Simulation of uncompressible fluid flow through a porous media

Recently, a great interest has been focused for investigations about transport phenomena in disordered systems. One of the most treated topics is fluid flow through anisotropic materials due to the importance in many industrial processes like fluid flow in filters, membranes, walls, oil reservoirs, etc. In this work is described the formulation of a 2D mathematical model to simulate the fluid flow behavior through a porous media (PM) based on the solution of the continuity equation as a function of the Darcy’s law for a percolation system; which was reproduced using computational techniques reproduced using a random distribution of the porous media properties (porosity, permeability and saturation). The model displays the filling of a partially saturated porous media with a new injected fluid showing the non-defined advance front and dispersion of fluids phenomena.     

Stochastic generalized porous media equations with reflection

A non-negative Markovian solution is constructed for a class of stochastic generalized porous media equations with reflection. To this end, some regularity properties and a comparison theorem are proved for stochastic generalized porous media equations, which are interesting by themselves. Invariant probability measures and ergodicity of the solution are also investigated.     

Harnack inequality and applications for stochastic evolution equations with monotone drifts

As a Generalization to Wang (Ann Probab 35:1333–1350, 2007) where the dimension-free Harnack inequality was established for stochastic porous media equations, this paper presents analogous results for a large class of stochastic evolution equations with general monotone drifts. Some ergodicity, compactness and contractivity properties are established for the associated transition semigroups. Moreover, the exponential convergence of the transition semigroups to invariant measure and the existence of a spectral gap are also derived. As examples, the main results are applied to many concrete SPDEs such as stochastic reaction-diffusion equations, stochastic porous media equations and the stochastic p-Laplace equation in Hilbert space.     

黑油模型

地下油气渗流的数学模型是十分复杂的。为了提高油气藏的采收率,有必要了解地下实际压力和流量等参数。本文仅就一些简化的情况,讨论数学模型的建立,以说明其基本思想。     

Uncertainty Quantification for Two-Phase Flow in Heterogeneous Porous Media

The simulation of flow and transport in porous media such as aquifers often involve dealing with complex heterogeneities. They are characterized by varying hydrogeological properties which differ strongly from the adjacent medium and often lead to significant changes of the flow behavior. However detailed information about the location of such heterogeneities is not always known. The deterministic models thus need to be extended stochastically to quantify uncertainties. As mathematical model we use the capillarity-free fractional flow formulation for two immiscible and incompressible fluid phases in a two-dimensional and partitioned domain. To cope with the randomly located heterogeneity interfaces we employ a stochastic Galerkin (SG) method [4]. The physical space of this system then is modelled by a central upwind finite volume scheme [5] in combination with mixed finite elements [7]. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Harnack inequality and strong Feller property for stochastic fast-diffusion equations

As a continuation to [F.-Y. Wang, Harnack inequality and applications for stochastic generalized porous media equations, Ann. Probab. 35 (2007) 1333-1350], where the Harnack inequality and the strong Feller property are studied for a class of stochastic generalized porous media equations, this paper presents analogous results for stochastic fast-diffusion equations. Since the fast-diffusion equation possesses weaker dissipativity than the porous medium one does, some technical difficulties appear in the study. As a compensation to the weaker dissipativity condition, a Sobolev-Nash inequality is assumed for the underlying self-adjoint operator in applications. Some concrete examples are constructed to illustrate the main results.     

考虑毛管形状的幂律流体等效渗透率计算方法(英文)

While studying the flow of oil and gas in the reservoir, it is not realistic that capillary with circular section is only used to express the pores. It is more representative to simulate porous media pore with kinds of capillary with triangle or rectangle section etc. In the condition of the same diameter, when polymer for oil displacement flows in the porous medium, there only exists shear flow which can be expressed with power law model. Based on fluid flow-pressure drop equation in single capillary, this paper gives a calculation method of equivalent permeability of power law fluid of single capillary and capillary bundles with different sections.     

