Two-region System Research Articles

Computation of large-scale parameters for dispersion in fissured porous medium using finite-volume method

In this work, we are interested in solute transport in fissured porous media. The medium is considered as a special case of a two-region system, and a two-equation model previously obtained from a volume averaging technique is used to derive large-scale dispersion coefficients. These coefficients are obtained as solutions of a set of closure problems and the main objective of this work is to present an efficient method to solve these closure problems. The method makes use of an unstructured grid and special techniques to take into account the fissure network. Results are compared with other existing methods on simple fissured media. Finally, the technique is applied to a complex structure.

A Numerical Investigation of Pressure Transient Analysis For Horizontal Wells In Composite Reservoirs Mimicking Thermal Recovery Situations

Abstract Horizontal wells have become quite popular for primary and enhanced oil recovery operations due to their well-documented advantages over vertical wells. Steam injection through horizontal wells has also been attempted at several places to improve heavy oil recovery. For horizontal wells undergoing steam injection, a steam chamber containing high mobility steam is established. This steam chamber, which can be of a complex shape, is surrounded by low mobility reservoir fluids. Such reservoir situations are referred to as composite reservoirs. An analytical solution for pressure transient tests for horizontal wells under composite reservoir situations with complex swept region shapes is not available yet. This study attempts a numerical investigation of pressure transient analysis for horizontal wells in two-region, composite reservoirs mimicking thermal recovery situations. A specialized three-dimensional, single-phase simulator was developed for this purpose. A closed, box-shaped reservoir was considered with a horizontal well. A detailed sensitivity study of transient pressure responses is presented with respect to grid size, well location in different directions, swept region shape, mobility ratio and storativity ratio. This study emphasizes the effects of the aforementioned factors on the swept volume estimation using the pseudosteadystate method. The pseudosteady-state method is based on an analysis of a Cartesian graph of pressure response versus time. This study establishes that for horizontal wells, the swept (or inner) region volume in a two-region system can be accurately estimated by the pseudosteady-state method for large mobility and storativity contrasts between the two regions. Introduction Horizontal wells have become quite popular in recent years due to their well-documented advantages over vertical wells(1). Steam injection through horizontal wells has also been attempted at several places to improve heavy oil recovery. Steam injection through horizontal wells includes scenarios such as steam-assisted gravity drainage process pioneered by Butler(2). Under such situations, a steam chamber containing high mobility steam is established around a horizontal well. A fall off test conducted on a horizontal injection well can be analysed using the pseudosteady-state (PSS) method. The PSS method was first proposed by Eggenschwiler et al.(3) to analyse thermal well tests conducted on vertical injection wells to estimate the steam chamber volume. Though a number of papers(4–9) have evaluated the applicability of the pseudosteadystate method for vertical injection wells, to the best of our knowledge, Reference (10) is the only study concentrating on the applicability of the pseudosteady-state method for horizontal injection wells. Issaka and Ambastha(10) used a numerical thermal simulator to generate fall off test data for horizontal, steam-injection wells. In this study, a specialized three-dimensional, single-phase, nonthermal simulator is used to study the applicability of the pseudosteady- state method for horizontal injection wells. In the next section, a brief description of the numerical model is presented followed by a section reporting our validation efforts for the numerical model. Thereafter, a detailed sensitivity study of transient pressure responses is presented with respect to various parameters, followed by concluding remarks.

Computer Simulations of the Goldman-Shen Experiment: Evaluation of Techniques for Minimizing the Influence of Spin-Lattice Relaxation

The magnetization of the mobile domain for a lamellar two-region system following the application of the Goldman–Shen pulse sequence was simulated under various conditions in order to evaluate the effectiveness of several suggested techniques for obtaining spin diffusion data separate from the effects of spin–lattice relaxation. It is shown that in general none of the methods are completely effective, but the conditions of applicability are discussed and simulated examples showing deviations from ideality are illustrated. © 1997 by John Wiley & Sons, Ltd.

Pressure Transient Behavior In Reservoirs With An Internal Circular Discontinuity

Abstract This paper presents the pressure transient response for reservoirs with an internal circular subregion. The solution is a generalization of that previously presented in the literature, which assumed that the subregion was either an impermeable barrier or a constant pressure circle. Two cases were considered, with the well located either inside or outside the circular discontinuity. When the well is located inside the circular subregion, the solution developed in the study is also a generalization of the well-known pressure response for a composite two-region system. The solution was obtained by using the concept of Green's function and by applying the Laplace transformation technique. The Laplace space response, although very complex, was inverted to real space with the Stehfest1 algorithm. When the well is located inside the circular subregion, the wellbore pressure response shows a behavior typical of radial composite systems, except for the effect of the well eccentricity, which acts as a skin factor. For the case where the active well is external to the subregion, three flow regimes are observed. The initial infinite-acting behavior and the late-time pseudoradial flow produce semilog straight lines which are parallel and reflect the transmissivity of the external region. The long-time effect of the discontinuity can be regarded as an apparent skin effect. The dependence of the solution on the diffusivity ratio, the mobility ratio, the geometric configuration and the size of the system was also investigated in the study. The solution can be used to simulate the pressure behavior for reservoirs which contain circular or approximately circular discontinuities, such as those caused by variations in rock and/or fluid properties due to the presence of shale lenses or fluid injection, for instance. The pressure solution can also be used to interpret single-well or interference tests in systems with these configurations.

