scholarly journals A New Way to Calculate Flow Pressure for Low Permeability Oil Well with Partially Penetrating Fracture

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
Xiong Ping ◽  
Liu Hailong ◽  
Hu Haixia ◽  
Wang Guan
Keyword(s):  
Separation Of Variables ◽  
Oil Well ◽  
Mathematical Methods ◽  
Laplace Domain ◽  
Linear Flow ◽  
Fourier Cosine ◽  
Fracture Length ◽  
Vertical Permeability ◽  
Flow Pressure ◽  
Boundary Width

In order to improve the validity of the previous models on calculating flow pressure for oil well with partially perforating fracture, a new physical model that obeys the actual heterogeneous reservoir characteristics was built. Different conditions, including reservoir with impermeable top and bottom borders or the reservoir top which has constant pressure, were considered. Through dimensionless transformation, Laplace transformation, Fourier cosine transformation, separation of variables, and other mathematical methods, the analytical solution of Laplace domain was obtained. By using Stephenson numerical methods, the numerical solution pressure in a real domain was obtained. The results of this method agree with the numerical simulations, suggesting that this new method is reliable. The following sensitivity analysis showed that the pressure dynamic linear flow curve can be divided into four flow streams of early linear flow, midradial flow, advanced spherical flow, and border controlling flow. Fracture length controls the early linear flow. Permeability anisotropy significantly affects the midradial flow. The degree of penetration and fracture orientation dominantly affect the late spherical flow. The boundary conditions and reservoir boundary width mainly affect the border controlling flow. The method can be used to determine the optimal degree of opening shot, vertical permeability, and other useful parameters, providing theoretical guidance for reservoir engineering analysis.

scholarly journals Quantization of a 3D Nonstationary Harmonic plus an Inverse Harmonic Potential System

2016 ◽  
Vol 2016 ◽  
pp. 1-6 ◽  
Author(s):  
Salim Medjber ◽  
Hacene Bekkar ◽  
Salah Menouar ◽  
Jeong Ryeol Choi
Keyword(s):  
Separation Of Variables ◽  
Type Equation ◽  
Invariant Operator ◽  
Wave Functions ◽  
Time Dependent ◽  
Harmonic Potential ◽  
Uncertainty Relations ◽  
Mathematical Methods ◽  
Potential System ◽  
Uvarov Method

The Schrödinger solutions for a three-dimensional central potential system whose Hamiltonian is composed of a time-dependent harmonic plus an inverse harmonic potential are investigated. Because of the time-dependence of parameters, we cannot solve the Schrödinger solutions relying only on the conventional method of separation of variables. To overcome this difficulty, special mathematical methods, which are the invariant operator method, the unitary transformation method, and the Nikiforov-Uvarov method, are used when we derive solutions of the Schrödinger equation for the system. In particular, the Nikiforov-Uvarov method with an appropriate coordinate transformation enabled us to reduce the eigenvalue equation of the invariant operator, which is a second-order differential equation, to a hypergeometric-type equation that is convenient to treat. Through this procedure, we derived exact Schrödinger solutions (wave functions) of the system. It is confirmed that the wave functions are represented in terms of time-dependent radial functions, spherical harmonics, and general time-varying global phases. Such wave functions are useful for studying various quantum properties of the system. As an example, the uncertainty relations for position and momentum are derived by taking advantage of the wave functions.

scholarly journals Using Homo-Separation of Variables for Solving Systems of Nonlinear Fractional Partial Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Abdolamir Karbalaie ◽  
Hamed Hamid Muhammed ◽  
Bjorn-Erik Erlandsson
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Separation Of Variables ◽  
General Purpose ◽  
New Method ◽  
Fractional Partial Differential Equations ◽  
Mathematical Methods ◽  
Partial Differential ◽  
Separation Of Variables Method ◽  
Linear And Nonlinear

A new method proposed and coined by the authors as the homo-separation of variables method is utilized to solve systems of linear and nonlinear fractional partial differential equations (FPDEs). The new method is a combination of two well-established mathematical methods, namely, the homotopy perturbation method (HPM) and the separation of variables method. When compared to existing analytical and numerical methods, the method resulting from our approach shows that it is capable of simplifying the target problem at hand and reducing the computational load that is required to solve it, considerably. The efficiency and usefulness of this new general-purpose method is verified by several examples, where different systems of linear and nonlinear FPDEs are solved.

