scholarly journals The Constructive Solution of the Fractal Composite Reservoir with Stress-Sensitivity Formation

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Wei Li ◽  
Songlin Zhang ◽  
Haohan Liu ◽  
Shunchu Li
Keyword(s):  
Traditional Approach ◽  
Perturbation Technique ◽  
Outer Boundary ◽  
Linear Equations ◽  
Stress Sensitivity ◽  
Constructive Method ◽  
Reservoir Model ◽  
Regular Perturbation ◽  
Pressure Derivative ◽  
Type Curves

The concentric two-zone composite reservoir model is a boundary value problems (BVPs) of modified Bessel equations. In this paper, we propose a constructive method to solve the BVPs for the system of modified Bessel equations with Robin mixed outer boundary condition and apply it to solve a two-zone fractal composite reservoir seepage model with stress-sensitivity formation. By using Pedrosa variable substitution, regular perturbation technique, Laplace transform, and Stehfest numerical inversion technique, the unified expression for the solutions of the reservoir model with three outer boundary (infinite, impermeable, and constant pressure) conditions is constructed. Type curves of bottom-hole pressure and pressure derivative are drawn, and sensitivity analysis of reservoir parameters are carried out. In comparison with the traditional approach, the solutions of this model are simple and regular, with continued fraction form, the constructive method is efficient and easy to operate. The application of this method avoids the complicated and trivial derivative operation and the use of Cramer’s rule to solve the system of linear equations. It can help to better understand the relationship between the solutions of the reservoir model and the inner and outer boundary conditions. The constructive method can be applied not only to solve the fractal composite reservoir model but also to solve more general reservoir model, BVPs of fluid diffusion, heat conduction, and so on.

Similar Constructive Method for Solving the Model of the Seepage in Multilayered Reservoir

2013 ◽  
Vol 739 ◽  
pp. 298-302
Author(s):  
Wei Li ◽  
Rong Jun Huang ◽  
Shun Chu Li ◽  
Dong Dong Gui
Keyword(s):  
Structural Equation ◽  
Outer Boundary ◽  
Constructive Method ◽  
Test Model ◽  
Well Test ◽  
Reservoir Model ◽  
Well Test Analysis ◽  
Test Analysis ◽  
The Laplace Transform ◽  
Well Test Model

A well test model analysis that based on the three outer boundary conditions (infinite boundary, closed boundary, constant value out boundary) is established for multilayered reservoir; The solutions to the distribution of reservoir pressure and the bottom-hole pressure are obtained in the Laplace space by the use of the Laplace transform; Though the analysis of solution expressions, the solutions to the reservoir model under the condition of three outer boundaries are found to have the same expression and a new method is obtained to solve the boundary value problem of such models of reservoirsimilar constructive method. The similar structural equation of the solution to the reservoir model ,which is obtained by the similar constructive method, is not only convenient for well test engineer to program the corresponding software for well test analysis but also has an important meaning to the theoretical analysis of the seepage regularity of reservoir.

A New Method and Applications of the Boundary Value Problem of Differential Equation

2014 ◽  
Vol 937 ◽  
pp. 695-699
Author(s):  
Hong E Li ◽  
Xiao Xu Dong ◽  
Shun Chu Li ◽  
Dong Dong Gui ◽  
Cong Yin Fan
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Kernel Function ◽  
Boundary Value ◽  
Outer Boundary ◽  
Seepage Flow ◽  
Petroleum Engineering ◽  
Constructive Method ◽  
Linear Homogeneous Differential Equation ◽  
The Mathematical Model

The similar structure of solution for the boundary value problem of second order linear homogeneous differential equation has been studied. Based on the analysis of the relationship between similar structure of solution, its kernel function, the equation and boundary conditions, similar constructive method (shortened as SCM) of solution is obtained. According to the SCM, the similar structure of solution and its kernel function are constructed for the mathematical model of homogeneous reservoir which considers the influence of bottom-hole storage and skin effect under the infinite outer boundary condition. The SCM is a new and innovative way to solve boundary value problem of differential equation and seepage flow theory, which is especially used in Petroleum Engineering.

scholarly journals New Understanding of Transient Pressure Response in the Transition Zone of Oil-Water and Gas-Water Systems

Geofluids ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-15 ◽  
Author(s):  
Wenbin Xu ◽  
Zhihui Liu ◽  
Jie Liu ◽  
Yongfei Yang
Keyword(s):  
Transition Zone ◽  
Outer Boundary ◽  
Water System ◽  
Pressure Response ◽  
Transient Pressure ◽  
Type Curves ◽  
Boundary Type ◽  
Study Results ◽  
Effective Mobility ◽  
Oil Water

