Porosity Fields Research Articles

Generation of overpressure and compaction‐driven fluid flow in a Plio‐Pleistocene growth‐faulted basin, Eugene Island 330, offshore Louisiana

The complex pressure and porosity fields observed in the Eugene Island (EI) 330 field (offshore Louisiana) are thought to result from sediment loading of low‐permeability strata. In this field, fluid pressures rise with depth from hydrostatic to nearly lithostatic, iso‐pressure surfaces closely follow stratigraphic surfaces which are sharply offset by growth‐faulting, and porosity declines with effective stress. A one‐dimensional hydrodynamic model simulates the evolution of pressure and porosity in this system. If reversible (elastic) compaction is assumed, sediment loading is the dominant source of overpressure (94%). If irreversible (inelastic) compaction and permeability reduction due to clay diagenesis are assumed, then thermal expansion of pore fluids and clay dehydration provide a significant component of overpressure (>20%). The model is applied to wells on the upthrown and downthrown sides of the major growth fault in the EI 330 field. Assuming that sediment loading is the only pressure source and that permeability is a function of lithology and porosity, the observed pressure and porosity profiles are reproduced. Observation and theory support a conceptual model where hydrodynamic evolution is intimately tied to the structural and stratigraphic evolution of this progradational deltaic system.

Mass and fluid flux during accretion at the Alaskan margin

Prestack depth migration of a seismic reflection transect across the Alaskan convergent margin clarified structure and provided an undistorted tectonic cross section. Between the trench floor and a backstop of eroded Eocene–Miocene rock, the accretionary prism displays three tectonic styles. Seaward of the deformation front, trench sediment is cut by incipient faults confined to the subsurface, and distributed deformation. In the 10 km landward of the deformation front, thrust faults develop into imbricate structures above a lower detachment. In the remaining 20 km, longer and more steeply dipping thrust units form a classical imbricate prism above an upper detachment. A velocity model at the scale of the thrust slices was derived from depth processing and seismic refraction data. The resulting true-scale tectonic cross section was balanced to estimate horizontal contraction. A porosity field was derived through the velocity-porosity relation in five nearby drill holes. It indicates porosity reduction patterns characteristic of each tectonic style. Seaward of the deformation front, the strain rate is high, and pore-fluid pressure elevates, but porosity reduction is minimal. In the 10 km landward of the deformation front, tectonism and dewatering are most rapid: sea-floor venting was observed only here. In the classic accretionary zone, strain rates decrease and dewatering is moderate. Here an upper detachment propagates across the lower parts of imbricate slices and doubles the underthrust mass. One-third of the trench sediment section is underthrust at the deformation front, whereas two-thirds of the sediment section is subducted beneath the backstop.

The Sensitivity of Tracer Concentration to Nonuniform Permeability and Porosity

Concentration of tracer at a producing well is a function of time and of the flow field that transports the tracer from one well to another. Small localized changes in the permeability and porosity fields result in small changes in the observed concentration history. Calculation of the effect of changes in the petrophysical parameter fields on observed concentration is an integral part of most approaches to solving the inverse problem, i.e. calculation of permeability and porosity fields that are consistent with observed tracer concentrations. This paper does not address the inverse problem directly, only the forward problem, but the results are presented as semi-analytic formulas which are intended to provide physical insight into the limitations of the inverse procedure. In particular, for a simple example of dipole flow between two wells in a nearly homogeneous infinite 2D porous medium, the width of the region of significant influence is shown to scale as √αd where α is the dispersivity and d is half the interwell distance. Also, for fixed injection and production rates the influence on concentration of variation in porosity and in log-permeability are shown to be similar in magnitude.

Three-Dimensional Reservoir Description From Multiwell Pressure Data and Prior Information

SummaryBayesian estimation techniques are applied to generate three-dimensional permeability and porosity fields conditioned to prior means, variograms, hard data and/or multiwell pressure data. A posteriori variances are generated to obtain a measure of the uncertainty in the rock property fields, or equivalently, a measure of the variability in realizations of the rock property fields. These a posteriori variances can also be used to quantify the value of data, i.e., to determine the reduction in uncertainty achieved by adding a particular type of data. A key ingredient of our methodology is the development and implementation of an efficient procedure to estimate sensitivity coefficients for three-dimensional single-phase flow problems.