scholarly journals Productivity Prediction Model of Vertical Artificial Fracture Well and Analysis of Influence Factor in Low Permeability Reservoir

2015 ◽  
Vol 8 (1) ◽  
pp. 420-426 ◽  
Author(s):  
Chengli Zhang ◽  
Dezhi Liang ◽  
Daiyin Yin ◽  
Guoliang Song
Keyword(s):  
Numerical Simulation ◽  
Pressure Gradient ◽  
Influence Factor ◽  
Low Permeability ◽  
Oil Well ◽  
Low Permeability Reservoir ◽  
Fracture Length ◽  
Capacity Model ◽  
Start Up ◽  
Artificial Fracture

Based on the analysis of seepage mechanism of fracturing wells in low permeability reservoir, this paper establishes the capacity model of the vertical fractured well production under the factors of Start-up pressure gradient, pressure sensitive effect and the artificial fracture length. The numerical simulation is compiled and software calculates the capacity model by using numerical simulation. This simulation technique verifies the validity of the model and numerical method. On this basis, we study the influence of the included angle of artificial fracture and well array direction, artificial fracture length, start-up pressure gradient and production pressure difference to the capacity of the oil well.

Using Rate Transient Analysis and Bayesian Algorithms for Reservoir Characterization in Unconventional Gas Wells during Linear Flow

2021 ◽  
pp. 1-19
Author(s):  
P. Yuhun ◽  
O. O. Awoleke ◽  
S. D. Goddard
Keyword(s):  
Hydraulic Fracturing ◽  
Marcellus Shale ◽  
Credible Interval ◽  
Production Data ◽  
Actual Case ◽  
Linear Flow ◽  
Fracture Length ◽  
Matrix Permeability ◽  
Rate Transient Analysis ◽  
Credible Intervals

Summary The main objective of this work is to improve robust, repeatable interpretation of reservoir characteristics using rate transient analysis (RTA). This is to generate probabilistic credible intervals for key reservoir and completion variables. This resulting data-driven algorithm was applied to production data from both synthetic and actual case histories. Synthetic production data from a multistage, hydraulically fractured horizontal completion in a reservoir modeled after the Marcellus Shale reservoir were generated using a reservoir model. The synthetic production data were analyzed using a combination of RTA and Bayesian techniques. First, the traditional log-log plot was produced to identify the linear flow production regime. Using the linear flow production data and traditional RTA equations, Bayesian inversion was carried out using two distinct Bayesian methods. The “rjags” and “EasyABC” packages in the open-source statistical software R were used for the traditional and approximate inversion, respectively. Model priors were based on (1) information available about the Marcellus Shale from technical literature and (2) results from a hydraulic fracturing forward model. Posterior distributions and credible intervals were produced for the fracture length, matrix permeability, and skin factor. These credible intervals were then compared with true reservoir and hydraulic fracturing data. The methodology was also repeated for an actual case in the Barnett shale. The most substantial finding was that for nearly all the investigated cases—including complicated scenarios (such as including finite fracture conductivity, fracturing fluid flowback, and heterogeneity in fracture length in the reservoir/hydraulic fracturing forward model)—the combined RTA-Bayesian model provided a 95% credible interval that encompassed the true values of the reservoir/hydraulic fracture parameters. We also found that the choice of the prior distribution did not affect the posterior distribution/credible interval in a significant manner as long as it was moderately concentrated and consistent with engineering science. Also, a comparison of the approximate Bayesian computation (ABC) and the traditional Bayesian algorithms showed that the ABC algorithm reduced computational time by at least an order of magnitude with minimal loss in accuracy. In addition, the production history used, the number of iterations, and the tolerance of fitting in the ABC analysis had a minimal impact on the posterior distribution after an optimal point, which were determined to be at least 1-year production history, 10,000 iterations, and 0.001, respectively. In summary, the RTA-Bayesian production analysis method was implemented using relatively user-friendly computational platforms [R and Excel® (Microsoft Corporation, Redmond, Washington, USA)]. This methodology provided reasonable characterization of all key variables such as matrix permeability, fracture length, and skin when compared to results obtained from analytical methods. This probabilistic characterization has the potential to enable better understanding of well performance ranges expected from shale gas wells. The methodology described here can also be generalized to shale oil systems during linear flow.

