monitoring of groundwater

The Gravity Recovery and Climate Experiment (GRACE) satellite mission is aimed at assessment of groundwater storage under different terrestrial conditions. The main objective of the presented study is to highlight the significance of aquifer complexity to improve the performance of GRACE in monitoring groundwater. Vidarbha region of Maharashtra, central India, was selected as the study area for analysis, since the region comprises a simple aquifer system in the western region and a complex aquifer system in the eastern region. Groundwater-level-trend analyses of the different aquifer systems and spatial and temporal variation of the terrestrial water storage anomaly were studied to understand the groundwater scenario. GRACE and its field application involve selecting four pixels from the GRACE output with different aquifer systems, where each GRACE pixel encompasses 50–90 monitoring wells. Groundwater storage anomalies (GWSA) are derived for each pixel for the period 2002 to 2015 using the Release 05 (RL05) monthly GRACE gravity models and the Global Land Data Assimilation System (GLDAS) land-surface models (GWSA_GRACE) as well as the actual field data (GWSA_Actual). Correlation analysis between GWSA_GRACE and GWSA_Actual was performed using linear regression. The Pearson and Spearman methods show that the performance of GRACE is good in the region with simple aquifers; however, performance is poorer in the region with multiple aquifer systems. The study highlights the importance of incorporating the sensitivity of GRACE in estimation of groundwater storage in complex aquifer systems in future studies.     

Groundwater sustainability and groundwater/surface-water interaction in arid Dunhuang Basin,northwest China

The Dunhuang Basin, a typical inland basin in northwestern China, suffers a net loss of groundwater and the occasional disappearance of the Crescent Lake. Within this region, the groundwater/surface-water interactions are important for the sustainability of the groundwater resources. A three-dimensional transient groundwater flow model was established and calibrated using MODFLOW 2000, which was used to predict changes to these interactions once a water diversion project is completed. The simulated results indicate that introducing water from outside of the basin into the Shule and Danghe rivers could reverse the negative groundwater balance in the Basin. River-water/groundwater interactions control the groundwater hydrology, where river leakage to the groundwater in the Basin will increase from 3,114?×?10⁴ m³/year in 2017 to 11,875?×?10⁴ m³/year in 2021, and to 17,039?×?10⁴ m³/year in 2036. In comparison, groundwater discharge to the rivers will decrease from 3277?×?10⁴ m³/year in 2017 to 1857?×?10⁴ m³/year in 2021, and to 510?×?10⁴ m³/year by 2036; thus, the hydrology will switch from groundwater discharge to groundwater recharge after implementing the water diversion project. The simulation indicates that the increased net river infiltration due to the water diversion project will raise the water table and then effectively increasing the water level of the Crescent Lake, as the lake level is contiguous with the water table. However, the regional phreatic evaporation will be enhanced, which may intensify soil salinization in the Dunhuang Basin. These results can guide the water allocation scheme for the water diversion project to alleviate groundwater depletion and mitigate geo-environmental problem.     

Characterising the vertical separation of shale-gas source rocks and aquifers across England and Wales (UK)

Shale gas is considered by many to have the potential to provide the UK with greater energy security, economic growth and jobs. However, development of a shale gas industry is highly contentious due to environmental concerns including the risk of groundwater pollution. Evidence suggests that the vertical separation between exploited shale units and aquifers is an important factor in the risk to groundwater from shale gas exploitation. A methodology is presented to assess the vertical separation between different pairs of aquifers and shales that are present across England and Wales. The application of the method is then demonstrated for two of these pairs—the Cretaceous Chalk Group aquifer and the Upper Jurassic Kimmeridge Clay Formation, and the Triassic sandstone aquifer and the Carboniferous Bowland Shale Formation. Challenges in defining what might be considered criteria for ‘safe separation’ between a shale gas formation and an overlying aquifer are discussed, in particular with respect to uncertainties in geological properties, aquifer extents and determination of socially acceptable risk levels. Modelled vertical separations suggest that the risk of aquifer contamination from shale exploration will vary greatly between shale–aquifer pairs and between regions and this will need to be considered carefully as part of the risk assessment and management for any shale gas development.     

