Bentheimer Sandstone Research Articles

Microstructural control on the anisotropy of elastic and transport properties in undeformed sandstones

Microstructural analysis of two undeformed sandstones, the Bentheim and Rothbach sandstones, supports the origin of the macroscopic anisotropies previously investigated (ultrasonic P-wave velocity, electrical conductivity and magnetic susceptibility). The shape and spatial distribution of both grains and voids were determined by computer image processing, measurement of magnetic anisotropy coupled with ferrofluids and stereological analysis. In addition, data on permeability measured in different directions are also provided for the two sandstones. The elastic models proposed in a previous paper are confirmed by the present study. In the Bentheim sandstone, the anisotropic behaviour was attributed to the non-spherical shape of the porosity, while in the Rothbach sandstone, an orientation-dependent cement radius at the intergranular contacts controls the observed anisotropy. The methods used for analysing the rock microstructure and its anisotropy are shown to be very useful and appropriate for detecting features such as pore or grain shape anisotropy, or the spatial distribution of grain contacts.

Pore fabric shape anisotropy in porous sandstones and its relation to elastic wave velocity and permeability anisotropy under hydrostatic pressure

To understand the relationship between pore space anisotropy and petrophysical properties, we developed a novel apparatus capable of simultaneously measuring permeability, porosity and ultrasonic velocities at hydrostatic pressures up to 100 MPa. First, we use magnetic susceptibilities and acoustic wave velocities to identify the principal anisotropy axes under ambient laboratory conditions. This directional anisotropy data is then used to guide experiments on two sandstones (Bentheim and Crab Orchard) under hydrostatic pressure from 5 to 90 MPa. We find the structural anisotropy formed by the void space is well described by velocity anisotropy in both cases. Under hydrostatic pressure, the acoustic anisotropy of Crab Orchard sandstone (COS) decreases from 3% and 7% at 5 MPa (P-wave and S-wave) to 1.5% and 1%, respectively, at effective pressures over 40 MPa; for Bentheim sandstone the decrease is considerably less. Permeability of COS is 125×10 −18 m 2, decreasing rapidly as effective pressure increases, with permeability parallel to bedding approximately twice that normal to bedding. In contrast, permeability of Bentheim sandstone is 0.86×10 −12 m 2, and varies little with effective pressure or coring direction. We relate many of our measurements made under hydrostatic pressure to the contrasting pore fabric between the two rock types, and infer that a critical pressure is required for the initiation of crack closure.

Stray field measurements of flow displacement distributions without pulsed field gradients

The probability distribution P ( ζ) of diffusive and advective molecular displacements is determined using a fixed field gradient (FFG) pulse sequence, on fluid flow through a Bentheimer sandstone, in the grossly inhomogeneous stray field of a super-conducting magnet. Two decades of q-space are scanned with stimulated echoes, using the gradient of the stray field and variable encoding times δ. The strength of the gradient permits the use of short encoding times, which is desirable for limiting the distorting effects produced by flow displacements through susceptibility induced field inhomogeneities. CPMG and CP echo trains are used to refocus separately the real and imaginary parts of the stimulated echo, for experimental efficiency.

Effects of bedding and foliation on mechanical anisotropy, damage evolution and failure mode

Abstract In this study we review recent advances in our understanding of anisotropy in rocks, focusing on dilatant and compactant failure in sandstones and in a foliated metamorphic rock. In sandstones, the anisotropy can be associated with bedding, as in the Rothbach sandstone, or it can also be due to shape anisotropy of the grains and/or the pores, as in the Bentheim sandstone. Two scenarios are proposed for the development of anisotropy in these two end members. In a metamorphic rock with strong foliation like the Four-mile gneiss, it has been commonly observed that the brittle strength is minimum at a foliation angle of about 30°–45°. A damage mechanics model is proposed that underscores the dominant role of biotite foliation in the development of microcracking. In contrast it is often observed in sandstones with strong bedding that the strength is minimum in the direction parallel to bedding. New results for the Rothbach sandstone showed that compared to parallel-to-bedding samples: (i) in the brittle faulting regime the perpendicular-to-bedding samples have both a higher strength and dilatancy stress; and (ii) in the cataclastic flow regime the compactive yield envelope for the perpendicular-to-bedding samples expands significantly towards higher stress values.

