Determination of Permeability Distributions Using NMR Velocity Imaging

This study develops a methodology for determining the absolute permeability distribution in a porous media sample using velocity data obtained from NMR imaging experiments. An objective function describing the discrepancy between observed and simulated data is reduced by iteratively updating the permeability. This parameter estimation scheme is based on an iterative method which uses optimal control theory to refine the estimates. Although this theory is developed for both isotropic and anisotropic porous media, the permeability reconstructions examined in this paper are restricted to the isotropic case. Synthetic data are used to investigate the impact of varying the noise in the experimental data, the degree of parameterization, the relative weighting of the regularization term in the objective function, and the amount and type of data required to obtain a satisfactory permeability reconstruction. These synthetic data are extracted from the solution of numerical experiments that have utilized an assumed permeability distribution. The methodology is also applied to data gathered in laboratory experiments for water flow in a sandstone sample.     

Determining Spatial Distributions of Permeability

Conventional laboratory experimentation provides an apparent value of hydraulic permeability which, at best, is representative of the entire sample. Nuclear magnetic resonance imaging (MRI) provides unique opportunities to probe spatial distributions of permeability at a much finer scale (Seto et al., Transp Porous Media 42:351–388, 2001). We advance the methodology for determining spatial distributions of permeability and provide, for the first time, laboratory determinations of permeability distributions with complete three-dimensional (3D) spatial resolution. We investigate new experimental designs that mitigate a possible lack of identifiability and provide for more accurate estimates of permeability. We demonstrate the application of MRI experiments and analyses that provide substantial improvements in the determination of the porosity distribution, an essential step for obtaining reliable measurements of spatially resolved velocity distributions. We investigate the use of global optimization to solve the associated inverse problem for determining permeability distributions from the measured velocity distributions. Our methodology is demonstrated with experimental data on sandstone and trabecular bone samples.     

A Procedure for Accurate One-Dimensional Strain Measurement Using the Grid Method

This paper deals with the accurate calculation of strain using the grid method. The strain field is first directly deduced from the fringe pattern without calculating the displacement field. This procedure is validated with two numerical examples. Two types of experiment are then carried out: a translation and a tensile test. It is observed that some additional fictitious strains appear in both cases. They are due to two main reasons which interact with each other: the grid defects and the displacement of the grid lines during testing. A suitable procedure is proposed to cancel out these fictitious strains. This procedure is successfully applied in two cases of fringe patterns.     

Three-Dimensional Characterization of Permeability at the Core Scale

In this article, we introduce an integrated method for characterizing permeability heterogeneity at the core scale. It combines the results of laboratory core flooding with already-developed field scale history matching techniques such as gradual deformation and pilot points. Prior to any experiment, X-ray computed tomography (CT) imaging techniques are used to obtain three-dimensional porosity distribution in cores. The samples are submitted to viscous, miscible displacement of water by water–glycerin mixture. The dynamic data collected during injection are the time variations in inlet–outlet pressure drop and three-dimensional CT-scan concentration maps of invading fluid collected at successive times. We develop an inversion or matching process which takes advantage of the available data to characterize the spatial distribution of permeability heterogeneities within core samples. Permeability is assumed to be related to porosity. This matching process involves two successive optimizations. First, an initial permeability guess derived from porosity is modified by varying deterministic parameters until the corresponding simulated pressure answer fits the measured pressure drop. Second, an extended optimization process with both deterministic and stochastic parameters is run to match pressure drop and concentration data. This methodology is applied to a synthetic example for which the permeability–porosity relation is known. It yields a three-dimensional permeability model reproducing the reference pressure and concentration maps. The methodology is also applied to experimental data. In this case, it provides three-dimensional permeability models leading to an improved, but perfectible data match. A major difficulty is the unknown relationship between permeability and porosity.     

