Pressure Drop in a Composite Reservoir

10.2118/19-pa ◽  
1961 ◽  
Vol 1 (03) ◽  
pp. 170-176 ◽  
Author(s):  
T.L. Loucks ◽  
E.T. Guerrero
Keyword(s):  
Pressure Drop ◽  
Pressure Distribution ◽  
Numerical Solutions ◽  
Drilling Fluid ◽  
Material Balance ◽  
Analytic Solutions ◽  
Unsteady State ◽  
Unpublished Manuscript ◽  
Variable Permeability ◽  
Small Times

Abstract Pressure drop characteristics in a system composed of two adjacent concentric regions of different permeability were studied. The differential equations for continuity of mass flow in the two regions were solved using the Laplace transformation and the necessary boundary conditions to give the pressure distribution in the composite reservoir. The resulting equation for pressure drop at the inner boundary was evaluated for a variety of composite reservoirs and compared with the results for a uniform reservoir. From this study it was found that under certain conditions the permeability in both zones, as well as the size of the inner zone, can be determined from the pressure drop curve. Introduction The theory for the pressure distribution and pressure build-up behavior of a well producing a single, slightly compressible fluid from infinite and finite homogeneous reservoirs was presented by Horner and Miller, Dyes and Hutchinson. Extensions on this original work to provide improved and extended interpretations and better agreement between theory and observed results have been made by Matthews, et al, van Everdingen, Gladfelter, et al, Stegemeier and Matthews, Hurst and Guerrero, and Perrine. More recently Lefkovits, et al, studied pressure build-up behavior in bounded reservoirs composed of stratified layers. Houpeurt has suggested various approaches to the general problem of variable permeability and porosity but presented no analytic solutions for particular permeability variations. Albert, Jaisson and Marion studied the finite composite reservoir and presented numerical solutions to the unsteady-state case and an analytical solution valid only for large times. They also studied the so-called pseudosteady state for several examples of radial permeability variations. Very similar examples have been treated in the unsteady state with application to pressure build-up by Loucks in an unpublished manuscript. More recently Hurst has presented the complete point-sink solution (valid for all times) for the infinite composite reservoir. He applied these solutions to interference between oil fields along with an even more elegant application of his explicit solution to the material-balance equation including water influx. Mortada approaches the same application by avoiding the point-source limitation but gives the solution for the aquifer region which is valid only for large times. Hopkinson, Natanson and Temple have treated both the finite and infinite composite reservoir obtaining the pressure distribution for the inner zone valid for large times. This paper presents a theoretical study of the pressure distribution in an infinite composite reservoir composed of two adjacent concentric regions of different permeability. The object was to determine the manner in which pressure drop at the inner boundary of a composite reservoir depends upon time, the permeability of each zone and the size of the inner zone. Expressions for the pressure distribution in both zones are developed which take into account the radius of the sink and are valid for small times as well as large times. It was felt that an understanding of the pressure drop behavior in various composite reservoirs would be of assistance in the interpretation of some pressure build-up curves which do not behave according to the theory derived for uniform systems. Often the region surrounding the wellbore is either more permeable or less permeable than the reservoir because of the various drilling and completion practices. The effects of reduced permeability due to drilling- fluid invasion and of increased permeability due to fracturing or acidizing need to be more carefully defined. Therefore, an equation for the pressure drop in a composite reservoir was developed, and the effects of both the permeability in each zone and the size of the inner zone were studied.

A Compact Model for Spherical Rough Contacts

2004 ◽  
Author(s):  
M. Bahrami ◽  
M. M. Yovanovich ◽  
J. R. Culham
Keyword(s):  
Pressure Distribution ◽  
Entire Range ◽  
Present Model ◽  
Numerical Solutions ◽  
Contact Radius ◽  
Bulk Deformation ◽  
Dimensional Parameters ◽  
Maximum Contact ◽  
Good Agreement ◽  
Key Parameter

The contact of rough spheres is of high interest in many tribological, thermal, and electrical fundamental analyses. Implementing the existing models is complex and requires iterative numerical solutions. In this paper a new model is presented and a general pressure distribution is proposed that encompasses the entire range of spherical rough contacts including the Hertzian limit. It is shown that the non-dimensional maximum contact pressure is the key parameter that controls the solution. Compact expressions are proposed for calculating the pressure distribution, radius of the contact area, elastic bulk deformation, and the compliance as functions of the governing non-dimensional parameters. The present model shows the same trends as those of the Greenwood and Tripp model. Correlations proposed for the contact radius and the compliance are compared with experimental data collected by others and good agreement is observed.

