Mathematical modeling of salt transport by coupled subsurface and surface water flows

A mathematical model is examined to describe the transport of salts by coupled flows of surface, soil, and subsoil waters for large-scale objects characterized by complex hydrogeologic conditions. Computational algorithms and computer programs developed for realization of the model are based on the use of finite-difference methods to distinguish between different physical processes and modeling regions. Results are presented from examples of calculations to illustrate characteristic features of the problem of mass transfer by coupled flows. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 4, pp. 118–126, July–August, 1998.     

Permeability Changes During the Evolution of a Geothermal Field Due to the Dissolution and Precipitation of Quartz

The changes in permeability and porosity associated with quartz deposition in an evolving geothermal reservoir are investigated. We review the processes associated with permeability changes in a geothermal reservoir and also review recent work on quartz solubility and deposition rates. Porosity and permeability changes are calculated for two reservoir models. The first is a generic model of a Taupo Volcanic Zone geothermal reservoir and the second is based on the model of the deep circulation system at Kakkonda published by Hanano. We find that when a reservoir experiences BPD conditions during part of its lifetime the lower reservoir becomes impermeable and a hydrothermal circulation system is only present in the upper part of the reservoir.     

Implementation of some higher-order convection schemes on non-uniform grids

A generalized formulation is applied to implement the quadratic upstream interpolation (QUICK) scheme, the second-order upwind (SOU) scheme and the second-order hybrid scheme (SHYBRID) on non-uniform grids. The implementation method is simple. The accuracy and efficiency of these higher-order schemes on non-uniform grids are assessed. Three well-known bench mark convection-diffusion problems and a fluid flow problem are revisited using non-uniform grids. These are: (1) transport of a scalar tracer by a uniform velocity field; (2) heat transport in a recirculating flow; (3) two-dimensional non-linear Burgers equations; and (4) a two-dimensional incompressible Navier-Stokes flow which is similar to the classical lid-driven cavity flow. The known exact solutions of the last three problems make it possible to thoroughly evaluate accuracies of various uniform and non-uniform grids. Higher accuracy is obtained for fewer grid points on non-uniform grids. The order of accuracy of the examined schemes is maintained for some tested problems if the distribution of non-uniform grid points is properly chosen.     

Improving the Estimations of Petrophysical Transport Behavior of Carbonate Rocks Using a Dual Pore Network Approach Combined with Computed Microtomography

Due to the intricate structure of carbonate rocks, relationships between porosity or saturation and petrophysical transport properties classically used for reservoir estimation and recovery strategies are either very complex or nonexistent. Thus, further understanding of the influence of the rock structure on the petrophysical transport properties becomes relevant. We therefore present a Dual Pore Network approach (D-PNM) applied to???-CT images of bimodal porous media. The major advantage of this method lies in the fact that it takes into account the real architecture of the connected macropore network as well as the microporosity unresolved by???-CT imaging. Whereas governing equations are solved in each individual macropore, transport behavior of microporosity is simulated by average quantities. Thus, D-PNM is particularly suited for the investigation of carbonate rocks, characterized by broad pore size distributions. We describe the principles of the image acquisition and network extraction procedure and the governing equations of D-PNM. The model is tested on three carbonate samples, two outcrop, and one reservoir carbonate. Calculated petrophysical transport properties are compared to experimental data and we show that D-PNM correctly reproduces conventional as well as unconventional electrical transport behavior. A major restriction of D-PNM is the requirement of a connected macropore network, that is, especially in the case of carbonates, not always available. Solutions to that are presented.     

Calculation of dynamic stall on an oscillating airfoil

A mathematical model of an unsteady separated flow around an oscillating airfoil is considered. This model is based on a viscid-inviscid approach. The points of separation and the intensity of vorticity displaced into the external flow are determined using boundary-layer equations in an integral form. Dynamic stall on an oscillating airfoil is studied. The mechanism and nature of antidamping are discovered. Novosibirsk State Technical University, Novosibirsk 630092. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 3, pp. 81–85, May–June, 2000.     

Instability of the quiescent state of an ideal conducting medium in a magnetic field

The linear stability of the quiescent states of an ideal compressible medium with infinite conductivity in a magnetic field is studied. It is shown by Lyapunov’s direct method that these quiescent states are unstable relative to small spatial perturbations, which decrease the potential energy (the sum of the internal energy of the medium and the energy of the magnetic field in this case). Two-sided exponential estimates of perturbation growth are obtained; the exponents in these estimates are calculated using the parameters of the quiescent states and the initial data for perturbations. A class of the most rapidly growing perturbations is separated and an exact formula to determine the rate of their increase is derived. An example is constructed of the quiescent states and the initial perturbations whose linear stage of evolution in time occurs in correspondence with the estimates. From the mathematical viewpoint, our results are preliminary, because the existence theorems for the solutions of the problems considered are not proved. Deceased. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Novosibirsk State University, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 2, pp. 148–155, March–April, 1999.     

