Anisotropic Beds Research Articles

Analytical Model of the Process of Steam-Assisted Gravity Drainage in an Anisotropic Bed

An analytical model of the process of steam-assisted gravity drainage in an element of the superviscous oil pool development has been constructed. The influence of the bed permeability anisotropy on the production rate of a producing horizontal well and cumulative steam–oil ratio has been investigated. A comparison of the calculated output of a horizontal well by the proposed model with the results of physical modeling of the process of steamassisted gravity drainage has been made.

Computational modeling for large wood dynamics with root wad and anisotropic bed friction in shallow flows

Abstract This study developed a computational model for large wood deposition patterns in shallow flows considering the effect of a root wad based on laboratory experiments. We used the Nays2DH depth-averaged two-dimensional model of iRIC to simulate shallow flows. A newly developed large wood simulation model was combined with the shallow flow model. The laboratory tests were performed by changing several hydraulic parameters. In shallow water with a depth similar to the diameter of large wood, the root wad decreased the draft for wood motion (the depth at which large wood contacts the river bed) by lifting the head of large wood. The experimental results showed that the large wood tends to move toward the side walls and deposit on the bed after passing an obstacle. Computational results reasonably showed that the proposed coupling model reproduced the fundamental and physical aspects of the phenomena.

Image processing and laser measurements for determination of the erosion and deposition of fine sediments by a gravity current

This paper reports a high-resolution, non-intrusive, low-cost technique to measure volumes of erosion and deposition of fine sediments. The combination of two techniques, photography and laser scanning, gives the advantage of simultaneous quantitative estimation of erosion and deposition and visualization of deposition patterns. The use of this technique is tested for recording the geomorphological changes induced by the passage of a gravity current over a erodible bed. Good results are obtained, indicating that the combination of both methods can be considered for a range of multi-purpose applications in which the measurement of erosion and deposition is required. Finally, results for the deposition patterns, which are created by the highly turbulent flow field of the gravity currents, are discussed, showing the potential of these techniques for studying anisotropic bed structures.

Problem on the temperature field in an anisotropic bed with sources in the presence of convection

Consideration has been given to the problem on the temperature field created by sources that are lumped in an anisotropic bed with a heat-conducting medium around when the liquid in this bed moves steadily in the radial direction. An “exact-on-average” asymptotic method was used for solution of this problem. Analytical solutions have been obtained for the zero approximation and the first expansion coefficient. It has been determined that the problem for the remainder term has a trivial solution.

Filtration pressure field in an inhomogeneous bed in constant drainage

UDC 532.546 Based on the modification of the "exact-on-average" method, simple analytical formulas have been found for calculation, in the zero and first asymptotic approximations, of pressure fields in an inhomogeneous anisot- ropic bed in constant drainage for the case of one-dimensional linear flow. The results of calculations of the fields from the obtained formulas have been given. Introduction. Problems on pressure fields appearing in filtration of liquids in porous media form the basis for the theory of mass transfer in these media and are of great practical importance for oil and gas production, hydrogeol- ogy, ecology, etc. (1). The problem on filtration of a liquid occupies a special place among the indicated problems be- cause of the variety of conditions of such filtration and its practical significance. Analytical dependences for description of pressure fields appearing in the indicated bed as a result of the liquid filtration in it can be obtained by modification of asymptotic methods whose capabilities have not been completely realized. Such modification is ef- fected through selection of the formal parameter of asymptotic expansion. It becomes necessary to construct the zero and first coefficients of this expansion, boundary-layer functions, and estimating expressions for the remainder term. The resulting zero coefficient of asymptotic expansion describes the averaged value of a physical parameter. Construc- tion of the first expansion coefficient requires supplementary conditions that are based on the trivial solution of the av- eraged problem for the remainder term. From this viewpoint, expressions for the zero and first approximations are called "exact-on-average" (2). This work seeks to determine the zero and first coefficients of asymptotic expansion that describe pressure fields in an inhomogeneous anisotropic bed with constant drainage from one-dimensional linear flow. In this case the zero coefficient describes pressure values averaged over the bed's thickness, whereas the first coefficient refines de- scription of the fields in the averaging zone and determines the steady-state field at large times. 1. Formulation of the Problem for Linear Flow in Constant Drainage. Figure 1 gives the flow geometry in a plane coordinate system (x, z) whose z axis coincides with the axis of the well. The medium is represented by three regions with plane boundaries z = 1; the covering bed and the underlying bed are assumed to be low-perme- able in the horizontal direction k 1x = 0; the central region −1 < z < 1 is highly permeable in the horizontal k x and in the vertical k z directions. For the sake of simplicity it is assumed that flow is one-dimensional and linear along the x axis, the surround- ing rocks are strongly anisotropic, and the vertical permeability k 1z dominates the horizontal permeability k 1x in them. This enables us to disregard the term with the second derivative with respect to the horizontal coordinate x for the sur- rounding medium. Next, we assume that the properties of the underlying and covering beds are identical. In accordance with this, we can simplify the formulation of the problem, using the symmetry condition ∂P ⁄ ∂z = 0 and z = 0. The considered problem is substantially simplified through the use of the so-called quasistationary approxima- tion without perceptible distortions of the basic regularities under study. This approximation is widely used in electro- dynamics for studying electromagnetic fields in electric circuits; its essence is neglect of the time derivative in the

