Stable hyperbolic singularities for three-phase flow models in oil reservoir simulation

We study the characteristic speeds of systems of two conservation laws representing three-phase flow in a porous medium with gravity taken into account.Generically hyperbolicity fails on open regions (elliptic regions) where the characteristic speeds assume complex values. The presence of such regions creates difficulties such as multiple solutions which indicate a modeling problem, according to some authors.The hyperbolicity of the models we study depends on the relative permeability functions. It is customary in oil engineering studies to suppose that the water and gas permeabilities depend only on their respective saturation, while the oil relative permeability changes with the gas and water saturations. Such a hypothesis on the oil relative permeability generically leads to elliptic regions.We define a set of three curves that surround elliptic regions of any model. By studying these curves, we indicate a procedure to locate the singularities and prove that for any choice of gravitational and viscosity parameters such regions shrinks to points where the characteristic speeds are real and equal, provided it is assumed that each relative permeability depends on its respective saturation only. Our results, together with a paper of Trangenstein, lead to the conclusion that in order to insure real wave speeds, such an assumption is necessary and sufficient when gravitational effects are considered in three-phase models.Research supported by Brazillian Government grant from CNPq under number 204395/88.7.     

Mathematical simulation of a reflux mass-transfer process

We study a numerical-analytic method of solving an initial-boundary value problem for a quasilinear system of differential equations of parabolic type with initial condition given by the Dirac delta function. One figure. Bibliography: 6 titles. Translated fromProblemy Matematicheskoi Fiziki, 1998, pp. 209–213.     

Numerical simulation of ice formation in a reservoir

The one-dimensional formulation is considered of a two-phase Stefan problem of heat flow with an unknown phase transition temperature that depends on the concentration of impurity. A numerical method is described for implementing the constraints that the heat and mass conservation are given on the unknown nonstationary boundary between the liquid and solid phases. Some examples are included of simulations for the sodium chloride solutions of different concentrations.     

Application of auxiliary space preconditioning in field-scale reservoir simulation

We study a class of preconditioners to solve large-scale linear systems arising from fully implicit reservoir simulation.These methods are discussed in the framework of the auxiliary space preconditioning method for generality.Unlike in the case of classical algebraic preconditioning methods,we take several analytical and physical considerations into account.In addition,we choose appropriate auxiliary problems to design the robust solvers herein.More importantly,our methods are user-friendly and general enough to be easily ported to existing petroleum reservoir simulators.We test the efciency and robustness of the proposed method by applying them to a couple of benchmark problems and real-world reservoir problems.The numerical results show that our methods are both efcient and robust for large reservoir models.     

Mathematical modeling of economic indices for oil field development

Under study is some mathematical model for quantitative evaluation of investment projects for development of oil fields at the stage of conceptual design. As the basis of such a model we suggest that the field is considered as a cluster of equitype elements of area pattern of oil wells. The model operates with the net present value as a continuous function of the process parameters and enables us to analyze a broad spectrum of possible options in implementing the investment project. Some important ratios between the technical and economic parameters are obtained in concise and practically suitable forms by application of operational calculus and the Laplace transform.     

Mathematical and numerical aspects of one-dimensional laminar flame simulation

The aim of this paper is to solve several mathematical and numerical questions related to the simulation of stationary and nonstationary premixed flat flames. Most of the results are obtained in the general context of complex chemical and diffusion mechanisms. The main mathematical results concern: (i) thea priori positivity of the mass fractions, and (ii) the sensitivity of the flame speed to the computational domain. The numerical method proposed for solving the stationary problem is a new combination of the pseudo-nonstationary approach, the Newton iterations, and the adaptive gridding. The computation of H₂-O₂-N₂ flames with various initial concentrations (including the chemical extinction zone) shows the efficiency of this method.     

Mathematical simulation of an irregular waveguide with reentering edges

A mathematical model is proposed to study an irregular waveguide with reentering edges. Theoretical estimates for the behavior of the solution near the reentrant corners are used to estimate the rate of convergence of the numerical solution to the exact one. The mode structure of the waveguide field is analyzed.     

Mathematical simulation of acoustic wave scattering in fractured media

The simulation of acoustic waves in fractured media is considered. A self-consistent field model is proposed that describes the formation of a scattered field and the attenuation of the incident field. For the total field, a wave equation with a complex velocity is derived and the corresponding dispersion equation is studied. A frequency-dependent field damping law and an energy variation law are established. An initial and a boundary value problem for waves in a fractured medium is addressed. A finite-difference scheme for the initial value problem is constructed, and a condition for its stability is established. Numerical results are presented.     

