Semi-analytical Solution Method Research Articles

Nonlocal quasi-3d vibration/ analysis of three-layer nanoplate surrounded by Orthotropic Visco-Pasternak foundation by considering surface effects and neutral surface concept

The present article deals with the application of refined plate theory and surface effects to investigate the nonlocal free vibration behavior of a functionally graded nano-plate composed of two piezoelectric face sheets resting on Orthotropic Visco-Pasternak. Surface and nonlocal effects are applied using the surface piezoelectricity and stress-strain gradient theories. Surface effects, refined quasi-3D higher-order theory, neutral surface position and in-plane mechanical preloads are considered to study piezoelectric sandwich nano-plate with a functionally graded core in this study for the first time, simultaneously. The size-dependent governing equations are derived based on 2D and 3D refined theory using Hamilton’s principle and different boundary conditions are studied and investigated according to the semi-analytical solution method. Comparing the results of the vibration behavior of 2D and quasi-3D models, local and non-local models, and also, examining different foundations simultaneously can be said as one of the innovations of the current research. Finally, a parametric study is presented to examine the effects of different parameters such as surface effects, small scale parameters, mechanical preloads, thickness stretching effects, neutral surface position, non-homogeneous coefficient, applied voltage, different foundation constants, boundary conditions, geometrical ratios and different theories in details. It is indicated that the inclusion of surface effects and thickness stretching effects leads to a significant influence on the vibration behavior of three-layered nano-scale plates. The presented results in this article may provide useful guidance for the design and development of the next generation of nano-devices.

A semi-analytical procedure to determine the ion recombination correction factor in high dose-per-pulse beams.

The conventional theories and methods of determining the ion recombination correction factor, such as Boag theory and the related two voltage method and Jaffé plot extrapolation, do not seem to yield accurate results in FLASH /high dose per pulse (DPP) beams ( 10 mGy DPP). This is due to the presence of a large free electron fraction that distorts the electric field inside the chamber sensitive volume. To understand the influence of these effects on the ion recombination correction factor and to develop new expressions for it, it is necessary to re-visit the underlyingphysics. To present a mathematical procedure to develop an analytical expression for the ion recombination correction factor. The expression is the basis for an extrapolation method so the correction factor can be determined in a clinical setting. A semi-analytical solution method, the homotopy perturbation method (HPM), is used to solve the partial differential equations(PDEs) describing the charge carrier physics, including space charge and free electrons. The electron velocity and attachment rate are modeled as functions of the electric field strength. An expression for the charge collection efficiency and ion recombination correction factor are developed. A fit procedure based on this expression is used to compare it to measured data from previously published articles. Another fit procedure using a general equationis also proposed and compared to the data. The series obtained for the charge collection efficiency and the ion recombination correction factor are determined to be asymptotic series and the optimal truncation established. The ion recombination correction factor exhibits a dependency due to the free electron presence. The fit using this expression agrees well with measured data as long as (1) the DPP is below 1 Gy for chambers with a 1 mm plate separation and (2) when the DPP is below 3 Gy for chambers with a 0.5 mm plate separation. In these DPP ranges, the deviation between measured and fit value did not exceed 6%. In both chamber cases the voltage range where the fit applies decreases as DPP increases. The general equationyielded comparable results. The HPM was shown to be applicable to a complex system of PDEs and generate meaningful and novel solutions, as they include both space charge and free electrons. The HPM also lends itself to other chamber geometries. The fit procedure was also shown to yield accurate results for the ion recombination correction up to the 1 Gy DPPlevel.

