Permeability identification of a weakly permeable partially saturated porous rock

The present paper deals with the determination of permeability in partially saturated conditions for weakly permeable porous continua such as argillites or deep clayey formations. The permeability can be deduced from measurements of transient weight loss of a sample submitted to a laboratory drying test: a decrease of relative humidity is imposed by saline solution in an hermetic chamber. Assumptions of constant gas pressure equal to atmospheric pressure and of negligible Fickean diffusive transport of vapour are adopted. The only transport phenomenon taken into account inside the sample is the Darcean advective transport of the water liquid. The forward problem is solved by following two modelling approaches: a linear one and a nonlinear one. The parameter identification procedure is based upon the solution of corresponding inverse problems. In the two cases, the Levenberg–Marquardt algorithm has been used for the minimization problem. In the linear approach, the solution of the forward problem is explicit. In the non linear approach, finite volume method for the spatial discretization combined with a Newton–Raphson algorithm has been used to solve the non linear forward problem. The identification method enables variations of permeability and capillary capacity to be estimated. Comparisons between linear and non linear approaches show that the first one is useful to give mean values and order of magnitude of permeability and capacity. A more complete information is deduced from the non linear approach as variations of equivalent capacity and permeability during a test are significant in most cases. The analysis of the obtained results shows that the basic modelling assumption of constant gas pressure inside the sample would not be relevant for lower range of relative humidities and liquid permeability than those investigated.     

Rarefied Pure Gas Transport in Non-isothermal Porous Media: Effective Transport Properties from Homogenization of the Kinetic Equation

Viscous flow, effusion, and thermal transpiration are the main gas transport modalities for a rarefied gas in a macro-porous medium. They have been well quantified only in the case of simple geometries. This article develops a model based on the homogenization of kinetic equations producing effective transport properties (permeability, Knudsen diffusivity, thermal transpiration ratio) in any porous medium sample, as described e.g., by a digitized 3D image. The homogenization procedure—neglecting the effect of gas density gradients on heat transfer through the solid—leads to macroscopic transfer relations, and to closure problems in for the obtention of effective properties. Coherence of the approach with previous literature on the subject is discussed. The asymptotic limits of the model (rarefied and continuum regimes) are also studied. One of the main results is that the effect of the geometry on thermal transpiration has to be described by a tensor, which is distinct from the permeability and Knudsen diffusion tensors.     

Finite Element-Based Characterization of Pore-Scale Geometry and Its Impact on Fluid Flow

We present a finite element (FEM) simulation method for pore geometry fluid flow. Within the pore space, we solve the single-phase Reynold’s lubrication equation—a simplified form of the incompressible Navier–Stokes equation yielding the velocity field in a two-step solution approach. (1) Laplace’s equation is solved with homogeneous boundary conditions and a right-hand source term, (2) pore pressure is computed, and the velocity field obtained for no slip conditions at the grain boundaries. From the computed velocity field, we estimate the effective permeability of porous media samples characterized by section micrographs or micro-CT scans. This two-step process is much simpler than solving the full Navier–Stokes equation and, therefore, provides the opportunity to study pore geometries with hundreds of thousands of pores in a computationally more cost effective manner than solving the full Navier–Stokes’ equation. Given the realistic laminar flow field, dispersion in the medium can also be estimated. Our numerical model is verified with an analytical solution and validated on two 2D micro-CT scans from samples, the permeabilities, and porosities of which were pre-determined in laboratory experiments. Comparisons were also made with published experimental, approximate, and exact permeability data. With the future aim to simulate multiphase flow within the pore space, we also compute the radii and derive capillary pressure from the Young–Laplace’s equation. This permits the determination of model parameters for the classical Brooks–Corey and van-Genuchten models, so that relative permeabilities can be estimated.     

