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Geomechanical Analysis with Rigorous Error Estimates for a Double-Porosity Reservoir Model

Description: A model of random polycrystals of porous laminates is introduced to provide a means for studying geomechanical properties of double-porosity reservoirs. Calculations on the resulting earth reservoir model can proceed semi-analytically for studies of either the poroelastic or transport coefficients. Rigorous bounds of the Hashin-Shtrikman type provide estimates of overall bulk and shear moduli, and thereby also provide rigorous error estimates for geomechanical constants obtained from up-scaling based on a self-consistent effective medium method. The influence of hidden (or presumed unknown) microstructure on the final results can then be evaluated quantitatively. Detailed descriptions of the use of the model and some numerical examples showing typical results for the double-porosity poroelastic coefficients of a heterogeneous reservoir are presented.
Date: April 11, 2005
Creator: Berryman, J G
Partner: UNT Libraries Government Documents Department

Effective Medium Theories for Multicomponent Poroelastic Composites

Description: In Biot's theory of poroelasticity, elastic materials contain connected voids or pores and these pores may be filled with fluids under pressure. The fluid pressure then couples to the mechanical effects of stress or strain applied externally to the solid matrix. Eshelby's formula for the response of a single ellipsoidal elastic inclusion in an elastic whole space to a strain imposed at a distant boundary is a very well-known and important result in elasticity. Having a rigorous generalization of Eshelby's results valid for poroelasticity means that the hard part of Eshelby's work (in computing the elliptic integrals needed to evaluate the fourth-rank tensors for inclusions shaped like spheres, oblate and prolate spheroids, needles and disks) can be carried over from elasticity to poroelasticity--and also thermoelasticity--with only relatively minor modifications. Effective medium theories for poroelastic composites such as rocks can then be formulated easily by analogy to well-established methods used for elastic composites. An identity analogous to Eshelby's classic result has been derived [Physical Review Letters 79:1142-1145 (1997)] for use in these more complex and more realistic problems in rock mechanics analysis. Descriptions of the application of this result as the starting point for new methods of estimation are presented, including generalizations of the coherent potential approximation (CPA), differential effective medium (DEM) theory, and two explicit schemes. Results are presented for estimating drained shear and bulk modulus, the Biot-Willis parameter, and Skempton's coefficient. Three of the methods considered appear to be quite reliable estimators, while one of the explicit schemes is found to have some undesirable characteristics.
Date: February 8, 2005
Creator: Berryman, J G
Partner: UNT Libraries Government Documents Department

Measures of microstructure to improve estimates and bounds on elastic constants and transport coefficients in heterogeneous media

Description: The most commonly discussed measures of microstructure in composite materials are the spatial correlation functions, which in a porous medium measure either the grain-to-grain correlations, or the pore-to-pore correlations in space. Improved bounds based on this information such as the Beran-Molyneux bounds for bulk modulus and the Beran bounds for conductivity are well-known. It is first shown here how to make direct use of this information to provide estimates that always lie between these upper and lower bounds for any microstructure whenever the microgeometry parameters are known. Then comparisons are made between these estimates, the bounds, and two new types of estimates. One new estimate for elastic constants makes use of the Peselnick-Meister bounds (based on Hashin-Shtrikman methods) for random polycrystals of laminates to generate self-consistent values that always lie between the bounds. A second new type of estimate for conductivity assumes that measurements of formation factors (of which there are at least two distinct types in porous media, associated respectively with pores and grains) are available, and computes new bounds based on this information. The paper compares and contrasts these various methods in order to clarify just what microstructural information and how precisely that information needs to be known in order to be useful for estimating material constants in random and heterogeneous media.
Date: October 7, 2004
Creator: Berryman, J G
Partner: UNT Libraries Government Documents Department

Bounds and self-consistent estimates for elastic constants of random polycrystals with hexagonal, trigonal, and tetragonal symmetries

