Generalized Equations Relating Pressure of Individual Wells and Grids in Reservoir Modeling

1969 ◽  
Vol 9 (03) ◽  
pp. 277-278
Author(s):  
A.S. Odeh ◽  
R. Al Hussainy
Keyword(s):  
Steady State ◽  
Dynamic Pressure ◽  
Initial Pressure ◽  
Drainage Area ◽  
Reservoir Modeling ◽  
Dimensionless Time ◽  
Generalized Equations ◽  
Reservoir Pressure ◽  
Steady State Flow ◽  
State Flow

In a recent paper, van Poollen et al. presented equations relating observed field pressures to those calculated by a numerical simulator. The equations are applicable to steady- and semi steady-state flow for wells draining circular areas and using an equivalent block radius. They implicitly assume that wells are located at the center of the drainage area. In this note we present equations that generalize the previous method and relate field to model pressures for various shapes of drainage area, well pressures for various shapes of drainage area, well location and grid configuration. Using the generalized equations of flow of Brons and Miller, and following a method of derivation similar to that reported by van Poollen et al., we obtain:For steady-state flow: .................(1) For semisteady-state flow: ...................(2) For unsteady-state flow: .....................(3) where subscripts m and f refer to the model and field, respectively, and P D theta of Eq. 3 is the dimensionless initial pressure. For a circular system A pi r, and CA = 31.6 from Ref. 3. If we substitute these values in Eqs. 1 and 2, we obtain the results of van Poollen et al. as represented by their Eqs. 14a and 14b. In a manner analogous to that of van Poollen et al. the average reservoir pressure can be related to the dynamic pressure by using a dimensionless time based on the drainage area A rather than radius. Also, the relation between the average reservoir pressure and the simulator pressure in Eqs. 1 and 2 can be based on producing time as well as shut-in time, since it is possible to generate one method from the other as was shown by Ramey. NOMENCLATURE A = area in sq ftB = formation volume factor, res. bbl/STBCA = shape factorDD =c = compressibility, 1/psih = thickness, ftk = permeability, md PD, m, t = dimensionless model pressure, [141.2kb/ PD, m, t = dimensionless model pressure, [141.2kb/(q B)]P PD = dimensionless average reservoir pressure PD = dimensionless average reservoir pressure P = pressure, psiq = production rate, STB/Drw = well radius, ftt = flow time, days= porosity fraction= viscosity, cp P. 277

Pressure Transient Performance of a Multilayered Reservoir with Crossflow

1962 ◽  
Vol 2 (04) ◽  
pp. 347-354 ◽  
Author(s):  
J.D. Pendergrass ◽  
V.J. Berry
Keyword(s):  
Steady State ◽  
Transient Flow ◽  
Natural Extension ◽  
Cross Flow ◽  
Reservoir Pressure ◽  
Transient Performance ◽  
Steady State Flow ◽  
Pressure Transient ◽  
State Flow ◽  
Flow Capacity