Determination of Dip and Anisotropy from Multi-frequency Tri-axial Induction Measurements

The tri-axial (or multi-component) induction log is used to measure not only the resistivity anisotropy of an anisotropic formation but also the relative dip of the tool with respect to the formation. When the tri-axial induction log is run at multiple frequencies, multi-frequency focusing can be applied to the measurements. The apparent dip and the apparent anisotropy are algebraically deflned from the frequency-focused tri-axial induction measurements. The apparent dip gives the true dip in thinly bedded formations, but a smaller dip than the true dip in a thick anisotropic bed when the anisotropy is small (the anisotropy efiect). The apparent anisotropy gives the true anisotropy in thick anisotropic formations but is afiected by the shoulder bed anisotropy when the formation is not thick (the shoulder bed efiect).

Modes of hill-slope failure under overburden loads: Insights from physical and numerical models

Using sand model experiments this paper investigates failure mechanism of bed materials along hill slopes due to overburden loading. The analysis takes into account three factors: 1) surface slope ( α), 2) loading pattern and 3) mechanical anisotropy of bed materials. In the experiments, external loading was simulated by moving down a rigid block extended either up to the brink of the slope (Type 1 loading) or beyond it (Type 2 loading). With progressive loading, the process of slope instability in sand models involves deformation localization in two modes: compaction (Mode 1) and shear failure (Mode 2). In isotropic models with Type 1 loading, Mode 1 deformation occurs in an elliptical region below the rigid block, which is flanked by a pair of shear bands of Mode 2. With increasing α ( α > 45°), Mode 2 localization propagates rapidly towards the sloping surface leading to collapse on slopes. The failure zones also propagate in the vertical direction, and its depth of penetration appears to be inversely related to α. With Type 2 loading, Mode 2 localization becomes dominant, forming multiple shear bands, and Mode 1 deformation decreases with increasing α. In case of anisotropic bed materials, the orientation of the anisotropy plane relative to that of the surface slope determines the mode of deformation localization. With Type 2 loading, Mode 1 is the dominant process of deformation when the anisotropy plane dips same as the surface slope ( α = 45°), whereas it is coupled with Mode 2 for the anisotropy planes dipping in the opposite directions. We complement the experimental findings with results from finite element models, and demonstrate the patterns of Mode 1 and Mode 2 localization considering the I 1 (= σ 1 + σ 2 + σ 3) stress invariant and Drucker–Prager yield criterion for elasto-plastic materials respectively.

Comparison of Coupled-Potentials and Field-Based Finite-Volume Techniques for Modeling of Borehole EM Tools

In this letter, we compare two finite-volume (FV) techniques, viz., the field-based formulation and the coupled vector-scalar potentials formulation, for the modeling of borehole electromagnetic tools in 3-D earth formations. Eccentric boreholes, anisotropic dipping beds, and high-contrast multilayered earth formations are considered. We compare the two FV formulations in terms of accuracy, convergence rate, and CPU time. Although these two formulations yield similar accuracy in the examples considered, their efficiencies in terms of convergence rate and CPU time depends on the scenario.

Introducing dimensionless parameters into the correlation of NMR relaxation time to transport properties of porous media

AbstractDimensionless parameters representing the viscous permeability (k) and NMR relaxation time (T1) of particle beds, while accounting also for the particle size, are shown to improve drastically the accuracy of k‐T1 correlations in the slow diffusion regime, in the absence of bulk relaxation effects. The finding is based on a regression analysis of numerical results for k and T1 in both random and ordered isotropic and anisotropic beds of fibers. Use of the formation factor (F) improves further the accuracy of the correlations only for the strongly anisotropic unidirectional arrays of fibers. A survey of related literature reveals an extensive effort in recent years in upgrading k‐T1 correlations, driven primarily by applications in petroleum and gas field exploration and recovery. © 2007 Wiley Periodicals, Inc. Concepts Magn Reson Part A 30A: 154–164, 2007.