Thermomechanical modeling of dissimilar-material interfaces in composite structures

Due to the heterogeneities in material properties of strongly-bonded multi-material composite structures, there exist singularities of the stress fields at the material interface as well as the interface corners on the bounding surfaces. The finite-element method is probably the only plausible option to analyze the stress around such interfaces between dissimilar materials subjected to combined thermo-mechanical loading. In this article, we propose a novel mathematical model for the computational analysis of coupled thermo-elastic fields with perfectly bonded interfaces of dissimilar isotropic/orthotropic composite-material structures. A scalar function of the spatial variables, namely, the displacement function, is used to integrate the temperature field and the associated displacement-function field to obtain two nonhomogeneous partial-differential equations of equilibrium that govern two thermo-mechanical sub-problems defined in terms of the appropriate conditions of orthogonal temperature gradients. For a specific temperature field, obtained as a steady-state solution of the given thermal problem, the two thermo-mechanical sub-problems are solved independently to obtain the full-field solution of the thermo-elastic problem using the method of superposition. As an alternative to the conventional computational approach, we employed a single variable semi-analytical method of solution to determine the thermo-mechanical response of the multi-material composite structures. The continuity of the temperature and the heat flux in the thermal field problem and the continuity of the displacement and the traction vectors are utilized in the elastic field problem to determine the appropriate algebraic equations suitable for nodal application at the interfaces as well as interface corners on the boundaries of the respective problems. The quantitative comparison of the present displacement-function solutions to several mixed-boundary-value problems with those obtained by the conventional finite-element method verifies the high accuracy and effectiveness of the present method. This computational model has potential applications in identifying near-interface high-stress locations as well as possible critical regions of the physical boundary of composite structures subjected to mixed modes of thermal and mechanical loadings.

Novel calculation method to predict gas–water two-phase production for the fractured tight-gas horizontal well

The prediction of production capacity in tight gas wells is greatly influenced by the characteristics of gas–water two-phase flow and the fracture network permeability parameters. However, traditional analytical models simplify the nonlinear problems of two-phase flow equations to a large extent, resulting in significant errors in dynamic analysis results. To address this issue, this study considers the characteristics of gas–water two-phase flow in the reservoir and fracture network, utilizes a trilinear flow model to characterize the effects of hydraulic fracturing, and takes into account the stress sensitivity of the reservoir and fractures. A predictive model for gas–water two-phase production in tight fractured horizontal wells is established. By combining the mass balance equation with the Newton–Raphson iteration method, the nonlinear parameters of the flow model are updated step by step using the average reservoir pressure. The accuracy of the model is validated through comparisons with results from commercial numerical simulation software and field case applications. The research results demonstrate that the established semi-analytical solution method efficiently handles the nonlinear two-phase flow problems, allowing for the rapid and accurate prediction of production capacity in tight gas wells. Water production significantly affects gas well productivity, and appropriate fracture network parameters are crucial for improving gas well productivity. The findings of this work could provide more clear understanding of the gas production performance from the fractured tight-gas horizontal well.

A Practical Model for Gas–Water Two-Phase Flow and Fracture Parameter Estimation in Shale

The gas flow in shale reservoirs is controlled by gas desorption diffusion and multiple flow mechanisms in the shale matrix. The treatment of hydraulic fracturing injects a large amount of fracturing fluids into shale reservoirs, and the fracturing fluids can only be recovered by 30~70%. The remaining fracturing fluid invades the reservoir in the form of a water invasion layer. In this paper, by introducing the concept of a water invasion layer, the hydraulic fracture network is di-vided into three zones: major fracture, water invasion layer and stimulated reservoir volume (SRV). The mathematical model considering gas desorption, the water invasion layer and gas–water two-phase flow in a major fracture is established in the Laplace domain, and the semi-analytical solution method is developed. The new model is validated by a commercial simulator. A field case from WY shale gas reservoir in southwestern China is used to verify the utility of the model. Several key parameters of major fracture and SRV are interpreted. The gas–water two-phase flow model established in this paper provides theoretical guidance for fracturing effectiveness evaluation and an efficient development strategy of shale gas reservoirs.

