Unrestricted Pressure-Transient Solutions for Homogeneous and Composite Elliptical Flow Models with Limited and Full Height Inner Boundary

2019 ◽  
Author(s):  
Leif Larsen
Keyword(s):  
Pressure Transient ◽  
Flow Models ◽  
Transient Solutions ◽  
Elliptical Flow ◽  
Full Height

Pressure-Transient Behavior of Partially Penetrating Inclined Fractures With a Finite Conductivity

SPE Journal ◽  
2018 ◽  
Vol 24 (02) ◽  
pp. 811-833 ◽  
Author(s):  
Bailu Teng ◽  
Huazhou Andy Li
Keyword(s):  
Fluid Flow ◽  
Transient Flow ◽  
Flow Behavior ◽  
Transient Behavior ◽  
Finite Conductivity ◽  
Linear Flow ◽  
Pressure Transient ◽  
Proposed Model ◽  
Elliptical Flow ◽  
The Matrix

Summary Field studies have shown that, if an inclined fracture has a significant inclination angle from the vertical direction or the fracture has a poor growth along the inclined direction, this fracture probably cannot fully penetrate the formation, resulting in a partially penetrating inclined fracture (PPIF) in these formations. It is necessary for the petroleum industry to conduct a pressure-transient analysis on such fractures to properly understand the major mechanisms governing the oil production from them. In this work, we develop a semianalytical model to characterize the pressure-transient behavior of a finite-conductivity PPIF. We discretize the fracture into small panels, and each of these panels is treated as a plane source. The fluid flow in the fracture system is numerically characterized with a finite-difference method, whereas the fluid flow in the matrix system is analytically characterized on the basis of the Green's-function method. As such, a semianalytical model for characterizing the transient-flow behavior of a PPIF can be readily constructed by coupling the transient flow in the fracture and that in the matrix. With the aid of the proposed model, we conduct a detailed study on the transient-flow behavior of the PPIFs. Our calculation results show that a PPIF with a finite conductivity in a bounded reservoir can exhibit the following flow regimes: wellbore afterflow, fracture radial flow, bilinear flow, inclined-formation linear flow, vertical elliptical flow, vertical pseudoradial flow, inclined pseudoradial flow, horizontal-formation linear flow, horizontal elliptical flow, horizontal pseudoradial flow, and boundary-dominated flow. A negative-slope period can appear on the pressure-derivative curve, which is attributed to a converging flow near the wellbore. Even with a small dimensionless fracture conductivity, a PPIF can exhibit a horizontal-formation linear flow. In addition to PPIFs, the proposed model also can be used to simulate the pressure-transient behavior of fully penetrating vertical fractures (FPVFs), partially penetrating vertical fractures (PPVFs), fully penetrating inclined fractures (FPIFs), and horizontal fractures (HFs).

scholarly journals Diffusive Mass Transfer and Gaussian Pressure Transient Solutions for Porous Media

Fluids ◽  
2021 ◽  
Vol 6 (11) ◽  
pp. 379
Author(s):  
Ruud Weijermars
Keyword(s):  
Porous Media ◽  
Mass Transport ◽  
Performance Testing ◽  
Step Function ◽  
Cumulative Probability ◽  
Hydraulic Fractures ◽  
Pressure Transient ◽  
Numerical Solution Methods ◽  
Transient Solutions ◽  
Pressure Depletion

This study revisits the mathematical equations for diffusive mass transport in 1D, 2D and 3D space and highlights a widespread misconception about the meaning of the regular and cumulative probability of random-walk solutions for diffusive mass transport. Next, the regular probability solution for molecular diffusion is applied to pressure diffusion in porous media. The pressure drop (by fluid extraction) or increase (by fluid injection) due to the production system may start with a simple pressure step function. The pressure perturbation imposed by the step function (representing the engineering intervention) will instantaneously diffuse into the reservoir at a rate that is controlled by the hydraulic diffusivity. Traditionally, the advance of the pressure transient in porous media such as geological reservoirs is modeled by two distinct approaches: (1) scalar equations for well performance testing that do not attempt to solve for the spatial change or the position of the pressure transient without reference to a well rate; (2) advanced reservoir models based on numerical solution methods. The Gaussian pressure transient solution method presented in this study can compute the spatial pressure depletion in the reservoir at arbitrary times and is based on analytical expressions that give spatial resolution without gridding-meaning solutions that have infinite resolution. The Gaussian solution is efficient for quantifying the advance of the pressure transient and associated pressure depletion around single wells, multiple wells and hydraulic fractures. This work lays the basis for the development of advanced reservoir simulations based on the superposition of analytical pressure transient solutions.

A Study on Elliptical Gas Flow in Double Porosity Gas Reservoirs

2013 ◽  
Vol 275-277 ◽  
pp. 576-583
Author(s):  
Xiao Tao Zhang ◽  
Ji Yuan Yang ◽  
Jin Xuan Xie
Keyword(s):  
Gas Flow ◽  
Mathieu Equation ◽  
Mathematic Model ◽  
Double Porosity ◽  
Flow Coefficient ◽  
Complete Analysis ◽  
Mathieu Functions ◽  
Gas Reservoirs ◽  
Flow Models ◽  
Elliptical Flow

Elliptical flow exists in the near vertical fracture area, or in anisotropic reservoirs, and the classical radial flow models cannot provide complete analysis for elliptical flow. This paper presents a mathematic model for gas elliptical flow in double porosity gas reservoirs. The differential equation of gas flow is written in Mathieu equation, so that the solution can be expressed also by Mathieu functions. The numerical calculation of relevant Mathieu functions , and its derivative is done to obtain the dimensionless pseudo pressure drop in Laplace space. Property sensitivities of double porosity systems including combination of , parameter related to anisotropy, storativity ratio , interporosity flow coefficient are studied by using the Laplace numerical inversion. The new solution not only includes factors considered for classic solutions in previous papers, but also incorporates the effect of reservoir anisotropy. It is kind of modification and supplementation to classic solutions.

