Analytical Solution In Laplace Domain Research Articles

Dynamic response of the half-space subjected to a moving point load and thermal stress

Dynamic transient response of the half-space subjected to a moving point load and thermal stress is investigated analytically in this study. By employing the Helmholtz decomposition and introducing a moving coordinate system, the corresponding modified partial differential equations of motion for the transient waves in the half-space are firstly obtained. With one-side and two-side Laplace transformation over the new time and space variables, the second-order partial differential equations of motion in the modified system are then simplified as the ordinary differential equations, whose solutions are thereafter obtained when the boundary condition is satisfied. To get the dynamic response in time domain, the analytical solutions in Laplace domain are inverted using the Cagniard-de Hoop method. Some examples are evaluated and discussed in details for the purpose of examining the effect of the moving load and thermal stress on the transient response of the half-space.

Analytical solution in Laplace domain of Smoluchowski equation for a flat potential with a rectangular sink

We propose a new method for finding the exact analytical solution in Laplace domain for the problem where a particle is diffusing on a flat potential in the presence of a rectangular sink of arbitrary width and height. In our model, diffusive motion is described by the Smoluchowski equation. Our method with this sink of rectangular shape is very general and can be used to deal with other potentials. We have derived exact analytical expression for rate constants using our model. This is the first model where the exact analytical solution in closed form is found in the case of a sink of arbitrary width. This model is more realistic for understanding reaction–diffusion systems that all other existing models available in literature.

A New Efficient Formula for the Thermal Diffusivity Estimation from the Flash Method Taking into Account Heat Losses in Rear and Front Faces

The flash method is a commonly used technique for estimating the thermal diffusivity of solid materials. In this work, we develop a new efficient formula for the thermal diffusivity calculation to process the flash method data in the case of heat losses at the rear and front faces. First, the transient heat equation is analytically solved using the Green functions under time-dependent convection boundary conditions. The property of the descending part of the analytical solution, containing series which converges very quickly for long times, allows to express the thermal diffusivity as a function of the slope of the natural logarithm of the temperature rise and the Biot numbers related to heat losses at the rear and front faces. These Biot numbers are calculated from the ratio between the integrals of the complete temperature trends of the two faces. For this purpose, we developed a novel way to evaluate these integrals using the analytical solutions in Laplace domain. To verify the accuracy of these formulas, we simulated experimental data adding a Gaussian noise to the theoretical temperature results of the flash method data.

Influence of Biot’s Compressibility Parameters on Transient Response of Saturated Poroelastic Media

In many branches of engineering, the effect of the compressibility parameters of solid and fluid are significant in describing the realistic behavior of porous media. The principle goal of the present work is to clarify the influence of the compressibility of each constituent on consolidation and dynamic behavior of porous media. In this research, an analytical solution in Laplace domain is obtained for dynamic 1-D problem, and further, the response in time domain is evaluated using the convolution quadrature method. By considering a wide range of compressibility parameters, the consolidation and wave propagation behavior are discussed in terms of pore pressure and solid displacement. These results offer some guidance in choosing appropriate compressibility parameters in practical problems.

Pressure transient analysis of multi-fractured horizontal wells in tight oil reservoirs with consideration of stress sensitivity

Multi-fractured horizontal well (MFHW) is an effective technique to develop unconventional reservoirs. Complex fracture network around the well and hydraulic fractures is formed during the fracturing process. Fracture network and hydraulic fractures are the main seepage channel which is sensitivity to the effective stress. However, most of the existing models do not take the effect of stress sensitivity into account. In this study, a new analytical model was established for MFHW in tight oil reservoirs based on the trilinear flow model. Fractal porosity and permeability were employed to describe the heterogeneous distribution of the complex fracture network, and the stress sensitivity of fracture was considered in the model. The Pedrosa substitution and perturbation method were applied to eliminate the nonlinearity of the model. By using the Laplace transformation method, the analytical solutions in Laplace domain were obtained. Then, validations were performed to show that the model is valid. Finally, sensitivity analysis was discussed. The presented model provides a new approach to the estimation of fracturing effect and can be also utilized for recognizing formation properties of tight oil reservoirs.

