Motivation
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| Aquifer Drive |
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| Aquifer Drive |
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Assumptions
| Transient flow in Radial Composite Reservoir |
Equations
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| Q^{\downarrow}_{AQ}= B \cdot \int_0^t W_{eD}(t - \tau) \dot p d\tau |
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| q^{\downarrow}_{AQ}(t)= \frac{dQ^{\downarrow}_{AQ}}{dt} |
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| W_{eD}(t)= \int_0^{t} \frac{\partial p_1}{\partial r_D} \bigg|_{r_D = 1} dt_D |
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| \frac{\partial p_1}{\partial t_D} = \frac{\partial^2 p_1}{\partial r_D^2} + \frac{1}{r_D}\cdot \frac{\partial p_1}{\partial r_D} |
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| p_1(t_D = 0, r_D)= 0 |
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| p_1(t_D, r_D=1) = 1 |
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| \frac{\partial p_1}{\partial r_D}
\bigg|_{(t_D, r_D=r_a/r_e)} = 0 |
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| Expand |
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Transient flow in Radial Composite Reservoir:
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| \frac{\partial p_a}{\partial t} = \chi \cdot \left[ \frac{\partial^2 p_a}{\partial r^2} + \frac{1}{r}\cdot \frac{\partial p_a}{\partial r} \right] |
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| p_a(t = 0, r)= p(0) |
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| p_a(t, r=r_e) = p(t) |
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| anchor | p1_PSS |
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| \frac{\partial p_a}{\partial r}
\bigg|_{(t, r=r_a)} = 0 |
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Consider a pressure convolution:
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| p_a(t, r) = p(0) + \int_0^t p_1 \left(\frac{(t-\tau) \cdot \chi}{r_e^2}, \frac{r}{r_e} \right) \dot p(\tau) d\tau |
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| \dot p(\tau) = \frac{d p}{d \tau} |
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One can easily check that | LaTeX Math Block Reference |
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honors the whole set of equations | LaTeX Math Block Reference |
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–| LaTeX Math Block Reference |
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and as such defines a unique solution of the above problem.The the water flowrate at interface with oil reservoir will be: | LaTeX Math Block |
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| q^{\downarrow}_{AQ}(t)= C_a \cdot \frac{\partial p_a(t,r)}{\partial r} \bigg|_{r=r_e} |
and cumulative flux: | LaTeX Math Block |
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| \frac{d Q^{\downarrow}_{AQ}}{dt} = q^{\downarrow}_{AQ}(t) |
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See Also
Petroleum Industry / Upstream / Subsurface E&P Disciplines / Field Study & Modelling / Aquifer Drive / Aquifer Drive Models
Reference
1. van Everdingen, A.F. and Hurst, W. 1949. The Application of the Laplace Transformation to Flow Problems in Reservoirs. Trans., AIME 186, 305.
2. Tarek Ahmed, Paul McKinney, Advanced Reservoir Engineering (eBook ISBN: 9780080498836)
3. Klins, M.A., Bouchard, A.J., and Cable, C.L. 1988. A Polynomial Approach to the van Everdingen-Hurst Dimensionless Variables for Water Encroachment. SPE Res Eng 3 (1): 320-326. SPE-15433-PA. http://dx.doi.org/10.2118/15433-PA