Fracture Parameters Optimization of BZ Oilfield Horizontal Well Integral Fracturing

2013 ◽  
Vol 457-458 ◽  
pp. 692-698
Author(s):  
Wen Jiang Xu ◽  
Yong Quan Hu ◽  
Jin Zhou Zhao ◽  
Zhi Qiang Li
Keyword(s):  
Mathematical Model ◽  
Optimization Design ◽  
Horizontal Well ◽  
Parameters Optimization ◽  
Production Performance ◽  
Oil Well ◽  
Fracture Conductivity ◽  
Fracture Length ◽  
Fracture Parameters ◽  
The Impact

Horizontal well technology has become an important technological means for offshore oilfield exploitation, but at present, most of the fracture parameters optimization of horizontal well fracturing are based on the single wells productivity after fracturing and pay less attention to consider the impact of injection wells.Therefore, aiming at injection and production development mode of BZ oilfield horizontal wells after fracturing, Integral fracturing physical model and productivity forecast mathematical model of horizontal well for the purpose of improving integrated exploitation benefit of the block is established respectively.Combining with reservoir parameters of BZ oilfield, a corresponding numerical simulator is developed by means of solving mathematical model to forecast production performance of oil well with different fracture number, fracture length, fracture conductivity. The best fracture parameters are obtained through analyzing the effect of fracture parameters on accumulative oil production, which provides theoretical foundation for integral fracturing optimization design of horizontal well of BZ oilfields, and has vital site guiding significance.

Study on the Production Decline Laws for Vertical Fracture Well in Low-Permeability Gas Reservoirs

2014 ◽  
Vol 527 ◽  
pp. 81-87
Author(s):  
Yi Ning Wang ◽  
Xiao Dong Wu ◽  
Rui He Wang ◽  
Feng Peng Lai ◽  
Bei Lin Qi ◽  
...  
Keyword(s):  
Superposition Principle ◽  
Stable State ◽  
Asymmetry Factor ◽  
Low Permeability ◽  
Oil Well ◽  
Gas Reservoirs ◽  
Well Productivity ◽  
Vertical Fracture ◽  
Fracture Length ◽  
Well Production

The vertical fracture was asymmetrical about the wellbore or two wings of a fracture are not certainly in a line for the complex geo-stress in the possession of fracturing of the gas reservoirs. In view of the low permeability reservoir after fracturing developing the asymmetrical vertical fracture and non-coplanar fractures, based on the non-steady seepage theory, using the potential function theory, superimposition principle and numerical analysis method, a performance prediction model for the vertical fracture in low-permeability gas reservoirs was deduced with pressure drop superposition principle. The production decline laws were analyzed by practical cases. The result shows that the initial production of the vertical fracture is relatively high but soon followed by a sharp decline. Then, the production keeps in a relatively stable state and declines slowly in the middle and later. The fracture asymmetry factor has little effect on the gas well productivity. The non-coplanar angles have greater effect on the oil well productivity in the initial stage. The more the fracture length and the bigger the flow conductivity, the higher the oil well production and the faster the decline rate will be. However, the increase amplitude will be getting smaller and smaller along with the fracture length and flow conductivity.

scholarly journals Analysis of bottom hole pressure of unsteady flow in dual-medium fracturing vertical well

2021 ◽  
Vol 11 (3) ◽  
pp. 1393-1401
Author(s):  
Liu Hailong
Keyword(s):  
Analytical Solution ◽  
Unsteady Flow ◽  
Separation Of Variables ◽  
Control Flow ◽  
Transitional Flow ◽  
Pressure Curve ◽  
Linear Flow ◽  
Hole Pressure ◽  
Bottom Hole Pressure ◽  
Bottom Hole

