Analytical Solutions for the Radial Flow Equation With Constant-Rate and Constant-Pressure Boundary Conditions in Reservoirs With Pressure-Sensitive Permeability

2009 ◽  
Author(s):  
Torsten Friedel ◽  
Hans-Dieter Voigt
Keyword(s):  
Boundary Conditions ◽  
Analytical Solutions ◽  
Radial Flow ◽  
Constant Pressure ◽  
Flow Equation ◽  
Pressure Sensitive ◽  
Constant Rate ◽  
Pressure Boundary Conditions

Interference Between Constant-Rate and Constant-Pressure Reservoirs Sharing a Common Aquifer

1985 ◽  
Vol 25 (03) ◽  
pp. 419-426 ◽  
Author(s):  
Abraham Sageev ◽  
Roland N. Horne
Keyword(s):  
Boundary Conditions ◽  
Line Source ◽  
Infinite System ◽  
Constant Pressure ◽  
Oil Field ◽  
Oil Fields ◽  
Internal Boundary ◽  
Finite Radius ◽  
Pressure Behavior ◽  
Constant Rate

Abstract A practical pressure transient analysis method is presented for interpreting interference between two oil fields or an oil field and a gas field sharing a common aquifer. One oil field is approximated as a constant-rate line source. The other interfering field is represented by a finite-radius circular source producing at constant rate or constant pressure. pressure. A rigorous application of the superposition principle is discussed, making use of a new model where a constant rate line source produces exterior to a circular boundary. Both constant pressure and impermeable internal boundaries are considered. Dimensionless pressure drop curves for both boundary conditions are presented. For the case of a line source producing near a constant-pressure internal boundary, producing near a constant-pressure internal boundary, dimensionless curves for the instantaneous rate and the cumulative injection from this internal boundary are given. These curves may be used to forecast the actual injection/production rate and the cumulative injection/ production at the interfering reservoir as a function of time. production at the interfering reservoir as a function of time. Introduction Pressure interference between hydrocarbon reservoirs Pressure interference between hydrocarbon reservoirs situated in a common aquifer is important in understanding and forecasting the behavior of these reservoirs under exploitation. The fluid driving energy stored in a reservoir is a function of its average pressure. Production in one reservoir causes a pressure drawdown at another reservoir and, hence, changes its deliverability and economic value over a long period of time. Bell and Shepherd I considered the pressure behavior of the Woodbine sand in east Texas, which contains several reservoirs. They presented a pressure loss map that shows that production from the east Texas field affected an extensive area of the Woodbine aquifer. Moore and Truby, using an electric analyzer, described the pressure behavior of five producing fields sharing a pressure behavior of five producing fields sharing a common aquifer. They presented pressure histories for each of the five reservoirs. Every pressure history consisted of five pressure drops. The first pressure drop at a reservoir was caused by its own production, to which four interfering pressure drops caused by the neighboring reservoirs were added. The interfering effect of the TXL field on the average pressure at Wheeler field was larger than the drawdown at Wheeler field caused by its own production. production. In describing interference between two reservoirs sharing a common infinite aquifer, some assumptions as to the shape of these reservoirs must be made. Theis presented the solution for a constant-rate line source in presented the solution for a constant-rate line source in an infinite system. Staliman modified this solution for a semi-infinite system bounded by a linear boundary. If the two reservoirs may be approximated by two line sources, their pressure effects may be superposed in space to yield the pressure interference between them. super-position in space is used to assemble the effects of several producing/injecting reservoirs in the same aquifer. producing/injecting reservoirs in the same aquifer. Carslaw and Jaeger presented solutions for a single finite-radius source in an infinite medium producing at either constant rate or constant pressure. Van Everdingen and Hurst applied those solutions to flow in reservoirs. Mortada used those solutions to describe interference between oil fields and, using superposition in space, calculated the pressure response of a reservoir to its own production and to production from an interfering production and to production from an interfering reservoir. If the reservoirs are of finite radii and are not approximated by line sources, the method of superposition in space must be used with care so that the inner boundary conditions are not violated. By superposing a finite-radius source in an infinite system onto another finite-radius source in an infinite system, the inner boundary conditions at both sources are violated. Mortada's results, therefore, are only approximate. Hursts presented a method for calculating pressure interference between finite-radius reservoirs that includes the material-balance equations. Hursts and Mortada also considered interference between oil fields connected to an aquifer with two permeability regions. Mueller and Witherspoon used the finite-radius constant-rate solution and normalized the time scale to describe interference pressure changes. They concluded that, for practical pressure changes. They concluded that, for practical purposes, interference points at a distance larger than 20 times purposes, interference points at a distance larger than 20 times the radius of the source have a line-source response. Uraiet and Raghavan presented interference log-log type curves for a finite-radius source producing at a constant pressure. In this study, two circular reservoirs in an infinite system are considered. One reservoir is approximated as a constant-rate line source. The other reservoir is considered to be a finite-radius source producing at either a constant rate or a constant pressure. Only single-step changes in rate or pressure are discussed, since they are the basis for superposition in time. SPEJ P. 419