Well test analysis requires a preselected model, which relies on the context input and the diagnostic result through the pressure logarithmic derivative curve. Transient pressure outer boundary response heavily impacts on the selection of such a model. Traditional boundary-type curves used for such diagnostic purpose are only suitable for single-phase flow in a homogeneous reservoir, while practical situations are often much more complicated. This is particularly true when transient pressure is derived during the field development phase, for example, from permanent down-hole gauge (PDG), where outer boundary condition such as an active aquifer with a transition zone above it plays a big role in dominating the late time pressure response. In this case, capillary pressure and the total mobility in the transition zone have significant effect on the pressure response. This effect is distinctly different for oil-water system and gas water system, which will result in the pressure logarithmic derivatives remarkably different from the traditional boundary-type curves. This paper presents study results derived through theoretical and numerical well testing approaches to solve this problem. The outcome of this study can help in understanding the reservoir behavior and guiding the management of mature field. According to the theoretical development by Thompson, a new approach was derived according to Darcy’s law, which shows that pressure response in the transition zone is a function of total effective mobility. For oil-water system, the total effective mobility increases with an increase in the radius of transition zone, while for gas-water system, the effect is opposite.

scholarly journals Perturbation Approach in the Dynamic Buckling of a Model Structure with a Cubic-quintic Nonlinearity Subjected to an Explicitly Time Dependent Slowly Varying Load

2019 ◽  
pp. 1-19
Author(s):  
A. M. Ette ◽  
I. U. Udo-Akpan ◽  
J. U. Chukwuchekwa ◽  
A. C. Osuji ◽  
M. F. Noah
Keyword(s):  
Perturbation Technique ◽  
Dynamic Buckling ◽  
Buckling Load ◽  
Time Dependent ◽  
Initial Time ◽  
Perturbation Approach ◽  
Elastic Model ◽  
Model Structure ◽  
Regular Perturbation ◽  
Long Run

This investigation is concerned with analytically determining the dynamic buckling load of an imperfect cubic-quintic nonlinear elastic model structure struck by an explicitly time-dependent but slowly varying load that is continuously decreasing in magnitude. A multi-timing regular perturbation technique in asymptotic procedures is utilized to analyze the problem. The result shows that the dynamic buckling load depends, among other things, on the first derivative of the load function evaluated at the initial time. In the long run, the dynamic buckling load is related to its static equivalent, and that relationship is independent of the imperfection parameter. Thus, once any of the two buckling loads is known, then the other can easily be evaluated using this relationship.

scholarly journals To the Solution of Geometric Inverse Heat Conduction Problems

2021 ◽  
Vol 24 (1) ◽  
pp. 6-12
Author(s):  
Yurii M. Matsevytyi ◽  
◽  
Valerii V. Hanchyn ◽  
Keyword(s):  
Heat Conduction ◽  
Iterative Process ◽  
Regularization Parameter ◽  
Outer Boundary ◽  
Linear Equations ◽  
The Body ◽  
Two Dimensional ◽  
Inverse Heat Conduction ◽  
System Of Linear Equations ◽  
Inverse Heat Conduction Problems

On the basis of A. N. Tikhonov’s regularization theory, a method is developed for solving inverse heat conduction problems of identifying a smooth outer boundary of a two-dimensional region with a known boundary condition. For this, the smooth boundary to be identified is approximated by Schoenberg’s cubic splines, as a result of which its identification is reduced to determining the unknown approximation coefficients. With known boundary and initial conditions, the body temperature will depend only on these coefficients. With the temperature expressed using the Taylor formula for two series terms and substituted into the Tikhonov functional, the problem of determining the increments of the coefficients can be reduced to solving a system of linear equations with respect to these increments. Having chosen a certain regularization parameter and a certain function describing the shape of the outer boundary as an initial approximation, one can implement an iterative process. In this process, the vector of unknown coefficients for the current iteration will be equal to the sum of the vector of coefficients in the previous iteration and the vector of the increments of these coefficients, obtained as a result of solving a system of linear equations. Having obtained a vector of coefficients as a result of a converging iterative process, it is possible to determine the root-mean-square discrepancy between the temperature obtained and the temperature measured as a result of the experiment. It remains to select the regularization parameter in such a way that this discrepancy is within the measurement error. The method itself and the ways of its implementation are the novelty of the material presented in this paper in comparison with other authors’ approaches to the solution of geometric inverse heat conduction problems. When checking the effectiveness of using the method proposed, a number of two-dimensional test problems for bodies with a known location of the outer boundary were solved. An analysis of the influence of random measurement errors on the error in identifying the outer boundary shape is carried out.