Vortex motions of liquid in a narrow channel

Quasi-linear integrodifferential equations that describe vortex flows of an ideal incomparessible liquid in a narrow curved channel in the Eulerian-Lagrangian coordinate system are considered. The necessary and sufficient conditions for hyperbolicity of the system of equations of motion are obtained for flows with a monotonic velocity depth profile. The propagation velocities of the characteristics and the characteristic form of the system are calculated. A particular solution is given in which the system of integrodifferential equations changes type with time. The solution of the Cauchy problem is given for linearized equations. An example of initial data for which the Cauchy problem is ill-posed is constructed. Novosibirsk State University, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 4, pp. 38–49, July–August, 1998.     

Exact solutions of the one-dimensional Russo-Smereka kinetic equation

We obtain new classes of invariant solutions of the integrodifferential equations describing the propagation of nonlinear concentration waves in a rarefied bubbly fluid. For all the solutions obtained, trajectories of particle motion in phase space are calculated. The stability of some flows is studied in a linear approximation. In several cases, the construction of solutions reduces to an integrodifferential equation of the second kind, which can be solved by the iteration method. Novosibirsk State University, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 4, pp. 21–32, July–August, 2000.     

二氧化碳在砂岩透镜体中充注封存的盖层岩石抗断裂性能分析

把CO2这一主要的温室气体注入到地下深处具有适当封闭条件的地层中进行封存和隔离,已被公认为是有效减少CO2排放量的一种比较安全的技术途径。砂岩透镜体油气藏具有良好的圈闭构造和储层物性,油气濒临枯竭的砂岩透镜体是较理想的CO2地质封存箱。基于币形裂纹模型和水力致裂原理,将纵向厚度和横向展布长度均远小于盖层岩石尺度的水平产状砂岩透镜体简化为盖层岩石中的I型币形裂纹,从岩石断裂力学角度分析封存箱盖层岩石的抗断裂性能。采用叠加原理给出了盖层岩石币形裂纹尖端（砂岩透镜体尖灭部位）应力强度因子的计算公式,在此基础上提出了断裂力学判别准则（K=KIC）和临界有效压应力判别准则（P=PC）,从岩石断裂力学角度为砂岩透镜体封存箱盖层岩石抗断裂性能分析和评价提供了一种新的研究思路。     

Finite element formulation for transport equations in a mixed co-ordinate system: An application to determine temperature effects on the single-well chemical tracer test

Heterogeneous equation systems in a pair of coupled co-ordinate systems are solved by a finite element method. The specific physical application studied is the effect of temperature on single-well chemical tracer (SWCT) tests to measure residual oil saturation (volume fraction of immobile oil phase) remaining after waterflooding of an oil reservoir. Since temperature effects are caused by injecting cooler surface fluid down a well into a warm reservoir, the vertical temperature profile in the wellbore as well as the temperature distribution in the porous oil-bearing layer must be considered. The entire system is modelled to account for the different transport mechanisms. However, it is expedient to divide the connected geometrical region into two model domains. The equations for each submodel are expressed in an appropriate set of co-ordinates. The variational formulation of each model is then discussed. A significant temperature effect on the estimation of residual oil saturation occurs when the radial temperature and concentration wave propagation speeds in the porous formation are about the same. In this case the temperature gradient is located across the chemical tracer bank, causing the chemical reaction rate to vary radially. The temperature effects are demonstrated for two actual field tests in complex reservoirs.     

Combining TOUGH2 and FLAC3D to Solve Problems in Underground Gas Storage

This paper describes modeling studies assessing the feasibility of increasing the maximum storage pressure in several underground natural gas storage reservoirs. This required an assessment of the potential for gas transport in the caprock and the geomechanical response to pressure change in the storage reservoir. To solve this problem in an efficient manner, two-phase flow (TOUGH2) and geomechanical (FLAC3D) models were combined in series. The TOUGH2 model was calibrated to fit pressure data collected on-site, from both the reservoir and caprock. The mechanical response of the caprock to increased storage pressure was modeled using FLAC3D, allowing assessment of the induced stresses in formations surrounding the reservoirs. We focused on two sites. In the first, field data were obtained from a deep borehole above the gas reservoir, which provided indirect observations of the geomechanical response of the caprock to pressure changes in the reservoir. In the second, open boreholes intersecting two thin caprock units immediately above the reservoir allowed gas flow to a shallower unit, significantly impacting the modeled fracture gradient.     

