Numerical Inversion Method Research Articles

Dynamic Response of a Step Loaded Cubic Cavity Embedded in a Partially Saturated Poroelastic Half-Space by the Boundary Element Method

The boundary element method is used to analyze the problem of dynamic loading acting inside a cubic cavity located in a partially saturated poroelastic halfspace. Defining relations of a Biot’s porous medium are used, which are written in Laplace representations for unknown functions of displacements of the skeleton and pore pressures of the fillers. The boundary element methodology is based on using boundary integral equations of the direct approach. Solutions in time are obtained using the stepped method of numerical inversion of Laplace transforms. Dynamic responses of displacements and pore pressures at points on the surface of the halfspace and the cavity have been constructed. The effect of the values of the saturation coefficient and of the depth of the location of the cavity on dynamic responses has been studied.

Numerical inversion method for the Laplace transform based on Boubaker polynomials operational matrix

We investigate through this research the numerical inversion technique for the Laplace transforms cooperated by the integration Boubaker polynomials operational matrix. The efficiency of the presented approach is demonstrated by solving some differential equations. Also, this technique is combined with the standard Laplace Homotopy Perturbation Method. The numerical results highlight that there is a very good agreement between the estimated solutions with exact solutions.

On reconstruction of the coefficient in complex Helmholtz’s equation

This note summarizes some preliminary results on the fast solution of the coefficient inverse problem for the Helmholtz equation, given measured pressure in a set of observation points. The Helmholtz equation is the model PDE for the harmonic problem of the linear theory of elasticity, and this work is a move in that direction. The problem has been the primary focus for several research areas, most notably seismic exploration. Still, practical problems are very challenging because they are non-linear and large. In this paper, we develop a novel numerical method for seismic full-waveform inversion based on Newton iterations. Its distinctive future is that it does not require the Jacobian of the target functional. Thus, in certain scenarios, it will perform only a fraction of computations comparing to the conventional Gauss-Newton algorithm. We present some early results on the Helmholtz equation in two dimensions.

Memory Response in Thermoelastic Plate with Three-Phase-Lag Model

In this article, using memory-dependent derivative (MDD) on three-phase-lag model of thermoelasticity, a new generalized model of thermoelasticity theory with time delay and kernel function is constructed. The governing coupled equations of the new generalized thermoelasticity with time delay and kernel function are applied to two dimensional problem of an isotropic plate. The two dimensional equations of generalized thermoelasticity with MDD are solved using state space approach. Numerical inversion method is employed for the inversion of Laplace and Fourier transforms. The displacements, temperature and stress components for different thermoelastic models are presented graphically and the effect of different kernel and time delay on the considered parameters are observed.

Analysis of flow characteristics of the dual media shale gas reservoir with the elastic outer boundary

In order to more accurately describe the seepage characteristics of shale gas reservoirs, in this paper, an elastic outer boundary condition is introduced, and a new dual-media shale gas seepage model is established to describe the seepage characteristics of shale gas reservoirs more accurately, while considering the adsorption and desorption process. Combining Laplace transformation and Similar Structure Method, the solution of the percolation model is obtained in Laplace space. Furthermore, the solutions are inverted into real space with the Stehfest numerical inversion method. Type curves of dimensionless pressure and dimensionless pressure derivative can be used to evaluate the reservoir characteristics. The results show that the conventional three kinds of outer boundary conditions (infinite, constant pressure and closed) are actually three special cases of elastic outer boundary. The introduction of elastic outer boundary conditions not only expands the scope of data interpretation, but also closer to the actual situation of the outer boundary of the reservoir. The theory of similar structure greatly simplifies the complex process of solving the model and will have a far-reaching impact on the development of well test analysis software in the future.

Magneto-thermoelastic interactions in generalized thermoelastic half-space for varying thermal and electrical conductivity

Significant variations are observed over the working temperature range of the various engineering problems such as designing the structures heated by sudden heat flux in engineering. At that time, major variations in the coefficient of the thermal conductivity of the material are observed. Electrical conductivity is also found to be directly dependent on the thermal conductivity. Therefore, present problem enlightens the influence of the variable thermal conductivity and electrical conductivity inside a half space medium permeated by the magnetic field in context of dual phase lag thermoelasticity. Thermal shock has been applied to the non-traction boundary of the medium. Analytical solutions are calculated by adopting Laplace transform technique. Physical fields such as conductive temperature, displacement as well stress of the medium are evaluated in time-domain numerically by using a suitable method of numerical inversion of Laplace transform. Further, to predict the utility of the present model, computational results are compared to the corresponding results of heat conduction model with single phase lag (Lord and Shulman model). By examining the numerical results that have been graphically represented, significant differences have been attributable to the studied fields due to the variable thermal conductivity and different heat conduction models.

