Summary
In this paper the threshold gradient effect in the nonsteady flow through a porous media is investigated and its role on the pressure and flow rate distributions in a flow system is evaluated. The system considered in this investigation is a linear oil reservoir. Two situations of practical interest, i.e. reservoir depleted at a constant pressure and at a constant flow rate, are considered. The solutions in a closed form for this flow system are obtained. A comparison between the pressure distributions obtained analytically, numerically and by the integral method is also presented. In order to illustrate the effect that threshold gradient would have on the flow behavior, several numerical examples are given. The results derived from this investigation show that the flow behavior in the presence of a threshold gradient depends strongly on two dimensionless parameters. These parameters are directly related to the threshold gradient, the physical properties of the porous medium and the flow parameters at the outface flow.
Zusammenfassung
In dieser Arbeit wird der Einfluß des Schwellgradienten in der instationären Strömung durch ein poröses Medium behandelt und seine Auswirkung auf Druck und Strömungsgeschwindigkeitsverteilung in einem Strömungssystem berechnet. Das in dieser Abhandlung betrachtete System ist ein linearer Ölbehälter. Zwei Fälle von praktischem Interesse, das sind Entleerung des Behälters bei konstantem Druck und bei konstanter Ausflußgeschwindigkeit, werden behandelt. Es werden geschlossene Lösungen für dieses Strömungssystem erhalten. Ein Vergleich zwischen den analytisch, numerisch und aus Integralgleichungen erhaltenen Druckverteilungen wird gebracht. Um den Einfluß zu zeigen, welchen der Schwellgradient auf das Strömungsverhalten haben kann, werden einige numerische Beispiele angeführt. Die aus dieser Untersuchung abgeleiteten Ergebnisse zeigen, daß das Strömungsverhalten unter einem Schwellgradienten stark von zwei dimensionslosen Parametern abhängt. Diese Parameter werden direkt auf den Schwellgradient, die physikalischen Eigenschaften des porösen Mediums und die Strömungsparameter der außenseitigen Strömung bezogen.
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Abbreviations
- a 2 :
-
a constant in Eqs. (11) and (12)
- F :
-
cross sectional area normal to flow
- c 0 :
-
oil compressibility coefficient
- c p :
-
porous medium compressibility coefficient
- k :
-
absolute permeability for visco-plastic flow
- \(\bar k\) :
-
absolute permeability for air flow
- l(t) :
-
front position
- L :
-
length of oil reservoir
- p(x, t) :
-
pressure distribution in system
- p 0 :
-
pressure at producing edge
- p k :
-
initial pressure in system
- Δp :
-
pressure dropp k−p 0
- Q 0 :
-
flow rate at producing edge
- R :
-
radial distance
- t :
-
time
- T :
-
dimensionless time
- T M :
-
dimensionless time required for front to reach closed boundary
- v :
-
velocity
- V :
-
volume of oil reservoir
- α0 :
-
threshold gradient
- ϱ:
-
specific mass
- ϕ:
-
porosity
- x :
-
linear distance
- τ0 :
-
shear stress at zero rate of shear
- ξ(t):
-
dimensionless front position
- μ:
-
viscosity
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Pascal, H. Nonsteady flow through porous media in the presence of a threshold gradient. Acta Mechanica 39, 207–224 (1981). https://doi.org/10.1007/BF01170343
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DOI: https://doi.org/10.1007/BF01170343