On the Buckley-Leverett Equation With Constant-Pressure Boundary Conditions

SPE Journal ◽  
2016 ◽  
Vol 21 (06) ◽  
pp. 2301-2307 ◽  
Author(s):  
Thormod E. Johansen ◽  
Lesley A. James ◽  
Xiaolong Liu
Keyword(s):  
Analytical Solution ◽  
Flow Rate ◽  
Constant Pressure ◽  
Analytical Determination ◽  
Constant Flow Rate ◽  
Displacement Front ◽  
Constant Rate ◽  
Volumetric Flux

Summary The classical Buckley-Leverett theory applies to 1D linear flow of two immiscible phases under the fundamental assumption that the flow rate (or total volumetric flux) is constant as a function of time. One phase is injected into the medium at a constant rate, thereby displacing the other phase. If the displacing phase is instead injected at constant pressure and the outlet pressure is also constant, the problem is still well-defined; however, the classical Buckley-Leverett theory does not apply. This is because the injected phase and the in-situ phase have different properties such as viscosities. If the boundary pressures are kept constant, the flow rate will, therefore, vary over time. The main result of this paper is to show that the solution of the displacement problem can be obtained from the constant-rate solution through an analytical determination of the flow rate as a function of time, given constant-pressure boundaries. The theory developed in this paper also provides an analytical solution for the location of the displacement front at any given time, the time for frontal breakthrough at the outlet end, and the pressure distribution as a function of time inside the medium. It is demonstrated through computed examples that the constant-flow-rate solution, in general, cannot be used to approximate the corresponding solution for constant-pressure boundaries because the variation in the flow rate is very significant. A standard numerical method has also been applied and compared with the analytical solutions, demonstrating that fine numerical-simulation grids are required for acceptable comparison with the analytical solution.

Analytical Determination of Generation-Recombination Rate In Amorphous Silicon

1986 ◽  
Vol 70 ◽  
Author(s):  
Jože Furlan ◽  
Slavko Amon
Keyword(s):  
Analytical Solution ◽  
Amorphous Silicon ◽  
Cross Sections ◽  
Recombination Rate ◽  
Exponential Model ◽  
Localized States ◽  
Analytical Determination ◽  
Capture Cross ◽  
Capture Cross Sections

ABSTRACTA general expression for generation-recombination rate in a-Si based on classical SRH theory including different electron and hole capture cross-sections for donor-like and acceptor-like centers inside the mobility gap is derived. Applying appropriate approximations and two-exponential model for localized states distribution two methods of analytical solution are presented and discussed.

scholarly journals Atmospheric CO<sub>2</sub>, CH<sub>4</sub>, and CO with the CRDS technique at the Izaña Global GAW station: instrumental tests, developments, and first measurement results

2019 ◽  
Vol 12 (4) ◽  
pp. 2043-2066 ◽  
Author(s):  
Angel J. Gomez-Pelaez ◽  
Ramon Ramos ◽  
Emilio Cuevas ◽  
Vanessa Gomez-Trueba ◽  
Enrique Reyes
Keyword(s):  
Flow Rate ◽  
Random Noise ◽  
Least Square ◽  
Spatial Inhomogeneity ◽  
Inlet Pressure ◽  
Processing Scheme ◽  
Linear Relationships ◽  
Least Square Fit

Abstract. At the end of 2015, a CO2/CH4/CO cavity ring-down spectrometer (CRDS) was installed at the Izaña Global Atmosphere Watch (GAW) station (Tenerife, Spain) to improve the Izaña Greenhouse Gases GAW Measurement Programme, and to guarantee the renewal of the instrumentation and the long-term maintenance of this program. We present the results of the CRDS acceptance tests, the raw data processing scheme applied, and the response functions used. Also, the calibration results, the implemented water vapor correction, the target gas injection statistics, the ambient measurements performed from December 2015 to July 2017, and their comparison with other continuous in situ measurements are described. The agreement with other in situ continuous measurements is good most of the time for CO2 and CH4, but for CO it is just outside the GAW 2 ppb objective. It seems the disagreement is not produced by significant drifts in the CRDS CO World Meteorological Organization (WMO) tertiary standards. The more relevant contributions of the present article are (1) determination of linear relationships between flow rate, CRDS inlet pressure, and CRDS outlet valve aperture; (2) determination of a slight CO2 correction that takes into account changes in the inlet pressure/flow rate (as well as its stability over the years), and attributing it to the existence of a small spatial inhomogeneity in the pressure field inside the CRDS cavity due to the gas dynamics; (3) drift rate determination for the pressure and temperature sensors located inside the CRDS cavity from the CO2 and CH4 response function drift trends; (4) the determination of the H2O correction for CO has been performed using raw spectral peak data instead of the raw CO provided by the CRDS and using a running mean to smooth random noise in a long water-droplet test (12 h) before performing the least square fit; and (5) the existence of a small H2O dependence in the CRDS flow and of a small spatial inhomogeneity in the temperature field inside the CRDS cavity are pointed out and their origin discussed.

scholarly journals The Lorenz model for single-mode homogeneously broadened laser: analytical determination of the unpredictable zone

