Unsteady Radial Flow of Gas Through Porous Media

1953 ◽  
Vol 20 (2) ◽  
pp. 210-214
Author(s):  
R. Jenkins ◽  
J. S. Aronofsky
Keyword(s):  
Porous Medium ◽  
Porous Media ◽  
Radial Flow ◽  
Transient Flow ◽  
Gas Reservoirs ◽  
Punch Card ◽  
Flow Rates ◽  
Pressure Distributions ◽  
History Of ◽  
Natural Gas Reservoirs

Abstract This paper presents a numerical method for describing the transient flow of gases radially inward or outward through a porous medium in which the initial and terminal pressures and/or rates are specified. Specific examples are worked out which have application in the study of natural-gas reservoirs. The computations were carried out by means of punch-card machines. The pressure distribution as a function of time has been calculated for various ratios of reservoir diameter to well diameter and for various dimensionless flow rates for a well penetrating the center of a homogeneous disk-shaped reservoir. A simple means of predicting the well pressure at any time in the history of such an idealized field has been developed. Flow rates and pressure distributions within the radial reservoir also have been calculated for the case in which the well pressure is suddenly lowered from its initial static value, and then held constant.

Statistics of Mathematical Two-Scale Closure of Momentum, Heat and Charge Transport Problems With Stochastic Orientation of Porous Medium Capillaries

2001 ◽  
Author(s):  
V. S. Travkin ◽  
K. Hu ◽  
I. Catton
Keyword(s):  
Heat Transfer ◽  
Porous Medium ◽  
Porous Media ◽  
Volume Averaging ◽  
Governing Equations ◽  
Rigorous Approach ◽  
Flow And Heat Transfer ◽  
Model Flow ◽  
History Of ◽  
Transport Problems

Abstract The history of stochastic capillary porous media transport problem treatments almost corresponds to the history of porous media transport developments. Volume Averaging Theory (VAT), shown to be an effective and rigorous approach for study of transport (laminar and turbulent) phenomena, is used to model flow and heat transfer in capillary porous media. VAT based modeling of pore level transport in stochastic capillaries results in two sets of scale governing equations. This work shows how the two scale equations could be solved and how the results could be presented using statistical analysis. We demonstrate that stochastic orientation and diameter of the pores are incorporated in the upper scale simulation procedures. We are treating this problem with conditions of Bi for each pore is in a range when Bi ≳ 0.1 which allows even greater distinction in assessing an each additional differential, integral, or integral-differential term in the VAT equations.

The Solution of Nonlinear Equations

1973 ◽  
Vol 13 (06) ◽  
pp. 348-356
Author(s):  
William Hurst
Keyword(s):  
Differential Equation ◽  
Porous Media ◽  
Radial Flow ◽  
Transient Flow ◽  
Second Order ◽  
Linear Functions ◽  
Small Computer ◽  
Perfect Gas ◽  
Desk Calculator ◽  
Linear Differential

Abstract This paper presents procedures for treating problems involving transient flow of gases in porous problems involving transient flow of gases in porous media. The methods involve stepwise calculations using linearized equations derived from the nonlinear, second-order equations that describe transient flow of gas. The distribution of pressure around a gas well at various times can be readily calculated with a desk calculator or a small computer. Equations and procedures are offered for both infinite and limited reservoirs. Solutions by these new technique's are shown to be in good agreement with computer solutions available in the literature. Also discussed is a procedure using relatively few image wells for treating problems in reservoirs with curved, irregular boundaries. Introduction This paper is concerned with solving problems involving transient flow of gas in porous systems. The main contribution of this paper is the development of a technique that permits the effective use of a desk calculator for the computations. The methods presented here will permit individual engineers not having access to one of the larger computers to solve many practical problems that heretofore would have been intractable. problems that heretofore would have been intractable. The equation describing unsteady-state flow of a perfect gas in a horizontal reservoir is reported by perfect gas in a horizontal reservoir is reported by Katz to be (1) This equation is obtained by combining the continuity equation, Darcy's law, and the following density relationship for the gas. =...............................(2) The intractableness of Eq. 1 stems from the fact that it is a nonlinear, second-order differential equation for which no analytical solution is known. In 1953, a pioneering use of computers was presented by Bruce et al., who approximated the presented by Bruce et al., who approximated the differential equation with difference equations and solved these numerically. Their solutions developed for horizontal flow of a perfect gas for circular reservoirs were presented in terms of dimensionless parameters in plots of pressure vs radius for various parameters in plots of pressure vs radius for various times and flow rates. Later, Aronofsky and Porter, using computers, solved radial flow problems for nonideal gases, permitting gas properties to vary as linear functions of pressure. Recently, advances in solving problems involving the flow of nonideal gas have been made by Ramey and his colleagues, who have developed equations and computer solutions for real gases in terms of pseudo-reduced pressures. In the present paper, a method of solving the flow equations for gas is developed for both infinite and limited reservoirs. The methods, which make use of the analytical solutions for the corresponding linear differential equation for radial flow, can be used with a desk calculator or a small computer to solve problems characterized by nonradial as well as by radial reservoir geometry. An example solution is given to illustrate the method for the flow of a perfect gas in a circular, horizontal reservoir; the results have been compared with those of Bruce. The technique can also be applied to the flow of nonideal gases in nonuniform systems, as well as to oil reservoirs above the bubble-point pressure and to aquifers. In all cases, the results of the example problems obtained by the procedures presented in the text are compared with procedures presented in the text are compared with available published and accepted numerical or analytical results. Finally, a discussion is offered with the intent of guiding the reader toward successful application of the technique in solving practical reservoir problems. problems. SOLUTION OF THE NONLINEAR PROBLEM MATHEMATICAL BACKGROUND In early tracts published on the transient flow of fluids in porous media, the following equation is derived for radial flow. SPEJ P. 348