Hybrid Gaussian-cubic radial basis functions for scattered data interpolation

Scattered data interpolation schemes using kriging and radial basis functions (RBFs) have the advantage of being meshless and dimensional independent; however, for the datasets having insufficient observations, RBFs have the advantage over geostatistical methods as the latter requires variogram study and statistical expertise. Moreover, RBFs can be used for scattered data interpolation with very good convergence, which makes them desirable for shape function interpolation in meshless methods for numerical solution of partial differential equations. For interpolation of large datasets, however, RBFs in their usual form, lead to solving an ill-conditioned system of equations, for which, a small error in the data can cause a significantly large error in the interpolated solution. In order to reduce this limitation, we propose a hybrid kernel by using the conventional Gaussian and a shape parameter independent cubic kernel. Global particle swarm optimization method has been used to analyze the optimal values of the shape parameter as well as the weight coefficients controlling the Gaussian and the cubic part in the hybridization. Through a series of numerical tests, we demonstrate that such hybridization stabilizes the interpolation scheme by yielding a far superior implementation compared to those obtained by using only the Gaussian or cubic kernels. The proposed kernel maintains the accuracy and stability at small shape parameter as well as relatively large degrees of freedom, which exhibit its potential for scattered data interpolation and intrigues its application in global as well as local meshless methods for numerical solution of PDEs.     

Convergence analysis of fixed stress split iterative scheme for anisotropic poroelasticity with tensor Biot parameter

We perform a convergence analysis of the fixed stress split iterative scheme for the Biot system modeling coupled flow and deformation in anisotropic poroelastic media with tensor Biot parameter. The fixed stress split iterative scheme solves the flow subproblem with all components of the stress tensor frozen using a multipoint flux mixed finite element method, followed by the poromechanics subproblem using a conforming Galerkin method in every coupling iteration at each time step. The coupling iterations are repeated until convergence and Backward Euler is employed for time marching. The convergence analysis is based on studying the equations satisfied by the difference of iterates to show that the fixed stress split iterative scheme for anisotropic poroelasticity with Biot tensor is contractive. We also demonstrate that the scheme is numerically convergent using the classical Mandel’s problem solution for transverse isotropy.     

A robust mesh optimisation method for multiphase porous media flows

Flows of multiple fluid phases are common in many subsurface reservoirs. Numerical simulation of these flows can be challenging and computationally expensive. Dynamic adaptive mesh optimisation and related approaches, such as adaptive grid refinement can increase solution accuracy at reduced computational cost. However, in models or parts of the model domain, where the local Courant number is large, the solution may propagate beyond the region in which the mesh is refined, resulting in reduced solution accuracy, which can never be recovered. A methodology is presented here to modify the mesh within the non-linear solver. The method allows efficient application of dynamic mesh adaptivity techniques even with high Courant numbers. These high Courant numbers may not be desired but a consequence of the heterogeneity of the domain. Therefore, the method presented can be considered as a more robust and accurate version of the standard dynamic mesh adaptivity techniques.     

Hydrogeology of a montane headwater groundwater system downgradient of a coal-mine waste rock dump: Elk Valley,British Columbia,Canada

Steelmaking-coal waste rock placed in mountain catchments in the Elk Valley, British Columbia, Canada, drain constituents of interest (CIs) to surface water downgradient of the waste rock dumps. The role of groundwater in transporting CIs in the headwaters of mountain catchments is not well understood. This study characterizes the physical hydrogeology of a portion of a 10-km² headwater catchment (West Line Creek) downgradient of a 2.7-km² waste rock dump placed over a natural headwater valley-bottom groundwater system. The study site was instrumented with 13 monitoring wells. Drill core samples were collected to determine subsurface lithology and geotechnical properties. The groundwater system was characterized using field testing and water-level monitoring. The valley-bottom sediments were composed of unconsolidated glacial and meltwater successions (<64 m thick) deposited as a series of cut and fill structures overlying shale bedrock. An unconfined basal alluvial aquifer located above fractured bedrock was identified as the primary conduit for groundwater flow toward Line Creek (650 m from the toe of the dump). Discharge through the basal alluvial aquifer was estimated using the geometric mean hydraulic conductivity (±1 standard deviation). These calculations suggest groundwater discharge could account for approximately 15% (ranging from 2 to 60%) of the total water discharged from the watershed. The residence time from the base of the waste rock dump to Line Creek was estimated at <3 years. The groundwater system was defined as a snowmelt (i.e., nival) regime dominated by direct recharge (percolation of precipitation) across the catchment.     

Impact of irrigated agriculture on groundwater-recharge salinity: a major sustainability concern in semi-arid regions

Intensive irrigated agriculture substantially modifies the hydrological cycle and often has major environmental impacts. The article focuses upon a specific concern—the tendency for progressive long-term increases in the salinity of groundwater recharge derived from irrigated permeable soils and replenishment of unconfined aquifers in more arid regions. This process has received only scant attention in the water-resource literature and has not been considered by agricultural science. This work makes an original contribution by analysing, from scientific principles, how the salinisation of groundwater recharge arises and identifies the factors affecting its severity. If not proactively managed, the process eventually will impact irrigation waterwell salinity, the productivity of agriculture itself, and can even lead to land abandonment. The types of management measure required for mitigation are discussed through three detailed case histories of areas with high-value groundwater-irrigated agriculture (in Spain, Argentina and Pakistan), which provide a long-term perspective on the evolution of the problem over various decades.