Influence of the polymer degradation on enhanced oil recovery processes

Polymer flooding is one of the most common and technically developed chemical Enhanced Oil Recovery (EOR) processes. Its main function is to increase the carrying phase’s (i.e., water or brine) viscosity in order to mobilize the remaining trapped oil. Many numerical simulators have been developed during the last 30 years considering the influence of the polymer molecules on the viscosity as well as on other physical parameters (e.g., diffusion, adsorption). Nevertheless, there are certain phenomena which were not previously considered, for instance, the interfacial effects of hydrophobically modified polymers. Furthermore, the degradation of the polymer molecules in a harsh environment such as the one found in porous media is well known. This causes a deterioration on the viscosifying properties, diminishing the efficiency of the method. It is important also to consider the effect of the polymer viscoelasticity on the microscopic sweeping efficiency, lowering the residual oil saturation, which has not been properly addressed. A new compositional 2D numerical simulator is presented for polymer flooding in a two-phase, three-component configuration, considering all these physical effects present in porous media and using a fully second-order accurate scheme coupled with total variation diminishing (TVD) functions. Results demonstrated that degradation cannot be considered negligible in any polymer EOR process, since it affected the viscoelastic and viscosifying properties, decreasing the sweeping efficiency at both micro- and macroscopic scales. This simulator will allow setting the desired designing properties for future polymers in relationship with the characteristics of the oil field to be exploited.     

Initial-boundary value problem of three phase flow in porous media

We study the mathematical model of three phase compressible flows through porous media. Under the condition that the rock, water and oil are incompressible, and the compressibility of gas is small, we present a finite element scheme to the initial-boundary value problem of the nonlinear system of equations, then by the convergence of the scheme we prove that the problem admits a weak solution.     

TWO PHASE COMPRESSIBLE FLOW IN POROUS MEDIA

We study the mathematical model of two phase compressible flows through porous media. Under the condition that the compressibility of rock, oil, and water is small, we prove that the initial-boundary value problem of the nonlinear system of equations admits a weak solution.     

The Global Random Attractor for a Class of Stochastic Porous Media Equations

We prove new L ²-estimates and regularity results for generalized porous media equations “shifted by” a function-valued Wiener path. To include Wiener paths with merely first spatial (weak) derivates we introduce the notion of “ζ-monotonicity” for the non-linear function in the equation. As a consequence we prove that stochastic porous media equations have global random attractors. In addition, we show that (in particular for the classical stochastic porous media equation) this attractor consists of a random point.     

Spatial Localization for Nonlinear Dynamical Stochastic Models for Excitable Media

Nonlinear dynamical stochastic models are ubiquitous in different areas. Their statistical properties are often of great interest, but are also very challenging to compute. Many excitable media models belong to such types of complex systems with large state dimensions and the associated covariance matrices have localized structures. In this article, a mathematical framework to understand the spatial localization for a large class of stochastically coupled nonlinear systems in high dimensions is developed. Rigorous \linebreak mathematical analysis shows that the local effect from the diffusion results in an exponential decay of the components in the covariance matrix as a function of the distance while the global effect due to the mean field interaction synchronizes different components and contributes to a global covariance. The analysis is based on a comparison with an appropriate linear surrogate model, of which the covariance propagation can be computed explicitly. Two important applications of these theoretical results are discussed. They are the spatial averaging strategy for efficiently sampling the covariance matrix and the localization technique in data assimilation. Test examples of a linear model and a stochastically coupled FitzHugh-Nagumo model for excitable media are adopted to validate the theoretical results. The latter is also used for a systematical study of the spatial averaging strategy in efficiently sampling the covariance matrix in different dynamical regimes.     

Irreducibility and strong Feller property for non-linear SPDEs

Under some non-degeneracy condition, the strong Feller property and irreducibility are studied for non-linear stochastic partial differential equations driven by multiplicative noise within the framework called ‘variational approach’. Our result for irreducibility can be applied to equations with locally monotone coefficients. In some special cases, we discuss the Hölder continuity of the associated Markov semigroups. The main results are applied to several examples such as stochastic Burgers equation, stochastic porous media equation and stochastic fast diffusion equation.     

Two‐phase flows in karstic geometry

Multiphase flow phenomena are ubiquitous. Common examples include coupled atmosphere and ocean system (air and water), oil reservoir (water, oil, and gas), and cloud and fog (water vapor, water, and air). Multiphase flows also play an important role in many engineering and environmental science applications. In some applications such as flows in unconfined karst aquifers, karst oil reservoir, proton membrane exchange fuel cell, multiphase flows in conduits, and in porous media must be considered together. Geometric configurations that contain both conduit (or vug) and porous media are termed karstic geometry. Despite the importance of the subject, little work has been performed on multiphase flows in karstic geometry. In this paper, we present a family of phase–field (diffusive interface) models for two‐phase flow in karstic geometry. These models together with the associated interface boundary conditions are derived utilizing Onsager's extremum principle. The models derived enjoy physically important energy laws. A uniquely solvable numerical scheme that preserves the associated energy law is presented as well. Copyright © 2013 John Wiley & Sons, Ltd.     

