General Transient Solution Research Articles

How graduate work with David T. Blackstock led to a lifelong obsession with wave equations

As one of David T. Blackstock’s first graduate students at the University of Rochester (1966), I quickly admired his elegant style of normalizing and solving equations and was baffled by his frequent allusions to great acousticians and mathematicians of the past. Shortly after my arrival, he submitted his transient solution to the viscous wave equation to J.A.S.A. We planned a summer research project (1967) to validate his solution experimentally. He also introduced me to advanced methods for solving equations. David told me there was no general transient solution to this equation, which inspired my determination to solve it. My data confirmed his predictions leading to my first published paper. I left Rochester with an M.S. and afterwards, I filled several notebooks with unsuccessful equations. Eventually, after ten years in ultrasound imaging, I chose the University of Bath for its excellent PhD program in nonlinear acoustics. David visited me there in 1990 and we discussed his 1985 general solution to the Burgers equation. Later at Bath, twenty-three years after beginning, aha moments led me to more general and causal solutions to both linear and nonlinear viscous wave equations for frequency power law media and papers, cited a thousand times.

Fluid flow in fractured reservoirs: Exact analytical solution for transient dual porosity model with variable rock matrix block size

Dual porosity reservoirs consist of two comparatively independent systems of fractures and matrix blocks with high and low permeability values, respectively. Semi-analytical and numerical studies on naturally fractured reservoirs have been already cited in the literature. The present study focuses on investigation of a linear double porosity model of a semi-infinite-acting naturally fractured reservoir using an exact analytical method. Different matrix block size distributions are embedded into a transient model to consider the effects of heterogeneity in fractured formations. In order to take into account transient fracture-matrix exchange, an analytical method is proposed to solve the coupled fracture and matrix equations. Gauss Legendre quadrature is employed to evaluate the double integral of general transient solution. Moreover, the corresponding shape factor was evaluated in a double porosity transient model coupled with variable matrix block size distribution. Results demonstrated that matrix block size distributions strongly affect the fluid transfer during the early time region. Also, the presented model was employed to generate interference test curves which in turn were studied to investigate the impacts of storativity ratio and matrix block size distributions on the fracture-matrix fluid transfer. Results illustrated that a series of obtained pressure data from an observational well along with the proposed model may be considered a robust method in fracture intensity assessment. Fast sensitivity analysis and very efficient computational cost are the benefit of the derived analytical solutions with respect to previous numerical solutions.

RETRACTED: Tracer transport in naturally fractured reservoirs: Analytical solutions for a system of parallel fractures

In naturally fractured reservoirs, modeling of mass transfer between matrix blocks and fractures is an important subject during gas injection or contaminant transport. This study focuses on developing an exact analytical solution to transient tracer transport problem along a discrete fracture in a porous rock matrix. Using Gauss-Legendre quadrature, an expression was obtained in the form of a double integral which is considered as the general transient solution. This solution has the ability to account the following phenomena: advective transport in fractures and molecular diffusion from the fracture to the matrix block. Certain assumptions are made which allow the problem to be formulated as two coupled, one-dimensional partial differential equations: one for the fracture and one for the porous matrix in a direction perpendicular to the fracture. Using the obtained analytical solution, tracer concentration in matrix block and fracture was calculated. The advective-diffusive equation in matrix and fracture was used for evaluation of the mass transfer shape factor. The derived analytical solution was used for analyzing early and late time periods of mass transport phenomenon in fractured porous media. Finally, validation of the analytical solution was done by comparing the obtained results with laboratory data adapted from a column tracer test conducted on a fractured till.

General transient solution of the one-step master equation in one dimension.

Exact analytical solutions of the master equation are limited to special cases and exact numerical methods are inefficient. Even the generic one-dimensional, one-step master equation has evaded exact solution, aside from the steady-state case. This type of master equation describes the dynamics of a continuous-time Markov process whose range consists of positive integers and whose transitions are allowed only between adjacent sites. The solution of any master equation can be written as the exponential of a (typically huge) matrix, which requires the calculation of the eigenvalues and eigenvectors of the matrix. Here we propose a linear algebraic method for simplifying this exponential for the general one-dimensional, one-step process. In particular, we prove that the calculation of the eigenvectors is actually not necessary for the computation of exponential, thereby we dramatically cut the time of this calculation. We apply our new methodology to examples from birth-death processes and biochemical networks. We show that the computational time is significantly reduced compared to existing methods.