Permeability evolution during localized deformation in Bentheim sandstone

Strain localization in porous sandstones may significantly impact the regional fluid flow. Previous laboratory studies that investigated permeability evolution with deformation concentrated primarily on shear localization and distributed cataclastic flow. Here we focus on compaction bands, a localized mode of deformation characterized by compaction and negligible shear. In this study we used Bentheim sandstone with porosity 23%, where discrete compaction bands have been observed to develop subperpendicular to the maximum principal stress. To investigate coupling between strain localization and permeability, we conducted permeability measurements during triaxial loading at confining pressures ranging from 10 to 350 MPa and microstructural observation on failed samples. Two types of failure were identified: shear localization and compaction localization at low and high effective pressures, respectively. For both failure modes the bulk permeability decreased with deformation. A dramatic decrease of more than one order of magnitude was associated with compaction localization, where permeability reduction occurred over a relatively narrow range of axial strain with the onset of shear‐enhanced compaction. Motivated by our microstructural observations, we modeled the permeability reduction of the failed sample as that of a layered medium with significant permeability contrast between the discrete bands and matrix. The model reproduced the experimental observations of permeability evolution during development of discrete compaction bands, with implications for the amount of localized strain and the permeability contrast. Permeability evolution during development of discrete and diffuse compaction bands suggests two different trends with strain, providing guidance for extrapolation of laboratory measurements to field settings.

Large-scale grid-enabled lattice Boltzmann simulations of complex fluid flow in porous media and under shear.

Well-designed lattice Boltzmann codes exploit the essentially embarrassingly parallel features of the algorithm and so can be run with considerable efficiency on modern supercomputers. Such scalable codes permit us to simulate the behaviour of increasingly large quantities of complex condensed matter systems. In the present paper, we present some preliminary results on the large-scale three-dimensional lattice Boltzmann simulation of binary immiscible fluid flows through a porous medium, derived from digitized X-ray micro-tomographic data of Bentheimer sandstone, and from the study of the same fluids under shear. Simulations on such scales can benefit considerably from the use of computational steering, and we describe our implementation of steering within the lattice Boltzmann code, called LB3D, making use of the RealityGrid steering library. Our large-scale simulations benefit from the new concept of capability computing, designed to prioritize the execution of big jobs on major supercomputing resources. The advent of persistent computational grids promises to provide an optimal environment in which to deploy these mesoscale simulation methods, which can exploit the distributed nature of computer, visualization and storage resources to reach scientific results rapidly; we discuss our work on the grid-enablement of lattice Boltzmann methods in this context.

Compaction localization in porous sandstones: spatial evolution of damage and acoustic emission activity

The coupled development of compaction and strain localization may significantly impact the stress field, strain partitioning and fluid transport. To investigate the phenomenon of compaction localization, mechanical deformation and microstructural observations were conducted on the Darley Dale, Rothbach, Berea, Bentheim and Diemelstadt sandstones with porosities ranging from 13 to 24%. The constitutive behavior, failure mode, acoustic emission (AE) activity and spatial distribution of damage were investigated. While our observations reveal a broad spectrum of geometric complexity associated with failure modes, two end-members can be distinguished: shear bands at relatively high angles and arrays of discrete compaction bands subperpendicular to the maximum compression direction, respectively. A hybrid localization mode involving high-angle shear bands and diffuse compaction bands was observed in sandstones with intermediate porosities. The discrete compaction bands in our laboratory samples are qualitatively similar to localization structures observed in the field. Whereas the development of high-angle shear bands is characterized by the continuous accumulation of AE, discrete bands are associated with episodic surges in AE that are characterized by an overall strain hardening trend punctuated by episodic stress drops.