Pressure Transient Behavior for Alternating Polymer Flooding in a Three-zone Composite Reservoir

2017 ◽  
Vol 25 (1) ◽  
pp. 1-10 ◽  
Author(s):  
Changyu Zhu ◽  
Shiqing Cheng ◽  
Youwei He ◽  
Engao Tang ◽  
Xiaodong Kang ◽  
...  
Keyword(s):  
History Matching ◽  
Petroleum Industry ◽  
Composite Model ◽  
Polymer Flooding ◽  
Analysis Model ◽  
High Concentration ◽  
Pressure Derivative ◽  
Type Curves ◽  
Formation Evaluation ◽  
Pressure Analysis

Alternating polymer flooding has achieved great attractions recently in oil industry, however, the research of pressure analysis in alternating polymer flooding reservoir is rare. This work presents a numerical pressure analysis method of three-zone composite model for formation evaluation. A new numerical pressure analysis model (three-zone composite model) is established by considering diffusion, convection, shear, and inaccessible pore volume, which is based on the rheology experiments. Based on this model, the type curves are then developed and sensitivity analysis is further conducted. The type curves have seven regimes in three-zone composite model. The characteristic is the obvious upturn of pressure derivative curve in transient regime between low concentration and high concentration polymer solution. Formation parameters can be interpreted by history matching and formation evaluation can be conducted based on this model. As an important part of formation evaluation, formation damage as a result of adsorption of polymers in porous media is evaluated by comparing the interpreted permeability with the original value before polymer flooding. The field test data proves that this proposed method can accurately evaluate reservoir characteristics in alternating polymer flooding reservoirs, which emphasizes the potential application of this method in petroleum industry.

scholarly journals Investigation on Interference Test for Wells Connected by a Large Fracture

2019 ◽  
Vol 9 (1) ◽  
pp. 206
Author(s):  
Guofeng Han ◽  
Yuewu Liu ◽  
Wenchao Liu ◽  
Dapeng Gao
Keyword(s):  
Double Porosity ◽  
Flow In Porous Media ◽  
Connection Form ◽  
Analysis Model ◽  
Pressure Derivative ◽  
Type Curve ◽  
Type Curves ◽  
Multi Stage ◽  
Interference Test ◽  
Straight Lines

Pressure communication between adjacent wells is frequently encountered in multi-stage hydraulic fractured shale gas reservoirs. An interference test is one of the most popular methods for testing the connectivity of a reservoir. Currently, there is no practical analysis model of an interference test for wells connected by large fractures. A one-dimensional equation of flow in porous media is established, and an analytical solution under the constant production rate is obtained using a similarity transformation. Based on this solution, the extremum equation of the interference test for wells connected by a large fracture is derived. The type-curve of pressure and the pressure derivative of an interference test of wells connected by a large fracture are plotted, and verified against interference test data. The extremum equation of wells connected by a large fracture differs from that for homogeneous reservoirs by a factor 2. Considering the difference of the flowing distance, it can be concluded that the pressure conductivity coefficient computed by the extremum equation of homogeneous reservoirs is accurate in the order of magnitude. On the double logarithmic type-curve, as time increases, the curves of pressure and the pressure derivative tend to be parallel straight lines with a slope of 0.5. When the crossflow of the reservoir matrix to the large fracture cannot be ignored, the slope of the parallel straight lines is 0.25. They are different from the type-curves of homogeneous and double porosity reservoirs. Therefore, the pressure derivative curve is proposed to diagnose the connection form of wells.

scholarly journals Pressure Analysis for Volume Fracturing Vertical Well considering Low-Velocity Non-Darcy Flow and Stress Sensitivity

Geofluids ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-10 ◽  
Author(s):  
Zhongwei Wu ◽  
Chuanzhi Cui ◽  
Japan Trivedi ◽  
Ning Ai ◽  
Wenhao Tang
Keyword(s):  
Flow Regime ◽  
Stress Sensitivity ◽  
Darcy Flow ◽  
Transition Flow ◽  
Low Velocity ◽  
Pressure Derivative ◽  
Threshold Pressure Gradient ◽  
Vertical Well ◽  
Dimensionless Pressure ◽  
Volume Fracturing