Homotopy analysis of wave transformation over permeable seabeds and porous structures

Based on the Homotopy Analysis Method (HAM), a semi-analytical solution method is proposed to solve the modified mild slope equation (MMSE) for waves propagation over permeable seabeds and porous structures. The present method overcomes the mathematical difficulties in treating the nonlinear dispersion equations introduced by the permeability effect, which limits the previous semi-analytical solutions to impermeable seabed and structures only. The proposed method is applied to both 1-D and 2-D axisymmetric wave transformation problems. The method is validated by systematically comparing the semi-analytical solutions of wave amplitudes and reflection coefficients against available experimental data and numerical results, demonstrating the method's capability for predicting wave transformation over both impermeable and permeable seabeds. It is ready to show that, for impermeable conditions, the present semi-analytical solutions degenerate into the earlier solutions without considering the porous effect. The semi-analytical solution method is further extended to examine the porous effects on wave transformation and reflection in various topographical conditions.

A Semi-Analytical Model for Gas–Water Two-Phase Productivity Prediction of Carbonate Gas Reservoirs

The productivity prediction of gas wells in carbonate gas reservoirs is greatly affected by the characteristics of gas–water two-phase flow and fracture seepage parameters. Compared with numerical simulation, the productivity prediction based on the analytical model is fast and widely used, but the traditional analytical model is fairly simplified while dealing with the nonlinear problem of the two-phase seepage equation, leading to a large discrepancy in the results of dynamic analysis. To solve this problem, this paper considers the characteristics of gas–water two-phase flow in the reservoir and fracture, uses the dual-medium model to characterize the stress sensitivity of the fracture and reservoir, and establishes a gas–water two-phase productivity prediction model for carbonate gas reservoirs. Combining the flowing material balance equation with the Newton iteration method, the nonlinear parameters of the percolation model are updated step by step with the use of average formation pressure, and the gas–water two-phase model is linearized through successive iterations to obtain the semi-analytical solution of the model. The accuracy of the model was verified using a comparison with the results of commercial numerical simulation software and field application, the gas–water two-phase productivity prediction curve was obtained, and the influence of sensitive parameters on productivity was analyzed. The results show that: (1) the semi-analytical solution method can efficiently deal with the gas–water two-phase nonlinear seepage problem and obtain the productivity prediction curve of carbonate gas wells rapidly and (2) the water production of the carbonate gas reservoir seriously affects the productivity of gas wells. During the development process, the production pressure difference should be reasonably controlled to reduce the negative impact of stress sensitivity on productivity performance.

Series Solution of the Time-Dependent Schrödinger–Newton Equations in the Presence of Dark Energy via the Adomian Decomposition Method

The Schrödinger–Newton model is a nonlinear system obtained by coupling the linear Schrödinger equation of canonical quantum mechanics with the Poisson equation of Newtonian mechanics. In this paper, we investigate the effects of dark energy on the time-dependent Schrödinger–Newton equations by including a new source term with energy density proportional to the cosmological constant Λ, in addition to the particle-mass source term. The resulting Schrödinger–Newton–Λ (S-N-Λ) system cannot be solved exactly, in closed form, and one must resort to either numerical or semianalytical (i.e., series) solution methods. We apply the Adomian Decomposition Method, a very powerful method for solving a large class of nonlinear ordinary and partial differential equations, to obtain accurate series solutions of the S-N-Λ system, for the first time. The dark energy dominated regime is also investigated in detail. We then compare our results to existing numerical solutions and analytical estimates and show that they are consistent with previous findings. Finally, we outline the advantages of using the Adomian Decomposition Method, which allows accurate solutions of the S-N-Λ system to be obtained quickly, even with minimal computational resources. The extensive use of the Adomian Decomposition Method in the field of quantum mechanics and quantum field theory may open new mathematical, and physical, perspectives on obtaining semi-analytical solutions for some complex problems of quantum theory.