Comparison of the anisotropic behaviour of undeformed sandstones under dry and saturated conditions

This article presents a systematic analysis of the anisotropic behaviours of the Bentheim and Rothbach sandstones using ultrasonic P-wave velocity, electrical conductivity and magnetic susceptibility measurements. For each sandstone, the data were obtained from three core samples drilled perpendicularly to each other and tested in dry- and water-saturated conditions. For acoustic and magnetic investigations, the same statistical analysis was applied in order to present the data on comparable stereoplots. Surprisingly, the Bentheim sandstone which appeared homogeneous at macroscopic scale showed a stronger elastic and electrical anisotropy than the Rothbach sandstone in which cross-laminations were clearly identified, as confirmed by a sedimentary magnetic fabric. A discussion on the velocity contrasts between dry and saturated samples led us to consider two different origins for the observed anisotropies. First, by comparing electrical and acoustic properties in the Bentheim sandstone, we conclude that the nature of the anisotropic behaviour is linked to the anisotropy of pore shape: the inclusion model developed by Kachanov (Kachanov, M., 1993. Elastic solids with many cracks and related problems. Advances in Applied Mechanics, vol. 30. Academic Press, Boston, MA, pp. 259–445) accounts for our observations if one considers that the pore space is made of parallel flat pores with moderate pore aspect ratio. Second, acoustic, electrical and magnetic properties indicate that the observed anisotropy in the Rothbach sandstone can be attributed to the matrix, and more specifically to cementation: we modified the Dvorkin and Nur (Geophysics 61 (5) (1996) 1363) model of cemented granular media by introducing a spatially variable contact length, and the model suggests that a very small variability of cemented contact length is enough to account for the observed P-wave velocity anisotropy. We emphasise the fact that combining several kinds of measurements is of great help in capturing the nature of the anisotropic behaviour of porous rocks.

Incremental propagation of discrete compaction bands: Acoustic emission and microstructural observations on circumferentially notched samples of Bentheim

To simulate how compaction localization may develop from structural and stress heterogeneity, we studied in the laboratory the influence of a stress concentration due to a V‐shaped circumferential notch in a cylindrical sample of Bentheim sandstone. Conventional triaxial experiments at confining pressure of 300 MPa were conducted. Acoustic emission activity was recorded, and each sample was deformed to a different stage and subsequently retrieved for microstructural observations. Our data indicate that compaction bands initiated from the notch tips and propagated by sequential increments as “anti‐cracks”. The transverse propagation of a compaction band was inferred to be faster than the axial displacement rate by 2 orders of magnitude. Energy dissipated for compaction band formation was estimated to be comparable to the shear fracture energy for shear band propagation.

Using MR techniques to probe permeability reduction in rock cores

AbstractPolymer solutions are used commonly in oil‐recovery applications to reduce water production. While the polymer treatment is known to modify the permeability of the rock, the mechanism by which this occurs is poorly understood, and, hence, the ability to optimize either the polymer solution or its application is restricted. In this work, the pulsed‐gradient spin‐echo nuclear magnetic resonance technique is used to determine displacement propagators (distributions) for brine solution flowing through a Bentheimer sandstone rock core, both before and after treatment of the rock core with a polyacrylamide polymer solution. A novel simulation strategy employing the lattice–Boltzmann method is then proposed to interpret the differences in these propagators in terms of the location of polymer retained within the pore structure of the rock following treatment. The results suggest that the polymer is preferentially trapped in comparatively low permeability pores.

A Model for the Mechanical Behaviour of Bentheim Sandstone in the Brittle Regime

The mechanical behaviour of Bentheim sandstone, a homogeneous quartz-rich sandstone with porosity of 22.8%, was investigated by triaxial compression tests conducted on dry samples. At confining pressures up to 35 MPa, the failure mode was characterized by a typical brittle deformation regime, as the samples showed dilatancy and failed by strain softening and brittle faulting. Previous studies have shown that the mechanical behaviour and failure mode of brittle porous granular rocks are governed by the time-dependent growth of microcracks. We analyse this process using the “Pore Crack Model” based on fracture mechanics analysis. It is consistent with the microstructure of porous granular rocks since it considers the growth of axial cracks from cylindrical holes in two dimensions. These cracks grow when their stress intensity factors reach the subcritical crack growth limit. Interaction between neighbouring cracks is introduced by calculating the stress intensity factor as the sum of two terms: a component for an isolated crack and an interaction term computed using the method of successive approximations. It depends on crack length, pore radius, pore density, and applied stresses. The simulation of crack growth from cylindrical holes, associated with a failure criterion based on the coalescence of interacting cracks, is used to compare the theoretical stress at the onset of dilatancy and at macroscopic rupture to the experimental determined values. Our approach gives theoretical results in good agreement with experimental data when microstructural parameters consistent with observations are introduced