Fully Coupled THMC Modeling of Wellbore Stability with Thermal and Solute Convection Considered

Wellbore stability analysis is an important topic in petroleum geomechanics. Analytical and numerical analysis of wellbore stability involves the study of interactions among pressure, temperature and chemical changes, and the mechanical response of the rock, a coupled thermal–hydraulic–mechanical–chemical (THMC) process. Thermal and solute convection have usually been overlooked in numerical models. This is appropriate for shales with extremely low permeability, but for shales with intermediate and high permeability (e.g., shale with a disseminated microfissure network), thermal and solute convection should be considered. The challenge of considering advection lies in the numerical oscillation encountered when implementing the traditional Galerkin finite element approach for transient advection–diffusion problems. In this article, we present a fully coupled THMC model to analyze the stress, pressure, temperature, and solute concentration changes around a wellbore. In order to overcome spurious spatial temperature oscillations in the convection-dominated thermal advection–diffusion problem, we place the transient problem into an advection– diffusion-reaction problem framework, which is then efficiently addressed by a stabilized finite element approach, the subgrid scale/gradient subgrid scale method (SGS/GSGS).     

Transient Heat and Moisture Flow Around Heat Source Buried in an Unsaturated Half Space

This article presents solutions for the transient heat and moisture transport due to both disk heat source and cylindrical heat source buried in an unsaturated half space. The solutions are presented in Hankel–Laplace transform domain and in dimensionless style. Coupled effect of thermally driven moisture transport is especially investigated because of its importance to alter the flow field in low-permeability medium. Parametric study has been performed to assess the effects of five independent dimensionless parameters on flow field. The stability and accuracy of the present solutions are demonstrated from the comparison between the results obtained from these solutions and those by using a well-established finite element code CODE_BRIGHT. Despite the simplified assumptions required in order to obtain analytical solutions in Hankel–Laplace transform domain, the results incorporate the main mechanisms involved in the coupled thermo-hydraulic (T-H) problem, and they may be eventually used for validation purposes.     

On Singularity Formation of a Nonlinear Nonlocal System

We investigate the singularity formation of a nonlinear nonlocal system. This nonlocal system is a simplified one-dimensional system of the 3D model that was recently proposed by Hou and Lei (Comm Pure Appl Math 62(4):501–564, 2009) for axisymmetric 3D incompressible Navier–Stokes equations with swirl. The main difference between the 3D model of Hou and Lei and the reformulated 3D Navier–Stokes equations is that the convection term is neglected in the 3D model. In the nonlocal system we consider in this paper, we replace the Riesz operator in the 3D model by the Hilbert transform. One of the main results of this paper is that we prove rigorously the finite time singularity formation of the nonlocal system for a large class of smooth initial data with finite energy. We also prove global regularity for a class of smooth initial data. Numerical results will be presented to demonstrate the asymptotically self-similar blow-up of the solution. The blowup rate of the self-similar singularity of the nonlocal system is similar to that of the 3D model.     

The vapour–liquid interface and stresses in dried bodies

The paper presents a contribution to modelling the problem of vapour–liquid interface receding into dried body and stresses induced by drying of capillary-porous bodies. A complex algorithm comprising the specific mechanisms of drying in the first and second periods of drying is constructed. It enables calculation and drawing of the body temperature and drying curves for the whole drying process and identification of the vapour–liquid interface receding into the body. The drying induced stresses caused by the receding vapour–liquid interface and the non-uniform distribution of moisture content and/or temperature are analyzed. Numerical calculations of the temperature and drying curves and the drying induced stresses are carried out for the example of a finite dimensional kaolin cylinder dried convectively.     

Rarefied Pure Gas Transport in Non-isothermal Porous Media: Validation and Tests of the Model

Viscous flow, effusion, and thermal transpiration are the main gas transport modalities for a rarefied gas in a macro-porous medium. They have been well quantified only in the case of simple geometries. This paper presents a numerical method based on the homogenization of kinetic equations producing effective transport properties (permeability, Knudsen diffusivity, thermal transpiration ratio) in any porous medium sample, as described, e.g. by a digitized 3D image. The homogenization procedure—neglecting the effect of gas density gradients on heat transfer through the solid—leads to closure problems in for the obtention of effective properties; they are then simplified using a Galerkin method based on a 21-element basis set. The kinetic equations are then discretized in space with a finite- volume scheme. The method is validated against experimental data in the case of a closed test tube. It shows to be coherent with past approaches of thermal transpiration. Then, it is applied to several 3D images of increasing complexity. Another validation is brought by comparison with other distinct numerical approaches for the evaluation of the Darcian permeability tensor and of the Knudsen diffusion tensor. Results show that thermal transpiration has to be described by an effective transport tensor which is distinct from the other tensors.     