Description: Peselnick, Meister, and Watt have developed rigorous methods for bounding elastic constants of random polycrystals based on the Hashin-Shtrikman variational principles. In particular, a fairly complex set of equations that amounts to an algorithm has been presented previously for finding the bounds on effective elastic moduli for polycrystals having hexagonal, trigonal, and tetragonal symmetries. The more analytical approach developed here, although based on the same ideas, results in a new set of compact formulas for all the cases considered. Once these formulas have been established, it is then straightforward to perform what could be considered an analytic continuation of the formulas (into the region of parameter space between the bounds) that can subsequently be used to provide self-consistent estimates for the elastic constants in all cases. These self-consistent estimates are easily shown (essentially by construction) to lie within the bounds for all the choices of crystal symmetry considered. Estimates obtained this way are quite comparable to those found by the Gubernatis and Krumhansl CPA (coherent potential approximation), but do not require any computations of scattering coefficients.
Date: September 16, 2004
Creator: Berryman, J G
Partner: UNT Libraries Government Documents Department

Estimates and Rigorous Bounds on Pore-Fluid Enhanced Shear Modulus in Poroelastic Media with Hard and Soft Anisotropy

Description: A general analysis of poroelasticity for hexagonal, tetragonal, and cubic symmetry shows that four eigenvectors are pure shear modes with no coupling to the pore-fluid mechanics. The remaining two eigenvectors are linear combinations of pure compression and uniaxial shear, both of which are coupled to the fluid mechanics. The analysis proceeds by first reducing the problem to a 2 x 2 system. The poroelastic system including both anisotropy in the solid elastic frame (i.e., with ''hard anisotropy''), and also anisotropy of the poroelastic coefficients (''soft anisotropy'') is then studied in some detail. In the presence of anisotropy and spatial heterogeneity, mechanics of the pore fluid produces shear dependence on fluid bulk modulus in the overall poroelastic system. This effect is always present (though sometimes small in magnitude) in the systems studied, and can be comparatively large (up to a maximum increase of about 20 per cent) in some porous media--including porous glass and Schuler-Cotton Valley sandstone. General conclusions about poroelastic shear behavior are also related to some recently derived product formulas that determine overall shear response of these systems. Another method is also introduced based on rigorous Hashin-Shtrikman-style bounds for nonporous random polycrystals, followed by related self-consistent estimates of mineral constants for polycrystals. Then, another self-consistent estimation method is formulated for the porous case, and used to estimate drained and undrained effective constants. These estimates are compared and contrasted with the results of the first method and a consistent picture of the overall behavior is found in three computed examples for polycrystals of grains having tetragonal symmetry.
Date: January 24, 2005
Creator: Berryman, J G
Partner: UNT Libraries Government Documents Department

Seismic Waves in Rocks with Fluids and Fractures

Description: Seismic wave propagation through the earth is often strongly affected by the presence of fractures. When these fractures are filled with fluids (oil, gas, water, CO{sub 2}, etc.), the type and state of the fluid (liquid or gas) can make a large difference in the response of the seismic waves. This paper will summarize some early work of the author on methods of deconstructing the effects of fractures, and any fluids within these fractures, on seismic wave propagation as observed in reflection seismic data. Methods to be explored here include Thomsen's anisotropy parameters for wave moveout (since fractures often induce elastic anisotropy), and some very convenient fracture parameters introduced by Sayers and Kachanov that permit a relatively simple deconstruction of the elastic behavior in terms of fracture parameters (whenever this is appropriate).
Date: February 6, 2006
Creator: Berryman, J G
Partner: UNT Libraries Government Documents Department

Role of Double-Porosity Dual-Permeability Models for Multi-Resonance Geomechanical Systems

Description: It is known that Biot's equations of poroelasticity (Biot 1956; 1962) follow from a scale-up of the microscale equations of elasticity coupled to the Navier-Stokes equations for fluid flow (Burridge and Keller, 1981). Laboratory measurements by Plona (1980) have shown that Biot's equations indeed hold for simple systems (Berryman, 1980), but heterogeneous systems can have quite different behavior (Berryman, 1988). So the question arises whether there is one level--or perhaps many levels--of scale-up needed to arrive at equations valid for the reservoir scale? And if so, do these equations take the form of Biot's equations or some other form? We will discuss these issues and show that the double-porosity dual-permeability equations (Berryman and Wang, 1995; Berryman and Pride, 2002; Pride and Berryman, 2003a,b; Pride et al., 2004) play a special role in the scale-up to equations describing multi-resonance reservoir behavior, for fluid pumping and geomechanics, as well as seismic wave propagation. The reason for the special significance of double-porosity models is that a multi-resonance system can never be adequately modeled using a single resonance model, but can often be modeled with reasonable accuracy using a two-resonance model. Although ideally one would prefer to model multi-resonance systems using the correct numbers, locations, widths, and amplitudes of the resonances, data are often inadequate to resolve all these pertinent model parameters in this complex inversion task. When this is so, the double-porosity model is most useful as it permits us to capture the highest and lowest detectable resonances of the system and then to interpolate through the middle range of frequencies.
Date: May 18, 2005
Creator: Berryman, J G
Partner: UNT Libraries Government Documents Department