PowerSAS.m—An Open-Source Power System Simulation Toolbox Based on Semi-Analytical Solution Technologies

In handling complex power system simulation tasks, semi-analytical solution (SAS) methods have proven to be numerically robust and computationally efficient. They provide a competitive alternative to traditional numerical approaches. Still, there is inadequate power system simulation software, especially the open-source tools, that implements this technology. This paper introduces PowerSAS.m, an open-source toolbox that closes this gap by providing SAS baseline simulation options for power system steady-state and dynamic simulations. At its core, it implements a novel SAS method and encloses various heuristics and simulation techniques to ensure enhanced computational performance. In case studies, we verify PowerSAS.m in benchmarking comparisons and demonstrate its functionalities in grid analysis scenarios.

A Novel Trilinear Flow Model for Capturing Dynamic Behavior of Water Flooding in Core Plug with Different Fracture Inclinations in Carbonate Reservoirs

Abstract The purpose of this study is to introduce a new three-linear flow model for capturing the dynamic behavior of water flooding with different fracture occurrences in carbonate reservoirs. Low-angle and high-angle fractures with different occurrences are usually developed in carbonate reservoirs. It is difficult to simulate the water injection development process and the law of water flooding is unclear, due to the large variation of the fracture dip. Based on the characteristics of water flooding displacement streamlines in fractured cores with different occurrences, the matrix is discretized into a number of one-dimensional linear subregions, and the channeling effect between each subregion is considered in this paper. The fractures are divided into the same number of fracture cells along with the matrix subregion, and the conduction effect between the fracture cells is considered. The fractured core injection-production system is divided into three areas of linear flow: The injected fluid flows horizontally and linearly from the matrix area at the inlet end of the core to the fracture and then linearly diverts from the fracture area. Finally, the matrix area at the outlet end of the core also presents a horizontal linear flow pattern. Thus, a trilinear flow model for water flooding oil in fractured cores with different occurrences is established. The modified BL equation is used to construct the matrix water-flooding analytical solution, and the fracture system establishes a finite-volume numerical solution, forming a high-efficiency semianalytical solution method for water-flooding BL-CVF. Compared with traditional numerical simulation methods, the accuracy is over 86%, the model is easy to construct, and the calculation efficiency is high. In addition, it can flexibly portray cracks at any dip angle, calculate various indicators of water flooding, and simulate the pressure field and saturation field, with great application effect. The research results show that the greater the fracture dip angle, the higher the oil displacement efficiency. When the fracture dip angle is above 45°, the fracture occurrence has almost no effect on the oil displacement efficiency. The water breakthrough time of through fractures is earlier than that of nonthrough fractures, and the oil displacement efficiency and injection pressure are more significantly affected by the fracture permeability. With the increase of fracture permeability, the oil displacement efficiency and the injection pressure of perforated fractured cores dropped drastically. The findings of this study can help for better understanding of the water drive law and optimizing its parameters in cores with different fracture occurrences. The three-linear flow model has strong adaptability and can accurately solve low-permeability reservoirs and high-angle fractures, but there are some errors for high-permeability reservoirs with long fractures.

Resilient Adaptive Parallel sImulator for griD (RAPID): An Open Source Power System Simulation Toolbox

This paper describes an open source power system simulation toolbox: Resilient Adaptive Parallel sImulator for griD (RAPID), a package of Python codes that implements an advanced power system dynamic simulation framework. The main function of RAPID is time-domain simulation, and it contains details of the solution process, including the quasi-steady-state power flow analysis, to solve nonlinear differential algebraic equations describing the detailed representation of network and transient stability models of dynamic devices in a computationally efficient manner. In particular, RAPID utilizes and incorporates emerging solution techniques; a novel “parallel-in-time” (Parareal) algorithm, adaptive model reduction, and semi-analytical solution methods as well as the standard numerical solution methods. Moreover, the whole simulation process for the transmission network has been coupled with OpenDSS, a widely used open-source distribution system simulator, to enable the co-simulation of integrated transmission and distribution systems. Basic structure and features of RAPID are presented to provide an easy-to-understand guideline for the usage of the tool and illustrate its unique capabilities. This paper also presents simulation results for a number of test scenarios and demonstrates RAPID’s values for advancing power systems research in dynamic modeling and simulation techniques toward researchers as well as educators and students.

