Adaptive multiscale model reduction with Generalized Multiscale Finite Element Methods

Modeling of multi-scale transport phenomena in shale gas production — A critical review

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Shales at all scales: Exploring coupled processes in mudrocks

An analytical model to couple gas storage and transport capacity in organic matter with noncircular pores

Multiscale model reduction for shale gas transport in a coupled discrete fracture and dual-continuum porous media

Organic-rich resource shales play an important role in global natural gas production. However, many uncertainties exist in an engineering analysis of gas transport and production such as the reservoir-scale flow simulation, history-matching, and optimization. In this work, we introduce a new set of governing equations to describe the characteristic features of porous structures of the organic-rich resource shale. We apply multi-scale analysis to mass balance equations, the equation of state (for free gas), and an adsorption isotherm. Using the macroscopic model, we study gas transport in shales, consisting of nanoporous organic material (kerogen) and the inorganic material. We conclude that both gas in-place and gas production rate depend on the amount of kerogen in the shale matrix. Adsorbed-phase transport by the organic pore walls is responsible for the increase in production rate. We investigate both Henry and Langmuir adsorption as well as different values of length scale ratio and diffusion coefficients.

Multi-scale Asymptotic Analysis of Gas Transport in Shale Matrix

The majority of processes in composite materials involve a wide range of scales. Because of the scale disparity in multi scale problem, it's often impossible to resolve the effect of small scales directly. In this paper we perform multi scale modeling in order to analyze properties of composite materials with periodical structure under temperature and stresses influence. Each component has its own thermal and mechanical (elastic) properties. We replace differential equations with rapidly oscillating coefficients by homogenized equations having effective parameters, incorporating multi scale structure and properties of any component. The influence of any component's properties and of the layer thickness on the effective properties of whole sample has been studied. We perform comparison of results, obtained with and without account for thermostresses.

Homogenization of thermoelasticity processes in composite materials with periodic structure of heterogeneities

In this paper we consider a generalization of Brinkman’s equation governing the flow of an incompressible fluid through a porous medium to take into account the variation of the viscosity with the pressure. There is overwhelming evidence that the viscosity depends on pressure and that this dependence is exponential. We also consider the problem of a barotropic fluid in which both the viscosity and the density depend upon the pressure. We use a homogenization procedure, that exploits a multiple scale structure possessed by a solid porous medium, a ‘micro-scale’ comparable to the pore size and a ‘macro-scale’ associated with the global size of the body, to carry out a multiple-scale asymptotic analysis. The ratio of the micro-scale length to the macro-scale length provides a small parameter that is a natural parameter with which to carry out a perturbation analysis. The perturbation procedure leads to a systematic methodology by means of which we can obtain equations at different orders of the small parameter. Specifically, we study some simple boundary value problems corresponding to one-dimensional flows. We find that while the dependence of the viscosity on the pressure can have a significant effect on the nature of the solution, so much so that the solution is completely different in comparison to the constant viscosity case Brinkman’s equation, we find that the dependence of the density on the pressure has little or no effect on the nature of the solution.

Homogenization of a generalization of Brinkman's equation for the flow of a fluid with pressure dependent viscosity through a rigid porous solid

Modeling porous flow in complex media is a challenging problem. Not only is the problem inherently multiscale but, due to high contrast in permeability values, flow velocities may differ greatly throughout the medium. To avoid complicated interface conditions, the Brinkman model is often used for such flows [O. Iliev, R. Lazarov, and J. Willems, Multiscale Model. Simul., 9 (2011), pp. 1350--1372]. Instead of permeability variations and contrast being contained in the geometric media structure, this information is contained in a highly varying and high-contrast coefficient. In this work, we present two main contributions. First, we develop a novel homogenization procedure for the high-contrast Brinkman equations by constructing correctors and carefully estimating the residuals. Understanding the relationship between scales and contrast values is critical to obtaining useful estimates. Therefore, standard convergence-based homogenization techniques [G. A. Chechkin, A. L. Piatniski, and A. S. Shamev, Homogen...

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Viktoria L. Savatorova

Papers

Multi-scale Asymptotic Analysis of Gas Transport in Shale Matrix

Homogenization of thermoelasticity processes in composite materials with periodic structure of heterogeneities

Homogenization of a generalization of Brinkman's equation for the flow of a fluid with pressure dependent viscosity through a rigid porous solid

Homogenization of High-Contrast Brinkman Flows

Mathematical modeling of gas transport in porous geological media with contrast of properties and irregular distribution of pores