The Method of Dynamic Reserves Prediction for a Constant Volume Gas Reservoir

2013 ◽  
Vol 275-277 ◽  
pp. 456-461
Author(s):  
Lei Zhang ◽  
Lai Bing Zhang ◽  
Bin Quan Jiang ◽  
Huan Liu
Keyword(s):  
Pressure Drop ◽  
Material Balance ◽  
Production Method ◽  
Gas Production ◽  
Constant Volume ◽  
Phase Method ◽  
Gas Reservoirs ◽  
Two Phase ◽  
Material Balance Method ◽  
Unit Pressure

The accurate prediction of the dynamic reserves of gas reservoirs is the important research content of the development of dynamic analysis of gas reservoirs. It is of great significance to the stable and safe production and the formulation of scientific and rational development programs of gas reservoirs. The production methods of dynamic reserves of gas reservoirs mainly include material balance method, unit pressure drop of gas production method and elastic two-phase method. To clarify the characteristics of these methods better, in this paper, we took two typeⅠwells of a constant volume gas reservoir as an example, the dynamic reserves of single well controlled were respectively calculated, and the results show that the order of the calculated volume of the dynamic reserves by using different methods is material balance method> unit pressure drop of gas production method >elastic two-phase method. Because the material balance method is a static method, unit pressure drop of gas production method and elastic two-phase method are dynamic methods, therefore, for typeⅠwells of constant volume gas reservoirs, when the gas wells reached the quasi-steady state, the elastic two-phase method is used to calculate the dynamic reserves, and when the gas wells didn’t reach the quasi-steady state, unit pressure drop of gas production method is used to calculate the dynamic reserves. The conclusion has some certain theoretical value for the prediction of dynamic reserves for constant volume gas reservoirs.

scholarly journals Determination of Avalanche Dynamics Friction Coefficients from Measured Speeds

1983 ◽  
Vol 4 ◽  
pp. 170-173 ◽  
Author(s):  
D. M. McClung ◽  
P. A. Schaerer
Keyword(s):  
Complex Terrain ◽  
Sliding Friction ◽  
Numerical Solutions ◽  
Analytic Solutions ◽  
Dynamics Model ◽  
Friction Coefficients ◽  
Turbulent Friction ◽  
Constant Friction ◽  
Avalanche Dynamics

An avalanche dynamics model, appropriate for complex terrain, for real avalanche paths was developed by Perla, Cheng and McClung in 1980. The model has two friction terms, one for sliding friction which is independent of speed, and one for turbulent friction which is proportional to V2, where V is the centre-of-mass speed along the incline. By introducing speed maxima for avalanches, along with start and stop reference positions, it is possible to determine the the two constant friction coefficients for the model. When this is done, it is found that speed data often exceed a model speed limit implied by the application of V = 0 at the start and stop positions. This effect is illustrated by analytic solutions of the relevant equations, as well as numerical solutions for actual avalanche paths. Some limitations and properties of the fundamental modelling are outlined and suggestions given for future use of such models.

New Analytical Approach to Operational Assessment of Fractured Well Productivity with Variable Permeability

2021 ◽  
Author(s):  
Evgeniy Viktorovich Yudin ◽  
George Aleksandrovich Piotrovskiy ◽  
Maria Vladimirovna Petrova ◽  
Alexey Petrovich Roshchektaev ◽  
Nikita Vladislavovich Shtrobel
Keyword(s):  
Hydraulic Fracturing ◽  
Hydraulic Fracture ◽  
Free Parameter ◽  
Numerical Solutions ◽  
Analytical Models ◽  
Productivity Index ◽  
Finite Conductivity ◽  
Well Productivity ◽  
Fracture Conductivity ◽  
Variable Permeability

