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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(t_D, r_D)}{\partial r_D}
\Bigg|_{r_D=r_{aD}} = 0 \quad {\rm or} \quad p_1(t_D, r_D = \infty) = 0 |
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which represents a specific unique function of dimensionless time
and distance
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One can easily check that
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honours the whole set of equations
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and as such defines a unique solution of the above problem
.Now having the pressure dynamics in hand one can calculate the water influx.
Water The water flowrate within
sector angle at
the interface with
oil reservoir will be net pay is:
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q^{\downarrow}_{AQ}(t)= \theta \cdot r_e \cdot h \cdot u(t,r_e) |
where
is flow velocity at
aquifer Net Pay ↔ Aquifer contact boundary, which is:
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where
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| body | --uriencoded--\displaystyle M = \frac%7Bk_w%7D%7B\mu_w%7D |
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is
aquifer Aquifer mobility.
Water flowrate becomesThe water flowrate is going to be:
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q^{\downarrow}_{AQ}(t)= \theta \cdot r_e \cdot h \cdot M \cdot \frac{\partial p_a(t,r)}{\partial r} \bigg|_{r=r_e} |
Cumulative and cumulative water flux is going to be:
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Q^{\downarrow}_{AQ}(t) = \int_0^t q^{\downarrow}_{AQ}(t) dt = \theta \cdot r_e \cdot h \cdot M \cdot \int_0^t \frac{\partial p_a(t,r)}{\partial r} \bigg|_{r=r_e} dt |
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W_{eD}(t) = \int_0^{t} \frac{\partial p_1}{\partial r_D} \bigg|_{r_D = 1} dt_D |
is universal a unique dimensionless function of dimensionless time time
which can be tabulated via numerical solution or approximated by
polynoms closed-form expression.
See Also
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Petroleum Industry / Upstream / Subsurface E&P Disciplines / Field Study & Modelling / Aquifer Drive / Aquifer Drive Models / Radial VEH Aquifer Drive @model
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