A note on criticality in the presence of diverging subsystem properties

The problem of the criticality eigenvalues and collision rates in a system containing a sub-volume that has some diverging parameter (∑ a or c) is addressed. First a comment is made on the finiteness of the criticality eigenvalue and the collision rates with diverging ∑ a. Then the case of a diverging multiplication factor is treated in a two-region system in a simplified transport model where scattering is of the forward-backward type (one-dimensional). The finiteness of the fundamental and higher order eigenvalues and eigenfunctions is demonstrated via exact solutions. Asyptotic values of the eigenvalues in one region, and corresponding asymptotic eigenfunctions, when the multiplication factor of the other region diverges, are given analytically. The so-called c-map (corresponding fundamental and higher order ( c 1, c 2) pairs), introduced recently by Garis et al. (1992), is calculated for high orders.

On T1 cancellation schemes in Goldman—Shen-type experiments

Modifications to Goldman—Shen-type spin-diffusion pulse sequences, designed to cancel spin—lattice relaxation effects by alternation of the magnetisation orientation during the spin-diffusion evolution period, are critically evaluated. In particular, the spin-diffusion-coupled relaxation behaviour of a three-region model, evaluated numerically, is used to show that such “ T 1 cancellation” schemes cannot, in general, separate T 1 and spin-diffusion effects. A further simple calculation shows that, even for a two-region system with uniform T 1 and no spin diffusion, a spatial magnetisation gradient is attenuated under such inversions giving the appearance of a spin-diffusion phenomenon. It is concluded that, in general, solutions of the full spin-diffusion-coupled relaxation behaviour must be undertaken if accurate morphological information is the aim of such experiments.

Magnetic spin lattice relaxation in the presence of spin-diffusion

The equations describing the spin-diffusion-coupled nuclear magnetic spin-lattice relaxation behaviour of a one-dimensional, two-region system are solved exactly. The solutions allow the calculation of the relaxation times and their associated amplitudes in terms of the values of the sizes, spin-diffusion coefficients and intrinsic relaxation times of each region. The morphology of the roots of the equation are represented graphically in a manner which illustrates their behaviour as a function of the system parameters. The results obtained from these exact calculations are compared with numerically simulated data which mimic experimental measurements and which are fitted to a maximum of four exponential processes. Good agreement is demonstrated, particularly for the longer relaxation components when there are several roots with significant amplitude. Preliminary consideration is given to the inverse problem of deriving system parameters from the observed relaxation behaviour.

Exchange processes in nmr

The exchange of particles and the (noncorporeal) exchange of magnetization in systems are considered where nuclei exist in various regions which differ in their magnetic resonance properties. We start with the stochastic theory of nuclear magnetic resonance and relaxation in such systems. Two-region systems are treated in some detail; here, also, the modified Bloch equations are introduced. Experimental examples concerning adsorbent-adsorbate systems are given. Finally, the influence of cross-relaxation on the relaxation of molecules adsorbed on interfaces is taken into account, the corresponding theory being treated. NMR methods which allow to distinguish between particle and spin exchange are discussed.

A molecular dynamics study of the overpopulation phenomena in a two-region system

By means of molecular dynamics, we have studied the time evolution of an isolated system composed of Lennard-Jones particles. The system was initially split into two regions, all the particles in the same region having the same speed. The dominant collisions leading to the overpopulation effects have been identified. It has also been shown that the more relevant features of the evolution of the system can be accurately described by means of a local equilibrium distribution function.

A model of population change with new and return migration.

A model of population change is applied to the analysis of new and return migration in a two-region system. The structure of the model is examined analytically given various assumptions about constancy in numbers of migrants and in migration rates. The model is applied to the numerical example of birthplace-specific movement between Great Britain and the rest of the world.

Energy corrections in pulsed neutron measurements for cylindrical geometry

A solution of the thermal neutron diffusion equation for a two-region concentric cylindrical system, with a constant neutron flux in the inner medium assumed, is given. The velocity-averaged dynamic parameters for thermal neutrons are used in the method. The corrections due to the diffusion cooling are introduced into the dynamic material buckling and into the velocity distribution of the thermal neutron flux. Detailed relations obtained for a hydrogenous moderator are given. Results of the measurements of the thermal neutron macroscopic absorption cross-sections for the samples in the two-region cylindrical systems are presented.

Solution of the thermal neutron diffusion equation for a two-region system by perturbation calculation

The paper presents a method of solving the diffusion equation for the thermal neutron flux in a heterogeneous medium. Perturbation calculation is successfully applied for a cylindrical concentric system after testing this method for a spherical concentric geometry, analytically solved by Czubek (1981). The method makes possible the calculation of the thermal neutron decay constant and the space distribution of the thermal neutron flux in a two-region system. The condition of a constant value of the neutron flux in the inner part of the system must be met in this method. This method has an application in the measurement of the thermal neutron absorption cross-section (Czubek 1981).