Shale-Gas Permeability and Diffusivity Inferred by Improved Formulation of Relevant Retention and Transport Mechanisms

A theoretically improved model incorporating the relevant mechanisms of gas retention and transport in gas-bearing shale formations is presented for determination of intrinsic gas permeability and diffusivity. This is accomplished by considering the various flow regimes according to a unified Hagen–Poiseuille-type equation, fully compressible treatment of gas and shale properties, and numerical solution of the non-linear pressure equation. The present model can accommodate a wide range of fundamental flow mechanisms, such as continuum, slip, transition, and free molecular flow, depending on the prevailing flow conditions characterized by the Knudsen number. The model indicates that rigorous determination of shale-gas permeability and diffusivity requires the characterization of various important parameters included in the present phenomenological modeling approach, many of which are not considered in previous studies. It is demonstrated that the improved model matches a set of experimental data better than a previous attempt. It is concluded that the improved model provides a more accurate means of analysis and interpretation of the pressure-pulse decay tests than the previous models which inherently consider a Darcian flow and neglect the variation of parameters with pressure.     

Development of a Generalized Version of the Poisson– Nernst–Planck Equations Using the Hybrid Mixture Theory: Presentation of 2D Numerical Examples

A numerical scheme for the transient solution of a generalized version of the Poisson–Nernst–Planck (PNP) equations is presented. The finite element method is used to establish the coupled non-linear matrix system of equations capable of solving the present problem iteratively. The PNP equations represent a set of diffusion equations for charged species, i.e. dissolved ions, present in the pore solution of a rigid porous material in which the surface charge can be assumed neglectable. These equations are coupled to the ‘internally’ induced electrical field and to the velocity field of the fluid. The Nernst–Planck equations describing the diffusion of the ionic species and Gauss’ law in use are, however, coupled in both directions. The governing set of equations is derived from a simplified version of the so-called hybrid mixture theory (HMT). The simplifications used here mainly concerns ignoring the deformation and stresses in the porous material in which the ionic diffusion occurs. The HMT is a special version of the more ‘classical’ continuum mixture theories in the sense that it works with averaged equations at macroscale and that it includes the volume fractions of phases in its structure. The background to the PNP equations can by the HMT approach be described by using the postulates of mass conservation of constituents together with Gauss’ law used together with consistent constitutive laws. The HMT theory includes the constituent forms of the quasistatic version of Maxwell’s equations making it suitable for analyses of the kind addressed in this work. Within the framework of HTM, constitutive equations have been derived using the postulate of entropy inequality together with the technique of identifying properties by Lagrange multipliers. These results will be used in obtaining a closed set of equations for the present problem.     

A New Coal-Permeability Model: Internal Swelling Stress and Fracture–Matrix Interaction

We have developed a new coal-permeability model for uniaxial strain and constant confining-stress conditions. The model is unique in that it explicitly considers fracture–matrix interaction during coal-deformation processes and is based on a newly proposed internal swelling stress concept. This concept is used to account for the impact of matrix swelling (or shrinkage) on fracture-aperture changes resulting from partial separation of matrix blocks by fractures that do not completely cut through the whole matrix. The proposed permeability model is evaluated using data from three Valencia Canyon coalbed wells in the San Juan Basin, where increased permeability has been observed during CH₄ gas production, as well as using published data from laboratory tests. Model results are generally in good agreement with observed permeability changes. The importance of fracture–matrix interaction in determining coal permeability, demonstrated in this study using relatively simple stress conditions, underscores the need for a dual-continuum (fracture and matrix) mechanical approach to rigorously capture coal-deformation processes under complex stress conditions, as well as the coupled flow and transport processes in coal seams.     

Numerical investigation of hypersonic nozzle flows at high reynolds numbers

The direct problem of viscous gas flow in a hypersonic nozzle of given geometry is solved on the basis of simplified Navier-Stokes equations. At a stagnation pressure of the order of several thousands of atmospheres, a compressibility factor is introduced into the equation of state. The gasdynamic parameter profiles and the Mach number distribution along the nozzle axis are obtained. The results of earlier calculations of profiled nozzles are revised. Novosibirsk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 161–164, January–February, 1999.     