Electro–magneto interaction in fractional Green-Naghdi thermoelastic solid with a cylindrical cavity

A unified mathematical model of Green–Naghdi’s thermoelasticty theories (GN), based on fractional time-derivative of heat transfer is constructed. The model is applied to solve a one-dimensional problem of a perfect conducting unbounded body with a cylindrical cavity subjected to sinusoidal pulse heating in the presence of an axial uniform magnetic field. Laplace transform techniques are used to get the general analytical solutions in Laplace domain, and the inverse Laplace transforms based on Fourier expansion techniques are numerically implemented to obtain the numerical solutions in time domain. Comparisons are made with the results predicted by the two theories. The effects of the fractional derivative parameter on thermoelastic fields for different theories are discussed.

Effects of two-temperature parameter and thermal nonlocal parameter on transient responses of a half-space subjected to ramp-type heating

Based upon the coupled thermoelasticity and Green and Lindsay theory, the new governing equations of two-temperature thermoelastic theory with thermal nonlocal parameter is formulated. To more realistically model thermal loading of a half-space surface, a linear temperature ramping function is adopted. Laplace transform techniques are used to get the general analytical solutions in Laplace domain, and the inverse Laplace transforms based on Fourier expansion techniques are numerically implemented to obtain the numerical solutions in time domain. Specific attention is paid to study the effect of thermal nonlocal parameter, ramping time, and two-temperature parameter on the distributions of temperature, displacement and stress distribution.

An Analytical Solution of Partially Penetrating Hydraulic Fractures in a Box-Shaped Reservoir

This paper presents a new method to give an analytical solution in Laplace domain directly that is used to describe pressure transient behavior of partially penetrating hydraulic fractures in a box-shaped reservoir with closed boundaries. The basic building block of the method is to solve diffusivity equation with the integration of Dirac function over the distance that is presented for the first time. Different from the traditional method of using the source solution and Green’s function presented by Gringarten and Ramey, this paper uses Laplace transform and Fourier transform to solve the diffusivity equation and the analytical solution obtained is accurate and simple. The effects of parameters including fracture height, fracture length, the position of the fracture, and reservoir width on the pressure and pressure derivative are fully investigated. The advantage of the analytical solution is easy to incorporate storage coefficient and skin factor. It can also reduce the amount of computation and compute efficiently and quickly.

Estimating first-order reaction rate coefficient for transport with nonequilibrium linear mass transfer in heterogeneous media

A travel-time based approach is developed for estimating first-order reaction rate coefficients for transport with nonequilibrium linear mass transfer in heterogeneous media. Tracer transport in the mobile domain is characterized by a travel-time distribution, and mass transfer rates are described by a convolution product of concentrations in the mobile domain and a memory function rather than predefining the mass transfer model. A constant first-order reaction is assumed to occur only in the mobile domain. Analytical solutions in Laplace domain can be derived for both conservative and reactive breakthrough curves (BTCs). Temporal-moment analyses are presented by using the first and second moments of conservative and reactive BTCs and the mass consumption of the reactant for an inverse Gaussian travel-time distribution. In terms of moment matching, there is no need for one to specify the mass transfer model. With the same capacity ratio and the mean retention time, all mass transfer models will lead to the same moment-derived reaction rate coefficients. In addition, the consideration of mass transfer generally yields larger estimations of the reaction rate coefficient than models ignoring mass transfer. Furthermore, the capacity ratio and the mean retention time have opposite influences on the estimation of the reaction rate coefficient: the first-order reaction rate coefficient is positively linearly proportional to the capacity ratio, but negatively linearly proportional to the mean retention time.

The transient two-dimensional flow through double porous media

This paper presents the analytical solutions in Laplace domain for two-dimensional nonsteady flow of slightly compressible liquid in porous media with double porosity by using the methods of integral transforms and variables separation. The effects of the ratio of storativities ω, interporosity flow parameter λ on the pressure behaviors for a vertically fractured well with infinite conductivity are investigated by using the method of numerical inversion. The new log-log diagnosis graph of the pressures is given and analysed.