Abstract Well pressure transient tests provide a means for directly obtaining information about formation pressure and reservoir flow capacity. Such tests have also been proposed for determining presence and location of faults or other reservoir closures and for measuring oil in place. For mathematical convenience, most theoretical studies have considered the reservoirs to be homogeneous. Definitive information is not yet available to show whether the actual presence of nonuniformities will make pressure transient behavior different from that of a uniform reservoir. The conclusions reached from actual transient tests are questionable, therefore, insofar as they rely on the original assumption of homogeneity. One type of nonuniformity commonly assumed to exist is that of stratification. In most reservoirs the strata are thought to be in vertical communication. Equations for the transient flow of a single-phase, compressible fluid in a one-well, bounded, circular reservoir have been solved for several situations involving cross flow between multiple strata of various thicknesses and permeabilities. The results show that except for the very early flow period, which usually is too short to be analyzed, the transient performance observed at the well is substantially identical with that of a homogeneous reservoir having the same dimensions and having the same steady-state flow capacity. Thus, stratification does not adversely affect interpretation of well transient tests. This conclusion holds for all commonly encountered combinations of reservoir thickness and external radius. Deviations are observed for unusually thick reservoirs whose outer radii are relatively small. The results of these studies also show that the presence and the amount of stratification cannot be simply diagnosed from reservoir pressure transient data when there is cross flow between strata. Introduction The last decade has brought wide acceptance of the transient well pressure test for determining reservoir parameters. Following the original work of Hurst and van Everdingen, the mathematical theory was thoroughly explored. Numerous authors have suggested how to determine static reservoir pressure, permeability-thickness product, original oil in place and reservoir limits for different reservoir geometry. The same mathematical techniques have been used to predict the transient performance of a reservoir over a long period of time. Most of the theoretical work has been for homogeneous, isotropic systems. Some results have also been presented for a homogeneous, anisotropic reservoir. Petroleum reservoirs are not homogeneous. The deposition process seems to favor creation of a stratified formation. This concept is sufficiently well accepted so that the most natural extension of the transient flow theory beyond the homogeneous case is to a stratified formation. Results for a stratified reservoir with no vertical communication between layers can be obtained from the results for a homogeneous reservoir. Lefkovitz, et al, have given a thorough treatment of the two-layer case. Two recent papers have treated the case of a two-layered reservoir with vertical communication or crossflow, between layers. Russell and Prats find that, after a relatively short time, the two-layered reservoir with cross flow exhibits a simple exponential pressure decline. From this time forward, the behavior is not distinguishable from the behavior of a homogeneous reservoir having the same steady-state flow capacity. The results of Katz and Tek are equivalent. Russell and Prats also speculate that a multilayered reservoir with crossflow will behave as a homogeneous system after long enough production time ".... providing the contrast in kh between layers is not too great". They also suggest that, at intermediate times, ".... the relative positions of the layers with respect to each other will have a great influence on the production behavior and on the time at which the previously mentioned large-time approximation might be valid".Katz and Tek remark upon the mathematical difficulty of treating a reservoir having many layers or strata. SPEJ P. 347^

Productivity-Index Optimization for Hydraulically Fractured Vertical Wells in a Circular Reservoir: A Comparative Study With Analytical Solutions

SPE Journal ◽  
2016 ◽  
Vol 21 (06) ◽  
pp. 2208-2219 ◽  
Author(s):  
Yunhu Lu ◽  
Kang Ping Chen
Keyword(s):  
Steady State ◽  
Analytical Solution ◽  
Performance Optimization ◽  
Ad Hoc ◽  
Drainage Area ◽  
Productivity Index ◽  
Steady State Flow ◽  
Fracture Conductivity ◽  
State Flow ◽  
Shape Approximation

Summary Productivity-index (PI) optimization by means of optimal fracture design for a vertical well in a circular reservoir is a canonical problem in performance optimization for hydraulically fractured wells. Recent availability of the exact analytical solution for the pseudosteady-state (PSS) flow of a vertically fractured well with finite fracture conductivity in an elliptical drainage area provides an opportunity to re-examine this fundamental problem in a more-rigorous manner. This paper first quantitatively estimates the shape-approximation-induced error in the PI when the exact solution for an elliptical drainage area is applied to a circular drainage area. It is shown that the shape-approximation-induced error in the PSS-flow PI is less than 1% for fracture penetration ratios up to 53%, and this error decreases significantly as the fracture conductivity is increased. PI optimization is then performed with the highly accurate analytical solution for this range of the penetration ratios. The results show that the optimal fracture conductivity increases linearly from 1.39 to 1.71 when the proppant number is increased from 0.0001 to 0.6. PI for the steady-state flow and a popular ad hoc PSS-flow PI are compared with the analytical PSS-flow PI. It is found that both the steady-state and the ad hoc PIs deviate significantly from the analytical PSS-flow PI. In particular, the optimal fracture conductivity for the steady-state flow and the ad hoc PIs decreases with the proppant number, opposite to the trend observed for the optimal fracture conductivity for the PSS flow. It is suggested that the ad hoc PI should be abandoned in favor of the more-rigorous analytical PSS-flow solution.