Poroelastic measurement schemes resulting in complete data sets for granular and other anisotropic porous media

Description: Poroelastic analysis usually progresses from assumed knowledge of dry or drained porous media to the predicted behavior of fluid-saturated and undrained porous media. Unfortunately, the experimental situation is often incompatible with these assumptions, especially when field data (from hydrological or oil/gas reservoirs) are involved. The present work considers several different experimental scenarios typified by one in which a set of undrained poroelastic (stiffness) constants has been measured using either ultrasound or seismic wave analysis, while some or all of the dry or drained constants are normally unknown. Drained constants for such a poroelastic system can be deduced for isotropic systems from available data if a complete set of undrained compliance data for the principal stresses are available - together with a few other commonly measured quantities such as porosity, fluid bulk modulus, and grain bulk modulus. Similar results are also developed here for anisotropic systems having up to orthotropic symmetry if the system is granular (i.e., composed of solid grains assembled into a solid matrix, either by a cementation process or by applied stress) and the grains are known to be elastically homogeneous. Finally, the analysis is also fully developed for anisotropic systems with nonhomogeneous (more than one mineral type), but still isotropic, grains - as well as for uniform collections of anisotropic grains as long as their axes of symmetry are either perfectly aligned or perfectly random.
Date: November 20, 2009
Creator: Berryman, J.G.
Partner: UNT Libraries Government Documents Department

Computing tomographic resolution matrices using Arnoldi`s itertive inversion algorithm

Description: Resolution matrices are useful in seismic tomography because they allow us to evaluate the information content of reconstructed images. Techniques based on the multiplicity of equivalent exact formulas that may be used to define the resolution matrices have been used previously by the author to design algorithms that avoid the need for any singular value decomposition of the ray-path matrix. An explicit procedure is presented for computing both model and data resolution matrices using Arnoldi`s algorithm for iterative inversion in seismic tomography. Arnoldi`s method differs from the Lanczos scheme by including explicit reorthogonalization of basic vectors. Some convenient notation is introduced to permit ready comparison of Arnoldi`s method with the Lanczos approach. Arnoldi`s method requires greater storage of basic vectors but completely overcomes the lack of basis vector orthogonality, which is the major practical limitation of the Lanczos method.
Date: September 1, 1994
Creator: Berryman, J.G.
Partner: UNT Libraries Government Documents Department

Nonlinear least squares and regularization

Description: A problem frequently encountered in the earth sciences requires deducing physical parameters of the system of interest from measurements of some other (hopefully) closely related physical quantity. The obvious example in seismology (either surface reflection seismology or crosswell seismic tomography) is the use of measurements of sound wave traveltime to deduce wavespeed distribution in the earth and then subsequently to infer the values of other physical quantities of interest such as porosity, water or oil saturation, permeability, etc. The author presents and discusses some general ideas about iterative nonlinear output least-squares methods. The main result is that, if it is possible to do forward modeling on a physical problem in a way that permits the output (i.e., the predicted values of some physical parameter that could be measured) and the first derivative of the same output with respect to the model parameters (whatever they may be) to be calculated numerically, then it is possible (at least in principle) to solve the inverse problem using the method described. The main trick learned in this analysis comes from the realization that the steps in the model updates may have to be quite small in some cases for the implied guarantees of convergence to be realized.
Date: April 1, 1996
Creator: Berryman, J.G.
Partner: UNT Libraries Government Documents Department

Double porosity modeling in elastic wave propagation for reservoir characterization