AbstractIn order to improve the validity of bottom hole pressure model, and simplify its calculation process, a mathematical model of instantaneous pressure for unsteady flow was established by considering the crossflow between the fractures and matrix. Different conditions, including the reservoir top has constant pressure, were considered. The basis for obtaining bottom hole pressure is to solve diffusivity equation with the integration of axisymmetric transformation and similar methods, which is presented for the first time. Different from the traditional method of using the Green’s function and source solution, this paper uses Laplace transformation, axisymmetric transformation and similar methods, separation of variables to obtain the analytical solution of Laplace domain. Then, the Stephenson Numerical method was used to obtain the numerical solution in a real domain. The results of this method agree with the numerical simulations and actual test data, suggesting the validity and accuracy of this method. Finally, the sensitivity analysis revealed that the pressure curve can be divided into eight stages, namely, early linear flow, continuous flow transition section, fracture linear flow, formation linear flow, crossflow, transitional flow, pseudo-radial flow and boundary control flow. The advantage of the analytical solution utilized in this paper is to incorporate exchange coefficient and skin factor efficiently, providing a theoretical basis for optimizing production pressure difference and determining the reasonable productivity.

Solutions for some diffusion processes with two barriers

1970 ◽  
Vol 7 (2) ◽  
pp. 423-431 ◽  
Author(s):  
A. L. Sweet ◽  
J. C. Hardin
Keyword(s):  
Diffusion Processes ◽  
Separation Of Variables ◽  
Probability Density Functions ◽  
First Passage Times ◽  
Mathematical Methods ◽  
First Passage ◽  
Forward Equation ◽  
Forward Equations ◽  
Two Barriers ◽  
Passage Times

Use is often made of the Wiener and Ornstein-Uhlenbeck (O.U.) processes in various applications of stochastic processes to problems of engineering interest. These applications frequently involve the presence of barriers. Although mathematical methods for solving Kolmogorov's forward equation for the above processes have previously been discussed ([1], [2]), many solutions for problems with two barriers do not seem to be available in the literature. Instead, one finds solutions for unrestricted processes or simulation used in place of analytical solutions in various applications ([3], [4], [5]). In this paper, solutions of Kolmogorov's forward equations in the presence of constant absorbing and/or reflecting barriers are obtained by means of separation of variables. This enables one to obtain expressions for the probability density functions for first passage times when absorbing barriers are present. The solution for the O.U. process is used to obtain a result of Breiman's [6] concerning first passage times.

The Symplectic System for Thermo-Viscoelastic Cylinders

2011 ◽  
Vol 250-253 ◽  
pp. 3662-3665
Author(s):  
Wei Xiang Zhang ◽  
Qi Xia Liu
Keyword(s):  
Boundary Conditions ◽  
Separation Of Variables ◽  
Solution Space ◽  
Governing Equations ◽  
Laplace Domain ◽  
General Solutions ◽  
Various Boundary Conditions ◽  
Temperature Boundary ◽  
The Time Domain ◽  
Symplectic System

An exact symplectic approach is presented for the isotropic viscoelastic solids subjected to external force and temperature boundary conditions. With the use of the method of separation of variables, all the general solutions of the governing equations are derived in the Laplace domain. These general solutions are expressed in concise analytical forms, and are easily to be transformed into the time domain. Accordingly, various boundary conditions can be conveniently described by the combination of the general solutions due to the completeness of the solution space. In the numerical example, the whole character of total creep of the viscoelastic solid is clearly exhibited.

Modal Representation of Second Order Flexible Structures With Damped Boundaries

2013 ◽  
Vol 135 (6) ◽  
Author(s):  
Lea Sirota ◽  
Yoram Halevi
Keyword(s):  
Initial Conditions ◽  
Separation Of Variables ◽  
Complete Solution ◽  
Flexible Structures ◽  
Second Order ◽  
Free Response ◽  
Laplace Domain ◽  
Boundary Case ◽  
Sum Of Products ◽  
Infinite Sum

The problem of obtaining a modal (i.e., infinite series) solution of second order flexible structures with viscous damping boundary conditions is considered. In conservative boundary systems, separation of variables is well established and there exist closed form modal solutions. However, no counterpart results exist for the damped boundary case and previous publications fall short of providing a complete solution for the series, in particular, its coefficients. The paper presents the free response of damped boundary structures to general initial conditions in the form of an infinite sum of products of spatial and time functions. The problem is attended via Laplace domain approach, and explicit expressions for the series components and coefficients are derived. The modal approach is useful in finite dimension modeling, since it provides a convenient framework for truncation. It is shown via examples that often few modes suffice for approximation with good accuracy.

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