Natural frequency responses of hybrid polymer/carbon fiber/FG-GNP nanocomposites paraboloidal and hyperboloidal shells based on multiscale approaches

This paper is devoted to obtain the vibrational behavior of doubly curved shells with paraboloidal and hyperboloidal geometries using an efficient semi-analytical solution method. To obtain the governing differential equations of the shells, the First-order Shear Deformation Theory (FSDT) is used. In addition, the shell structure is composed of a new hybrid three phases nanocomposite material. The material includes three parts: (1) polymer matrix, (2) carbon macroscale fiber, and (3) Graphene NanoPlatelets (GNP) nanoscale filler. Three different multiscale approaches including (1) bridging model, (2) Mori-Tanaka scheme, and (3) generalized self-consistent model are employed for homogenization of hybrid matrix and macroscale carbon fibers. The governing equation of motions are obtained using Hamilton's principles and Green-Gauss theory. Afterwards, an efficient semi-analytical solution procedure entitled Generalized Differential Quadrature Method (GDQM) is used for solving the governing differential equations of the structure. Finally, the frequency parameter of the structure is achieved for different states of boundary conditions and geometric properties. Moreover, the results are verified with the responses of other references obtained for a paraboloidal (Cap) shell structure. It is worth mentioning that the effect of homogenization approach, boundary conditions, volume fraction of carbon fibers and distribution pattern of GNP within the polymer matrix are studied numerically. It is concluded that the solution method is accurate, efficient and capable to obtain the natural frequency of doubly curved shells.

Study on seismic response characteristics of the interlayer isolation structure

Based on Timoshenko beam theory and considering the layer bending deformation and shear deformation of the structure, a semi analytical solution method (SAM) is proposed for the interlayer isolation structure system. The equivalent mechanical model of cantilever beam is established by using the distributed parameter system, on which the dynamic partial differential equation is derived; the boundary condition and continuity condition are used to solve the equation to obtain the natural vibration characteristics; the orthogonal conditions are deduced by Betti’s law and the seismic response of the structure can be solved by the mode superposition method. A 10 story isolated structure is designed as well as analysed by the semi analytical method (SAM) and the ETABS finite element simulation (FESE) separately. The results show that: the existence of the isolation layer may reduce the response of the superstructure and amplify the response of the substructure, and the amplification can be solved by increasing the mass, bending stiffness of the superstructure and shear stiffness of the isolation layer. In regular interlayer isolation structure, the optimal frequency ratio of the superstructure and substructure increases with the rising of isolation layer position. The results also prove the correctness and accuracy of the SAM, which enriches the analysis theory of interlayer isolation structure and provides reference for the accurate analysis of structural response in engineering.

Vibration of FG-CNT and FG-GNP sandwich composite coupled Conical-Cylindrical-Conical shell

In this paper, free vibration analysis of sandwich composite joined conical-cylindrical-conical shells is implemented using First-order Shear Deformation Theory (FSDT). The sandwich structure is made of three layers including two face sheets and one core. These layers are reinforced with functionally graded carbon nanotubes (FG-CNTs) and functionally graded graphene nanoplatelets (FG-GNPs). Eight patterns are considered for distribution of CNTs and GNPs throughout the thickness of layers. To obtain the equivalent mechanical properties of each layer, the well-known rule of mixture is utilized. The coefficients of material matrix is calculated using the Equivalent Single Layer (ESL) theory. Moreover, the Donnell's approach is employed for obtaining the equations of motion associated to the conical and cylindrical shells. The principle of Hamilton is used for achieve the governing system of partial differential equation (PDE) of joined shells. Afterwards, this system of PDE is solved by using an efficient semi-analytical solution method, namely Generalized Differential Quadrature Method (GDQM). Different cases of boundary conditions are investigated in this study. By comparing the obtained results with the reference solutions, the correctness and efficiency of the proposed formulation are proved. Furthermore, several new, complicated and applicable examples are considered to study the effect of different parameters of geometric, material and boundary conditions on the natural frequency of the joined shells as well.