Description: Phenomenological equations for the poroelastic behavior of a double porosity medium have been formulated and the coefficients in these linear equations identified. The generalization from a single porosity model increases the number of independent coefficients from three to six for an isotropic applied stress. In a quasistatic analysis, the physical interpretations are based upon considerations of extremes in both spatial and temporal scales. The limit of very short times is the one most relevant for wave propagation, and in this case both matrix porosity and fractures behave in an undrained fashion. For the very long times more relevant for reservoir drawdown,the double porosity medium behaves as an equivalent single porosity medium At the macroscopic spatial level, the pertinent parameters (such as the total compressibility) may be determined by appropriate field tests. At the mesoscopic scale pertinent parameters of the rock matrix can be determined directly through laboratory measurements on core, and the compressibility can be measured for a single fracture. We show explicitly how to generalize the quasistatic results to incorporate wave propagation effects and how effects that are usually attributed to squirt flow under partially saturated conditions can be explained alternatively in terms of the double-porosity model. The result is therefore a theory that generalizes, but is completely consistent with, Biot`s theory of poroelasticity and is valid for analysis of elastic wave data from highly fractured reservoirs.
Date: June 1998
Creator: Berryman, J. G.
Partner: UNT Libraries Government Documents Department

Transversely isotropic elasticity and poroelasticity arising from thin isotropic layers

Description: Since the classic work of Postma [1955] and Backus [1962], much has been learned about elastic constants in vertical transversely isotropic (VTI) media when the anisotropy is due to fine layering of isotropic elastic materials. However, new results are still being discovered. For example, the P-wave anisotropy parameter c{sub 11}/c{sub 33} lies in the range 1/4 {<=} c{sub 11}/c{sub 33} {<=} <{lambda}+2{mu}><1/({lambda}+2{mu})>, when the layers are themselves composed of isotropic elastic materials with Lame constants {lambda} and {mu} and the vertical average of the layers is symbolized by <{center_dot}>. The lower bound corrects a result of Postma. For porous layers, a connected solid frame forms the basis of the elastic behavior of a poroelastic medium in the presence of confining forces, while connected pores permit a percolating fluid (if present) to influence the mechanical response of the system from within. For isotropic and anisotropic poroelastic media, we establish general formulas for the behavior of transversely isotropic poroelasticity arising from laminations of isotropic components. The Backus averaging method is shown to provide elementary means of constructing general formulas. The results for confined fluids are then compared with the more general Gassmann [1951] formulas that must be satisfied by any anisotropic poroelastic medium and found to be in complete agreement. Such results are important for applications to oil exploration using AVO (amplitude versus offset) since the presence or absence of a fluid component, as well as the nature of the fluid, is the critical issue and the ways in which the fluid influences seismic reflection data still need to be better understood.
Date: July 1, 1997
Creator: Berryman, J.G.
Partner: UNT Libraries Government Documents Department

Variational structure of inverse problems in wave propagation and vibration

Description: Practical algorithms for solving realistic inverse problems may often be viewed as problems in nonlinear programming with the data serving as constraints. Such problems are most easily analyzed when it is possible to segment the solution space into regions that are feasible (satisfying all the known constraints) and infeasible (violating some of the constraints). Then, if the feasible set is convex or at least compact, the solution to the problem will normally lie on the boundary of the feasible set. A nonlinear program may seek the solution by systematically exploring the boundary while satisfying progressively more constraints. Examples of inverse problems in wave propagation (traveltime tomography) and vibration (modal analysis) will be presented to illustrate how the variational structure of these problems may be used to create nonlinear programs using implicit variational constraints.
Date: March 1, 1995
Creator: Berryman, J.G.
Partner: UNT Libraries Government Documents Department

Pore Fluid Effects on Shear Modulus in a Model of Heterogeneous Rocks, Reservoirs, and Granular Media