Physics Inspired Machine Learning for Solving Fluid Flow in Porous Media: A Novel Computational Algorithm for Reservoir Simulation

2021 ◽  
Author(s):  
Chico Sambo ◽  
Yin Feng
Keyword(s):  
Machine Learning ◽  
Porous Media ◽  
Fluid Flow ◽  
Boundary Conditions ◽  
Constant Pressure ◽  
Flow In Porous Media ◽  
Reservoir Engineering ◽  
Flow Boundary ◽  
Pressure Boundary Conditions ◽  
Source And Sink

Abstract The Physics Inspired Machine Learning (PIML) is emerging as a viable numerical method to solve partial differential equations (PDEs). Recently, the method has been successfully tested and validated to find solutions to both linear and non-linear PDEs. To our knowledge, no prior studies have examined the PIML method in terms of their reliability and capability to handle reservoir engineering boundary conditions, fractures, source and sink terms. Here we explored the potential of PIML for modelling 2D single phase, incompressible, and steady state fluid flow in porous media. The main idea of PIML approaches is to encode the underlying physical law (governing equations, boundary, source and sink constraints) into the deep neural network as prior information. The capability of the PIML method in handling reservoir engineering boundary including no-flow, constant pressure, and mixed reservoir boundary conditions is investigated. The results show that the PIML performs well, giving good results comparable to analytical solution. Further, we examined the potential of PIML approach in handling fluxes (sink and source terms). Our results demonstrate that the PIML fail to provide acceptable prediction for no-flow boundary conditions. However, it provides acceptable predictions for constant pressure boundary conditions. We also assessed the capability of the PIML method in handling fractures. The results indicate that the PIML can provide accurate predictions for parallel fractures subjected to no-flow boundary. However, in complex fractures scenario its accuracy is limited to constant pressure boundary conditions. We also found that mixed and adaptive activation functions improve the performance of PIML for modeling complex fractures and fluxes.

New Treatment of Pressure Boundary Conditions for DSMC Method in Micro-Channel Flow Simulations

2007 ◽  
Author(s):  
J. J. Ye ◽  
J. Yang ◽  
J. Y. Zheng ◽  
W. Z. Li ◽  
S. Z. He ◽  
...  
Keyword(s):  
Boundary Conditions ◽  
Pressure Distribution ◽  
Analytical Solutions ◽  
Stokes Equations ◽  
New Method ◽  
Micro Channel ◽  
Micro Channels ◽  
Micro Flows ◽  
New Treatment ◽  
Pressure Boundary Conditions

Using DSMC to simulate micro flows in micro-channels, the numerical treatment of boundary conditions is very important. In this paper, several previous numerical treatments of boundary conditions are discussed with their merits and demerits, and a new treatment method based on the assumption of certain pressure distribution in the cells for boundary conditions is proposed. As comparable validity tests, it is applied in the DSMC simulations for the Poiseuille micro flows in micro-channels with four types of classical pressure boundary conditions. The dimensionless velocity profiles are shown and compared with analytical solutions derived from the Navier-Stokes equations with slip boundary conditions. The pressure distributions along the centerline of the micro-channel with the different boundary conditions are presented, and the simulation solutions agree well with the slip analytical solutions. As the Knudsen number increased, a strong linearity of the pressure distribution can be evidently predicted by the new method. Compared with the inlet and outlet velocity distribution, it is shown that the new method has better efficiency than the previous methods in the convergence.