Investigation of a turbulent boundary layer on a hypersonic aircraft model

An algorithm for calculation of a spatial compressible turbulent boundary layer on the surface of a pointed body is developed. The algorithm is based on the numerical solution of three-dimensional equations and algebraic models of turbulence. The flow around a hypersonic aircraft model is calculated, and the resultant Stanton numbers are compared with experimental data. The influence of the Mach number, the angle of attack, and the Reynolds number on the boundary-layer parameters is studied. It is shown that the change in the location of the transition zone has a weak effect on the skin-friction coefficient in the region of developed turbulent flow. Institute of Theoretical and Applied Mechanics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090.¹Technical University, Delft, the Netherlands. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 1, pp. 115–125, January–February, 1999.     

Ellipticity conditions of the static equations of nonlinear elasticity

The relations of the nonlinear model of the theory of elasticity are considered. The Cauchy and the strain gradient tensors are taken to be the characteristics of the stress-strain state of a body. Sufficient conditions under which the static equations of elasticity are of elliptic type are established. These conditions are expressed in the form of constraints imposed on the derivatives of the elastic potential with respect to the strain-measure characteristics. The cases of anisotropic and isotropic bodies are treated, including the case where the Almansi tensor is taken to be the strain measure. The plane strain of a body is investigated using actual-state variables. Novosibirsk State University, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 2, pp. 196–203, March–April, 1999.     

Stability of non-newtonian fluid flows

The stability of non-Newtonian fluid films moving on inclined planes is studied within the framework of the two-parameter Ostwald-de Waele model taking into account surface tension and van der Waals forces. The problem is solved analytically in the linear formulation, and the evolution of finite-amplitude perturbations is determined numerically. Novosibirsk Military Institute, Novosibirsk 630117. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 3, pp. 75–80, May–June, 2000.     

Examination of wall damping for the k-ε turbulence model using direct simulations of turbulent channel flow

Handler, Hendricks and Leighton have recently reported results for the direct numerical simulation (DNS) of a turbulent channel flow at moderate Reynolds number. These data are used to evaluate the terms in the exact and modelled transport equations for the turbulence kinetic energy k and the isotropic dissipation function ε. Both modelled transport equations show significant imbalances in the high-shear region near the channel walls. The model for the eddy viscosity is found to yield distributions for the production terms which do not agree well with the distributions calculated from the DNS data. The source of the imbalance is attributed to the wall-damping function required in eddy viscosity models for turbulent flows near walls. Several models for the damping function are examined, and it is found that the models do not vary across the channel as does the damping when evaluated from the DNS data. The Lam-Bremhorst model and the standard van Driest model are found to give reasonable agreement with the DNS data. Modification of the van Driest model to include an effective origin yields very good agreement between the modelled production and the production calculated from the DNS data, and the imbalance in the modelled transport equations is significantly reduced.     

Microconvection in a vertical layer

In the case of very weak gravity the classical Oberbeck-Boussinesq approximation is not valid for describing thermal gravitational convection. This was pointed out in [1] where a new model was proposed under the assumption that the fluid is isothermal and incompressible. In this model the velocity vector is no longer solenoidal. Below, on the basis of this model we analyze the convective motion in a vertical layer, on the rigid boundaries of which a heat flux that depends on time only is prescribed. It is found that the nonsolenoidal character of the velocity does not lead to considerable restructuring of the steady-state convection. At the same time, the patterns of the unsteady, in particular, periodic convective flow calculated within the framework of the classical and the new models differ significantly.Novosibirsk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 76–84, September–October, 1994.     

Three-dimensional mathematical model and its application in the neighborhood of the Three Gorges Reservoir dam in the Yangtze River

The calculation of flow and sediment transport is one of the most important tasks in river engineering. The task is particularly difficult because a number of complex physical phenomena should be accounted for more realistically in a model with a predictive power. Three-dimensional calculations of river flow and suspended sediment transport are performed in this paper with application in the Three Gorges Reservoir in the Yangtze River. A period of 76 years after the dam is built is simulated and the results are compared with laboratory measurements obtained by Tsinghua University whereby the model is verified and calibrated. Generally speaking, the calculated results agree well with the experiments, demonstrating that the present model can be used for flow and sediment transport prediction in rivers. The project supported by the National Natural Science Foundation of China (50009004)     

Calculation of vibrations of thin plates by partitioning into elastically connected undeformable units

A mathematical model of a thin plate in the form of a system of elastically connected undeformable rectangular units is proposed and substantiated. With necessary additions, the basic statements of the method can be extended to the case of a thick plate. Novosibirsk State Engineering University, Novosibirsk 630092. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 5, pp. 159–168, September–October, 1998.