In general, there is stress sensitivity damage in tight reservoirs and fractures. Furthermore, the flow in tight reservoirs is the low-velocity non-Darcy flow. Currently, few researches of pressure analysis for volume fracturing vertical well are conducted simultaneously considering the low-velocity non-Darcy flow and stress sensitivity. In the paper, a novel flow model of a volume fractured vertical well is proposed and solved numerically. Firstly, the threshold pressure gradient, permeability modulus, and experimental data are, respectively, utilized to characterize the low-velocity non-Darcy flow, matrix stress sensitivity, and fracture stress sensitivity. Then, a two-region composite reservoir is established to simulate the vertical well with volume fracturing. After that, the logarithm meshing method is used to discrete the composite reservoir, and the flow model is solved by the method of finite difference and IMPES. Finally, the model verification is conducted, and the effects of the low-velocity non-Darcy flow and stress sensitivity on the pressure and pressure derivative are analyzed. The six flow regimes are identified by the dimensionless pressure and pressure derivative curve. They are, respectively, the fracture linear flow regime, early transition flow regime, radial flow regime, crossflow regime, advanced transition flow regime, and boundary controlling flow regime. The stress sensitivity and threshold pressure gradient have a great effect on the dimensionless pressure and pressure derivative. With the increase of reservoir stress sensitivity, the pressure and pressure derivative are upward at the advanced transition flow and boundary controlling regimes. However, the pressure and pressure derivative are downward at the advanced transition flow and boundary controlling regimes when the fracture sensitivity increases. An increase in the threshold pressure gradient results in a high dimensionless pressure and pressure derivative. This work reveals the effects of low-velocity non-Darcy flow and stress sensitivity on pressure and provides a more accurate reference for reservoir engineers in pressure analysis when developing a tight reservoir by using the volume fracturing vertical well.

scholarly journals Sedimentation of a surfactant-laden drop under the influence of an electric field

2018 ◽  
Vol 849 ◽  
pp. 277-311 ◽  
Author(s):  
Antarip Poddar ◽  
Shubhadeep Mandal ◽  
Aditya Bandopadhyay ◽  
Suman Chakraborty
Keyword(s):  
Surface Tension ◽  
Electric Field ◽  
Surfactant Concentration ◽  
Perturbation Technique ◽  
Internal Flow ◽  
Dielectric Fluid ◽  
Regular Perturbation ◽  
Spherical Drop ◽  
Leaky Dielectric ◽  
Marangoni Stress

The sedimentation of a surfactant-laden deformable viscous drop acted upon by an electric field is considered theoretically. The convection of surfactants in conjunction with the combined effect of electrohydrodynamic flow and sedimentation leads to a locally varying surface tension, which subsequently alters the drop dynamics via the interplay of Marangoni, Maxwell and hydrodynamic stresses. Assuming small capillary number and small electric Reynolds number, we employ a regular perturbation technique to solve the coupled system of governing equations. It is shown that when a leaky dielectric drop is sedimenting in another leaky dielectric fluid, the Marangoni stress can oppose the electrohydrodynamic motion severely, thereby causing corresponding changes in the internal flow pattern. Such effects further result in retardation of the drop settling velocity, which would have otherwise increased due to the influence of charge convection. For non-spherical drop shapes, the effect of Marangoni stress is overcome by the ‘tip-stretching’ effect on the flow field. As a result, the drop deformation gets intensified with an increase in sensitivity of the surface tension to the local surfactant concentration. Consequently, for an oblate type of deformation the elevated drag force causes a further reduction in velocity. For similar reasons, prolate drops experience less drag and settle faster than the surfactant-free case. In addition to this, with increased sensitivity of the interfacial tension to the surfactant concentration, the asymmetric deformation about the equator gets suppressed. These findings may turn out to be of fundamental significance towards designing electrohydrodynamically actuated droplet-based microfluidic systems that are intrinsically tunable by varying the surfactant concentration.

Production Performance Analysis for Multi-Branched Horizontal Wells in Composite Coal Bed Methane Reservoir Considering Stress Sensitivity

2020 ◽  
Vol 142 (7) ◽  
Author(s):  
Ruizhong Jiang ◽  
Xiuwei Liu ◽  
Yongzheng Cui ◽  
Xing Wang ◽  
Yue Gao ◽  
...  
Keyword(s):  
Superposition Principle ◽  
Horizontal Wells ◽  
Stress Sensitivity ◽  
Field Application ◽  
Model Verification ◽  
Production Performance ◽  
Coal Bed ◽  
Coal Bed Methane ◽  
Element Discretization ◽  
Type Curves

Abstract Coal bed methane (CBM) significantly contributes to unconventional energy resources. With the development of the drilling technology, multi-branched horizontal wells (MBHWs) have been put into the exploitation of CBM. In this paper, a semi-analytical mathematical model is introduced to study the production characteristics of MBHWs in the composite CBM reservoir. Stress sensitivity, composite reservoir, and complex seepage mechanisms (desorption, diffusion, and Darcy flow) are taken into consideration. Through Pedrosa transformation, Perturbation transformation, Laplace transformation, Finite cosine transformation, element discretization, superposition principle, and Stehfest numerical inversion, pseudo-pressure dynamic curves and production decline curves are plotted and 13 flow regimes are divided. Then, the sensitivity analysis of related parameters is conducted to study the influences of these parameters based on these two type curves. Model verification and field application are introduced which shows that the model is reliable. The model proposed in this paper and relevant results analysis can provide some significant guidance for a better understanding of the production behavior of MBHWs in the composite CBM reservoir.

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