Abstract Requirements of targeted optimization are imposed on the hydraulic fracturing operations carried out in the conditions of borderline economic efficiency of fields taking into account geological and technological features. Consequently, the development of new analytical tools foranalyzing and planning the productivity of fractured wells, taking into account the structuralfeatures of the productive reservoir and inhomogeneous distribution of the fracture conductivity, is becoming highly relevant. The paper proposes a new approach of assessing the vertical hydraulic fracture productivityin a rectangular reservoir in a pseudo-steady state, based on reservoir resistivity concept described in the papers of Meyer et al. However, there is a free parameter in the case of modeling the productivity of a hydraulic fracture by the concept. The parameter describes the distribution of the inflow along the plane of the fracture. This paper presents a systematic approach to determining of the parameter. The resulting model allows to conduct an assessment of the influence of various complications in the fracture on the productivity index. During the research a method of determining the free parameter was developed,it was based on the obtained dependence of the inflow distribution on the coordinate along the fracture of finite conductivity. The methodology allowed to refine existent analytical solution of the Meyer et al. model, which, in turn, allowed to assess the influence of different fracture damages in the hydraulic fracture on the productivity index of the well. The work includes the cases of the presence of fracture damages at the beginning and at the end of the fracture. A hydraulic fracture model was built for each of the types of damages, it was based on the developed method, and also the solution of dimensionless productivity ratio was received. The results of the obtained solution were confirmed by comparison with the numerical solutions of commercial simulators and analytical models available in the literature. The advantage of the methodology is the resulting formulas for well productivity are relatively simple, even for exotic cases ofvariable conductivity fractures. The approaches and algorithms described in the paper assume the calculation of the productivity of a hydraulic fracture with variable conductivity and the presence of other complicatingfactors.The methodology of the paper can be used for analysis and diagnosis problems with formation hydraulic fracturing. The efficiency of the calculations allows using the presented methodology to solve inverse problems of determining the efficiency of the hydraulic fracturing operation.

scholarly journals Resonant Stellar Orbits in Spiral Galaxies

1975 ◽  
Vol 69 ◽  
pp. 237-244
Author(s):  
P. O. Vandervoort
Keyword(s):  
Excellent Agreement ◽  
Spiral Galaxy ◽  
Harmonic Balance ◽  
Spiral Galaxies ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Analytic Solutions ◽  
Stellar Orbits ◽  
Method Of Harmonic Balance ◽  
Resonance Phenomena

This paper reviews a series of investigations of the orbits of stars in the regions of the Lindblad resonances of a spiral galaxy. The analysis is formulated in an epicyclic approximation. Analytic solutions of the epicyclic equations of motion are obtained by the method of harmonic balance of Bogoliubov and Mitropolsky. These solutions represent the resonance phenomena exhibited by the orbits in generally excellent agreement with numerical solutions.

The Solution of Lateral Heat Loss Problem Using Collocation Method with Cubic B-Splines Finite Element

2011 ◽  
Vol 110-116 ◽  
pp. 3184-3190
Author(s):  
Necdet Bildik ◽  
Duygu Dönmez Demir
Keyword(s):  
Finite Element ◽  
Stability Analysis ◽  
Heat Loss ◽  
Collocation Method ◽  
Numerical Solutions ◽  
Analytic Solutions ◽  
B Splines ◽  
Stability Method ◽  
The Stability ◽  
Lateral Heat

This paper deals with the solutions of lateral heat loss equation by using collocation method with cubic B-splines finite elements. The stability analysis of this method is investigated by considering Fourier stability method. The comparison of the numerical solutions obtained by using this method with the analytic solutions is given by the tables and the figure.

scholarly journals Causal cosmology with braneworld gravity including Gauss Bonnet coupling

2020 ◽  
Vol 35 (26) ◽  
pp. 2050216
Author(s):  
Partha Sarathi Debnath
Keyword(s):  
Numerical Solutions ◽  
Analytic Solutions ◽  
Causal Theory ◽  
Type Ii ◽  
Dissipative Effects ◽  
Exponential Expansion ◽  
Linear Field Equation ◽  
The Universe ◽  
Special Case ◽  
Linear Field

Causal cosmological evolutions in Randall Sundrum type II (RS) braneworld gravity with Gauss Bonnet coupling and dissipative effects are discussed here. Causal theory of dissipative effects are illustrated by Full Israel Stewart theory are implemented. We consider the numerical solutions of evolutions and analytic solutions as a special case for extremely non-linear field equation in Randall Sundrum type II braneworld gravity with Gauss Bonnet coupling. Cosmological models admitting Power law expansion, Exponential expansion and evolution in the vicinity of the stationary solution of the universe are investigated for Full Israel Stewart theory. Stability of equilibrium or fixed points related to the dynamics of evolution in Full Israel Stewart theory in Randall Sundrum type II braneworld gravity together with Gauss Bonnet coupling are disclosed here.

scholarly journals Internal Gravity Waves in a Saturated Moist Neutral Atmosphere

2013 ◽  
Vol 70 (12) ◽  
pp. 3693-3709 ◽  
Author(s):  
David J. Muraki ◽  
Richard Rotunno
Keyword(s):  
Numerical Solutions ◽  
Internal Gravity Waves ◽  
Analytic Solutions ◽  
Neutral Atmosphere ◽  
Atmospheric Flow ◽  
Start Up ◽  
Exact Analytic Solutions ◽  
Downward Displacement ◽  
Neutral Flow ◽  
Saturated Atmosphere