Radiation Transport in One-Dimensional Finite Systems—Part II: Gamma-Ray Transport Studies with ASFIT

Here we present the results of extensive calculations on the gamma-ray transport in finite heterogeneous systems using the technique developed in Part I of this paper. Systems studied are water, concrete, aluminum, iron, and lead, and the data presented are: evolution of spectra with collisions, reflected and transmitted spectra, spectra at different depths, and buildup factors for different source energies and different thicknesses of each medium. Similar data are also presented for two-region systems of water-iron, iron-lead, and water-lead. Several interesting results have been observed, among which are: (a) a step structure and a second peak in the low energy part of the back-scattered spectra from light media, (b) progressive buildup of a peak in the low energy region of transmitted flux, and (c) significant dip in the build-up factors near the outer boundary of the systems. The results are discussed on the basis of the physical processes involved. A comparison of the present results with those published in literature is made.

Elementary models for population growth and distribution analysis

Abstract A version of the Lotka-Volterra interaction model is adapted to describe population growth and migration processes in a two-region system. The regions are identified as a metropolis and its non-metropolitan hinterland. Several conditions on growth and migration regimes are imposed. The time behavior of the systems are analyzed, noting especially situations where total depopulation or population explosion eventually occur in one or both populations. Neither growth control nor migration control alone results in a condition of long-run stability in both regions. If at least a momentary condition of zero growth is achieved in both regions, it is possible to maintain finite populations if each population follows a logistic natural growth process and migration flow is proportional to the volume of interaction. It is necessary also that the natural increase limitation is strong relative to migration rates. This result holds even if one population has a net migration advantage over the other.

Investigation of two-region systems by pulsed neutron method

Investigation of two-region systems by pulsed neutron method

One-Speed Neutron Wave Propagation with a Source off the Interface

One-speed diffusion and transport theory is used to solve the problem of neutron wave propagation in a two-region system. The source is at a finite distance “a” to the left of the interface of the adjacent half-spaces. The solution in both models corresponds to the interpretation of the classical wave problem. The waves propagate till they reach the interface. At the interface, there is a component which is reflected, whereas another component is transmitted. Two coupled singular integral equations were solved simultaneously to evaluate the coefficients. One of the equations is of the Riemann-Hilbert type with discontinuous coefficients. There exists a discrete separation constant for the diffusion model for all oscillation frequencies. For the transport model, it is seen that there is a limiting frequency above which the discrete waves apparently disappear. Thus, a mixture of discrete and continuum waves, incident to an interface from the left medium may give rise to purely continuum or to a mixture of discrete and continuum waves on the right. Also, it is seen that a purely continuum incident wave may give rise to both discrete and/or continuum transmitted components.

Two-dimensional transport models for the lower layers of the atmosphere

This paper describes two boundary value problems whose solutions assist in the interpretation of two-dimensional heat and mass transfer in the lower layers of the atmosphere. One model considers the transient diffusion of mass into a semi-infinite two-region system; one layer is finite in thickness and the other one is infinitely thick. The diffusivities of the two layers are different. Mass is suddenly and continuously added to the boundary of the finite layer causing diffusion throughout the whole system. A second model considers two-dimensional heat or mass transfer in a semi-infinite convection system where the boundary heat or mass flux distribution is sinusoidal with downwind distance. Uniform velocity and eddy diffusivity profiles are postulated. The analytical results obtained for the two diffusion models are evaluated and compared to some observed phenomena for the atmosphere. Applications of the results to practical atmospheric diffusion problems in air pollution and climatology are also noted.

Experimental and Theoretical Studies of Heavy-Water Homogeneous Two-Region System

A series of critical experiments and theoretical analyses has been made on two-region systems in which a heavy water solution of uranyl sulfate with 235U enrichment of about 20% is surrounded by heavy water, poisoned with B or mixed with ThO2. D-to-235U atomic ratios in the core solutions ranged 4,500–1,000, depending on the core diameter and the blanket concentration. The theoretical effective multiplication factor is decreased by treating the leakage of fast neutrons from the core rigorously, and is increased by using spatial-dependent effective cross sections. These treatments are justified by the agreement found between the calculated and measured values for the Cd ratio and the thermal flux distribution. The disagreement seen in the effective multiplication factor between the theoretical and experimental values may be attributed to uncertainty in the resonance integral of 235U. The discrepancy in the effective multiplication factors does not exceed 1%, when the accepted value of νI f -I a is reduced ...

Experimental and Theoretical Studies of Heavy-Water Homogeneous Two-Region System

Experimental and Theoretical Studies of Heavy-Water Homogeneous Two-Region System

Two-Group Material Bucklings in a Two-Region System

"Two-Group Material Bucklings in a Two-Region System." Nuclear Science and Engineering, 19(1), p. 131