Analysis of the ferrofluid microstructure based on the static magnetic measurements

In this work, we applied regularized inversion method [New J. Phys. 19, 073012 (2017)] for microstructure analysis of 10 ferrofluid samples with different ferroparticle concentrations, which were obtained by diluting of the first one. The experimental measurements of the magnetization M(H) of all samples were subjected to numerical inversion using 120 ferroparticle fractions. Since volume concentration of ferrofluid samples does not exceed 3%, the properties of each fraction were described using Langevin theory. The magnetogranulomertic analysis confirmed that all 10 samples have the same magnetic moment distribution function, which has two peaks. The first (small) peak in the magnetic moment distribution function demonstrates the presence of a certain number of aggregates or correlated ferroparticle structures in the samples, whose magnetic moments have been compensated. The second peak corresponds to ordinary single particles distributed in the sample. The main advantage of using numerical inversion method is the absence of an a priori fixed form of the distribution function instead of a standard log-normal or Γ-distribution.

Consolidation of Viscoelastic Soil With Vertical Drains for Continuous Drainage Boundary Conditions Incorporating a Fractional Derivative Model

In geotechnical engineering, vertical drainage is the most economical method for accelerating the consolidation of a large area of soft ground. In this study, we analyze the viscoelasticity of the soil and the actual drainage conditions on the top surface of the soil, and then we introduce continuous drainage boundary conditions and adopt a fractional derivative model to describe the viscoelasticity of the soil. With the use of a viscoelasticity model, the governing partial differential equation for vertical drains under continuous drainage boundary conditions is obtained. With the application of the Crump numerical inversion method, the consolidation solution for vertical drains is also obtained. Further, the rationality of the proposed solution is verified by several examples. Moreover, some examples are provided to discuss the influence of interface drainage parameters on the top surface of soil and the viscoelasticity parameters of soil on the consolidation behavior of vertical drains. The proposed method can be applied in the fields of transport engineering to predict the consolidation settlement of a foundation reinforced by vertical drains.

Determination of Unsaturated Hydraulic Properties of Seepage Flow Process in Municipal Solid Waste

The unsaturated hydraulic characteristics of waste soil are an essential basis for predicting and evaluating leachate migration and distribution in landfills. The saturated water content and permeability coefficient were measured, and a multi-step drainage monitoring experiment was conducted indoors at different dry densities, particle sizes, and degradation ages. Single and dual permeability models were adopted to determine the unsaturated hydraulic characteristic parameters of waste. Results show that dry density and particle size are the key factors affecting the saturated water content and permeability of waste. A single degradation age has little effect on it. Respectively, the saturated water content has a linear relationship with dry density, and permeability has an exponential relationship with dry density under limited experimental data. The overflow numerical inversion method can accurately obtain the unsaturated hydraulic characteristic parameters of wastes and summarizes the values of the unsaturated hydraulic characteristic parameters of wastes with different attributes in the literature and the results of this study. The dual-permeability model performed significantly better than the single-permeability model for water movement, suggesting that a dual-domain description is required for water flow in landfills.

A semi-analytical solution of finite-conductivity multi-wing fractured well in naturally fractured reservoirs by boundary element method

Large-scale vertical well fracturing leads to formation of multi-wing fractures, which can effectively reduce fluid seepage resistance and increase single well production. As one of the post-fracture evaluation methods, pressure transient analysis is used widely in oil/gas field evaluation by engineers. It is of great significance for petroleum engineers to choose the correct and appropriate well test model for the accurate evaluation of formation parameters. Since oil/gas outer boundary is often rectangle or arbitrarily shape, a semi-analytical model of multi-wing fractured well (MWFW) with arbitrarily shaped outer boundary gas reservoirs is presented. The reservoirs and hydraulic fracture mathematical models are established and solved. Coupling reservoirs and hydraulic fracture solution, and then discreting hydraulic fracture and outer boundary. The wellbore pressure solution with arbitrarily shaped outer boundary is obtained by pressure drop superposition, boundary element method (BEM) and Stehfest numerical inversion method. The bi-wing symmetrical hydraulic fracture and MWFW with finite conductivity are verified respectively by using numerical solution and comparison results reaches a good match, which certifies the correctness of the model and solving method. Typical curves of wellbore pressure and rate decline are discussed and influence of conductivity, hydraulic fracture number and hydraulic fracture distribution style et al. on wellbore pressure and rate decline is analyzes by combining with rate distribution of different time and segment. Influence of other parameters (Storativity ratio, interporosity coefficient and boundary shape size) on wellbore pressure and rate decline is also discussed in the end of paper.