一种基于流线方法的CO2混相驱数学模型

采用改进的黑油模型研究思路，建立起一种实用的CO2混相驱数学模型，模型中利用混相流的油、气相对渗透率及有效粘度的调整来实现混相过程的模拟．为了实现生产动态的快速预测，用流线方法替代传统的有限差分法求解该模型．在结合边界元方法确定复杂边界条件下稳态渗流场流线分布的基础上，采用显式全变差递减法对流管内的一维渗流问题求解．同时，利用该模型讨论了开采方式、溶剂段塞尺寸、注入周期等对CO2驱开采效果的影响规律．建立的模型的优点在于：输入参数较少、计算快捷，适应于对任意形状边界条件下各种井网配置的CO2驱替动态进行计算，为CO2混相驱油田的早期筛选及油藏动态管理提供了有效的工具．     

能源数值模拟计算方法的理论和应用

１引言石油是国民经济和社会发展的重要支柱,油田的勘探和开发是发展石油工业的关键．我国开发的油田均进入了二次采油期,大多数已进入注水开发中后期,特别是大庆油田和胜利油田,若继续单纯采用注水开采,产量每年将减少数百万吨．稳定石油产量的唯一方法是采用三次采油新技术,开发尚滞留在地下约５０％以上已探明的储量．若能平均提高３０％的采收率,即相当于再生了同等规模的油田．所谓油藏数值模拟,就是用电子计算机模拟地下油藏十分复杂的化学、物理及流体流动的真实过程,以便选出最佳的开采方案,监控措施．对于三次采油新技术…     

Contamination and remediation waves in a filtration model

We propose a model for the filtration of suspended particles in porous media and we examine some of its mathematical properties. The model includes a variable porosity that depends on the volume of particles retained through filtration and a kinetics law that allows both a positive and negative rate of particle accretion. We characterize the properties of accretion rates that lead to contamination and remediation wave fronts in the model.     

Strong Solutions of Stochastic Generalized Porous Media Equations: Existence,Uniqueness, and Ergodicity

Explicit conditions are presented for the existence, uniqueness, and ergodicity of the strong solution to a class of generalized stochastic porous media equations. Our estimate of the convergence rate is sharp according to the known optimal decay for the solution of the classical (deterministic) porous medium equation.     

Plastic Structural Analysis Under Stochastic Uncertainty

Problems from limit load or shakedown analysis are based on the convex, linear or linearized yield/strength condition and the linear equilibrium equation for the generic stress vector. Having to take into account, in practice, stochastic variations of the model parameters (e.g., yield stresses, plastic capacities) and external loadings, the basic stochastic plastic analysis problem must be replaced by an appropriate deterministic substitute problem. Instead of calculating approximatively the probability of failure based on a certain choice of failure modes, here, a direct approach is presented based on the costs for missing carrying capacity and the failure costs (e.g., costs for damage, repair, compensation for weakness within the structure, etc.). Based on the basic mechanical survival conditions, the failure costs may be represented by the minimum value of a convex and often linear program. Several mathematical properties of this program are shown. Minimizing then the total expected costs subject to the remaining (simple) deterministic constraints, a stochastic optimization problem is obtained which may be represented by a “Stochastic Convex Program (SCP) with recourse”. Working with linearized yield/strength conditions, a “Stochastic Linear Program (SLP) with complete fixed recourse” is obtained. In case of a discretely distributed probability distribution or after the discretization of a more general probability distribution of the random structural parameters and loadings as well as certain random cost factors one has a linear program (LP) with a so-called “dual decomposition data” structure. For stochastic programs of this type many theoretical results and efficient numerical solution procedures (LP-solver) are available. The mathematical properties of theses substitute problems are considered. Furthermore approximate analytical formulas for the limit load factor are given.     

Density-driven compactional flow in porous media

In the mathematical modelling of compactional flow in porous media, the constitutive relation is typically modelled in terms of a nonlinear relationship between effective pressure and porosity, and compaction is essentially poroelastic. However, at depths deeper than 1 km where the pressure is high, compaction becomes more akin to a viscous one. Two mathematical models of compaction in porous media are formulated and the nonlinear equations are then solved numerically. The essential features of numerical profiles of poroelastic and viscous compaction are thus compared with asymptotic solutions. Two distinguished styles of density-driven compaction in fast and slow compacting sediments are analysed and shown in this paper.