Description: To provide quantitative measures of the importance of fluid effects on shear waves in heterogeneous reservoirs, a model material called a ''random polycrystal of porous laminates'' is introduced. This model poroelastic material has constituent grains that are layered (or laminated), and each layer is an isotropic, microhomogeneous porous medium. All grains are composed of exactly the same porous constituents, and have the same relative volume fractions. The order of lamination is not important because the up-scaling method used to determine the transversely isotropic (hexagonal) properties of the grains is Backus averaging, which--for quasi-static or long-wavelength behavior--depends only on the volume fractions and layer properties. Grains are then jumbled together totally at random, filling all space, and producing an overall isotropic poroelastic medium. The poroelastic behavior of this medium is then analyzed using the Peselnick-Meister-Watt bounds (of Hashin-Shtrikman type). We study the dependence of the shear modulus on pore fluid properties and determine the range of behavior to be expected. In particular we compare and contrast these results to those anticipated from Gassmann's fluid substitution formulas, and to the predictions of Mavko and Jizba for very low porosity rocks with flat cracks. This approach also permits the study of arbitrary numbers of constituents, but for simplicity the numerical examples are restricted here to just two constituents. This restriction also permits the use of some special exact results available for computing the overall effective stress coefficient in any two-component porous medium. The bounds making use of polycrystalline microstructure are very tight. Results for the shear modulus demonstrate that the ratio of compliance differences R (i.e., shear compliance changes over bulk compliance changes when going from drained to undrained behavior, or vice versa) is usually nonzero and can take a wide range of values, both above and below the value R = 4/15 ...
Date: March 23, 2005
Creator: Berryman, J G
Partner: UNT Libraries Government Documents Department

Target characterization using decomposition of the time-reversal operator: electromagnetic scattering from small ellipsoids

Description: Decomposition of the time-reversal operator for an array, or equivalently the singular value decomposition of the multistatic response matrix, has been used to improve imaging and localization of targets in complicated media. Typically, each singular value is associated with one scatterer even though it has been shown in several cases that a single scatterer can generate several singular values. In earlier papers Chambers and Berryman [1, 2] showed that a small spherical scatterer can generate up to six singular values depending on the array geometry and sphere composition. It was shown that the existence and characteristics of multiple singular values for each scatterer can, in principle, be used to determine certain properties of the scatterers, e.g. conducting or non-conducting material. In this paper, we extend this analysis to non-spherical targets and show how orientation information about the target may be obtained from the spectrum of singular values. The general properties of the decomposition for small non-spherical dielectric (and possibly conductive) targets in an electromagnetic field are derived and detailed results are obtained for the specific cases of non-magnetic and perfectly conducting needles and disks. It is shown that scatterer orientation can be estimated by tracking the singular values of a linear array as it is rotated around its midpoint.
Date: May 18, 2006
Creator: Chambers, D H & Berryman, J G
Partner: UNT Libraries Government Documents Department

Frequency-dependent viscous flow in channels with fractal rough surfaces

Description: The viscous dynamic permeability of some fractal-like channels is studied. For our particular class of geometries, the ratio of the pore surface area-to-volume tends to {infinity} (but has a finite cutoff), and the universal scaling of the dynamic permeability, k({omega}), needs modification. We performed accurate numerical computations of k({omega}) for channels characterized by deterministic fractal wall surfaces, for a broad range of fractal dimensions. The pertinent scaling model for k({omega}) introduces explicitly the fractal dimension of the wall surface for a range of frequencies across the transition between viscous and inertia dominated regimes. The new model provides excellent agreement with our numerical simulations.
Date: May 1, 2010
Creator: Cortis, A. & Berryman, J.G.
Partner: UNT Libraries Government Documents Department

Quasi-static analysis of elastic behavior for some systems having higher fracture densities.

Description: Elastic behavior of geomechanical systems with interacting (but not intersecting) fractures is treated using generalizations of the Backus and the Schoenberg-Muir methods for analyzing layered systems whose layers are intrinsically anisotropic due to locally aligned fractures. By permitting the axis of symmetry of the locally anisotropic compliance matrix for individual layers to differ from that of the layering direction, we derive analytical formulas for interacting fractured regions with arbitrary orientations to each other. This procedure provides a systematic tool for studying how contiguous, but not yet intersecting, fractured domains interact, and provides a direct (though approximate) means of predicting when and how such interactions lead to more dramatic weakening effects and ultimately to failure of these complicated systems. The method permits decomposition of the system elastic behavior into specific eigenmodes that can all be analyzed, and provides a better understanding about which of these specific modes are expected to be most important to the evolving failure process.
Date: October 15, 2009
Creator: Berryman, J.G. & Aydin, A.
Partner: UNT Libraries Government Documents Department

Inverse problem in anisotropic poroelasticity: Drained constants from undrained ultrasound measurements