Extension of Hoshen–Kopelman algorithm to non-lattice environments

We extend the Hoshen–Kopelman (HK) algorithm for cluster labeling to non-lattice environments, where sites are placed at random at non-lattice points. This extension is useful for continuum systems and disordered networks. Our extension of the HK algorithm relies on several data structures that describe network connectivity regardless of its dimensionality. Just as for the classic HK algorithm on lattices, our extension is completed in a single pass through the sites of the network and cluster relabeling operates on a vector whose size is much smaller than the size of the network. Our extension of the HK algorithm works for any environment (lattice or non-lattice) of any dimensionality, type (sites, bonds or both), and with arbitrary connectivity between the sites. The proposed extension is illustrated through a simple network consisting of 16 sites and 24 bonds, and applied to a complex network extracted from a 3D micro-focused X-ray CT image of Bentheimer sandstone consisting of 3677 sites and 8952 bonds.

Developing nuclear magnetic resonance imaging for engineering applications

Nuclear magnetic resonance (NMR) is a powerful, noninvasive measurement method which can be used to obtain spatially resolved information about fluid distributions within media. In the present work, an approach for quantitatively analyzing magnetic resonance imaging (MRI) data is extended for use with multiple spatial dimensions. A relaxation model is introduced and then used in an inverse problem to obtain local estimates of relaxation distributions. These distributions are then used to obtain spatial distributions of the intrinsic magnetization, the key quantity for quantification of NMR images. The inverse problem utilizes B-spline basis functions to represent the unknown relaxation distribution and a regularization term to stabilize the solution. The impact of the regularization is controlled through the use of a regularization parameter which is chosen using a data driven, computerized method based on nonparametric regression. The approach is demonstrated on laboratory data obtained from a Bentheimer sandstone sample to obtain one- and two-dimensional porosity distributions.

Experiments and Modeling of High-Velocity Pressure Loss in Sandstone Fractures

Summary High-velocity pressure loss in Bentheimer sandstone fractures of varying fracture widths was studied. Direct measurements of the roughness showed that the fractures are self-affine. The new results support a rough fracture high-velocity pressure-loss model. The high-velocity pressure loss was described by a Forchheimer equation with a dominating square term. The square term is a power law in fracture width, and the power is given by a roughness exponent. For low velocities, the pressure loss was not described by the Forchheimer equation. In agreement with theory, a weak-inertia-flow regime exists that separates the Darcy-flow regime from the Forchheimer-flow regime. An expression for the incremental high-velocity skin of a pinched-out hydraulic fracture was derived. Asymptotically for small fracture widths the skin is a power law in fracture width.

Mechanical behaviour and failure mode of bentheim sandstone under triaxial compression

Hydrostatic and triaxial compression tests have been conducted on nominally dry samples of Bentheim sandstone, a homogeneous quartz-rich sandstone with porosity of about 23%. A broad range of confining pressures were used to observe the transition from the brittle faulting to cataclastic flow regime. Mechanical data for the brittle strength and compactive yield stress can be fitted with empirical envelopes that have been shown to be applicable to other porous sandstones. However, the Bentheim sandstone is somewhat unusual in that quasiductile failure (characterized by an overall hardening trend punctuated by episodic strain softening and compaction band formation) was observed over a wide range of confining pressures from 120 MPa to 300 MPa. Since this failure mode is similar to observations in honeycombed cellular solids, it is speculated that the prevalence of quasiductile failure in the Bentheim sandstone arises from its relatively homogeneous mineralogy and grain size. Compaction band formation may be inhibited in other sandstones with higher fractions of feldspar and clay, as well as more disperse grain sizes.

Probing the gelation of polymers within a Bentheimer sandstone by 1H-PFG n.m.r.