Linear vs. Radial Boundary-Dominated Flow: Implications for Gas-Well-Decline Analysis

SPE Journal ◽  
2015 ◽  
Vol 20 (05) ◽  
pp. 1053-1066 ◽  
Author(s):  
Pichit Vardcharragosad ◽  
Luis F. Ayala ◽  
Miao Zhang
Keyword(s):  
Radial Flow ◽  
Transient Flow ◽  
Transient Behavior ◽  
Late Time ◽  
Gas Reservoirs ◽  
Linear Flow ◽  
Unconventional Resources ◽  
Flow Geometry ◽  
Fast Pace ◽  
The Impact

Summary Linear flow is a fundamental reservoir-flow geometry typically associated with production from unconventional resources stimulated by means of hydraulic fracturing. Recently, linear flow has been intensively studied following the fast pace of development of unconventional resources. Previous studies have mainly focused on early transient behavior and behavior of composite linear-flow systems. In this work, a density-based analysis method is extended to study decline behavior of the linear-flow system in boundary-dominated flow (BDF). In this study, we first discuss traditional approaches used to model linear flow in gas reservoirs. Second, we show the applicability of the density-based method for gas linear flow both analytically and numerically. Next, late-time solutions are discussed, and the analytical forecasting solution that best describes the BDF behavior is selected for long-term decline-behavior studies. Previously reported results on radial flow as well as early transient-flow effect are also incorporated to provide a more complete understanding of decline behavior and the impact of flow geometry. We show that boundary-dominated responses in linear-flow scenarios fully develop at much later stages of reservoir depletion compared with radial-flow scenarios. As a result, and in marked contrast with radial flow, purely hyperbolic decline behavior may be completely lost in linear-flow scenarios during boundary-dominated conditions. It is demonstrated that most of the recoverable hydrocarbons are produced during the early transient period for linear-flow conditions, whereas most of them are recovered during the BDF period for radial flow. These results suggest that the availability of accurate early transient models is much more critical for the formulation of linear-flow-decline models than had been traditionally necessary for radial-flow-decline models.

Analysis of Factors Influencing Mobility and Adsorption in the Flow of Polymer Solution Through Porous Media

1974 ◽  
Vol 14 (04) ◽  
pp. 337-346 ◽  
Author(s):  
G.J. Hirasaki ◽  
G.A. Pope
Keyword(s):  
Porous Medium ◽  
Porous Media ◽  
Polymer Solution ◽  
Polymer Adsorption ◽  
Rock Properties ◽  
Dimensionless Number ◽  
Permeability Reduction ◽  
Flow Rates ◽  
System Properties ◽  
Effective Pore Radius