Second Order Adaptive Boundary Conditions for Exterior Flow Problems: Non-Symmetric Stationary Flows in Two Dimensions

We consider the problem of solving numerically the stationary incompressible Navier–Stokes equations in an exterior domain in two dimensions. For numerical purposes we truncate the domain to a finite sub-domain, which leads to the problem of finding so called “artificial boundary conditions” to replace the boundary conditions at infinity. To solve this problem we construct – by combining results from dynamical systems theory with matched asymptotic expansion techniques based on the old ideas of Goldstein and Van Dyke – a smooth divergence free vector field depending explicitly on drag and lift and describing the solution to second and dominant third order, asymptotically at large distances from the body. The resulting expression appears to be new, even on a formal level. This improves the method introduced by the authors in a previous paper and generalizes it to non-symmetric flows. The numerical scheme determines the boundary conditions and the forces on the body in a self-consistent way as an integral part of the solution process. When compared with our previous paper where first order asymptotic expressions were used on the boundary, the inclusion of second and third order asymptotic terms further reduces the computational cost for determining lift and drag to a given precision by typically another order of magnitude. Peter Wittwer: Supported in part by the Fonds National Suisse.     

A phenomenological and extended continuum approach for modelling non-equilibrium flows

This paper presents a new technique that combines Grad’s 13-moment equations (G13) with a phenomenological approach to rarefied gas flows. This combination and the proposed solution technique capture some important non-equilibrium phenomena that appear in the early continuum-transition flow regime. In contrast to the fully coupled 13-moment equation set, a significant advantage of the present solution technique is that it does not require extra boundary conditions explicitly; Grad’s equations for viscous stress and heat flux are used as constitutive relations for the conservation equations instead of being solved as equations of transport. The relative computational cost of this novel technique is low in comparison to other methods, such as fully coupled solutions involving many moments or discrete methods. In this study, the proposed numerical procedure is tested on a planar Couette flow case, and the results are compared to predictions obtained from the direct simulation Monte Carlo method. This test case highlights the presence of normal viscous stresses and tangential heat fluxes that arise from non-equilibrium phenomena, which cannot be captured by the Navier–Stokes–Fourier constitutive equations or phenomenological modifications.      

Conduction–radiation interaction in 3D irregular enclosures using the finite volume method

Interaction of conduction and radiation in 3D enclosures is carried out with a gray participating media. Application of block structured grid is shown with the finite volume method (FVM). Radiation modeling is performed with the FVM and is coupled with an ‘in-house’ code to solve the set of transport equations. The detailed numerical results are presented for a cubical and a cylindrical enclosure as these results are not available in the literature. The numerical simulation for the cylindrical enclosure is performed using a block-structured ‘O’ grid. Two additional geometries are considered in order to show the applicability of the present work. Results of temperature, radiative heat flux and total heat flux distributions are presented for different optical thicknesses, scattering albedoes, emissivities and conduction–radiation parameters. The 3D results are validated with the available 2D results or results with pure radiation problems as limiting cases.     

An Initial and Boundary Value Problem Modeling of Fish-like Swimming

In this paper we consider an initial and boundary value problem that models the self-propelled motion of solids in a bidimensional viscous incompressible fluid. The self-propelling mechanism, consisting of appropriate deformations of the solids, is a simplified model of the propulsion mechanism of fish-like swimmers. The governing equations consist of the Navier–Stokes equations for the fluid, coupled to Newton’s laws for the solids. Since we consider the case in which the fluid–solid system fills a bounded domain we have to tackle a free boundary value problem. The main theoretical result in the paper asserts the global existence and uniqueness (up to possible contacts) of strong solutions of this problem. The second novel result of this work is the provision of a numerical method for the fluid–solid system. This method provides a simulation of the simultaneous displacement of several swimmers and is tested on several examples.     

