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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

Tomographic resolution without singular value decomposition

Description: An explicit procedure is presented for computing both model and data resolution matrices within a Paige-Saunders LSQR algorithm for iterative inversion in seismic tomography. These methods are designed to avoid the need for an additional singular value decomposition of the ray-path matrix. The techniques discussed are completely general since they are based on the multiplicity of equivalent exact formulas that may be used to define the resolution matrices. Thus, resolution matrices may also be computed for a wide variety of iterative inversion algorithms using the same ideas.
Date: June 1, 1994
Creator: Berryman, J. G.
Partner: UNT Libraries Government Documents Department

Scale-up in Poroelastic System and Applications to Reservoirs

Description: A fundamental problem of heterogeneous systems is that the macroscale behavior is not necessarily well-described by equations familiar to us at the meso- or microscale. In relatively simple cases like electrical conduction and elasticity, it is hue that the equations describing macroscale behavior take the same form as those at the microscale. But in more complex systems, these simple results do not hold. Consider fluid flow in porous media where the microscale behavior is well-described by Navier-Stokes' equations for liquid in the pores while the macroscale behavior instead obeys Darcy's equation. Rigorous methods for establishing the form of such equations for macroscale behavior include multiscale homogenization methods and also the volume averaging method. In addition, it has been shown that Biot's equations of poroelasticity follow in a scale-up of the microscale equations of elasticity coupled to Navier-Stokes. Laboratory measurements have shown that Biot's equations indeed hold for simple systems but heterogeneous systems can have quite different behavior. So the question arises whether there is yet another level 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 equations play a special role in the scale-up to equations describing reservoir behavior, for fluid pumping, geomechanics, as well as seismic wave propagation.
Date: July 1, 2003
Creator: Berryman, J G
Partner: UNT Libraries Government Documents Department

Bounds and Estimates for Transport Coefficients of Random and Porous Media with High Contrasts

Description: Bounds on transport coefficients of random polycrystals of laminates are presented, including the well-known Hashin-Shtrikman bounds and some newly formulated bounds involving two formation factors for a two-component porous medium. Some new types of self-consistent estimates are then formulated based on the observed analytical structure both of these bounds and also of earlier self-consistent estimates (of the CPA or coherent potential approximation type). A numerical study is made, assuming first that the internal structure (i.e., the laminated grain structure) is not known, and then that it is known. The purpose of this aspect of the study is to attempt to quantify the differences in the predictions of properties of a system being modeled when such organized internal structure is present in the medium but detailed spatial correlation information may or (more commonly) may not be available. Some methods of estimating formation factors from data are also presented and then applied to a high-contrast fluid-permeability data set. Hashin-Shtrikman bounds are found to be very accurate estimates for low contrast heterogeneous media. But formation factor lower bounds are superior estimates for high contrast situations. The new self-consistent estimators also tend to agree better with data than either the bounds or the CPA estimates, which themselves tend to overestimate values for high contrast conducting composites.
Date: September 24, 2004
Creator: Berryman, J G
Partner: UNT Libraries Government Documents Department

Bounds on Transport Coefficients of Porous Media

Description: An analytical formulation of conductivity bounds by Bergman and Milton is used in a different way to obtain rigorous bounds on the real transport coefficients (electrical conductivity, thermal conductivity, and/or fluid permeability) of a fluid-saturated porous medium. These bounds do not depend explicitly on the porosity, but rather on two formation factors--one associated with the pore space and the other with the solid frame. Hashin-Shtrikman bounds for transport in random polycrystals of porous-material laminates will also be discussed.
Date: March 21, 2005
Creator: Berryman, J G
Partner: UNT Libraries Government Documents Department

Poroelastic Analysis of Thomsen Parameters in Finely Layed VTI Media

Description: Thomsen's anisotropy parameters for weak elastic and poroelastic anisotropy are now commonly used in exploration, and can be conveniently expressed in terms of the layer averages of Backus. Although there are five effective shear moduli for any layered VTI medium, only one effective shear modulus for the layered system contains all the dependence of pore fluids on the elastic or poroelastic constants that can be observed in vertically polarized shear waves in VTI media. The effects of the pore fluids on this effective shear modulus can be substantial when the medium behaves in an undrained fashion, as might be expected at higher frequencies such a sonic and ultrasonic for well-logging or laboratory experiments, or at seismic frequencies for lower permeability regions of reservoirs.
Date: March 17, 2003
Creator: Berryman, J G
Partner: UNT Libraries Government Documents Department

Bounds on Elastic Constants for Random Polycrystals of Laminates

Description: A well-known result due to Hill provides an exact expression for the bulk modulus of any multicomponent elastic composite whenever the constituents are isotropic and the shear modulus is uniform throughout. Although no precise analog of Hill's result is available for the opposite case of uniform bulk modulus and varying shear modulus, it is shown here that some similar statements can be made for shear behavior of random polycrystals composed of laminates of isotropic materials. In particular, the Hashin-Shtrikman-type bounds of Peselnick, Meister, and Watt for random polycrystals composed of hexagonal (transversely isotropic) grains are applied to the problem of polycrystals of laminates. An exact product formula relating the Reuss estimate of bulk modulus and an effective shear modulus (of laminated grains composing the system) to products of the eigenvalues for quasi-compressional and quasi-uniaxial shear eigenvectors also plays an important role in the analysis of the overall shear behavior of the random polycrystal. When the bulk modulus is uniform in such a system, the equations are shown to reduce to a simple form that depends prominently on the uniaxial shear eigenvalue - as expected from physical arguments concerning the importance of uniaxial shear in these systems. One application of the analytical results presented here is for benchmarking numerical procedures used for estimating elastic behavior of complex composites.
Date: April 30, 2004
Creator: Berryman, J G
Partner: UNT Libraries Government Documents Department