A Novel Infinite-Acting-Radial-Flow Analysis Procedure for Estimating Permeability Anisotropy From an Observation-Probe Pressure Response at a Vertical, Horizontal, or Inclined Wellbore

2014 ◽  
Vol 17 (02) ◽  
pp. 152-164 ◽  
Author(s):  
M.. Onur ◽  
P.S.. S. Hegeman ◽  
I.M.. M. Gök
Keyword(s):  
Radial Flow ◽  
Flow Analysis ◽  
Single Layer ◽  
Flow Equation ◽  
Observation Point ◽  
Analysis Procedure ◽  
Pressure Transient ◽  
Flow Equations ◽  
Vertical Permeability ◽  
Inclination Angles

Summary This paper presents a new infinite-acting-radial-flow (IARF) analysis procedure for estimating horizontal and vertical permeability solely from pressure-transient data acquired at an observation probe during an interval pressure-transient test (IPTT) conducted with a single-probe, dual-probe, or dual-packer module. The procedure is based on new infinite-acting-radial-flow equations that apply for all inclination angles of the wellbore in a single-layer, 3D anisotropic, homogeneous porous medium. The equations for 2D anisotropic cases are also presented and are derived from the general equations given for the 3D anisotropic case. It is shown that the radial-flow equation presented reduces to Prats' (1970) equation assuming infinite-acting radial flow at an observation point along a vertical wellbore in isotropic or 2D anisotropic formations of finite bed thickness. The applicability of the analysis procedure is demonstrated by considering synthetic and field packer/probe IPTT data. The synthetic IPTT examples include horizontal- and slanted-well cases, but the field IPTT is for a vertical well. The results indicate that the procedure provides reliable estimates of horizontal and vertical permeability solely from observation-probe pressure data during radial flow for vertical, horizontal, and inclined wellbores. Most importantly, the analysis does not require that both spherical and radial flow prevail at the observation probe during the test.

An Approximate Method for Transient Radial Flow

1962 ◽  
Vol 2 (03) ◽  
pp. 225-256 ◽  
Author(s):  
G. Rowan ◽  
M.W. Clegg
Keyword(s):  
Infinite Series ◽  
Analytical Solutions ◽  
Radial Flow ◽  
Numerical Solutions ◽  
Basic Theory ◽  
Variable Pressure ◽  
Flow Problems ◽  
Disturbed Zone ◽  
Approximate Analytical Solutions ◽  
Wide Range

Abstract The basic equations for the flow of gases, compressible liquids and incompressible liquids are derived and the full implications of linearising then discussed. Approximate solutions of these equations are obtained by introducing the concept of a disturbed zone around the well, which expands outwards into the reservoir as fluid is produced. Many important and well-established results are deduced in terms of simple functions rather than the infinite series, or numerical solutions normally associated with these problems. The wide range of application of this approach to transient radial flow problems is illustrated with many examples including; gravity drainage of depletion-type reservoirs; multiple well systems; well interference. Introduction A large number of problems concerning the flow of fluids in oil reservoirs have been solved by both analytical and numerical methods but in almost all cases these solutions have some disadvantages - the analytical ones usually involve rather complex functions (infinite series or infinite integrals) which are difficult to handle, and the numerical ones tend to mask the physical principles underlying the problem. It would seem appropriate, therefore, to try to find approximate analytical solutions to these problems without introducing any further appreciable errors, so that the physical nature of the problem is retained and solutions of comparable accuracy are obtained. One class of problems will be considered in this paper, namely, transient radial flow problems, and it will be shown that approximate analytical solutions of the equations governing radial flow can be obtained, and that these solutions yield comparable results to those calculated numerically and those obtained from "exact" solutions. It will also be shown that the restrictions imposed upon the dependent variable (pressure) are just those which have to be assumed in deriving the usual diffusion-type equations. The method was originally suggested by Guseinov, whopostulated a disturbed zone in the reservoir, the radius of which increases with time, andreplaced the time derivatives in the basic differential equation by its mean value in the disturbed zone. In this paper it is proposed to review the basic theory leading to the equations governing the flow of homogeneous fluids in porous media and to consider the full implications of the approximation introduced in linearising them. The Guseinov-type approximation will then be applied to these equations and the solutions for the flow of compressible and incompressible fluids, and gases in bounded and infinite reservoirs obtained. As an example of the application of this type of approximation, solutions to such problems as production from stratified reservoirs, radial permeability discontinuities; multiple-well systems, and well interference will be given. These solutions agree with many other published results, and in some cases they may be extended to more complex problems without the computational difficulties experienced by other authors. THEORY In order to review the basic theory from a fairly general standpoint it is proposed to limit the idealising assumptions to the minimum necessary for analytical convenience. The assumptions to be made are the following:That the flow is irrotational.That the formation is of constant thickness.Darcy's Law is valid.The formation is saturated with a single homogeneous fluid. SPEJ P. 225^