Elastic coefficients of anisotropic foundation beds

The influence exerted by anisotropy of the deformation properties of soil foundation beds on the elastic characteristics used in dynamic analyses of machine-bearing foundations is investigated. A model of an anisotropic foundation bed in the form of a transversally isotropic elastic half space is used to estimate the extent of this influence. It is suggested that correction factors dependent on the degree of anisotropy of the soil be used in the analyses.

Sliding over anisotropic beds

AbstractMany glacier beds are anisotropic, by which is meant that the dominant wavelengths are different in the two map-plane directions. A largely unexplored consequence of Nye-Kamb sliding theory is the fact that an anisotropic bed can produce a sliding velocity not parallel to the tangential traction vector. This has important consequences, since observations of non-parallel flow are often taken as indications that the shallow-ice approximation has broken down, whereas this need not be the case with an anisotropic bed.Mathematically, this effect can be incorporated through the use of a sliding tensor. The mathematical properties of this tensor are outlined, and the correct "invariant" for the sliding law, a quadratic form, is deduced. Nye-Kamb theory for anisotropic beds is discussed. Flow on the infinite plane and the properties of surface-topography diffusion are elucidated. The properties of kinematic waves and shock waves are discussed. Kinematic waves can have a lateral component. Numerical computations of ice-sheet flow on beds with anisotropic roughness are presented, with emphasis placed on how this affects divide-ridge structure It is suggested that cold-based ice sheets, which have an anisotropic bed affecting the shear layer, may also show non-parallelism of surface slope and velocity.

Induction Log Shoulder-Bed Corrections to Anisotropic Formations and the Effect of Shale Anisotropy in Thinly Laminated Sand/Shale Sequences

Summary We examine induction log responses to layered, dipping, and anisotropic formations analytically. The analytical model is especially helpful in understanding induction log responses to thinly laminated binary formations, such as sand/shale sequences that exhibit macro-scopically anisotropic resistivity. We discuss two applications of the analytical model. In one application, we examine special induction log shoulder-bed corrections for use when thin anisotropic beds are encountered. It is known that thinly laminated sand/shale sequences act as macroscopically anisotropic formations. Hydrocarbon-bearing formations also act as macroscopically anisotropic formations when they consist of alternating layers of different grainsize distributions. When such formations are thick, induction logs accurately read the macroscopic conductivity, from which the hydrocarbon saturation in the formations can be computed. When the laminated formations are not thick, proper shoulderbed corrections (or thinbed corrections) should be applied to obtain the true macroscopic formation conductivity and to estimate the hydrocarbon saturation more accurately. We use the analytical model to calculate the thinbed effect and to evaluate the shoulder-bed corrections. We show that the formation resistivity and, hence, the hydrocarbon saturation are greatly overestimated when the anisotropy effect is not accounted for and conventional shoulder-bed corrections are applied to the log responses from such laminated formations. In another application, we examine the effect of shale anisotropy in thinly laminated sand/shale sequences. How do induction logs respond to such laminated formations when shale laminae are anisotropic? How accurate are the estimates of the sand-laminae resistivity and of the hydrocarbon saturation in these sand laminae? To answer these questions, we used the analytical model to examine the effect of shale lamina anisotropy on induction log responses in thinly laminated formations. We learned that the macroscopic conductivity of a laminated formation is determined uniquely by the sandlaminae conductivity and the anisotropic shale conductivity, if the sand fraction is known. We also found that the sand resistivity estimate depends on the value of shale resistivity and the dip angle, if the shale is assumed to be isotropic and if the sand fraction is known. When the sand fraction is not known, the sand resistivity and the sand fraction can be estimated if the set of horizontal and vertical macroscopic resistivities is known. In this case, the sand resistivity tends to be overestimated if the shale anisotropy is not accounted for. On the other hand, the sand fraction estimate is generally insensitive to the shale anisotropy, except for low sand-laminae resistivity (Rsd/Rsh &amp;lt; 5).

Experimental Design for Surface-to-Borehole Hydrocarbon Applications

We discuss in a simplified surface-to-borehole example the resolution of the coefficient of anisotropy λ for an anisotropic layer sandwiched in an isotropic half-space as a function of transmitter-receiver configuration and magnetic field components. Our analysis supposes that the horizontal resistivity of the anisotropic bed has been established by previous logging, and that only horizontal magnetic fields will be used to determine the vertical resistivity. The resolution analysis shows that for the model studied the coefficient of anisotropy λ can be resolved within 10% for a broad range of transmitter-receiver configurations if ∂B x /∂t and ∂B y /∂t are used jointly in the interpretation. Using only ∂B x /∂t or ∂B y /∂t individually, λ can only be resolved for a limited range of transmitter-receiver configurations. A joint interpretation of both horizontal magnetic field components facilitates a better resolution of the coefficient of anisotropy compared to a single interpretation.