The Fenske Conservation Method for Pressure Transient Solutions with Storage

1985 ◽  
Vol 25 (03) ◽  
pp. 437-444
Author(s):  
R.N. Horne
Keyword(s):  
Diffusion Equation ◽  
Closed Form ◽  
Approximate Method ◽  
Simple Approximation ◽  
Tabular Data ◽  
Pressure Transient ◽  
Type Curves ◽  
System Pressure ◽  
Wellbore Storage ◽  
Transient Solutions

Abstract A method for determining pressure transient solutions for wells with storage (and skin) is presented, based upon an approach developed by Fenske in the ground water literature. By specifically conserving the entire mass of the system, including the wellbore or wellbores, it is possible to generate many common type curves. The procedure can be developed in an exact manner, in which case it is equivalent to existing techniques, however, a simple approximation step makes possible the generation of common solutions without recourse to numerical or ornate analytical techniques. In some cases the approximation provides equivalent solutions with substantially less computation, in other cases the Solutions are significantly in error, although perhaps still usable. The achievement of computationally rapid, closed-form solutions is of significant advantage for current developments in computerized interpretation. Examples shown in the paper demonstrate the application of the approximate Fenske method, and its exact generalization. The approximate method gives agreement within 2% of the standard single-well, storage and skin type curves. However, as the storage becomes small, as for example in the cylindrical wellbore case (where it is zero), the accuracy becomes unacceptable at early time. As an example of how the method can be applied to configurations other than those developed by Fenske the derivation of the method for the slug test problem is demonstrated. The solution obtained using the approximate method is within acceptable accuracy. The Fenske method is applicable to a wide variety of problems with storage and skin, probably including problems as yet unsolved. An approximate form is available for fast calculation, and the more correct form is equivalent to standard analytical methods. Introduction From a practical standpoint, the derivation of pressure transient solutions often involves intricate mathematics, even for quite simple configurations. As a result, the development of type curves for a new situation becomes a major research project which may involve the analytic or numerical inversion of complex Laplace transforms, numerical integration of Green's or Source Functions, or finite difference techniques. The use of such methods, although mathematically elegant, also makes it difficult to describe the reservoir behavior in a sufficiently closed form that it may be implemented in a computerized interpretation procedure. Such procedures can be designed to handle tabular data, but can operate procedures can be designed to handle tabular data, but can operate much more effectively if the model response (and its derivatives with respect to the unknown reservoir parameters) can be given in the form of analytical expressions. From a philosophical viewpoint, it must be acknowledged that the diffusion equation, which is almost always used to develop pressure transient solutions, does not, in fact, correctly represent pressure transient solutions, does not, in fact, correctly represent the behavior of the physical system. Pressure responses are not received, even infinitesimally, at great distance when a well is opened. Rather, there is a wave component of the response which transmits the signal through the system. Thus, since the diffusion equation is itself only an approximate solution to the problem, it seems opportune to seek a different approximation, hopefully one which does not involve the complexities of Bessel functions, Hankel transforms and series solutions that may be present in solutions to the diffusion equation. A method presented by Fenske in the groundwater literature in 1977 provides an intriguing possibility in the search for alternative solution methods. This method is particularly useful in problems involving wellbore storage, and, as will be shown in this paper, operates by replacing the Green's function of the problem by a simple approximation. By specifically conserving the mass of the system, the Fenske approach is capable of closely approximating known solutions to wellbore storage problems.

A Semianalytical Approach To Model Pressure Transients in Heterogeneous Reservoirs

2010 ◽  
Vol 13 (02) ◽  
pp. 341-358 ◽  
Author(s):  
F.. Medeiros ◽  
E.. Ozkan ◽  
H.. Kazemi
Keyword(s):  
Numerical Models ◽  
Boundary Element Methods ◽  
Transient Responses ◽  
Pressure Transient ◽  
Pressure Transients ◽  
Heterogeneous Reservoirs ◽  
Transient Solutions ◽  
Heterogeneous Formations ◽  
Complex Forms ◽  
Locally Homogeneous

Summary Pressure-transient responses of wells in a heterogeneous reservoir are usually computed with numerical models by using fine gridding and very short timesteps. An exceptions to this practice has been the use of analytical, semianalytical, and boundary-element methods for relatively simpler forms of heterogeneity, such as layering or the existence of natural fractures. This paper presents a semianalytical approach to compute pressure transients for more-complex forms of heterogeneity including composite, layered, and compartmentalized reservoirs. In this approach, the reservoir is divided into blocks corresponding to locally homogeneous substructures, and analytical pressure-transient solutions for adjacent blocks are coupled at the boundaries. This approach is consistent with the averaging effect of pressure transients and provides an alternative to full numerical modeling of pressure-transient responses in heterogeneous formations. The validation of the approach is demonstrated in comparison to the analytical solution for horizontal wells in a homogeneous reservoir. Application examples highlight the physical consistency of the approach and demonstrate its capability to model different types of reservoir heterogeneity.

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