Corrected equations for membrane transport characterization by manometric techniques

Equations for characterization of membrane permeability and diffusivity via pressure decay measurements in closed and open systems for three different configurations are discussed in terms of time lag and general transient solutions. The solutions presented correct errors in the earlier work of Nguyen et al. [J. Chem. Soc., Faraday Trans. 88 (1992) 3553].

Analytical solutions for ore fluid transport in fractured rocks: a case study on ore-forming processes of the stratabound skarn copper deposits in China

The Dongguashan skarn copper deposit can be considered as the product of the transport-chemical reaction coupling process of ore-forming materials (for example, complexes of copper) in discrete, parallel fractures in a porous medium system. A mathematical model of metallogenesis has been established and the accurate analytical solutions for depicting the transport of ore-forming materials have been worked out. In establishing the model of metallogenesis and working out the analytical solutions, the following aspects have been taken into consideration: (1) advective transport along fractures; (2) diffusion and longitudinal mechanical dispersion of ore-forming materials (solutes) along the fracture axis; (3) diffusion of ore-forming materials from the fractures to the wall media; (4) adsorption of ore-forming materials on the surface of wall-rock matrices; (5) adsorption of ore-forming materials within the wall-rock matrices; (6) reduction of the concentrations of solutes due to the chemical reactions between ore fluids and wall-rock matrices and the precipitation of ore-forming materials. The general transient solution takes the form of a double integral, which can be evaluated using the Gauss-Legendre quadrature. By comparing the steady-state solutions in the special case of D = 0 (without dispersion) and of D ¬= 0 (with dispersion), a simple criterion can be established, with which one can assess the importance of longitudinal dispersion along the fracture system. Case studies showed that the developing extent of fractures in the system would exert a great influence on the transport rate and distance of ore-forming materials. In case that fractures are developed at small intervals, ore-forming materials will be transported along the fracture system over larger distance because of the limited capability of the wall rocks to store ore-forming materials. That is to say, larger orebodies would be formed. In the case of higher transport rates of ore fluids along the fracture system the longitudinal mechanical dispersion is negligible, but in the reversed case, the longitudinal mechanical dispersion would exert a great influence on fluid transport. Under such circumstances, the longitudinal mechanical dispersion could not be neglected. In the normal case of D ¬= 0, the steady-state solutions could provide a potential approach to predicting the ultimate distance over which ore fluids of a certain concentration level penetrate along the fracture system, that is, how long an orebody would extend. Analytical solutions can rationally interpret the spatial distribution characteristics of strata-bound skarn copper deposits, the rules governing spatial variations in ore texture and ore grade, as well as other important geological characteristics of the ore deposits. This work will provide a better understanding of the metallogenic mechanism of strata-bound skarn copper deposits widely spread along the Middle-Lower Reaches of the Yangtze River Valley and its adjacent areas in China.

Transient current distribution in a sphere by feeding a current on the surface from a loop of arbitrary shape

The complete electromagnetic field of an arbitrary transient current distribution in a sphere can be represented by a superposition of a transverse electric and a transverse magnetic field. In this paper, the transient current distribution in a permeable and conducting hollow sphere is calculated for an exciting current of arbitrary time dependence which is fed into the sphere via the surface from an open loop of arbitrary shape. The original configuration is separated into two characteristic configurations whose current distributions are derived from a transverse electric and a transverse magnetic second order potential. The transverse magnetic field is determined by the position of the feeding points on the surface of the sphere. The transverse electric field is influenced only by the contour of the exciting loop in the nonconducting space. If the exciting loop is closed outside the hollow sphere, the transverse magnetic field vanishes and the solution for the transverse electric field can be used to describe the shielding of the transient electromagnetic field by a spherical screen. The general transient solution is proved for the special case of a sinusoidal time dependence and compared with the direct solution of the differential equation for the complex amplitudes of field quantities. An energy balance is introduced which describes the energy flow during the transient process.