EVOLUTION OF POROSITY DISTRIBUTION FOR ONE-DIMENSIONAL PROBLEM OF VISCOUS SINTERING

SUMMARY Viscous sintering of a porous ball with various initial distributions of porosity versus radius is considered. For the solution of the corresponding boundary-value problems of the evolution of porosity and flow velocity fields during sintering, the numerical algorithms based on the diAerential quadrature method (DQM) and an arbitrary Eulerian‐Lagrangian version of the finite element method (FEM) (the permeable element method) are elaborated. A comparative analysis of the calculation results is carried out. The question of the influence of non-uniformity of porosity distribution on the localization of densification is discussed. #1997 by John Wiley & Sons, Ltd. Commun. numer. meth. engng., 13, 355‐372 (1997). No. of Figures: 9 No. of Tables: 1 No. of

Failure of a Porous Solid from a Deep Notch

A finite strain finite element method is used to examine the stress state near the tip of a deep notch in an elastic-plastic porous solid. The notch is loaded in mode I plane strain tension and small scale yielding is assumed. Two rate independent strain hardening material models are used: a version of the Gurson model (1977) and the more recent FKM model developed by Fleck, Kuhn and McMeeking (1992). Under increasing KI, void growth is initially stable and independent of mesh dimension. Localization of plastic flow sets in at a finite value Ki, and the deformation field is mesh-size dependent thereafter. The initiation of crack growth at the notch root is assumed to occur when a critical level of porosity is attained. The results show that the shape of the plastic zone for both the Gurson and the FKM material is highly dependent on the initial porosity. In the case of low initial porosity, the plastic zone shape is similar to that of a fully dense material; at higher initial porosities the plastic zone is concentrated ahead of the notch tip. The effect of the initial void volume fraction on the porosity field and the critical stress intensity factor is studied, and the mesh-size dependence of the results is discussed. The analysis is useful for prediction of the notched strength of porous metals.

Fragmentation and heave modelling using a coupled discrete element gas flow code

Abstract Experimental evidence indicates that a large percentage of the crack development and subsequent fragmentation and heave in the blasting process is due to the effects of the high pressure gas generated by the explosive in the blasthole. An explicit finite/discrete element code has been developed to dynamically model stress field development, crack generation and crack growth as well as the motion and stacking of the rock fragments. Cracking in the current model is limited to failure under tension (Rankine). This failure is due to both the stress and the gas flow which is treated by coupling the discrete element porosity field to a finite volume gas flow model. Several examples are discussed which show striking qualitative differences between blasting with stemming to contain the gases in the blastholes, and without stemming.

On the continuum modelling of porous media containing fluid: a molecular viewpoint with particular attention to scale

Mass conservation and linear momentum balance relations for a porous body and any fluid therein, valid at any given length scale in excess of nearest-neighbour molecular separations, are established in terms of local weighted averages of molecular quantities. The mass density field for the porous body at a given scale is used to identify its boundary at this scale, and a porosity field is defined for any pair of distinct length scales. Specific care is paid to the interpretation of the stress tensor associated with each of the body and fluid at macroscopic scales, and of the force per unit volume each exerts on the other. Consequences for the usual microscopic and macroscopic viewpoints are explored.

Correlation of geometry effects with fracture toughness by damage equivalence

The near crack tip porosity fields in different fracture specimens, single edge notched, single edge notched loaded in centre of ligament, three point bending specimens, and small scale yielding (SSY) mode have been studied by the finite deformation finite element method. The presence and subsequent growth of smaller-scale voids were taken into account by using Gurson's model to describe the constitutive behaviour of the material. Based on damage equivalence at a characteristic position in the SSY mode and actual fracture specimens, the ratio of scaling parameters,J values, in both modes was obtained, and was used to eliminate the geometry dependence of fracture toughness through correlations to the small scale yielding mode.

Upscaling Capillary Pressure‐Saturation Curves in Heterogeneous Porous Media

Numerical models of flow in the vadose zone require capillary pressure‐saturation relationships for the grid blocks discretizing the medium. However, the scale of these grid blocks is generally much larger than the sample scale at which capillary pressure‐saturation relationships are measured. This study investigates the upscaling of capillary curves under static conditions of capillary‐gravity equilibrium. The averaging regions are horizontal layers, parallel to the phreatic surface, as in one‐dimensional models of infiltration. At the point or sample scale it is assumed that all capillary curves can be normalized using the Leverett (1941∥ approach to a common dimensionless form described by the Corey‐Brooks model. Static water saturation at any point in the medium is then expressed as a parametric function of three regionalized variables, permeability, porosity, and irreducible water saturation. Water saturations at the layer scale are obtained by spatial averaging of point‐scale values. With certain assumptions it is shown that the nondimensional relationship between capillary pressure and saturation at the layer scale is the same as at the sample scale. In general, the nondimensional scaling of layer‐averaged capillary pressure requires exhaustive knowledge of the spatial distribution of the permeability‐porosity ratio within the averaging region. However, when permeability and porosity are jointly lognormal, this spatial distribution may be approximated by a lognormal model requiring only two parameters for each averaging region. The developments are illustrated using published field data from the Apache Leap tuff site. Exact and estimated upscaled capillary curves are compared in simulated coregionalized fields of permeability, porosity, and irreducible water saturations. Results show that the upscaled capillary pressure‐saturation curves accurately reproduce the vertical saturation profile obtained by exhaustive spatial averaging of point‐scale water saturations.