Abstract Displacement of oil by polymer solution has several unique characteristics that are not present in normal waterflooding. These include non-Newtonian effects, permeability reduction, and polymer adsorption. polymer adsorption. The rheological behavior of the flow of polymer solution through porous media could be Newtonian at low flow rates, pseudoplastic at intermediate flow rates, and dilatant at high flow rates. The pseudoplastic behavior is modeled with the pseudoplastic behavior is modeled with the Blake-Kozeny model for power-law model fluids. The dilatant behavior is modeled with the viscoelastic properties of the polymer solution. properties of the polymer solution. The reduction in permeability is postulated to be due to an adsorbed layer of polymer molecular coils that reduces the effective size of the pores. A dimensionless number has been formulated to correlate the permeability reduction factor with the polymer, brine, and rock properties. This polymer, brine, and rock properties. This dimensionless number represents the ratio of the size of the polymer molecular coil to an effective pore radius polymer molecular coil to an effective pore radius of the porous medium.A model has been developed to represent adsorption as a function of polymer, brine, and rock properties. The model assumes that the polymer is properties. The model assumes that the polymer is adsorbed on the surface of the porous medium as a monolayer of molecular coils that have a segment density greater than the molecular coil in dilute solution. Introduction Displacement of oil by polymer solutions has several unique characteristics that are not present in normal waterflooding. These include non-Newtonian effects, permeability reduction, and polymer adsorption. In principle, the effects could polymer adsorption. In principle, the effects could be measured experimentally for each fluid-rock system of interest over the entire range of flow conditions existing in the reservoir. However, there are seldom complete data on all systems of interest. A correlation that represents these effects as a function of the polymer, brine, rock properties, and flow conditions would result in a more accurate evaluation of systems that may not have been measured in the laboratory at the desired conditions. Moreover, if the dependence of these effects on the system properties were known, it would aid the search for an optimal system. A model is proposed for representing the effects as a function of the system properties. The model is consistent with a number of experimental observations but enough data have not yet been acquired to determine the extent of applicability of a correlation. It is hoped that the presentation of these models will encourage further research to verify or improve the models. MODEL FOR PSEUDOPLASTIC FLOW THROUGH POROUS MEDIA The Blake-Kozeny model represents the porous medium as a bundle of capillary tubes with a length that is greater than the length of the porous medium by a tortuosity factor, tau. The equivalent radius of the capillary tubes can be related to the particle diameter of a packed bed from the hydraulic radius concept or to the permeability and porosity by comparison with Darcy's law for Newtonian fluids.The modified Blake-Kozeny models represents the flow of a power-law fluid in the capillaries. The relationship between the pressure drop and flow rate can be expressed as a product of the friction factor and Reynolds number.(1) This expression can be related to the apparent viscosity and the rock permeability and porosity through the following relationships:(2) where(3) SPEJ P. 337

Conservative solute transport with periodic velocity and sinusoidal source concentration in semi-infinite porous media: An analytical solution

2014 ◽  
Vol 6 (1) ◽  
pp. 1024-1031
Author(s):  
R R Yadav ◽  
Gulrana Gulrana ◽  
Dilip Kumar Jaiswal
Keyword(s):  
Porous Medium ◽  
Porous Media ◽  
Laplace Transformation ◽  
Seepage Velocity ◽  
Transformation Technique ◽  
Numerical Examples ◽  
One Dimensional ◽  
Porous Domain ◽  
Conservative Solute Transport ◽  
Source Concentration

The present paper has been focused mainly towards understanding of the various parameters affecting the transport of conservative solutes in horizontally semi-infinite porous media. A model is presented for simulating one-dimensional transport of solute considering the porous medium to be homogeneous, isotropic and adsorbing nature under the influence of periodic seepage velocity. Initially the porous domain is not solute free. The solute is initially introduced from a sinusoidal point source. The transport equation is solved analytically by using Laplace Transformation Technique. Alternate as an illustration; solutions for the present problem are illustrated by numerical examples and graphs.

Capillary Imbibition Dynamics in Porous Media

2014 ◽  
Author(s):  
Swayamdipta Bhaduri ◽  
Pankaj Sahu ◽  
Siddhartha Das ◽  
Aloke Kumar ◽  
Sushanta K. Mitra
Keyword(s):  
Porous Medium ◽  
Porous Media ◽  
Paper Strip ◽  
Capillary Imbibition ◽  
Flow Physics ◽  
Fundamental Understanding ◽  
Paper Based Microfluidics ◽  
To Come

The phenomenon of capillary imbibition through porous media is important both due to its applications in several disciplines as well as the involved fundamental flow physics in micro-nanoscales. In the present study, where a simple paper strip plays the role of a porous medium, we observe an extremely interesting and non-intuitive wicking or imbibition dynamics, through which we can separate water and dye particles by allowing the paper strip to come in contact with a dye solution. This result is extremely significant in the context of understanding paper-based microfluidics, and the manner in which the fundamental understanding of the capillary imbibition phenomenon in a porous medium can be used to devise a paper-based microfluidic separator.