Radionuclide Transport in Fractured Porous Media—Analytical Solutions for a System of Parallel Fractures with a Kinetic Solubility-Limited Dissolution Model

Analytical solutions are developed for the problem of radionuclide transport in a system of parallel fractures situated in a porous rock matrix. A kinetic solubility-limited dissolution model is used as the inlet boundary condition. The solutions consider the following processes: (a) advective transport in the fractures, (b) mechanical dispersion and molecular diffusion along the fractures, (c) molecular diffusion from a fracture to the porous matrix, (d) molecular diffusion within the porous matrix in the direction perpendicular to the fracture axis, (e) adsorption onto the fracture wall (f) adsorption within the porous matrix, and (g) radioactive decay: The solutions are based on the Laplace transform method. The general transient solution is in the form of a double integral that is evaluated using composite Gauss-Legendre quadrature. A simpler transient solution that is in the form of a single integral is also presented for the case that assumes negligible longitudinal dispersion along the fractures. The steady-state solutions are also provided. A number of examples are given to illustrate the effects of the following important parameters: (a) fracture spacings, (b) dissolution-rate constants, (c) fracture dispersion coefficient, (d) matrix retardation factor, and (e) fracture retardation factor.

Radionuclide Transport in Fractured Porous Media—Analytical Solutions for a System of Parallel Fractures with a Flux-Type Boundary Condition

An analytical solution is developed for the problem of radionuclide transport in a system of planar parallel fractures situated in a porous rock matrix. The flux at the inlet boundary of a fracture is assumed to decrease exponentially with time. The solution considers the following processes: (a) advective transport in the fractures, (b) mechanical dispersion and molecular diffusion along the fractures, (c) molecular diffusion from a fracture to the porous matrix, (d) adsorption onto the fracture wall, (e) adsorption within the porous matrix, and (f) radioactive decay. The solution is based on the Laplace transform method. The general transient solution is in the form of a double integral that is evaluated using composite Gauss-Legendre quadrature. A simpler transient solution that is in the form of a single integral is also presented for the case that assumes negligible longitudinal dispersion along the fractures. A few examples are given to illustrate the effect of various fracture spacings and groundwater velocities, a 1% penetration distance, and the effect of neglecting the longitudinal dispersion in the fractures.

Radionuclide transport in fractured porous media—Analytical solutions for a system of parallel fractures with a constant inlet flux

Analytical solutions are developed for the problem of radionuclide transport in a system of parallel fractures situated in a porous rock matrix. A constant flux is used as the inlet boundary condition. The solutions consider the following processes: (a) advective transport along the fractures; (b) mechanical dispersion and molecular diffusion along the fractures; (c) molecular diffusion from a fracture to the porous matrix; (d) molecular diffusion within the porous matrix in the direction perpendicular to the fracture axis; (e) adsorption onto the fracture wall; (f) adsorption within the porous matrix, and (g) radioactive decay. The solutions are based on the Laplace transform method. The general transient solution is in the form of a double integral that is evaluated using composite Gauss-Legendre quadrature. A simpler transient solution that is in the form of a single integral is also presented for the case that assumes negligible longitudinal dispersion along the fractures. The steady-state solutions are also provided. A number of examples are given to illustrate the effects of various important parameters, including: (a) fracture spacing; (b) fracture dispersion coefficient; (c) matrix diffusion coefficient; (d) fracture width; (e) groundwater velocity; (f) matrix retardation factor; and (g) matrix porosity.