Bias in estimating fractal dimension with the rescaled-range (R/S) technique

Fractal geostatistics are being applied to subsurface geological data as a way of predicting the spatial distribution of hydrocarbon reservoir properties. The fractal dimension is the controlling parameter in stochastic methods to produce random fields of porosity and permeability. Rescaled range (R/S)analysis has become a popular way of estimating the fractal dimension, via determination of the Hurst exponent (H). A systematic investigation has been undertaken of the bias to be expected due to a range of factors commonly inherent in borehole data, particularly downhole wireline logs. The results are integrated with a review of previous work in this area. Small datasets. overlapping samples, drift and nonstationariry of means can produce a very large bias, and convergence of estimates of H around 0.85–0.90 regardless of original fractal dimension. Nonstationarity can also account for H>1, which has been reported in the literature but which is theoretically impossible for fractal time series. These results call into question the validity of fractal stochastic models built using fractal dimensions estimated with the R/Smethod.

Horizontal Permeability Anisotropy Developed from the Upscaling of Anisotropic Facies Patterns: ABSTRACT

Traditionally, the size and shape of grid blocks owe more to the limitations of numerical simulation procedures than to any consideration of geological heterogeneity. Geological systems, and hence associated permeability and porosity fields, are inherently complex, being characterized by nested levels of heterogeneity. Moreover, for directionally dependent variables, such as permeability, a heterogeneous nature at small scales will lead directly to anisotropy in average (or measured) values at larger scales. The representation of geologically complex volumes as individual horizontal and vertical permeability values has long been recognized as a necessary simplification. Recent advances in stochastic modeling have vastly improved the representation of complex geology in unknown interwell regions. The high level of detail provided by such models often requires several million grid blocks. This degree of detail cannot, however, be used in flow simulations, and hence, scaling up is necessary. The difficulties in scaling up of rock-property data are based on the conflicting requirements of significantly reducing the number of grid blocks, while maintaining a reasonable degree of detail in the representation of heterogeneity. To analyze whether grid blocks adequately represent complex heterogeneities the authors constructed complex, geologically realistic channel-levee complexes, using an object-modeling (Boolean) approach. This approach has themore » advantage over reservoir, or outcrop, derived data in so much as the 3-D architecture is fully known. The disadvantages are that the realizations are too simplistic, not necessarily geologically realistic, and contain only one scale of heterogeneity. In the present context, however, simplification of both geological and scale-related heterogeneities can be seen as advantageous in the interpretation of the results.« less

Applying Simple Geostatistical Techniques to a Routine Production Geology Problem--A Case Study: ABSTRACT

A production geology reservoir description was shown to represent poorly the known dynamic reservoir behavior. The permeability field was originally generated from a general porosity-permeability law applied to a contoured porosity field. This resulted in unrealistically high permeability values in some layers and early water breakthrough in flow simulations. Furthermore, known well values of permeability were not honored. To improve the model, a geostatistical approach was used. The first step involved the use of a porosity-permeability law based on total layer porosity values. That is, with no cutoff applied, thereby allowing obtainment of values of k closer to the known reservoir values. This alone produces a permeability field, which although better represents the absolute permeability values is still too smooth-preferential pathways occurring as artifacts of contouring. A second step, therefore, involves the [open quotes]unsmoothing[close quotes] of the permeability field. At each point of the permeability field the permeability values are resampled from a distribution centered around the original value. This is a form of Monte-Carlo replacement, conditioned to well data and honoring the original trend to the data. In this case no correlation was included due to lack of information. This simple Gaussian simulation approach provides multiple realizations based onmore » deterministic information. The advantages are clear, more realistic images (no artificial pathways), improved match of water breakthrough, and the honoring of all deterministic data (wells and trend). A subsequent step was the incorporation of further [open quotes]soft[close quotes] information. Geological analysis suggested that there was a degradation of reservoir properties along the east-west axis. This new information was rapidly assimilated into the model to produce final images of the reservoir.« less

Correlation model of transport in random fields

This paper is an investigation of the correlation model for transport of nonreactive solutes in media with random porosity and permeability. The method of perturbation is used to obtain a second‐order approximate, nonlocal (integrodifferential) equation for mean concentration. An approximate method of localization is used to convert to with the same order of approximation differential equations of transport. Exact averaged equations for one‐dimensional transport are examined, and the question of the consistency and asymptotic behavior for approximate averaged equations is discussed. A detailed investigation of transport in a stratified system has been carried out. The second moment of concentration is examined, the variance of the concentration is computed, and cross‐correlation moments are obtained for fields of porosity and velocity, including solute concentration.