Abstract This work is motivated by an unusual feature associated with the start-up of a moist nearly neutral atmospheric flow over a mountain ridge that was previously observed in a full-physics numerical model. In that study, the upstream propagation of a wave of subsidence precluded the establishment of upward-displaced and saturated flow that might be expected upstream of the topography. This phenomenon was hypothesized to be a consequence of the peculiar property of saturated moist neutral flow: an upward air parcel displacement produces zero buoyancy, while a downward displacement desaturates the air parcel and produces a positive buoyancy anomaly. In the present study, this hypothesis is confirmed within numerical solutions to a reduced system of equations that incorporates the saturated-atmosphere property in a particularly simple manner. The relatively uncomplicated nature of these solutions motivates the numerical solution of a further simplified initial-value problem for both nonhydrostatic and hydrostatic flow. Exact analytic solutions are developed for the latter hydrostatic case, which explains the upstream-propagating wave of subsidence as a shock phenomenon.

Calculation Methods of the Dynamic Reserves for Gas Wells in a Low-Permeability Gas Reservoir

2013 ◽  
Vol 295-298 ◽  
pp. 3243-3248
Author(s):  
Lei Zhang ◽  
Lai Bing Zhang ◽  
Jun Jie Zhang ◽  
Feng Lan ◽  
Pan Deng
Keyword(s):  
Pressure Drop ◽  
Material Balance ◽  
Production Method ◽  
Gas Production ◽  
Phase Method ◽  
Low Permeability ◽  
Gas Reservoir ◽  
Two Phase ◽  
Material Balance Method ◽  
Unit Pressure

Accurately calculating dynamic reserves for single well in a low-permeability gas reservoir has an important guiding significance to high efficiency development of the gas reservoir. During the development of the gas reservoir, dynamic analysis methods were often used to calculate dynamic reserves. Dynamic analysis methods mainly include the material balance method, the gas production method in unit pressure drop, the flexible two-phase method and the production unstable method. Dynamic reserves for four types of gas wells in a low-permeability gas field were calculated using these four methods. Calculation results show that dynamic reserves from big to small are respectively obtained using material balance method, gas production method in unit pressure drop, flexible two-phase method and production unstable method. Calculating dynamic reserves obtained by flexible two-phase method and production unstable method are utilized to production dynamic data of gas well, and those obtained by material balance method and gas production method in unit pressure drop are utilized to the reservoir parameters of different state. Therefore, the values of dynamic reserves obtained using flexible two-phase method and production unstable method in the low-permeability gas reservoir may be more accurate than those obtained using the other methods.

Cassava Mash Dewatering Parameters

2007 ◽  
Vol 3 (1) ◽  
Author(s):  
Oladele Peter Kolawole ◽  
Leo A.S. Agbetoye ◽  
A. S. Ogunlowo
Keyword(s):  
Pressure Drop ◽  
Pressure Distribution ◽  
Steel Plate ◽  
Pressure Gauge ◽  
Galvanized Steel ◽  
Hydraulic Jack ◽  
Experimental Equipment ◽  
Face Area ◽  
Measuring Cylinder ◽  
Medium Resistance

A study was conducted to evaluate the parameters affecting the dewatering of cassava mash during processing. First, studies on the pressure distribution within the mash during the dewatering were carried out. Experimental equipment consisting of tyre tube filled with water, a copper tube, and a pressure gauge was designed and fabricated to measure pressure used in expressing the juice contained in the grated cassava mash. It also included a cylindrical dewatering tank made of galvanized steel plate and a sack which was used as control. The tank had 7mm holes drilled at the base to allow the flow of juice. The volume of juice was measured using a measuring cylinder and the stopwatch measured the time. IITA TMS 4(2) 1425 variety of cassava at three levels of maturity age of 9, 12 and 15 months was utilized in the study. The dewatering pressure is from hydraulic jack used to press the grated mash. The dewatering parameters investigated were pressure drop, face area of the filter medium and mash resistance. The results showed that mash resistance varied with the age of the cassava with the highest value of 54,000,000,000 m/kg recorded. Medium Resistance also varied with the age, 33,000,000,000/m was the highest value recorded for 15 months old sample. 0.00371m3 volume of filtrate was obtained from the 12 months old sample with 0.0945 kg mash cake deposit on the filtering medium as the highest deposit. The Kozemy constant value for TMS 4(2) 1425 variety of cassava was found to be 11400000 and Porosity 0.0181, the result presents the distribution and values of identified parameters numerically for equipment designers use.

Sign in / Sign up

Export Citation Format

Share Document