A Simple Model for the Variation of Permeability due to Partial Saturation in Dual Scale Porous Media

The main focus of this work is to model macroscopically the effects of partial saturation upon the permeability of dual scale fibrous media made of fiber bundles when a Newtonian viscous fluid impregnates it. A new phenomenological model is proposed to explain the discrepancies between experimental pressure results and analytical predictions based on Darcy's law. This model incorporates the essential features of relative permeability but without the necessity of measuring saturation of the liquid for its prediction. The model is very relevant for the small scale industrial systems where a liquid is forced to flow through a fibrous porous medium. It requires four parameters. Two of them are the two permeability values based on the two length scales. One length scale is of the order of magnitude of the individual fiber radius and corresponds to the permeability of the completely staurated medium, the other is of the order of magnitude of the distance between the fiber bundles and corresponds to the permeability of the partially saturated medium. The other two parameters are the lengths of the two partially saturated regions of the flow domain. The two lengths of the partially saturated region and the permeability of the fully saturated flow domain can be directly measured from the experiments. The excellent agreement between the model and the experimental results of inlet pressure profile with respect to time suggests that this model may be used to describe the variation of the permeability behind a moving front in such porous media for correct pressure prediction. It may also be used to characterize the fibrous medium by determining the two different permeabilities and the relative importance of the unsaturated portion of the flow domain for a given architecture.     

On the Determination of Nonlinear Response Distributions for Oscillators with Combined Harmonic and Random Excitation

The analysis of two different nonlinear systems, both subject to an excitation that comprises a harmonic and a random component, is presented in this paper. Both systems are known to exhibit different coexisting attractors for a purely harmonic forcing. The random part causes a disturbance of the response of these systems: Even though a dominating effect of the attractors for the deterministic case is still visible, the random disturbance also leads to occasional jumps between the areas surrounding the different attractors. To access the likelihood of the system being found in a specific state, probability density functions are approximated numerically by means of a localized statistical linearization.     

Double Time Scale Analysis of the Consolidation of a Two-Strata Poroelastic Layer with High Permeability Contrast

The consolidation of a saturated porous medium submitted to a vertical load is a classical problem, and has been given a complete analytical solution within the framework of linear poroelasticity. For a layer constituted by several homogeneous strata, the problem can be dealt with by means of analytical as well as numerical methods, provided that the contrast between the properties of the strata remain moderate. In this paper, we consider a layer constituted by two strata and discuss the situation in which the contrast between the properties of the strata becomes very large and fluid conduction exhibits very different characteristic times in each stratum. The method used consists in introducing a double time scale, and gives an analytical solution, which makes it possible to discuss the influence of the ratio between strata thicknesses on the overall consolidation time of the layer.     

An Experimental Study on the Influence of Sub-Core Scale Heterogeneities on CO<Subscript>2</Subscript> Distribution in Reservoir Rocks

This article presents the results of CO₂/brine two-phase flow experiments in rocks at reservoir conditions. X-ray CT scanning is used to determine CO₂ saturation at a fine scale with a resolution of a few pore volumes and provide 3D porosity and saturation maps that can be use to correlate CO₂ saturations and rock properties. The study highlights the strong influence of sub-core scale heterogeneities on the spatial distribution of CO₂ at steady state and provides useful relative permeability data on a sample originated from an actual storage site (CO2CRC-Otway project, Victoria, South-West Australia). Two different samples tested, although different in nature, present strong heterogeneities, but differ in the detail of the connectivity of high porosity layers. In both samples, the results of the investigations show that sub-core scale heterogeneities control the sweep efficiency and may cause channeling through the porous medium. In one of the samples, CO₂ saturation appears uncorrelated to porosity close to the outlet end of the core. This observation is understood as a result of the position and the orientation of high porosity layers with respect to the inlet face of the core. Finally, in the operating conditions of the two experiments, the saturation maps demonstrate that gravity does not play a major role since no detectable buoyancy driven flow is observed.     

