Unsteady-State Flow of Flexible Polymers in Porous Media

2001 ◽  
Vol 234 (2) ◽  
pp. 269-283 ◽  
Author(s):  
Pacelli L.J. Zitha ◽  
Guy Chauveteau ◽  
Liliane Léger
Keyword(s):  
Porous Media ◽  
Unsteady State ◽  
Flexible Polymers ◽  
State Flow

Flow of Non-Newtonian Power-Law Fluids Through Porous Media

1979 ◽  
Vol 19 (03) ◽  
pp. 155-163 ◽  
Author(s):  
A.S. Odeh ◽  
H.T. Yang
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Porous Media ◽  
Test Data ◽  
Power Law ◽  
Newtonian Fluids ◽  
Unsteady State ◽  
Partial Differential ◽  
State Flow ◽  
Power Law Fluids

Abstract The partial differential equation that describes the flow, of non-Newtonian, power-law, slightly compressible fluids in porous media is derived. An approximate solution, in closed form, is developed for the unsteady-state flow behavior and verified by. two different methods. Using the unsteady-state solution, a method for analyzing injection test data is formulated and used to analyze four injection tests. Theoretical results were used to derive steady-state equations of flow, equivalent transient drainage radius, and a method for analyzing isochronal test data. The theoretical fundamentals of the flow, of non-Newtonian power-law fluids in porous media are established. Introduction Non-Newtonian power-law fluids are those that obey the relation = constant. Here, is the viscosity, e is the shear rate at which the viscosity is measured, and n is a constant. Examples of such fluids are polymers. This paper establishes the theoretical foundation of the flow of such fluids in porous media. The partial differential equation describing this flow is derived and solved for unsteady-state flow. In addition, a method for interpreting isochronal tests and an equation for calculating the equivalent transient drainage radius are presented. The unsteady-state flow solution provides a method for interpreting flow tests (such as injection tests).Non-Newtonian power-law fluids are injected into the porous media for mobility control, necessitating a basic porous media for mobility control, necessitating a basic understanding of the flow behavior of such fluids in porous media. Several authors have studied the porous media. Several authors have studied the rheological properties of these fluids using linear flow experiments and standard viscometers. Van Poollen and Jargon presented a theoretical study of these fluids. They described the flow by the partial differential equation used for Newtonian fluids and accounted for the effect of shear rate on viscosity by varying the viscosity as a function of space. They solved the equation numerically using finite difference. The numerical results showed that the pressure behavior vs time differed from that for Newtonian fluids. However, no methods for analyzing flow-test data (such as injection tests) were offered. This probably was because of the lack of analytic solution normally required to understand the relationship among the variables.Recently, injectivity tests were conducted using a polysaccharide polymer (biopolymer). The data showed polysaccharide polymer (biopolymer). The data showed anomalies when analyzed using methods derived for Newtonian fluids. Some of these anomalies appeared to be fractures. However, when the methods of analysis developed here were applied, the anomalies disappeared. Field data for four injectivity tests are reported and used to illustrate our analysis methods. Theoretical Consideration General Consideration The partial differential equation describing the flow of a non-Newtonian, slightly compressible power-law fluid in porous media derived in Appendix A is ..........(1) where the symbols are defined in the nomenclature. JPT P. 155

scholarly journals Analytical Solution of Unsteady-state Forchheimer Flow Problem in an Infinite Reservoir: The Boltzmann Transform Approach

2021 ◽  
Vol 2 (3) ◽  
pp. 225-233
Author(s):  
Temitayo Sheriff Adeyemi
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Porous Media ◽  
Analytical Solution ◽  
Analytical Approach ◽  
Unsteady State ◽  
Mathematical Relationship ◽  
State Flow ◽  
Infinite Reservoir ◽  
Forchheimer Flow