Study on the effect of polymer viscosity and Darcy velocity on relative permeability curves in polymer flooding

Relative permeability curves in polymer flooding are important parameters to reflect the two-phase flow performance in reservoir rocks, which have important application in numerical simulation, reservoir engineering and prediction of crude oil production. Although the researches on the relative permeability curves in polymer flooding have made some progress at present, the relative permeability curves used in these researches are generally obtained by traditional experiment methods (steady-state method and unsteady-state method) or microscopic pore-scale modeling method. However, these methods are incapable of considering the physicochemical properties (i.e. rheology, residual resistance factor, adsorption and diffusion) of polymer completely. In this paper, a numerical inversion method based on automatic history matching is used to determine relative permeability curves in polymer flooding. The method performs automatic history matching by Levenberg-Marquardt (LM) algorithm on the unsteady-state experimental results and the predicted results of the numerical simulation model. As the physic-chemical properties of polymer solution have been fully taken into account in the numerical simulation model by far, the method makes up for the shortcomings of the traditional methods. Based on the inversed relative permeability curves, the effects of polymer viscosity and Darcy velocity in the rheo-thinning regime on relative permeability curves are then studied respectively. The results show that polymer viscosity and Darcy velocity in the rheo-thinning regime have little influence on the shape of oil relative permeability curve, but have obvious influence on the shape of water relative permeability curve. The deviation between the water relative permeability curve of polymer flooding and the one of water flooding increases linearly with the increase of polymer viscosity and decreases in power function with the increase of Darcy velocity in the rheo-thinning regime. At last, the prediction model of relative permeability curves under the influence of polymer viscosity and Darcy velocity in the rheo-thinning regime is established by multiple nonlinear regression method. The verification results show that this prediction model is reliable and can be used to predict the relative permeability curves at different polymer viscosities and Darcy velocities in the rheo-thinning regime.

Improvement of Experimental Equipment Based on the One‐Step Outflow Test and Research on Rapid Determining SWCC

The soil‐water characteristic curve (SWCC) is an important hydraulic parameter for modeling water flow and contaminant transport in the unsaturated soil. However, direct measurement of the SWCC is still difficult. The commonly used measuring instrument in the laboratory is the pressure plate extractor. In this paper, the original pressure plate extractor has been improved based on the one‐step outflow test. Compared with the original pressure plate extractor, three devices, water storage, data acquisition system, and bubbles scouring device, are added to the improved pressure plate extractor. The improved equipment eliminates the problems of long test time consumption and inaccurate test data. A one‐step outflow test is performed utilizing the improved pressure plate extractor. The test data (the amount of spilled water and the time of pressurization) are substituted into the one‐dimensional moisture migration analysis software HYDRUS‐1D. The parameters (α, m, and n) are identified by the numerical inversion method in the inversion module of the HYDRUS‐1D, which are associated with the VG model to fit SWCC. The fitted SWCC is highly consistent with the measured SWCC, which is obtained from the conventional test of the original pressure plate extractor. Results confirm that the improved pressure plate extractor not only saves considerable time but also effectively measures the SWCC. The improved pressure plate extractor also involves a simple operation. The influence factors of controlling the geometry of SWCC are also discussed, the results of the discussion confirm that the geometry of SWCC is directly controlled by pore distribution, and the consolidation pressure is an indirect factor.

An accurate numerical method for inversion of Laplace transforms with applications in process dynamics and control

AbstractLaplace transforms are frequently used in the design of control structures for chemical engineering processes. In this paper, we have developed a novel numerical method to invert Laplace transforms, which involves a particular system of linear algebraic equations. We have proven that the error of our method is bounded, and the proposed formula asymptotically converges to the exact intended Laplace inverse function. A set of numerical simulations have been carried out to demonstrate the efficiency and reliability of our method. As a salient advantage, in addition to its high accuracy, our method requires the Laplace transform function to be defined only on the real axis.