Effect of Non-Darcy Flow on the Constant-Pressure Production of Fractured Wells

1981 ◽  
Vol 21 (03) ◽  
pp. 390-400 ◽  
Author(s):  
K.H. Guppy ◽  
Heber Cinco-Ley ◽  
Henry J. Ramey
Keyword(s):  
Constant Pressure ◽  
Darcy Flow ◽  
Flow In Porous Media ◽  
Graphical Form ◽  
Finite Conductivity ◽  
Fracture Conductivity ◽  
Type Curves ◽  
Pressure Behavior ◽  
Apparent Conductivity ◽  
Constant Rate

Abstract In many low-permeability gas reservoirs, producing a well at constant rate is very difficult or, in many cases, impossible. Constant-pressure production is much easier to attain and more realistic in practice. This is seen when production occurs into a constant-pressure separator or during the reservoir depletion phase, when the rate-decline period occurs. Geothermal reservoirs, which produce fluids that drive backpressure turbines, and open-well production both incorporate the constant-pressure behavior. For finite-conductivity vertically fractured systems, solutions for the constant-pressure case have been presented in the literature. In many high-flow-rate wells, however, these solutions may not be useful since high velocities are attained in the fracture, which results in non-Darcy effects within the fracture. In this study, the effects of non-Darcy flow within the fracture are investigated. Unlike the constant-rate case, it was found that the fracture conductivity does not have a constant apparent conductivity but rather an apparent conductivity that varies with time. Semianalytical solutions as well as graphical solutions in the form of type curves are presented to illustrate this effect. An example is presented for analyzing rate data by using both solutions for Darcy and non-Darcy flow within the fracture. This example relies on good reservoir permeability from prefracture data to predict the non-Darcy effect accurately. Introduction To fully analyze the effects of constant-bottomhole-pressure production of hydraulically fractured wells, it is necessary that we understand the pressure behavior of finite-conductivity fracture systems producing at constant rate as well as the effects of non-Darcy flow on gas flow in porous media. Probably one of the most significant contributions in the transient pressure analysis theory for fractured wells was made by Gringarten et al.1,2 In the 1974 paper,2 general solutions were made for infinite-conductivity fractures. Cinco et al.3 found a more general solution for the case of finite-conductivity fractures and further extended this analysis in 1978 to present a graphical technique to estimate fracture conductivity.4 For the case of constant pressure at the wellbore, solutions were presented in graphical form by Agarwal et al.5 In his paper, a graph of log (1/qD) vs. log (tDxf) can be used to determine the conductivity of the fracture by using type-curve matching. Although such a contribution is of great interest, unique solutions are difficult to obtain. More recently, Guppy et al.6 showed that the Agarwal et al. solutions may be in error and presented new type curves for the solution to the constant-pressure case assuming Darcy flow in the fracture. That paper developed analytical solutions which can be applied directly to field data so as to calculate the fracture permeability-width (kfbf) product.

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