Calibration 7" – Cutthroat Flume as New Size for Discharge Measurement at Free Flow Condition

2020 ◽  
Vol 38 (12A) ◽  
pp. 1783-1789
Author(s):  
Jaafar S. Matooq ◽  
Muna J. Ibraheem
Keyword(s):  
Steady State ◽  
Flow Condition ◽  
Laboratory Experiments ◽  
Free Flow ◽  
Experimental Program ◽  
Steady State Flow ◽  
Empirical Formulas ◽  
State Flow ◽  
Operation Conditions ◽  
Throat Width

 This paper aims to conduct a series of laboratory experiments in case of steady-state flow for the new size 7 ̋ throat width (not presented before) of the cutthroat flume. For this size, five different lengths were adopted 0.535, 0.46, 0.40, 0.325 and 0.27m these lengths were adopted based on the limitations of the available flume. The experimental program has been followed to investigate the hydraulic characteristic and introducing the calibrated formula for free flow application within the discharge ranged between 0.006 and 0.025 m3/s. The calibration result showed that, under suitable operation conditions, the suggested empirical formulas can accurately predict the values of discharge within an error ± 3%.

A Graphical Method for Storage Coefficient Determination from Quasi-Steady State Flow Data

1996 ◽  
Vol 27 (4) ◽  
pp. 247-254 ◽  
Author(s):  
Zekâi Şen
Keyword(s):  
Steady State ◽  
Graphical Method ◽  
Pumping Test ◽  
Quasi Steady State ◽  
Flow Data ◽  
Steady State Flow ◽  
Storage Coefficient ◽  
State Flow ◽  
Aquifer Test

A simple, approximate but practical graphical method is proposed for estimating the storage coefficient independently from the transmissivity value, provided that quasi-steady state flow data are available from a pumping test. In the past, quasi-steady state flow distance-drawdown data have been used for the determination of transmissivity only. The method is applicable to confined and leaky aquifers. The application of the method has been performed for various aquifer test data available in the groundwater literature. The results are within the practical limits of approximation compared with the unsteady state flow solutions.

Steady-State Flow Behavior of CO2 Foam

2004 ◽  
Author(s):  
J.S. Kim ◽  
Y. Dong ◽  
W.R. Rossen
Keyword(s):  
Steady State ◽  
Flow Behavior ◽  
Steady State Flow ◽  
State Flow ◽  
Co2 Foam

Dependency of Unsteady Time-Linearized Flutter Investigations on the Steady State Flow Field

2011 ◽  
Author(s):  
Michael Blocher ◽  
Markus May ◽  
Harald Schoenenborn
Keyword(s):  
Steady State ◽  
Steady State Solution ◽  
Elastic Stability ◽  
Compressor Cascade ◽  
Steady State Flow ◽  
State Flow ◽  
Flow Parameters ◽  
Pressure Distributions ◽  
German Aerospace ◽  
École Polytechnique

The influence of the steady state flow solution on the aero-elastic stability behaviour of an annular compressor cascade shall be studied in order to determine sensitivities of the aero-dynamic damping with respect to characteristic flow parameters. In this context two different flow regimes — a subsonic and a transonic case — are subject to the analysis. The pressure distributions, steady as well as unsteady, on the blade surface of the NACA3506 profile are compared to experimental data that has been gained by the Institute of Aeroelasticity of the German Aerospace Center (DLR) during several wind tunnel tests at the annular compressor cascade facility RGP-400 of the Ecole Polytechnique Fe´de´rale de Lausanne (EPFL). Whereas a certain robustness of the unsteady CFD results can be stated for the subsonic flow regime, the transonic regime proves to be very sensitive with respect to the steady state solution.

Sign in / Sign up

Export Citation Format

Share Document