The gelation of a polyvinylalcohol–glutaraldehyde–water solution confined in a Bentheimer sandstone was characterized by carrying out 1H-n.m.r. spin-lattice relaxation rate measurement (1/ T 1) and pulse field gradient diffusion ( D) measurements at 67°C. At any time during the gel reaction neither the longitudinal magnetization versus storage time nor the echo-amplitude versus gradient strength (squared) could be described by single exponential functions. In order to characterize these multi-exponential decay curves by a minimum number of parameters a gaussian type of distribution function (Rayleigh distribution) in 1/ T 1 and D were adopted. When implementing these distribution functions and fitting all spin-lattice relaxation data and diffusion data simultaneously, in order to constrain the fitting more effectively, the two n.m.r. derived parameters (1/ T 1 and D) were found to give consistent results. During gelation the average relaxation rate and the average diffusion coefficient versus reaction time were found to be described by a first order rate process with a rate constant equal to 18×10 −5 s −1. Also, the widths of the two distribution functions were found to decrease with reaction time. Moreover, the gelation rate within the Bentheimer sandstone was found to be significantly faster compared to the gelation rate of the bulk solution.

NMR characterization of hydrocarbon gas in porous media

Laboratory investigations of nuclear magnetic resonance (NMR) relaxation properties of methane gas in bulk and in porous media are reported. Measurements were performed for methane alone and together with water in glass bead packs and Bentheimer sandstone. Results indicate that surface relaxation effects can be significant and that diffusion effects dominate observed T 2 relaxation.

Fluid transport in glass beads phantoms: spatial velocity measurements and confirmation of the stochastic model

A stochastic model of fluid flow in porous rocks has been previously developed to explain the measured distribution of local velocity. The theoretical predictions of this model agree well with experimental results obtained from the magnetic resonance imaging-based measurements of the spatial variation of velocity of water permeating through Bentheimer and Clashach sandstones. To further verify previous results, we have performed new velocity measurement experiments using an efficient velocity encoded π-echo planar imaging sequence on glass bead phantoms that exhibit more regular pore size distribution than rocks. The results show that velocity distributions in glass bead phantoms also exhibit Gaussian profiles and the linear relationship between the velocity variance and the mean velocity (the Mansfield-Issa equation).

3-D Pore-Scale Modelling of Sandstones and Flow Simulations in the Pore Networks

Summary A new method for generating realistic homogenous and heterogeneous 3-D pore-scale sandstone models is presented. The essence of our method is to build sandstone models which are analogs of actual sandstones by numerically modelling the results of the main sandstone-forming geological processes - sandgrain sedimentation, compaction, and diagenesis. The input data for the modelling are obtained from image analyses of thin section images of the actual sandstone. The spatial continuity of the sandstone model in the X, Y, and Z directions is determined using a scale-independent invasion percolation based algorithm. The resulting spatial continuity function, which is an ellipsoid, may be used as a heterogeneity descriptor for the sandstone model. Heterogeneity analyses show that compaction reduces the spatial continuity in the horizontal direction more rapidly than in the vertical one. The architecture and geometry of the network representation of the pore space are determined by applying various 3-D image analysis algorithms directly on the fully characterised sandstone model. A 3-D pore network which was generated from thin section data from a strongly water wet Bentheimer sandstone is used as input to a two-phase network flow simulator. Simulated transport properties for the sandstone model are in good agreement with those determined experimentally.

Fluid Transport in Porous Rocks. II. Hydrodynamic Model of Flow and Intervoxel Coupling

In a preceding paper [P. Mansfield and B. Issa,J. Magn. Reson. A122, 137–148 (1996)], a stochastic model of fluid flow in porous rocks based upon the experimental observation of water flow through a Bentheimer sandstone core was proposed. The flow maps were measured by NMR-imaging techniques. The stochastic theory led to a Gaussian velocity distribution with a mean value in accord with Darcy's law. Also predicted was a linear relationship between flow variance and mean fluid flow through rock, the Mansfield–Issa equation, originally proposed as an empirical relationship. In the present work a flow coupling mechanism between voxels is proposed. Examination of the flow coupling between isolated voxel pairs leads to a complementary explanation of the Gaussian velocity distribution, and also gives further details of the Mansfield–Issa equation. These details lead to a new expression for the connectivity, 〈C〉, between voxels with an experimental value of 〈C〉 = 5.64 × 10−9for Bentheimer sandstone.