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Abstract

In this study we investigate the influences of pressure-dependent fluid properties and pressure-dependent fluid properties and stress-sensitive rock properties on pressure transient analysis. Single-phase flow through the porous medium is considered. These pressure-dependent terms are accounted for by use of a dimensionless pseudopressure mD(1, tD); a large variety of rates, pseudopressure mD(1, tD); a large variety of rates, geometries, and inner and outer boundary conditions are studied. For all practical production rates and most boundary conditions the pseudopressure solutions, in terms of mD(1, tD) are essentially the same as the conventional solutions using pD(1, tD) that have been documented for constant-property liquid flow. The only exceptions occur when the system studied has a closed outer boundary, and the pressure transient bas occurred for a long enough time to be affected by this boundary. in these cases, a simple adjustment can be made to the mD(1, tD) solutions to bring them in line with the pD(1, tD) solutions.

Introduction

As early as 1928, it was recognized that porous media are not always rigid and nondeformable. porous media are not always rigid and nondeformable. This problem is usually handled by means of properly chosen "average" properties. This method properly chosen "average" properties. This method only reduces the errors involved and generally does not eliminate these errors.

Classical treatments solve the diffusivity equation, which assumes that the diffusivity is a constant independent of pressure. When both pressure and property changes are small, the constant-property property changes are small, the constant-property assumption is justified. If, instead, rock and fluid property changes are important over the pressure property changes are important over the pressure range of interest, these changes cannot be neglected and a variable-property solution should be obtained.

Raghavan et al. derived a flow equation considering that rock and fluid properties vary with pressure. This equation, when expressed as a pressure. This equation, when expressed as a function of a pseudopressure, m(p), resembles the diffusivity equation. They studied pressure drawdown tests in radial bounded reservoirs produced at a constant mass rate. These test results were obtained for only a specific set of rock and fluid properties. Because this work included only a few cases, there was a need to study the use of the m(p) method for a greater variety of flow conditions.

This paper presents the results of an investigation of the application of the m(p) method to drawdown, buildup, injection, and falloff testing. Results were obtained for five different sets of rock and fluid properties. The investigation also includes the effects of wellbore damage and wellbore storage.

MATHEMATICAL FORMULATION

To formulate the mathematical model, the assumptions usually used in well-testing theory are applied. We assume horizontal flow with no gravity effects, a fully penetrating well, isothermal single-phase fluid obeying Darcy's law, an isotropic and homogeneous formation, and purely elastic rock properties. That is, physical property changes with properties. That is, physical property changes with stress changes are reversible. In the radial systems, allowance is made for a region of reduced or improved permeability near the wellbore.

The assumption of horizontal flow is not quite valid because of changes in porosity and height with pressure. However, it is easily shown that the vertical component of flow is negligible for practical rock properties and can properly be practical rock properties and can properly be neglected. Gavalas and Seinfeld also made this assumption.

The rock properties needed in the flow equation porosity, permeability, and pore compressibility as porosity, permeability, and pore compressibility as functions of pressure - are found in the literature only for sandstones. It has been concluded that, in consolidated sandstones, the deformations are usually purely elastic. Therefore, only these are considered in this study. Other rocks may not behave as elastic materials.

SPEJ

P. 140

AIME, Artificial Intelligence, compressibility, constant mass rate, deviation, diffusivity, Drillstem Testing, drillstem/well testing, equation of state, flow capacity, fluid modeling, fluid property, graph, outer boundary, permeability, porous media, pressure transient analysis, pressure transient testing, pseudopressure, Reservoir Characterization, reservoir geomechanics, straight line, Upstream Oil & Gas, Van Everdingen, wellbore storage

SPE Disciplines:

- Reservoir Description and Dynamics > Reservoir Characterization > Reservoir geomechanics (1.00)
- Reservoir Description and Dynamics > Formation Evaluation & Management > Pressure transient analysis (1.00)
- Reservoir Description and Dynamics > Formation Evaluation & Management > Drillstem/well testing (1.00)
- Reservoir Description and Dynamics > Fluid Characterization > Fluid modeling, equations of state (1.00)