Open Physics ◽  
2014 ◽  
Vol 12 (3) ◽  
Author(s):  
Samia Ayadi ◽  
Olivier Haeberlé
Keyword(s):  
Analytical Solution ◽  
Analytical Procedure ◽  
Dissipative System ◽  
Period Doubling ◽  
Single Mode ◽  
Analytical Determination ◽  
Pumping Rate ◽  
Fifth Order ◽  
Doubling Sequence

AbstractWe have applied harmonic expansion to derive an analytical solution for the Lorenz-Haken equations. This method is used to describe the regular and periodic self-pulsing regime of the single mode homogeneously broadened laser. These periodic solutions emerge when the ratio of the population decay rate ℘ is smaller than 0:11. We have also demonstrated the tendency of the Lorenz-Haken dissipative system to behave periodic for a characteristic pumping rate “2C P”[7], close to the second laser threshold “2C 2th ”(threshold of instability). When the pumping parameter “2C” increases, the laser undergoes a period doubling sequence. This cascade of period doubling leads towards chaos. We study this type of solutions and indicate the zone of the control parameters for which the system undergoes irregular pulsing solutions. We had previously applied this analytical procedure to derive the amplitude of the first, third and fifth order harmonics for the laser-field expansion [7, 17]. In this work, we extend this method in the aim of obtaining the higher harmonics. We show that this iterative method is indeed limited to the fifth order, and that above, the obtained analytical solution diverges from the numerical direct resolution of the equations.

scholarly journals Analisis Performance Sumur X Menggunakan Metode Standing Dari Data Pressure Build Up Testing

2016 ◽  
Vol 5 (2) ◽  
pp. 57-61
Author(s):  
Novrianti Novrianti
Keyword(s):  
Flow Rate ◽  
Maximum Flow ◽  
Maximum Flow Rate ◽  
Well Test ◽  
Time Data ◽  
Constant Flow Rate ◽  
Optimal Flow ◽  
Production Wells ◽  
Inflow Performance

The number of production wells refers to the performance of the well, which is shown in the graph of inflow performance relationship (IPR). Reservoir characteristics influence on performance of the well, type of welltest and methods that be used in the determination of IPR. By using the IPR curves, maximum flow rate and the optimal flow rate of the well will be known. Pressure Build Up test is used to know performance and a maximum flow rate of the X well. Well test conducted for 15 hours. The well produced at a constant flow rate than close the wellhead. The Pressure data and time data obtained from the well test. The result of Pressure build-up testing analysis among permeability, skin and flow efficiency. After analyzing the Pressure build-up testing permeability obtained 190 mD, skin + 1,68 and 0,83 flow efficiency. Based on the value of flow efficiency Standing method is the most appropriate method is used to analyze the productivity of X well. Standing appropriate method for wells with skin ≠ 0 and flow efficiency ≠ 1. The maximum flow rate of the X well using Standing Method on the 0,83 flow efficiency was 13,91 MMSCFD

scholarly journals DETERMINATION OF THE GAS STREAM FLOW RATE WHEN CONDUCTING HYDRODYNAMIC STUDIES OF WELLS (PART 2)

2015 ◽  
pp. 43-54
Author(s):  
M. S. Rogalev ◽  
N. V. Saranchin ◽  
V. N. Maslov ◽  
A. B. Derendyaev
Keyword(s):  
Analytical Solution ◽  
Flow Rate ◽  
Compression Ratio ◽  
Stream Flow ◽  
Critical Flow ◽  
Flow Mode ◽  
Technical Literature ◽  
Basic Assumptions ◽  
Scientific Technical

The review and analysis of the formula suggested in scientific technical literature for calculation of the compression ratio of gas stream passing through a diaphragm in the critical flow mode were conducted. The basic assumptions were formulated which lead to its incorrect determination and directly introduce the error in the calculation of the medium flow rate. The analytical solution id is suggested for obtaining a reliable value of the compression ratio of gas stream passing through a diaphragm in the critical flow mode. A practical approval of the obtained analytical solution is presented

scholarly journals Analytical solution of Buckley-Leverett equation for gas flooding including the effect of miscibility with constant-pressure boundary

2019 ◽  
Vol 37 (3) ◽  
pp. 960-991 ◽  
Author(s):  
Lingyu Mu ◽  
Xinwei Liao ◽  
Zhiming Chen ◽  
Jiandong Zou ◽  
Hongyang Chu ◽  
...  
Keyword(s):  
Analytical Solution ◽  
Numerical Solution ◽  
Relative Permeability ◽  
Constant Pressure ◽  
Gas Production ◽  
Influential Factors ◽  
Fractional Flow ◽  
Pressure Profiles ◽  
Gas Flooding ◽  
Volumetric Flux

This paper presents an analytical solution of Buckley-Leverett equation for gas flooding with constant-pressure boundary including the effect of miscibility on the viscosity and relative permeability. First, a relative permeability model and a viscosity model with consideration of miscibility are used to describe the variations of relative permeability and viscosity of oil and gas. Then, based on the fractional-flow theory, the Buckley-Leverett equation for gas flooding with constant-pressure boundary including the effect of miscibility is constructed and solved analytically. From the analytical solution, the saturation and pressure profiles, the total volumetric flux and the breakthrough time are determined. To verify the theory, the analytical solution is compared with the numerical solution. The comparison shows that the analytical solution is in reasonable agreement with numerical solution. Through the study on the influential factors, it can be concluded that total volumetric flux is increasing with the increases of permeability and pressure and decrease of gas viscosity. The increase of total volumetric flux accelerates breakthrough of the injected gas. Furthermore, with the pressure increase, there are remarkable reduction in residual oil saturation and improvement of relative permeability, resulting in higher gas saturation and oil displacement efficiency. The analytical solution presented in this paper provides guidance on analyzing the distribution of saturation and pressure profiles, predicting the gas production and oil recovery efficiency of oil well.