scholarly journals Incremental and acceleration production estimation and their effect on optimization of well infill locations in tight gas reservoirs

2021 ◽  
Author(s):  
Atheer Dheyauldeen ◽  
Omar Al-Fatlawi ◽  
Md Mofazzal Hossain
Keyword(s):  
Transient Flow ◽  
Development Planning ◽  
Main Role ◽  
Gas Reservoirs ◽  
Type Curves ◽  
Field Development ◽  
Drilling Performance ◽  
Well Location ◽  
Infill Drilling ◽  
Tight Formations

AbstractThe main role of infill drilling is either adding incremental reserves to the already existing one by intersecting newly undrained (virgin) regions or accelerating the production from currently depleted areas. Accelerating reserves from increasing drainage in tight formations can be beneficial considering the time value of money and the cost of additional wells. However, the maximum benefit can be realized when infill wells produce mostly incremental recoveries (recoveries from virgin formations). Therefore, the prediction of incremental and accelerated recovery is crucial in field development planning as it helps in the optimization of infill wells with the assurance of long-term economic sustainability of the project. Several approaches are presented in literatures to determine incremental and acceleration recovery and areas for infill drilling. However, the majority of these methods require huge and expensive data; and very time-consuming simulation studies. In this study, two qualitative techniques are proposed for the estimation of incremental and accelerated recovery based upon readily available production data. In the first technique, acceleration and incremental recovery, and thus infill drilling, are predicted from the trend of the cumulative production (Gp) versus square root time function. This approach is more applicable for tight formations considering the long period of transient linear flow. The second technique is based on multi-well Blasingame type curves analysis. This technique appears to best be applied when the production of parent wells reaches the boundary dominated flow (BDF) region before the production start of the successive infill wells. These techniques are important in field development planning as the flow regimes in tight formations change gradually from transient flow (early times) to BDF (late times) as the production continues. Despite different approaches/methods, the field case studies demonstrate that the accurate framework for strategic well planning including prediction of optimum well location is very critical, especially for the realization of the commercial benefit (i.e., increasing and accelerating of reserve or assets) from infilled drilling campaign. Also, the proposed framework and findings of this study provide new insight into infilled drilling campaigns including the importance of better evaluation of infill drilling performance in tight formations, which eventually assist on informed decisions process regarding future development plans.

scholarly journals Data-Driven Reduced-Order Modeling of Convective Heat Transfer in Porous Media

Fluids ◽  
2021 ◽  
Vol 6 (8) ◽  
pp. 266
Author(s):  
Péter German ◽  
Mauricio E. Tano ◽  
Carlo Fiorina ◽  
Jean C. Ragusa
Keyword(s):  
Heat Transfer ◽  
Porous Medium ◽  
Porous Media ◽  
Steady State ◽  
Convective Heat Transfer ◽  
Convective Heat ◽  
Data Driven ◽  
Reduced Order ◽  
Medium Formulation ◽  
Flow Resistances

This work presents a data-driven Reduced-Order Model (ROM) for parametric convective heat transfer problems in porous media. The intrusive Proper Orthogonal Decomposition aided Reduced-Basis (POD-RB) technique is employed to reduce the porous medium formulation of the incompressible Reynolds-Averaged Navier–Stokes (RANS) equations coupled with heat transfer. Instead of resolving the exact flow configuration with high fidelity, the porous medium formulation solves a homogenized flow in which the fluid-structure interactions are captured via volumetric flow resistances with nonlinear, semi-empirical friction correlations. A supremizer approach is implemented for the stabilization of the reduced fluid dynamics equations. The reduced nonlinear flow resistances are treated using the Discrete Empirical Interpolation Method (DEIM), while the turbulent eddy viscosity and diffusivity are approximated by adopting a Radial Basis Function (RBF) interpolation-based approach. The proposed method is tested using a 2D numerical model of the Molten Salt Fast Reactor (MSFR), which involves the simulation of both clean and porous medium regions in the same domain. For the steady-state example, five model parameters are considered to be uncertain: the magnitude of the pumping force, the external coolant temperature, the heat transfer coefficient, the thermal expansion coefficient, and the Prandtl number. For transient scenarios, on the other hand, the coastdown-time of the pump is the only uncertain parameter. The results indicate that the POD-RB-ROMs are suitable for the reduction of similar problems. The relative L2 errors are below 3.34% for every field of interest for all cases analyzed, while the speedup factors vary between 54 (transient) and 40,000 (steady-state).

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