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.

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

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.

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