A Model for a Multistage Fractured Horizontal Well with Rectangular SRV in a Shale Gas Reservoir

Although great success has been achieved in the shale gas industry, accurate production dynamic analyses is still a challenging task. Long horizontal wells coupling with mass hydraulic fracturing has become a necessary technique to extract shale gas efficiently. In this paper, a comprehensive mathematical model of a multiple fractured horizontal well (MFHW) in a rectangular drainage area with a rectangular stimulated reservoir volume (SRV) has been established, based on the conceptual model of “tri-pores” in shale gas reservoirs. Dimensionless treatment and Laplace transformation were employed in the modeling process, while the boundary element method was used to solve the mathematical model. The Stehfest numerical inversion method and computer programing techniques were employed to obtain dimensionless type curves, production rate, and cumulative production. Results suggest that 9 flow stages can be observed from the pseudopressure derivative type curve when the reservoir and the SRV are large enough. The number of fractures, SRV permeability, and reservoir permeability have no effect on the total production when the well is abandoned. As SRV and reservoir permeability increases, the production rate is much higher in the middle production stage. Although the SRV scale and its permeability are very important for early and intermediate production rates, the key factors restricting the shale gas production rate are the properties of the shale itself, such as adsorbed gas content, natural fractures, and organic content. The proposed model is useful for analyzing production dynamics with stimulated horizontal wells in shale gas reservoirs.

Rate Decline Analysis of Horizontal Wells with Multiple Variable Conductivity and Uneven Distributed Fractures

In this paper, a new rate decline analysis model of horizontal wells with variable conductivity and uneven distribution of multiple fractures is proposed. By Laplace transformation, point source integration, and superposition principle, solutions of multiple infinite conductivity fractures in closed reservoirs are obtained. By coupling Fredholm integral equation of variable conductivity, linear equations of variable conductivity fractures in Laplace space are obtained. Gauss-Newton iteration, Duhamel convolution, and Stehfest numerical inversion method are used to obtain the bottom hole production solution. The accuracy of the results is verified by comparing with Eclipse software simulation. Then, the influence of some important reservoir and fracture parameters on the production is analyzed. The calculative results show that the smaller the fracture spacing is, the earlier the fracture begins to decline, the more the production will decrease; the change of different fracture length with the total fracture length unchanged has almost no effect on the production; the angle between fracture and x -axis has an important effect on the production; the smaller the angle between fracture and x -axis is, the stronger the interference between fractures is, the higher the production; the initial fracture conductivity affects the early production behavior, and the higher the initial fracture conductivity, the higher the production; the larger the fracture declines index, the lower the production, but the decreasing range gradually decreases with the increase of the decline index; the larger the reservoir drainage radius, the later the energy depletion stage, the higher the production. At last, a good fitting effect is obtained by fitting an example of oil field. The model proposed in this paper enriches the model base of rate decline analysis of fractured horizontal wells and lays a theoretical foundation for efficient development and practice of tight reservoirs.

Radiative heat transfer enhancement in MHD porous channel flow of an Oldroyd-B fluid under generalized boundary conditions

This study explains the transient free convection phenomenon in a vertical porous channel subject to nonlinear thermal radiation. The infinite vertical channel encloses magnetohydrodynamic (MHD) flow of an Oldroyd-B fluid. The left channel wall possesses time-dependent velocity , while the right wall exhibits no motion. The momentum and temperature field equations are developed on the bases of momentum conservation law and Fourier’s principle of heat transfer. Laplace transformation technique and Durbin’s numerical inversion method are jointly incorporated to compute the solutions of the formulated problem. The influences of flow and material parameters on heat transfer and fluid velocity are graphically scrutinized with physical aspects. The numerical computations for skin friction and temperature gradient are tabularized to comprehensively examine the wall shear stress and heat transfer rate. Finally, velocity fields for Maxwell fluid, second grade fluid, and viscous fluid are traced out as limiting cases and their comparison is drawn with the velocity field of an Oldroyd-B fluid. Besides this, some newly published results are also deduced from the acquired solutions. It is observed that increasing the magnitude of radiation parameter Rd rapidly enhances the rate of heat transfer at the right channel wall while an inverse behavior of Nusselt number is witnessed at the left channel wall. The Maxwell fluid and second grade fluid indicate the swiftest and slowest channel flow rates respectively. The shear stress specifies dual nature for relaxation and retardation parameters subject to static and moving wall. Additionally, it is found that the flow of an Oldroyd-B fluid is retarded by a magnetic field.