Transient solutions for the buffer behavior in statistical multiplexing

In this paper we present time-dependent (or transient) solutions for a mathematical model of statistical multiplexing. The problem is motivated by the need to better understand the performance of fast packet switching in asynchronous transfer mode (ATM), which will be adopted in the broadband ISDN. The transient solutions will be of critical value in understanding dynamic behavior of the multiplexer, and loss probabilities at the cell (or packet) level. We use the double Laplace transform method, and reduce the partial differential equation that governs the multiplexer behavior to the eigenvalue problem of a matrix equation in the Laplace transform domain. We derive important properties of these eigenvalues, by extending earlier results discussed by Anick, Mitra and Sondhi (1982) for the equilibrium solutions. A most critical step in our analysis is to identify sets of linear equations that uniquely determine the time-dependent probability distributions at the buffer boundaries. These boundary conditions are in turn used to solve the general transient solutions. For the infinite buffer case, we show that a closed form solution is given in terms of explicitly identified eigenvalues and eigenvectors. When the buffer capacity is finite, the determination of boundary conditions requires us to solve a matrix equation. We also observe that the statistical multiplexing not only achieves the effective bandwidth gain (i.e., a multiplexing gain), but also reduces the system's packet loss probability and shorten transient periods. We present some numerical results to illustrate our solution technique. A potential application of the time-dependent solution is in the area of preventive congestion control in a high speed network.

General transient solutions for the four-coupled Bloch-type equations for a two-level system

When the collisional dynamics of two states under consideration for a radiative transition is different, the polarization and population relaxations in gas phase at low pressure are governed by four-, instead of three-, coupled Bloch-type equations. The general transient solutions of these four-coupled equations are solved by using the Laplace transformation technique. It has been found that an additional exponential term appears because of the effect. This is also responsible for the non-exponential character in the decay signal of a transient microwave pulse experiment, besides other factors reported earlier.

A transient analytical model for predicting the redistribution of injected interstitials

A general formal transient solution to the linear boundary value problem which governs the transport of interstitials injected into an arbitrary bounded domain from oxidizing segments on the surface is presented. An explicit expression for the Green's function which appears as the kernel in the integral representation of the general transient solution is given for a simple domain of interest in the study of oxidation-enhanced diffusion (OED). By combining this explicit expression for the Green's function with the general formal solution, a transient analytical model for predicting the redistribution of interstitials injected into the simple domain from a single oxidizing segment is obtained. An illustrated example important to OED is considered and results obtained with the model and by approximations to the model are compared. Also, the results obtained with the SUPREM-IV program are presented for comparison. >

On the theory of stress-assisted diffusion, II

Similarity and general steady-state solutions of a recently developed strees-assisted diffusion theory are derived. General transient solutions are obtained analytically for certain classes of stress distributions. For general stress distributions a perturbation method is employed to produce transient solutions. Under appropriate conditions the derived solutions are reduced to previous formulae that have unsystematically appeared in the literature. Examples of interesting crack problems involving stress singularities as well as the elimination of singularities are considered. An equilibrium solution is utilized together with a straightforward physical argument to produce rationally two empirical formulae previously proposed in the literature to model embrittlement and stress corrosion cracking phenomena. This solution is further used to model embrittlement and stress corrosion cracking data more successfully than previous attempts. An appendix on certain preliminary elastodiffusive fracture criteria is given.

Contaminant transport in fractured porous media: Analytical solutions for a system of parallel fractures

An exact analytical solution is developed for the problem of transient contaminant transport in discrete parallel fractures situated in a porous rock matrix. The solution takes into account advective transport in the fractures, molecular diffusion and mechanical dispersion along the fracture axes, molecular diffusion from the fracture to the porous matrix, adsorption onto the face of the matrix, adsorption within the matrix, and radioactive decay. The general transient solution is in the form of a double integral that is evaluated using Gauss‐Legendre quadrature. A transient solution is also presented for the simpler problem that assumes negligible longitudinal dispersion along the fracture. This assumption is usually reasonable when the advective flux in a fracture is large. A comparison between two steady state solutions, one with dispersion and one without, permits a criterion to be developed that is useful for assessing the significance of longitudinal dispersion in terms of the overall system response. Examples of the solutions demonstrate that penetration distances along fractures can be substantially larger through multiple, closely spaced fractures than through a single fracture because of the limited capability of the finite matrix to store solute.