Description: Poroelastic analysis has traditionally focused on the relationship between dry or drained constants which are assumed known and the saturated or undrained constants which are assumed unknown. However, there are many applications in this field of study for which the main measurements can only be made on the saturated/undrained system, and then it is uncertain what the eects of the uids were on the system, since the drained constants remain a mystery. The work presented here shows how to deduce drained constants from undrained constants for anisotropic systems having symmetries ranging from isotropic to orthotropic. Laboratory ultrasound data are then inverted for the drained constants in three granular packings: one of glass beads, and two others for distinct types of more or less angular sand grain packings. Experiments were performed under uniaxial stress, which resulted in hexagonal (transversely isotropic) symmetry of the poroelastic response. One important conclusion from the general analysis is that the drained constants are uniquely related to the undrained constants, assuming that porosity, grain bulk modulus, and pore uid bulk modulus are already known. Since the resulting system of equations for all the drained constants is linear, measurement error in undrained constants also propagates linearly into the computed drained constants.
Date: November 20, 2009
Creator: Berryman, J.G. & Nakagawa, S.
Partner: UNT Libraries Government Documents Department

Analysis of the growth of strike-slip faults using effective medium theory

Description: Increases in the dimensions of strike-slip faults including fault length, thickness of fault rock and the surrounding damage zone collectively provide quantitative definition of fault growth and are commonly measured in terms of the maximum fault slip. The field observations indicate that a common mechanism for fault growth in the brittle upper crust is fault lengthening by linkage and coalescence of neighboring fault segments or strands, and fault rock-zone widening into highly fractured inner damage zone via cataclastic deformation. The most important underlying mechanical reason in both cases is prior weakening of the rocks surrounding a fault's core and between neighboring fault segments by faulting-related fractures. In this paper, using field observations together with effective medium models, we analyze the reduction in the effective elastic properties of rock in terms of density of the fault-related brittle fractures and fracture intersection angles controlled primarily by the splay angles. Fracture densities or equivalent fracture spacing values corresponding to the vanishing Young's, shear, and quasi-pure shear moduli were obtained by extrapolation from the calculated range of these parameters. The fracture densities or the equivalent spacing values obtained using this method compare well with the field data measured along scan lines across the faults in the study area. These findings should be helpful for a better understanding of the fracture density/spacing distribution around faults and the transition from discrete fracturing to cataclastic deformation associated with fault growth and the related instabilities.
Date: October 15, 2009
Creator: Aydin, A. & Berryman, J.G.
Partner: UNT Libraries Government Documents Department

Evaluating Bounds and Estimators for Constants of Random Polycrystals Composed of Orthotropic Elastic Materials

Description: While the well-known Voigt and Reuss (VR) bounds, and the Voigt-Reuss-Hill (VRH) elastic constant estimators for random polycrystals are all straightforwardly calculated once the elastic constants of anisotropic crystals are known, the Hashin-Shtrikman (HS) bounds and related self-consistent (SC) estimators for the same constants are, by comparison, more difficult to compute. Recent work has shown how to simplify (to some extent) these harder to compute HS bounds and SC estimators. An overview and analysis of a subsampling of these results is presented here with the main point being to show whether or not this extra work (i.e., in calculating both the HS bounds and the SC estimates) does provide added value since, in particular, the VRH estimators often do not fall within the HS bounds, while the SC estimators (for good reasons) have always been found to do so. The quantitative differences between the SC and the VRH estimators in the eight cases considered are often quite small however, being on the order of ±1%. These quantitative results hold true even though these polycrystal Voigt-Reuss-Hill estimators more typically (but not always) fall outside the Hashin-Shtrikman bounds, while the self-consistent estimators always fall inside (or on the boundaries of) these same bounds.
Date: March 1, 2012
Creator: Berryman, J. G.
Partner: UNT Libraries Government Documents Department

Mechanics of layered anisotropic poroelastic media with applications to effective stress for fluid permeability