Abstract

Well test analyses of unsteady-state liquid flow have been based primarily on the linearized diffusivity equation for idealized reservoirs. Studies of pressure behavior of heterogeneous reservoirs have been highly restricted, and no general correlations have been developed for systems in which reservoir porosity, permeability and compressibility, together with fluid density and viscosity, are treated as functions of pressure. A second-order, nonlinear, partial-differential equation results when variations of the above parameters are considered. in the present study, this equation was reduced by a change of variables to a form similar to the diffusivity equation, but with a pressure- (or potential-) dependent diffusivity. pressure- (or potential-) dependent diffusivity. By making this transformation, the solutions to the linear diffusivity equation may be used to obtain solutions to nonlinear flow equations in which reservoir and fluid properties are pressure dependent. This paper provides correlations in terms of dimensionless potential and dimensionless time for a closed radial-flow system producing at a constant rate. The solutions obtained have been correlated with the conventional van Everdingen and Hurst solutions. It also has been shown that the solutions can be correlated with the transient drainage concept introduced by Aronofsky and Jenkins, even though no theoretical basis exists whereby their validity can be proved. In fact, the latter correlation provides a better approximation to the nonlinear provides a better approximation to the nonlinear equation than the van Everdingen and Hurst solutions for large values of dimensionless time. Substitution of the potential described has many important consequences in addition to those already mentioned. Usually, the second-degree pressure gradient term is neglected by assuming that pressure gradients in the reservoir are small. in the present study, these gradients are handled rigorously. Moreover, the selection of parameters such as "average reservoir compressibility" is avoided.

Introduction

The concept that the porous medium is absolutely rigid and nondeformable is a valid assumption for a wide range of problems of practical interest. It has been long realized that in many problems this assumption leads to certain discrepancies, however, and that the use of "average" properties of the medium would reduce these errors. Considerable research effort has been made to study the effect of pressure-dependent rock characteristics (compressibility, pressure-dependent rock characteristics (compressibility, porosity, permeability) and fluid properties porosity, permeability) and fluid properties using analytical and /or numerical techniques. As a result, numerous methods of solution have been outlined in principle, and a larger number of particular problems have been solved by means of particular problems have been solved by means of high-speed digital computers. Rowan and Clegg give a thorough review of the basic equations governing fluid flow in porous media, showing how the form of the equation changes depending on which of the parameters are taken as functions of pressure of space variables. They also discuss the implications of linearizing the basic equations. Bixel et al. have treated problems involving a single linear and a single problems involving a single linear and a single radial discontinuity. Mueller has considered the transient response of nonhomogeneous aquifers in which permeability and other properties vary as functions of space coordinates. Carter and Closmann and Ratliff have considered the problem of composite reservoirs and studied pressure response and oil production.

SPEJ

p. 267

boundary, compressibility, constant rate, correlation, diffusivity equation, dimensionless time, Drillstem Testing, drillstem/well testing, equation, flow in porous media, flow rate, Fluid Dynamics, fluid property, general utility, investigation, numerical method, production control, production monitoring, production time, pseudopressure, radial flow, real gas, Reservoir Surveillance, Upstream Oil & Gas, Van Everdingen, variation

SPE Disciplines:

The effect of variations of pressure-dependent viscosity and gas lawdeviation factor on the flow of real gasses through porous media has beenconsidered. A rigorous gas flow equation was developed which is a second order,non-linear partial differential equation with variation coefficients. Thisequation was reduced by a change of variable to a form similar to thediffusivity equation, but with potential-dependent diffusivity. The change ofvariable can be used as a new pseudo-pressure for gas flow which replacespressure or pressure-squared as currently applied to gas flow.

Substitution of the real gas pseudo-pressure has a number of importantconsequences. First, second degree pressure gradient terms which have commonlybeen neglected under the assumption that the pressure gradient is smalleverywhere in the flow system, are rigorously handled. Omission of seconddegree terms leads to serious errors in estimated pressure distributions fortight formations. Second, flow equations in terms of the real gaspseudo-pressure do not contain viscosity or gas law deviation factors, and thusavoid the need for selection of an average pressure to evaluate physicalproperties. Third, the real gas pseudo-pressure can be determined numericallyin terms of pseudo-reduced pressures and temperatures from existing physicalproperty correlations to provide generally useful information. The real gaspseudo-pressure was determined by numerical integration and is presented inboth tabular and graphical form in this paper. Finally, production of real gascan be correlated in terms of the real gas pseudo-pressure and shown to besimilar to liquid flow as described by diffusivity equation solutions.

Application of the real gas pseudo-pressure to radial flow systems undertransient, steady-state or approximate pseudo-steady-state injection orproduction have been considered. Superposition of the linearized real gas flowsolutions to generate variable rate performance was investigated and foundsatisfactory. This provides justification for pressure build-up testing. It isbelieved that the concept of the real gas pseudo-pressure will lead to improvedinterpretation of results of current gas well testing procedures, both steadyand unsteady-state in nature, and improved forecasting of gas production.

application, boundary, correlation, Drillstem Testing, drillstem/well testing, equation, flow equation, flow in porous media, Fluid Dynamics, gas flow, ideal gas, natural gas, permeability, physical property, pressure gradient, production control, production monitoring, real gas, real gas pseudo-pressure, Reservoir Surveillance, superposition, transformation, Upstream Oil & Gas, variation

Abstract

Previous gas well test analyses have been based mainly upon linearizationsof ideal gas flow results, although a method for drawdown analysis based uponreal gas flow results has been proposed. Linearizations based upon ideal gasflow require estimation of gas physical properties at some sort of averagepressure, and implicitly involve the assumption that pressure gradients aresmall everywhere in the reservoir. A new real gas flow equation has beendeveloped by means of a substitution which couples pressure, viscosity and gaslaw deviation factor. This substitution has been called the real gaspseudo-pressure. Use of the real gas pseudo-pressure leads to simple equationsdescribing real gas flow which do not contain pressure-dependent gasproperties, and which do not require the assumption of small pressure gradientseverywhere in the flow system.