For several decades, attempts had been made by several authors to develop models suitable for predicting the effects of Forchheimer flow on pressure transient in porous media. However, due to the complexity of the problem, they employed numerical and/or semi-analytical approach, which greatly affected the accuracy and range of applicability of their results. Therefore, in order to increase accuracy and range of applicability, a purely analytical approach to solving this problem is introduced and applied. Therefore, the objective of this paper is to develop a mathematical model suitable for quantifying the effects of turbulence on pressure transient in porous media by employing a purely analytical approach. The partial differential equation (PDE) that governs the unsteady-state flow in porous media under turbulent condition is obtained by combining the Forchheimer equation with the continuity equation and equations of state. The obtained partial differential equation (PDE) is then presented in dimensionless form (by defining appropriate dimensionless variables) in order to enhance more generalization in application and the method of Boltzmann Transform is employed to obtain an exact analytical solution of the dimensionless equation. Finally, the logarithms approximation (for larger times) of the analytical solution is derived. Moreover, after a rigorous mathematical modeling and analysis, a novel mathematical relationship between dimensionless time, dimensionless pressure, and dimensionless radius was obtained for an infinite reservoir dominated by turbulent flow. It was observed that this mathematical relationship bears some similarities with that of unsteady-state flow under laminar conditions. Their logarithm approximations also share some similarities. In addition, the results obtained show the efficiency and accuracy of the Boltzmann Transform approach in solving this kind of complex problem. Doi: 10.28991/HEF-2021-02-03-04 Full Text: PDF

Revisiting Current Techniques for Analyzing Reservoir Performance: A New Approach for Horizontal-Well Pseudosteady-State Productivity Index

SPE Journal ◽  
2018 ◽  
Vol 24 (01) ◽  
pp. 71-91 ◽  
Author(s):  
Salam Al-Rbeawi
Keyword(s):  
Porous Media ◽  
Pressure Drop ◽  
Transient State ◽  
Darcy Flow ◽  
Analytical Models ◽  
Productivity Index ◽  
Pressure Derivative ◽  
Starting Time ◽  
State Flow ◽  
State Pressure

Summary The objective of this paper is to revisit currently used techniques for analyzing reservoir performance and characterizing the horizontal-well productivity index (PI) in finite-acting oil and gas reservoirs. This paper introduces a new practical and integrated approach for determining the starting time of pseudosteady-state flow and constant-behavior PI. The new approach focuses on the fact that the derivative of PI vanishes to zero when pseudosteady-state flow is developed. At this point, the derivative of transient-state pressure drop and that of pseudosteady-state pressure drop become mathematically identical. This point indicates the starting time of pseudosteady-state flow as well as the constant value of pseudosteady-state PI. The reservoirs of interest in this study are homogeneous and heterogamous, single and dual porous media, undergoing Darcy and non-Darcy flow in the drainage area, and finite-acting, depleted by horizontal wells. The flow in these reservoirs is either single-phase oil flow or single-phase gas flow. Several analytical models are used in this study for describing pressure and pressure-derivative behavior considering different reservoir configurations and wellbore types. These models are developed for heterogeneous and homogeneous formations consisting of single and dual porous media (naturally fractured reservoirs) and experiencing Darcy and non-Darcy flow. Two pressure terms are assembled in these models; the first pressure term represents the time-dependent pressure drop caused by transient-state flow, and the second pressure term represents time-invariant pressure drop controlled by the reservoir boundary. Transient-state PI and pseudosteady-state PI are calculated using the difference between these two pressures assuming constant wellbore flow rate. The analytical models for the pressure derivatives of these two pressure terms are generated. Using the concept that the derivative of constant PI converges to zero, these two pressure derivatives become mathematically equal at a certain production time. This point indicates the starting time of pseudosteady-state flow and the constant behavior of PI. The outcomes of this study are summarized as the following: Understanding pressure, pressure derivative, and PI behavior of bounded reservoirs drained by horizontal wells during transient- and pseudosteady-state production Investigating the effects of different reservoir configurations, wellbore lengths, reservoir homogeneity or heterogeneity, reservoirs as single or dual porous media, and flow pattern in porous media whether it has undergone Darcy or non-Darcy flow Applying the concept of the PI derivative to determine the starting time of pseudosteady-state stabilized PI The novel points in this study are the following: The derivative of the PI can be used to precisely indicate the starting time of pseudosteady-state flow and the constant behavior of PI. The starting time of pseudosteady-state flow determined by the convergence of transient- and pseudosteady-state pressure derivative or by the PI curve is always less than that determined from the curves of total pressure drop and its derivative. Non-Darcy flow may significantly affect the transient-state PI, but pseudosteady-state PI is slightly affected by non-Darcy flow. The starting time of pseudosteady-state flow is not influenced by non-Darcy flow. The convergence of transient- and pseudosteady-state pressure derivatives is affected by reservoir configurations, wellbore lengths, and porous-media characteristics.

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