Transport in porous packings: Statistical characterization of transport, role of fluctuation and data analysis

AbstractThe study of dynamics of transport in porous packings (in particular chromatographic columns) represents an illuminating step towards the statistical and probabilistic analysis of transport phenomena and can be generalized to other transport conditions of environmental interest. This paper addresses the use of Monte Carlo simulation in the analysis of transport in porous media, the local characterization of porosity fields in disordered models of pore networks, and the analysis of dispersion in terms of velocity fluctuations, and presents a simplified treatment of experimental data arising from transport problems.

A three-dimensional numerical investigation of fracture initiation by ductile failure mechanisms in a 4340 steel

Fracture initiation in ductile metal plates occurs due to substantial tunneling of the crack in the interior of the specimen followed by final failure of side ligaments by shear lip formation. The tunneled region is characterized by a flat, fibrous fracture surface. This phenomenon is clearly exhibited in a recent experimental investigation [8] performed on pre-notched plates of a ductile heat treatment of 4340 carbon steel. Experimental evidence obtained in [8] suggests that tunneling begins at an average value of J which is significantly lower than the J value at which gross initiation is observed on the free surface. In the present work, fracture initiation in the 4340 steel specimens used in [8] is analyzed by performing a 3-dimensional numerical simulation. A damage accumulation model that accounts for the ductile failure mechanisms of void nucleation, growth, and void coalescence is employed. Results indicate that incipient Cmaterial failure at the center-plane of the 3-dimensional specimen is predicted quite accurately by this computation. Also, good agreement between results obtained at the center-plane of the 3-dimensional specimen and a plane strain analysis, suggests that a local definition of J can be used to characterize fracture initiation in the center-plane of the specimen. Finally, radial and thickness variations of the stress and porosity fields are examined with view of understanding the subsequent propagation of the failure zone.

The determination of fracture toughness for a porous elastic-plastic solid

The near tip stress and porosity fields in a plane strain mode I crack under small scale yielding (s.s.y.) conditions have been studied using a constitutive model for porous elastic-plastic solids. These fields have been used in conjunction with criteria for fracture of tensile specimens to predict values of KIc for a mild steel over a range of temperatures. The regimes of dominance of fracture by slip-induced cleavage and void coalescence mechanisms have been determined. The predictions are shown to compare well with experimental data.

Statistical characteristics of the motion of a fluid particle in a medium with random porosity

In the study of flow of a neutral admixture in a porous medium, it is most often assumed in the stochastic formulation that the porosity is constant and a determinate quantity, and the velocity is a random function [1–4]. The velocity distribution is usually regarded as known. Flow in a porous medium with random porosity has been studied to a far lesser extent. We note [5], which studies the averaged equations obtained within the framework of the correlation approximation. We consider the model problem of one-dimensional motion of a fluid particle (position of the front for flow of a neutral admixture in a porous medium) in a medium with random porosity. For a particular form of random porosity field, expressions are obtained for the one- and two-point densities of the distribution of the position of the particle. A study is made of the dependences of the first four moments and the correlation function of the position of the particle as functions of the time. It is shown that for large values of the time the motion of the particle is asymptotically similar to Brownian motion. It is shown by means of numerical modeling that the results obtained transfer to the case of an arbitrary random porosity field.

Anisotropy and microstructure analysis by magnetic measurement

Analysis of digital data from the approach-to-saturation region of the hysteresis loop has been made convenient through the use of a microprocessor-controlled hysteresigraph. To evaluate microstructure and magnetocrystalline anisotropy effects on the high field part of the magnetization curve three analysis techniques are considered. For the situation where magnetic saturation is incomplete, the theory is modified in a manner which permits the determination of the a and b coefficients of the approach-to-saturation equation from the intercept and slope of a single curve. With each method, experimental results yielded reasonably accurate estimates of the effective porosity and anisotropy fields from a variety of polycrystalline magnetic materials.

Electrostatically Enhanced Granular Bed Filters

A mathematical model for the filtration of dust in granular filters under the influence of a Coulombic force is developed. The paper considers the effect of filter porosity and external field orientation on the filtration efficiency of inertialess, inertial, and diffusional aerosols. Calculated filtration efficiencies may be used to design electrostatically enhanced granular filters.