Mathematical simulation of acoustic wave refraction near a caustic

Issues related to the computation of wave fields in an acoustic medium near caustics are considered. A boundary condition on a caustic is established, and the Green’s function of a boundary value problem for the general case of a varying speed of sound is constructed. For this purpose, an auxiliary Goursat problem is considered and a system of its particular solutions is constructed using hyper-geometric functions. A Volterra integral equation for the Green’s function is obtained, and an algorithm for its expansion with respect to smoothness is described. A finite difference scheme approximating the solution of the differential problem with an unbounded coefficient is proposed. Numerical results are presented.     

An improved reservoir simulation model in the petroleum industry

Sensible prediction of petroleum production from a well reliesheavily on the accurate determination of reservoir constantsduring pressure drawdown or build-up tests. Models for simulatinga fractured reservoir have been developed to accommodate theslope changes in the measured pressure curves. This paper presentsa modification to Barenblatt et al.s (1960) prototype, suitablefor the simulation of a fractured reservoir. The modificationis intended to provide an improved interpretation of fluid flowin fractured porous media, as compared to the traditional conceptualization.The significant differences in terms of reservoir pressure predictionbetween this model and the one suggested by Warren & Root(1963) are identified and shown to depend on the compressibilitymagnitudes of fractures and matrix blocks. The analytical solutionsof this improved model have been obtained for both the caseof radial flow with quasi-steady matrix flow, and for the caseof one-dimensional linear flow where transient matrix flow isretained.     

Mathematical programming time-based decomposition algorithm for discrete event simulation

Mathematical programming has been proposed in the literature as an alternative technique to simulating a special class of Discrete Event Systems. There are several benefits to using mathematical programs for simulation, such as the possibility of performing sensitivity analysis and the ease of better integrating the simulation and optimisation. However, applications are limited by the usually long computational times. This paper proposes a time-based decomposition algorithm that splits the mathematical programming model into a number of submodels that can be solved sequentially to make the mathematical programming approach viable for long running simulations. The number of required submodels is the solution of an optimisation problem that minimises the expected time for solving all of the submodels. In this way, the solution time becomes a linear function of the number of simulated entities.     

Mathematical modeling and computer simulation of a robotic rat pup

In this paper, we describe the mathematics and computer implementation of a robotic rat pup simulation. Our goal is to understand neurobehavioral principles in a mammalian model organism—the Norway rat pup (Rattus norvegicus). Our approach is unique in that animal, simulation, and robot studies occur in parallel and inform each other. Behavior is dependent on the nervous system, body morphology, physiology, environment, and the interactions among these elements. Autonomous robotics hardware models and their associated simulations allow the possibility of systematically manipulating variables in each of these elements in ways that would be impossible using live animals. Specifically, we describe the development and validation of a Newtonian-dynamics-based simulation of a robotic rat pup, including mathematical formulation and computer implementation. The computer simulation consists of three distinct components that interact to simulate robotic behavior: (1) dynamics of the robotic rat pup itself, including sensors and actuators, (2) environmental coupling dynamics of the robot arena with the robotic rat pup, and (3) the robot control algorithms as implemented on the physical robot. The mathematical formulation, software implementation, model identification, model validation, and an application example are all described.     

Mathematical analysis and numerical simulation of pattern formation under cross-diffusion

Cross-diffusion driven instabilities have gained a considerable attention in the field of population dynamics, mainly due to their ability to predict some important features in the study of the spatial distribution of species in ecological systems. This paper is concerned with some mathematical and numerical aspects of a particular reaction–diffusion system with cross-diffusion, modeling the effect of allelopathy on two plankton species. Based on a stability analysis and a series of numerical simulations performed with a finite volume scheme, we show that the cross-diffusion coefficient plays a important role on the pattern selection.     

Mathematical programming formulations for approximate simulation of multistage production systems

Mathematical programming representation has been recently used to describe the behavior of discrete event systems as well as their formal properties. This new way of representing discrete event systems paves the way to the creation of simpler mathematical programming models that reduce the complexity of the system analysis. The paper proposes an approximate representation for a class of production systems characterized by several stages, limited buffer capacities and stochastic production times. The approximation exploits the concept of a time buffer, modeled as a constraint that put into a temporal relationship the completion times of two customers in a sample path. The main advantage of the proposed formulation is that it preserves its linearity even when used for optimization and, for such a reason, it can be adopted in simulation–optimization problems to reduce the initial solution space. The approximate formulation is applied to relevant problems such as buffer capacity allocation in manufacturing systems and control parameters setting in pull systems.     