Equations required to determine flow capacity, well condition and staticformation pressure from pressure drawdown and build-up tests with the real gaspseudo-pressure concept are presented. Also shown are applications of the realgas pseudo-pressure to back-pressure testing, the gas materials balance andrigorous determination of average gas properties for previous gas flowequations. Included are sample calculations for well test analyses.

application, approximation, average value, back-pressure curve, Carter, constant rate, determination, Drillstem Testing, drillstem/well testing, equation, flow capacity, gas flow solution, gas property, physical property, production control, production monitoring, real gas, real gas pseudo-pressure, Reservoir Surveillance, skin effect, Substitution, Tech, Upstream Oil & Gas

Abstract

Drawdown and buildup data in a homogeneous, uniform, closed, cylindrical reservoir containing oil and gas and producing by solution gas drive at a constant surface oil rate were investigated. The well was assumed to be located at the center of the reservoir. Gravity effects were not included. Though the reservoir systems studied were assumed to be homogeneous, the effect of a damaged region in the vicinity of the wellbore was examined.

Recently, alternate expressions for describing multiphase flow through porous media have been presented. These expressions incorporate changes presented. These expressions incorporate changes in effective permeability and fluid properties (formation volume factor, viscosity, gas solubility) with pressure by means of a pseudopressure function. The validity of applying the pseudopressure-function concept to drawdown and pseudopressure-function concept to drawdown and buildup testing for multiphase-flow situations was investigated. The pseudopressure function for analyzing drawdown behavior is calculated difrerently from that required to analyze buildup data. Consequently, two pseudopressure functions are required for analysis of well behavior in multiphase-flow systems.

Dimensionless groups are used to extend the results to other situations having different permeabilities, spacing, reservoir thickness, well permeabilities, spacing, reservoir thickness, well radii, porosity, etc., provided the PVT relations and relative-permeability characteristics are identical to those used in this study. The pseudopressure-function concept used to analyze pseudopressure-function concept used to analyze drawdown and buildup behavior extends the applicability of the results to a wide range of PVT relations and relative-permeability characteristics.

Introduction

During the past 30 years, more than 300 publications have considered various problems publications have considered various problems pertaining to well behavior. Except for a few (about pertaining to well behavior. Except for a few (about 10), most papers examining transient pressure behavior assume that the fluids in the reservoir obey the diffusivity equation. This implies the use of a single-phase, slightly compressible fluid. The reason for the popularity of this approach is twofold: (1) the ease with which the diffusivity equation can be solved for a wide variety of problems, and (2) the demonstration by some problems, and (2) the demonstration by some workers that, for some multiphase-flow situations, single-phase flow results may be used provided appropriate modifications are made. The necessary modifications are summarized in Ref. 1.

The main objective of this study is to present a method for rigorously incorporating changes in fluid properties and relative-permeability effects in the properties and relative-permeability effects in the analysis of pressure data when two phases of oil and gas are flowing. This should enable the engineer to calculate the absolute formation permeability rather than the effective permeability to each of the flowing phases. This method is based on an idea suggested by Fetkovich, who proposed that if an expression similar to the real gas pseudopressure is defined, then equations describing pseudopressure is defined, then equations describing simultaneous flow of oil and gas through porous media may be simplified considerably. The validity of the equations and methods for calculating the pseudopressure function, however, was not presented pseudopressure function, however, was not presented by Fetkovich.

LITERATURE REVIEW AND THEORETICAL CONSIDERATIONS

General equations of motion describing multiphase flow in porous media have been known since 1936. These equations, and the assumptions involved in deriving them, are discussed thoroughly in the literature and will not be considered here.

Equations for two-phase flow were first solved by Muskat and Meres for a few special cases. Evinger and Muskat studied the effect of multiphase flow on the productivity index of a well and examined the steady radial flow of oil and gas in a porous medium. Under conditions of steady radial porous medium. Under conditions of steady radial flow the oil flow rate is given by

(1)

SPEJ

P. 196

buildup data, buildup testing, correlation, correspond, dimensionless time, equation, flow in porous media, flow rate, Fluid Dynamics, Formation Permeability, gas saturation, GOR, permeability, pressure drop, pressure transient analysis, pressure transient testing, procedure, pseudopressure function, relative-permeability characteristic, reservoir, saturation, solution gas drive, Upstream Oil & Gas

SPE Disciplines:

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