An Experimental Method for One Dimensional Permeability Characterization of Heterogeneous Porous Media at the Core Scale

Many reservoir simulator inputs are derived from laboratory experiments. Special core analysis techniques generally assume that core samples are homogeneous. This assumption does not hold for porous media with significant heterogeneities. This paper presents a new method to characterize core scale permeability heterogeneity. The method is validated by both numerical and experimental results. The leading idea consists in injecting a high viscosity miscible fluid into a core sample saturated with a low viscosity fluid. In such conditions, the fluid displacement is expected to be piston-like. We investigate the evolution of the pressure drop as a function of time. A continuous permeability profile is estimated along flow direction from the pressure drop assuming that the core sample is a stack of infinitely thin cross sections perpendicular to flow direction. Thus, we determine a permeability value for each cross section. Numerical and laboratory experiments are carried out to validate the method. Flow simulations are performed for numerical models representing core samples to estimate the pressure drop. The selected models are sequences of plugs with constant permeabilities. In addition, laboratory displacements are conducted for both low permeability and high permeability core samples. To investigate whether there is dispersion inside the porous medium, CT scan measurements are performed during fluid displacement: the location of the front is observed at successive time intervals. The results validate the methodology developed in this paper as long as heterogeneity is one dimensional.     

Analysis of Variance of Porosity and Heterogeneity of Permeability at the Pore Scale

Transport in Porous Media - We present a new statistical variance approach for characterizing heterogeneities related to pore spaces in reservoir rocks. Laboratory-based computer microtomography...     

A New Cleat Volume Compressibility Determination Method and Corresponding Modification to Coal Permeability Model

Coal is known as a dual-porosity media composed of cleat and matrix pore. Methane can be stored in the cleats or adsorbed on the inner surface of matrix pore. While fluid mobility is mainly controlled by the developed cleat network, methane desorption has a significant effect on cleat deformation. In the process of coalbed methane recovery, both reservoir compaction and matrix shrinkage will occur and have opposite effects on permeability evolutions. A variety of analytical permeability models have been developed to describe the transient characteristics of permeability in coals. In this study, three common permeability models are first revisited and evaluated against the experimental data under uniaxial strain condition. Shi–Durucan (S&D) model demonstrates the best performance among these models. However, constant cleat volume compressibility was used to assume for S&D model, and the generalization of S&D model is significantly limited. For ease of generalization, the relation between cleat volume compressibility and effective horizontal stress is re-derived and introduced to the derivation of permeability model. Since coal reservoirs usually demonstrate strong anisotropy and heterogeneity, the influences of elastic and adsorption properties are further tested to reveal the overall trend of permeability. The results show that S&D model and its modification with the main variable of effective horizontal stress have the best performances in matching the experimental data under uniaxial strain. The relationship between cleat volume compressibility and effective horizontal stress can be better reflected by the inverse proportional function. In addition, the strengths of reservoir compaction effect relative to matrix shrinkage effect in different models only vary with Poisson’s ratio, while their magnitudes are also affected by Young’s modulus. For a typical coal reservoir, the C&B and P&M models will observe a stronger permeability decline at the initial, while the improved P&M model will receive an earlier and more rapid rebound than the S&D and W&Z models.     

Resolution Effect: An Error Correction Model for Intrinsic Permeability of Porous Media Estimated from Lattice Boltzmann Method

Transport in Porous Media - In digital rock physics, the intrinsic permeability of a porous rock sample can be evaluated from its micro-computed tomography ($$\upmu$$-CT) image through lattice...     

A Model for Tracking Fronts of Stress-Induced Permeability Enhancement

Using an analogy to the classical Stefan problem, we construct evolution equations for the fluid pore pressure on both sides of a propagating stress-induced damage front. Closed form expressions are derived for the position of the damage front as a function of time for the cases of thermally-induced damage as well as damage induced by over-pressure. We derive expressions for the flow rate during constant pressure fluid injection from the surface corresponding to a spherically shaped subsurface damage front. Finally, our model results suggest an interpretation of field data obtained during constant pressure fluid injection over the course of 16 days at an injection site near Desert Peak, NV.     