Mathematical simulation of deformation and wave processes in multilayered structures

The deformation and wave processes induced by collisions of an impactor with deformable layered targets of various configurations are analyzed. The numerical solution of such problems is associated with an adequate treatment of wave processes in a continuous medium, which is an especially difficult task in the case of layered targets. To deal with the former problem, it is proposed to use adaptive Lagrangian triangular meshes. Wave processes are simulated using the grid-characteristic method, which can serve as a basis for algorithms that do not fail near the boundary of the computational domain and at numerous material interfaces. Additionally, hybrid and hybridized grid-characteristic schemes are applied that substantially improve numerical solutions with steep gradients (discontinuous solutions). These methods provide an adequate treatment of wave processes in layered targets (wave reflection and refraction at contact surfaces, secondary-wave interaction, changes in the conditions on these boundaries, etc.).     

Large scale petroleum reservoir simulation and parallel preconditioning algorithms research

Solving large scale linear systems efficiently plays an important role in a petroleum reservoir simulator, and the key part is how to choose an effective parallel preconditioner. Properly choosing a good preconditioner has been beyond the pure algebraic field. An integrated preconditioner should include such components as physical background, characteristics of PDE mathematical model, nonlinear solving method, linear     

Mathematical description of the properties of muscle tissue

Theoretical research on the problem of describing the properties of muscle tissue is briefly reviewed. The principal approaches to the problem are indicated, their shortcomings are noted, and the requirements that must be satisfied by an up-to-date continuum model are formulated. The procedure for constructing such a model is described, together with the principal and secondary hypotheses, and the conclusions that follow from a comparison of theory and experiment are noted.     

Large scale petroleum reservoir simulation and parallel preconditioning algorithms research

Solving large scale linear systems efficiently plays an important role in a petroleum reservoir simulator, and the key part is how to choose an effective parallel preconditioner. Properly choosing a good preconditioner has been beyond the pure algebraic field. An integrated preconditioner should include such components as physical background, characteristics of PDE mathematical model, nonlinear solving method, linear solving algorithm, domain decomposition and parallel computation. We first discuss some parallel preconditioning techniques, and then construct an integrated preconditioner, which is based on large scale distributed parallel processing, and reservoir simulation-oriented. The infrastructure of this preconditioner contains such famous preconditioning construction techniques as coarse grid correction, constraint residual correction and subspace projection correction. We essentially use multi-step means to integrate totally eight types of preconditioning components in order to give out the final preconditioner. Million-grid cell scale industrial reservoir data were tested on native high performance computers. Numerical statistics and analyses show that this preconditioner achieves satisfying parallel efficiency and acceleration effect.     

Mathematical simulation of laser induced melting and evaporation of multilayer materials

Using the laser induced remelting of a three-layer target Al + Ni + Cr as an example, the use of the dynamic adaptation for solving the multifront Stefan problem with explicit tracking of the melting and evaporation fronts is considered. The dynamic adaptation is used to construct quasi-uniform grids in regions with moving boundaries. The characteristic size of those regions may vary by several orders of magnitude in the process of computations. The algorithm used to construct the grids takes into account the varying size of the region and the velocity of the boundary motion, which makes it possible to automatically distribute the grid points without using fitting parameters. The mathematical simulation of the doping process using the melt with respect to the thick substrate and thin doping layers showed the importance of the sequencing of coatings. The computations showed that if the upper exposed layer is chromium, then it can completely evaporate or sublimate by the end of the pulse due to its heat-transfer properties. This can be easily changed if the doping layers are arranged according to the scheme Al + Cr + Ni. Then, the upper exposed layer is nickel, which is not so easily evaporated.     

Robustness analysis in Multi-Objective Mathematical Programming using Monte Carlo simulation

In most multi-objective optimization problems we aim at selecting the most preferred among the generated Pareto optimal solutions (a subjective selection among objectively determined solutions). In this paper we consider the robustness of the selected Pareto optimal solution in relation to perturbations within weights of the objective functions. For this task we design an integrated approach that can be used in multi-objective discrete and continuous problems using a combination of Monte Carlo simulation and optimization. In the proposed method we introduce measures of robustness for Pareto optimal solutions. In this way we can compare them according to their robustness, introducing one more characteristic for the Pareto optimal solution quality. In addition, especially in multi-objective discrete problems, we can detect the most robust Pareto optimal solution among neighboring ones. A computational experiment is designed in order to illustrate the method and its advantages. It is noteworthy that the Augmented Weighted Tchebycheff proved to be much more reliable than the conventional weighted sum method in discrete problems, due to the existence of unsupported Pareto optimal solutions.