Comparison of Three FE-FV Numerical Schemes for Single- and Two-Phase Flow Simulation of Fractured Porous Media

We benchmark a family of hybrid finite element–node-centered finite volume discretization methods (FEFV) for single- and two-phase flow/transport through porous media with discrete fracture representations. Special emphasis is placed on a new method we call DFEFVM in which the mesh is split along fracture–matrix interfaces so that discontinuities in concentration or saturation can evolve rather than being suppressed by nodal averaging of these variables. The main objective is to illustrate differences among three discretization schemes suitable for discrete fracture modeling: (a) FEFVM with volumetric finite elements for both fractures and porous rock matrix, (b) FEFVM with lower dimensional finite elements for fractures and volumetric finite elements for the matrix, and (c) DFEFVM with a mesh that is split along material discontinuities. Fracture discontinuities strongly influence single- and multi-phase fluid flow. Continuum methods, when used to model transport across such interfaces, smear out concentration/saturation. We show that the new DFEFVM addresses this problem producing significantly more accurate results. Sealed and open single fractures as well as a realistic fracture geometry are used to conduct tracer and water-flooding numerical experiments. The benchmarking results also reveal the limitations/mesh refinement requirements of FE node-centered FV hybrid methods. We show that the DFEFVM method produces more accurate results even for much coarser meshes.     

Stability of a Steady Viscous Incompressible Flow Past an Obstacle

We derive a sufficient condition for stability of a steady solution of the Navier–Stokes equation in a 3D exterior domain Ω. The condition is formulated as a requirement on integrability on the time interval (0, +∞) of a semigroup generated by the linearized problem for perturbations, applied to a finite family of certain functions. The norm of the semigroup is measured in a bounded sub-domain of Ω. We do not use any condition on “smallness” of the basic steady solution.      

Geometric and Hydrodynamic Characteristics of Three-dimensional Saturated Prefractal Porous Media Determined with Lattice Boltzmann Modeling

Fractal and prefractal geometric models have substantial potential for contributing to the analysis of flow and transport in porous media such as soils and reservoir rocks. In this study, geometric and hydrodynamic parameters of saturated 3D mass and pore–solid prefractal porous media were characterized using the lattice Boltzmann model (LBM). The percolation thresholds of the 3D prefractal porous media were inversely correlated with the fraction of micro-pore clusters and estimated as 0.36 and 0.30 for mass and pore–solid prefractal porous media, respectively. The intrinsic permeability and the dispersivity of the 3D pore–solid prefractals were larger than those of the 3D mass prefractals, presumably because of the occurrence of larger solid and pore cluster sizes in the former. The intrinsic permeability and dispersivity of both types of structure increased with increasing porosity, indicating a positive relationship between permeability and dispersivity, which is at odds with laboratory data and current theory. This discrepancy may be related to limitations of the convection dispersion equation at the relatively high porosity values employed in the present study.     

Explicit Rate–Time Solutions for Modeling Matrix–Fracture Flow of Single Phase Gas in Dual-Porosity Media

One of the widely used methods for modeling matrix–fracture fluid exchange in naturally fractured reservoirs is dual porosity approach. In this type of modeling, matrix blocks are regarded as sources/sinks in the fracture network medium. The rate of fluid transfer from matrix blocks into fracture medium may be modeled using shape factor concept (Warren and Root, SPEJ 3:245–255, 1963); or the rate–time solution is directly derived for the specific matrix geometry (de Swaan, SPEJ 16:117–122, 1976). Numerous works have been conducted to study matrix–fracture fluid exchange for slightly compressible fluids (e.g. oil). However, little attention has been taken to systems containing gas (compressible fluid). The objective of this work is to develop explicit rate–time solutions for matrix–fracture fluid transfer in systems containing single phase gas. For this purpose, the governing equation describing flow of gas from matrix block into fracture system is linearized using pseudopressure and pseudotime functions. Then, the governing equation is solved under specific boundary conditions to obtain an implicit relation between rate and time. Since rate calculations using such an implicit relation need iterations, which may be computationally inconvenient, an explicit rate–time relation is developed with the aid of material balance equation and several specific assumptions. Also, expressions are derived for average pseudopressure in matrix block. Furthermore, simplified solutions (originated from the complex general solutions) are introduced applicable in infinite and finite acting flow periods in matrix. Based on the derived solutions, expressions are developed for shape factor. An important observation is that the shape factor for gas systems is the same as that of oil bearing matrix blocks. Subsequently, a multiplier is introduced which relates rate to matrix pressure instead of matrix pseudopressure. Finally, the introduced equations are verified using a numerical simulator.