Transient pressure behavior of horizontal well in gas reservoirs with arbitrary boundary

Transient pressure analysis is a crucial tool to forecast the production performance during the exploration and production process in gas reservoirs. Usually, a regular shaped outer boundary is assumed in previous studies for well-testing analysis, which is just a simplification of practical cases and cannot reflect the actual boundaries of reservoirs. In this paper, a mathematical model is established to analyze the transient pressure behaviors of a horizontal well in an arbitrarily shaped gas reservoir. Dimensionless treatment, Laplace transformation, and boundary element method are applied in solving the model, which is verified by comparing with the results from the source function method. Based on the Stehfest numerical inversion method, the models of single-porosity media and dual-porosity media are solved respectively. Then, the time-domain curves of pseudo pressure and its derivative are obtained, and the flow regimes are identified. Finally, the impacts of some critical parameters on pressure transient behaviors are analyzed, including storativity ratio, interporosity coefficient, well length, and well orientation. This paper presents an effective way to handle complex external boundary problems in gas reservoirs.

A CNN-LSTM Model for Tailings Dam Risk Prediction

Tailings ponds are places for storing industrial waste. Once the tailings pond collapses, the villages nearby will be destroyed and the harmful chemicals will cause serious environmental pollution. There is an urgent need for a reliable forecasting model, which could investigate the tendency in saturation line and issue early warnings. In order to fill the gap, this work presents a hybrid network - Long-Short-Term Memory (LSTM) and Convolutional Neural Network (CNN), namely CNN-LSTM network for predicting the tailings pond risk. Firstly, the nonliear data processing method was composed to impute the missing value with the numerical inversion (NI) method, which combines correlation analysis, sensitivity analysis, and Random Forest (RF) algorithms. Secondly, a new forecasting model was proposed to monitor the saturation line, which is the lifeline of the tailings pond and can directly reflect the stability of the tailings pond. The CNN was used to identify and learn the spatial structures in the time series, then followed by LSTM cells for detecting the long-term dependence. Finally, different experiments were conducted to evaluate the effectiveness of the model by comparing it with other state-of-the-art algorithms. The results showed that combing CNN with LSTM layers achieves the best score in mean absolute error (MAE), root-mean-square error (RMSE) and coefficient of determination (R <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ).

ЧИСЛЕННОЕ МОДЕЛИРОВАНИЕ ДИНАМИЧЕСКОГО ОТКЛИКА В ЧАСТИЧНО НАСЫЩЕННОМ ПОРОУПРУГОМ ПОЛУПРОСТРАНСТВЕ ОТ ДЕЙСТВИЯ НАГРУЗКИ ВНУТРИ КУБИЧЕСКОЙ ПОЛОСТИ

Computer modeling based on the boundary element method is performed for the problem of loading in terms of the Heaviside step function inside a cubic cavity located in a partially saturated poroelastic half-space. A poroelastic medium is represented by a heterogeneous material-based model consisting of an elastic matrix phase and two phases of fillers – liquid and gas filling the pore system. The material model corresponds to a three-component medium. The constitutive relations of poroelastic medium written in terms skeleton displacements and pore pressures of fillers are considered. The original initial-boundary value problem is reduced to a boundary value problem by using the formal application of the Laplace transform. The research technique is based on the direct approach boundary integral equations of 3D isotropic linear theory of poroelasticity. Boundary integral equations corresponding to the boundary value problem are solved by the boundary element method in combination with the collocation method. In this study 8-noded elements have been adopted to discretize the boundary of poroelastic half-space. It is assumed that the element is linear with respect to displacements and pore pressures, while only one central node is used to represent tractions and fluxes. Algorithms for eliminating singularities, decreasing the order and subdividing elements are employed to compute the integral coefficients of a discrete analogue of the boundary integral equation. Regular integrals are calculated using the Gauss quadrature formula. The solution in time is obtained by numerical inversion of the Laplace transform. The numerical inversion method relies on quadrature formulas for computing the convolution integral. The time dependences of unknown displacement functions and pore pressures at points on the surface of the half-space and the cavity are plotted. The corresponding graphs are given. The influence of the cavity depth and degree of saturation on dynamic responses is investigated. The solution obtained by using the model of a fully saturated poroelastic material is compared to that of partially saturated poroelastic material. It is noted that the model used for solving this problem leads to an underestimation of displacement and overestimation of pore pressure estimates.

Modeling of memory-dependent derivatives with the state-space approach

PurposeThe purpose of this paper is to deal with a new generalized model of thermoelasticity theory with memory-dependent derivatives (MDD).Design/methodology/approachThe two-dimensional equations of generalized thermoelasticity with MDD are solved using a state-space approach. The numerical inversion method is employed for the inversion of Laplace and Fourier transforms.FindingsThe solutions are presented graphically for different values of time delay and kernel function.Originality/valueThe governing coupled equations of the new generalized thermoelasticity with time delay and kernel function, which can be chosen freely according to the necessity of applications, are applied to a two-dimensional problem of an isotropic plate.