Effects of the Gas Feed on Bubble Formation in a Microfluidic T-Junction: Constant-Pressure versus Constant-Flow-Rate Injection

2019 ◽  
Vol 58 (23) ◽  
pp. 10092-10105 ◽  
Author(s):  
Sheng Mi ◽  
Netsanet Tesfaye Weldetsadik ◽  
Zafar Hayat ◽  
Taotao Fu ◽  
Chunying Zhu ◽  
...  
Keyword(s):  
Flow Rate ◽  
Constant Pressure ◽  
Bubble Formation ◽  
Constant Flow ◽  
Constant Flow Rate

On elastic regime of filtration in hydraulic fracture

2016 ◽  
Vol 11 (2) ◽  
pp. 156-166
Author(s):  
V.Sh. Shagapov ◽  
Z.M. Nagaeva
Keyword(s):  
Pressure Drop ◽  
Rheological Properties ◽  
Flow Rate ◽  
Analytical Solutions ◽  
Hydraulic Fracture ◽  
Constant Pressure ◽  
Constant Flow ◽  
Exact Analytical Solutions ◽  
Constant Flow Rate ◽  
Pressure Evolution

Fluid recovery from a well in the modes of constant pressure drop and constant flow rate is considered basing on a theoretical model of filtration in a hydraulic fracture. Exact analytical solutions are obtained that allowed analyzing the effect of the reservoir and fracture properties (such as porosity, permeability and crack width) as well as the rheological properties of the saturating fluid upon: pressure evolution in the fracture, well flow rate at constant pressure drop, and pressure evolution in the wellbore. On the basis of theoretical models describing the filtration for a crack in an oil or gas reservoirs, the considered problem on the selection of fluids from the well in modes constant differential pressure and constant flow. For the considered tasks are received exact analytical solutions, based on which we analyzed the influence of reservoir characteristics of the formation and fractures (for example, their porosity, permeability and width of cracks) and the rheological properties of the saturating fluid on the evolution of the pressure in the fracture, the production rate at a constant differential to the dynamics of the pressure in the well.

scholarly journals Effects of Calcium Entry Blocker Emopamil on Postischemic Energy Metabolism of the Isolated Perfused Rat Brain

1987 ◽  
Vol 7 (4) ◽  
pp. 489-496 ◽  
Author(s):  
Gerhard Wilhelm Bielenberg ◽  
Heinz Haubruck ◽  
Josef Krieglstein
Keyword(s):  
Energy Metabolism ◽  
Flow Rate ◽  
Constant Pressure ◽  
High Energy ◽  
Calcium Entry ◽  
High Energy Phosphates ◽  
Calcium Entry Blocker ◽  
Constant Flow ◽  
Constant Flow Rate ◽  
Entry Blocker

The effects of emopamil on postischemic energy metabolism and electroencephalographic (EEG) recovery were investigated in the isolated rat brain perfused at either constant pressure or, alternatively, at constant flow rate. Flow rate and perfusion pressure were monitored continuously. The brains were perfused with a fluorocarbon emulsion for 30 min, and after 30 min of ischemia, perfusion was reinstituted for 5, 30, or 60 min. Global cerebral perfusion rate was increased by emopamil throughout the perfusion period and, accordingly, in brains perfused at a constant flow rate, perfusion pressure was reduced by the drug. At constant pressure perfusion, after 5 min after ischemia, cortical levels of creatine-phosphate, adenosine triphosphate (ATR), glucose, glucose-6-phosphate, and fructose-6-phosphate were higher in emopamil-treated brains than in controls, although the levels of adenosine diphosphate (ADP) and adenosine monophosphate (AMP) were reduced. When brains were perfused at constant flow rate, however, emopamil exhibited no effect on brain energy metabolism in the early reperfusion period. Postischemic restoration of high-energy phosphates proved to depend on the flow rate used. After 30 min of postischemic reperfusion, cortical levels of lactate were lower in emopamil-treated brains compared to controls at both constant pressure and constant volume perfusion. Postischemic lactate levels were independent of flow rate and were also reduced when emopamil was only present during reperfusion. The postischemic restoration of cortical EEG activity was improved by the calcium entry blocker. The results suggest that the restoration of high-energy phosphates during the early postischemic recovery can be mainly attributed to the vascular effect of emopamil, whereas the lower lactate levels are caused by a direct action of the calcium entry blocker on brain parenchyma.

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