A Pore-Scale Numerical Simulation Method for Estimating the Permeability of Sand Sediment

A numerical method system to estimate the permeability of sand sediments, at a microscopic scale, was developed. Initially, 3D geometrical representations of the sand grains are reconstructed from a series of 2D X-ray CT scans of real sand grains. 2D cross-sectional slices of the grain outlines are combined together to produce 3D objects via spherical harmonics series expansions. Then, the reconstructed sand grains are packed randomly inside a cubic, microscopic, domain by a combination of a growth method and a simulated annealing method to achieve a predefined porosity. Finally, a single-phase water flow within the domain was simulated numerically, using the lattice Boltzmann method. The calculated permeability of these systems compares well with the values provided by conventional theoretical models. One of the contributions of this study is to show that it is possible to predict the permeability of sand sediments of variable porosities, using sand grains from CT images with changing size distributions and orientations.     

A Unified Solution for the Free Vibration Analysis of Simply Supported Cylindrical Shells with General Structural Stress Distributions

Most existing studies on the vibration analysis of cylindrical shells with structural stress are limited to uniform stress distribution. However, non-uniform stress distributions are encountered in many engineering applications. In this study, a unified solution for the vibration analysis of cylindrical shells with a general stress distribution is presented using the Flu¨gge shell theory and modal orthogonality simplification. The obtained analytical solution can be applied to a structure with arbitrary distributed stress, thus it has a wider range of applications than previous methods. The accuracy and advantage of the proposed method are validated by comparing with the finite element method results.     

Determination of the Effective Permeability of Doubly Porous Materials by a Two-Scale Homogenization Approach

Transport in Porous Media - The present work is dedicated to determining the effective permeability of doubly porous materials made of a solid phase comprising a network of interconnected pores at...     

A Numerical Procedure for the Dynamic Plastic Response of Beams with Rotatory Inertia and Transverse Shear Effects

ABSTRACT

A numerical procedure is used to examine the influence of transverse shear forces in the yield criterion and rotatory inertia on the dynamic plastic response of beams. Various results are presented for a long beam impacted by a mass and a simply supported beam loaded impulsively, both of which are made from a rigid perfectly plastic material with yielding controlled by the Ilyushin-Shapiro ield criterion.

Transverse shear effects lead to a dramatic reduction in the slopes of the deformed profiles for both beam problems. Moreover, the slope of the deformed profile underneath the striker in the impact problem is quite sensitive to the actual shape of a yield curve, while the maximum transverse displacement is less sensitive. The retention of rotatory inertia in the basic equations leads to further reductions up to 17 and 10% in the slopes and maximum transverse displacements, respectively.     

Determination of Nanoparticle Macrotransport Coefficients from Pore Scale Processes

Nanoparticle transport in porous media is modeled using a hierarchical set of differential equations corresponding to pore scale and macroscale. At the pore scale, movement and interaction of a single particle with a solid matrix is modeled using the advection–dispersion–sorption equation. A single nanoparticle entering the space encounters viscous, diffusion and surface forces. Surface forces (electrostatic and van der Waals forces) between nanoparticles and mineral grains appear as sorption propensity on solid matrix boundary condition. These local events are then transformed into a macroscale continuum by imposing periodic boundary conditions for contiguous unit cells representing porous media and using a scheme of moment analysis. At the macroscale, propagation and retention of particles are characterized by three position-independent coefficients: mean nanoparticle velocity vector \({\bar{\mathbf{U}}}^*\), macroscopic dispersion coefficient \({\bar{\mathbf{D}}}^*\), and mean nanoparticle retention rate constant \({\bar{K}}^*\). The modeling results are validated with a set of nanoparticle transport tests in porous microchips. We also present simulations of realistic porous media, where an actual image of sandstone samples is processed into binary tones. The representative unit cells are constructed from the resulting binary image by searching for areas within the sample with maximum similarities to the whole sample in terms of porosity and specific surface area, which are found to show strong correlations with the resulting \({\bar{\mathbf{U}}}^*\) and \({\bar{K}}^*\), respectively.