On the generalized model of shell structures with functional cross-sections

In the present study, a single general formulation has been presented for the analysis of various shell-shaped structures. The proposed model is comprehensive and a variety of theories can be used based on it. The cross-section of the shell structure can be arbitrarily analyzed with the presented equations. In other words, various types of shell structures, including cylindrical, conical, spherical, elliptical, hyperbolic, parabolic, and any non-geometric structure with functional cross-section, can be modeled mechanically with only one partial differential equation system. The obtained equations have been solved by applying SAPM semi-analytical solution method. In order to present a comprehensive research, dynamic nonlinear analysis is considered. The variation of material properties through the thickness has been assumed as functionally graded and its effect on the strength of the shell structure with the functional cross-section has been investigated. The numerical results have been compared with available papers and also with FEM results for some structures that there is no paper available for validation. Different types of shell structures have been studied in terms of cross-sectional shape and properties. Finally, the effects of some important factors on the results such as boundary conditions, nonlinear analysis, dynamic analysis, and rotation of the structure around its central axis have been conducted thoroughly. This study and its original governing equations can be considered as a comprehensive reference for mechanical analysis of various shell structures with functional cross-sectional shape.

Examination of Semi-Analytical Solution Methods in the Coarse Operator of Parareal Algorithm for Power System Simulation

With continuing advances in high-performance parallel computing platforms, parallel algorithms have become powerful tools for development of faster than real-time power system dynamic simulations. In particular, it has been demonstrated in recent years that parallel-in-time (Parareal) algorithms have the potential to achieve such an ambitious goal. The selection of a fast and reasonably accurate coarse operator of the Parareal algorithm is crucial for its effective utilization and performance. This paper examines semi-analytical solution (SAS) methods as the coarse operators of the Parareal algorithm and explores performance of the SAS methods to the standard numerical time integration methods. Two promising time-power series-based SAS methods were considered; Adomian decomposition method and Homotopy analysis method with a windowing approach for improving the convergence. Numerical performance case studies on 10-generator 39-bus system and 327-generator 2383-bus system were performed for these coarse operators over different disturbances, evaluating the number of Parareal iterations, computational time, and stability of convergence. All the coarse operators tested with different scenarios have converged to the same corresponding true solution (if they are convergent) and the SAS methods provide comparable computational speed, while having more stable convergence to the true solution in many cases.

Improvement of structural characteristics of composite thin-walled beams using variable stiffness concept via curvilinear fiber placement

This study presents the use of variable stiffness concept via curvilinear fiber placement to achieve improved structural characteristics in composite thin-walled beams (TWBs). The TWB used in the study is constructed in circumferentially asymmetric stiffness (CAS) configuration. The variation of fiber angles along the span and the width of the TWB is included by defining two fiber path functions. A parametric study is performed to investigate the effects of different fiber paths on the structural performance metrics including frequency spacing, unit twist, and critical buckling load. For this purpose, a semi-analytical solution method is developed to conduct free vibration, deformation, and buckling analyses of the TWB with curvilinear fibers. The semi-analytical method is validated with several finite element (FE) analyses performed using ABAQUS. Elastic stress analyses of TWBs with selected fiber paths subjected to simplified distributed loading are also conducted using the FE method, and a ply failure criterion is applied to evaluate the strength of these TWBs. Overall results show that curvilinear fiber placement varied along the span leads to greater structural performance for a composite TWB than the straight fiber configuration.