Description: The mechanics of vertically layered porous media has some similarities to and some differences from the more typical layered analysis for purely elastic media. Assuming welded solid contact at the solid-solid interfaces implies the usual continuity conditions, which are continuity of the vertical (layering direction) stress components and the horizontal strain components. These conditions are valid for both elastic and poroelastic media. Differences arise through the conditions for the pore pressure and the increment of fluid content in the context of fluid-saturated porous media. The two distinct conditions most often considered between any pair of contiguous layers are: (1) an undrained fluid condition at the interface, meaning that the increment of fluid content is zero (i.e., {delta}{zeta} = 0), or (2) fluid pressure continuity at the interface, implying that the change in fluid pressure is zero across the interface (i.e., {delta}p{sub f} = 0). Depending on the types of measurements being made on the system and the pertinent boundary conditions for these measurements, either (or neither) of these two conditions might be directly pertinent. But these conditions are sufficient nevertheless to be used as thought experiments to determine the expected values of all the poroelastic coefficients. For quasi-static mechanical changes over long time periods, we expect drained conditions to hold, so the pressure must then be continuous. For high frequency wave propagation, the pore-fluid typically acts as if it were undrained (or very nearly so), with vanishing of the fluid increment at the boundaries being appropriate. Poroelastic analysis of both these end-member cases is discussed, and the general equations for a variety of applications to heterogeneous porous media are developed. In particular, effective stress for the fluid permeability of such poroelastic systems is considered; fluid permeabilities characteristic of granular media or tubular pore shapes are treated in some detail, as ...
Date: June 1, 2010
Creator: Berryman, J. G.
Partner: UNT Libraries Government Documents Department

Poroelastic response of orthotropic fractured porous media

Description: An algorithm is presented for inverting either laboratory or field poroelastic data for all the drained constants of an anisotropic (specifically orthotropic) fractured poroelastic system. While fractures normally weaken the system by increasing the mechanical compliance, any liquids present in these fractures are expected to increase the stiffness somewhat, thus negating to some extent the mechanical weakening influence of the fractures themselves. The analysis presented quantifies these effects and shows that the key physical variable needed to account for the pore-fluid effects is a factor of (1 - B), where B is Skempton's second coe#14;fficient and satisfies 0 {<=} #20; B < 1. This scalar factor uniformly reduces the increase in compliance due to the presence of communicating fractures, thereby stiffening the fractured composite medium by a predictable amount. One further goal of the discussion is to determine how many of the poroelastic constants need to be known by other means in order to determine the rest from remote measurements, such as seismic wave propagation data in the field. Quantitative examples arising in the analysis show that, if the fracture aspect ratio a{sub f} ~ 0.1 and the pore fluid is liquid water, then for several cases considered Skempton's B ~ 0:9, so the stiffening effect of the pore-liquid reduces the change in compliance due to the fractures by a factor 1-B ~ 0.1, in these examples. The results do however depend on the actual moduli of the unfractured elastic material, as well as on the pore-liquid bulk modulus, so these quantitative predictions are just examples, and should not be treated as universal results. Attention is also given to two previously unremarked poroelastic identities, both being useful variants of Gassmann's equations for homogeneous -- but anisotropic -- poroelasticity. Relationships to Skempton's analysis of saturated soils are also noted. The paper ...
Date: December 1, 2010
Creator: Berryman, J.G.
Partner: UNT Libraries Government Documents Department

Pore-fluid effects on seismic waves in vertically fractured earth with orthotropic symmetry

Description: For elastically noninteracting vertical-fracture sets at arbitrary orientation angles to each other, a detailed model is presented in which the resulting anisotropic fractured medium generally has orthorhombic symmetry overall. Some of the analysis methods and ideas of Schoenberg are emphasized, together with their connections to other similarly motivated and conceptually related methods by Sayers and Kachanov, among others. Examples show how parallel vertical-fracture sets having HTI (horizontal transversely isotropic) symmetry transform into orthotropic fractured media if some subsets of the vertical fractures are misaligned with the others, and then the fractured system can have VTI (vertical transversely isotropic) symmetry if all of the fractures are aligned randomly or half parallel and half perpendicular to a given vertical plane. An orthotropic example having vertical fractures in an otherwise VTI earth system (studied previously by Schoenberg and Helbig) is compared with the other examples treated and it is finally shown how fluids in the fractures affect the orthotropic poroelastic system response to seismic waves. The key result is that fracture-influence parameters are multiplied by a factor of (1-B), where 0 {le} B &lt; 1 is Skempton's second coefficient for poroelastic media. Skempton's B coefficient is itself a measurable characteristic of fluid-saturated porous rocks, depending on porosity, solid moduli, and the pore-fluid bulk modulus. For heterogeneous porous media, connections between the present work and earlier related results of Brown and Korringa are also established.
Date: May 15, 2010
Creator: Berryman, J.G.
Partner: UNT Libraries Government Documents Department