Effects of anisotropic poroelastic beds on wave propagation in shallow water

Wave propagation in seabeds is modeled by the propagator matrix derived from the Biot theory of anisotropic poroelastic media. The theoretical calculations are compared with measurements of wave‐induced pore pressure distributions in sand beds in wave tanks as well as in the field. It was found that a strong coupling between the shear waves and the two compressional waves exists in anisotropic beds which is absent in isotropic beds. The intrinsic coupling in the anisotropic beds is responsible for strong wave attenuation and wave reflection. [Work sponsored by ONR, Code 425UA].

Effects of formation anisotropy on resistivity‐logging measurements

Some developments in the mathematical analysis of resistivity well‐logging measurements in anisotropic beds are presented. The no‐borehole case in which two thick, anisotropic beds meet at a plane interface for bedding planes parallel to the interface has been discussed previously by Kunz and Moran (1958). The treatment here is extended to include arbitrary orientation of the bedding planes. Relations are derived for finding the potential distributions in the two media produced by a point current source in one of them. Numerically evaluated apparent‐resistivity profiles across the boundary between two anisotropic beds are shown for a normal resistivity‐logging device for various conditions of anisotropy and dip. The effect on the responses of a normal device of introducing a borehole perpendicular to the bedding planes of a thick anisotropic bed was also treated by Kunz and Moran (1958). The added effect of eccentering has been studied following the methods of Gianzero and Rau (1977) but, since the results added nothing of interest, no details are given. For the borehole axis not perpendicular to the bedding planes, no progress can be reported in the analysis. When alternating currents are used, the solutions acquire characteristics dependent on the frequency. Appropriate relations are developed starting from Maxwell’s equations. For electrode devices using alternating current, such as a normal or a Laterolog™, solutions are derived in terms of the vector potential for the case of a borehole penetrating a thick anisotropic bed normal to the bedding planes and for the no‐borehole case where the sonde axis is perpendicular to a sequence of beds with all bedding planes parallel to the bed boundaries. Details of the frequency effects are considered only for the case of a homogeneous medium, but indications of the method of solution for heterogeneous media are given. One principal result is that the “paradox of anisotropy” (see Kurtz and Moran, 1958) still remains valid. For induction‐logging devices, the transmitter‐coil sources of the electromagnetic (EM) field are treated as (alternating) magnetic dipoles. When the source‐ and receiver‐coil axes both are oriented perpendicularly to the bedding planes, only the component of resistivity parallel to the bedding planes affects the responses. With the addition of coils oriented parallel to the bedding planes, it is theoretically possible to determine formation dip from the out‐of‐phase (reactive) voltages in the receiver coils. Analyses are outlined for a homogeneous medium, for a thin bed, and for borehole cases usually considered. Values of the horizontal and vertical conductivities (and coefficient of anisotropy) can, in principle, be derived from the measured values of the induction‐logging conductivity signal and the out‐of‐phase signal from the formation. A difficulty with the method is the effect of heterogeneities. When true horizontal conductivity changes across a bed boundary, a plot of computed apparent values of horizontal and vertical resistivities across the boundary shows undesirable artifacts, particularly on the vertical resistivity curve. In borehole cases, the values found also are strongly effected when hole‐fluid conductivity is much larger than the horizontal conductivity of the formation.

Percolation in deep-seated anisotropic fractured beds with a common character of nonlinear inheritance

Percolation in deep-seated anisotropic fractured beds with a common character of nonlinear inheritance

SOME EFFECTS OF FORMATION ANISOTROPY ON RESISTIVITY MEASUREMENTS IN BOREHOLES

It is shown that a wide class of potential problems involving anisotropic media can be transformed into equivalent problems involving only isotropic media. By means of such transformations it is possible, in a large number of cases, to determine the apparent resistivities which would be observed in anisotropic formations, using electrode‐type resistivity logging devices. Discussion is given of an infinite, anisotropic medium with and without borehole, of two semi‐infinite anisotropic beds (without borehole), and of a thin isotropic bed bounded by anisotropic adjacent formations (without borehole). An interpretation chart for the normal device is presented for thick, non‐invaded, anisotropic beds penetrated by a borehole.