Homotopy Analysis-Based Hybrid Genetic Algorithm and Secant Method to Solve IVP and Higher-Order BVP

The analytical solution of the Initial Value Problem (IVP) and the Boundary Value Problem (BVP) is an essential issue in numerous engineering applications. However, it cannot usually be realized by simple methods. Accordingly, utilizing semi-analytical solution methods such as the homotopy analysis method (HAM) is an alternative. However, it depends on calculating a dedicated convergence parameter. Different algorithms were proposed to realize this parameter. Thus, converting the residual function into an optimization problem represented a promising solution. The resulted optimization problem can be solved either using traditional analytical methods or meta-heuristic algorithms. In this paper, a new algorithm called Homotopy Analysis Method-based hybrid Genetic algorithm and Secant method (HAMGS) is proposed for solving such optimization problems. The proposed hybridization is based on utilizing the Secant method in the crossover process of the genetic algorithm to improve and accelerate the convergence. Moreover, this hybridization will protect the secant method from divergence in searching for new individuals. As a result, the convergence control parameter can be detected algebraically without drawing the h-curves. This facilitates identifying the minimum number of HAM terms to reach the solution, which reduces the computational burden remarkably. The presumed improvement of pairing the genetic algorithm and Secant method for improving the HAM solver is verified via four IVPs and three higher-order BVPs. The results corroborate the competence of the proposed algorithm and its ability to solve such problems efficiently.

Mechanical simulation of artificial gravity in torus-shaped and cylindrical spacecraft

Large deformations and stress analyses in two types of space structures that are intended for people to live in space have been studied in this research. The structure under analysis is assumed to rotate around the central axis to create artificial gravitational acceleration equal to the gravity on the Earth's surface. The analysis is fully dynamic, which is formulated based on the energy method by using the first-order shear deformation shell theory in two systems, cylindrical and torus. Also, the nonlinear von Kármán strain field has been assumed. The obtained set of partial differential equations has been solved using the semi-analytical polynomial solution method (SAPM). The main purpose of this paper is to study the effects of unusual conditions in the space outside the Earth's atmosphere (which is a complete vacuum environment without pressure) on the strength of the analyzed structure. The numerical results of the governing equations have been evaluated using those of other studies and the simulation efficiency performed in this research has been proven. Finally, the effect of important parameters on the numerical results, including the angular velocity of the structure (which causes artificial gravity), the amount of imposed mechanical and hygro-thermal loads, the structure size and material specifications have been investigated in more detail.

A two-step semi-analytical solution method for the pressure response during cold water injection for a vertical well in a warm bounded reservoir

Development of a mathematical solution for injection/falloff problem of an oil-water system for transient data analysis is complicated due to the presence of two mobile phases that create a moving discontinuity. The solutions that have been developed in the past are based on a series of assumptions that make the models limited. The previous models have neglected one or more of the key reservoir and well parameters such as the variation of saturation in the invaded zone, temperature effects, wellbore storage, skin and outer boundary conditions. In this work, a generalized two-step semi-analytical solution method is presented to precisely simulate the pressure transient behaviour for injection and falloff tests following cold water injection for a fully-penetrating vertical well in a warm and bounded oil reservoir in the presence of skin and wellbore storage. The formulated model uses two simplistic steps to solve the complex mathematical problem and improve computational efficiency. The usage of the Laplace-transform finite-difference method to solve the system of equations eliminates the need for temporal discretization, thereby resolving issues related to convergence and time iterations by allowing an unlimited timestep size without the loss of stability and accuracy. The accuracy and consistency of the proposed solution method are assessed by comparing the results with that of ECLIPSE 100 and an excellent agreement is observed. The results suggest that the injection period pressure behaviour is vastly influenced by the injected water propagation under cold conditions, whereas, the falloff period pressure behaviour provides information of the water bank under cold conditions and the uninvaded zone under hot conditions. The comparison of the non-isothermal and isothermal pressure behaviour shows that the isothermal model can be improved using modified fluid viscosities to match the non-isothermal pressure behaviour that not only could minimize the mathematical complications, but also the computation time.