Consider a system of net hydrocarbon pay and finite or infinite volume Aquifer as a radial composite reservoir with inner composite area being a Net Pay Area and outer composite area an Aquifer
(see schematic and notations on Fig. 1 at Radial VEH Aquifer Drive @model).
The Aquifer's outer boundary may be "no-flow" for finite-volume Aquifer or constant pressure for infinite-volume Aquifer.
The transient pressure diffusion in the outer (Aquifer) composite area is going to honour the following equation:
| Driving equation | Initial condition |
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Homogeneous Aquifer reservoir with | Initial Aquifer pressure is considered to be the same as Net Pay Area |
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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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| Inner Aquifer boundary | Outer Aquifer boundary is one of the two below: |
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| Pressure variation at the contact with Net Pay Are | "No-flow" | "Constant pressure" |
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| p_a(t, r=r_e) = p(t) |
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p1 | \frac{\partial p_a}{\partial r}
\bigg|_{(t, r=r_a)} = 0 |
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| anchor | pconst |
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| alignment | left |
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| p_a(t, r = \infty) = 0 |
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Consider dimensionless solution
of the following equation:
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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 |
or | LaTeX Math Block |
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| \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
.
...
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:
...
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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\xi = \tau + \frac{r_e^2}{\chi} \cdot t_D \rightarrow t_D = \frac{(\xi-\tau)\chi}{r_e^2} \rightarrow d\xi = \frac{r_e^2}{\chi} \cdot dt_D |
and flux cumulative influx becomes:
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Q^{\downarrow}_{AQ}(t) = \theta \cdot h \cdot M \cdot \frac{r_e^2}{\chi} \cdot \int_0^t \dot p(\tau) d\tau \int_0^{(t-\tau)\chi/r_e^2}
\frac{\partial p_1( t_D, r_D)}{\partial r_D} \Bigg|_{r_D=1} dt_D = B \cdot \int_0^t \dot p(\tau) d\tau \int_0^{(t-\tau)\chi/r_e^2}
\frac{\partial p_1( t_D, r_D)}{\partial r_D} \Bigg|_{r_D=1} dt_D |
where
is
water influx constant and which leads to
:...
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Q^{\downarrow}_{AQ}= B \cdot \int_0^t W_{eD} \left( \frac{(t-\tau)\chi}{r_e^2}\right) \dot p(\tau) d\tau |
where
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W_{eD}(t) = \int_0^{t} \frac{\partial p_1}{\partial r_D} \bigg|_{r_D = 1} dt_D |
is a unique dimensionless function of dimensionless time
which can be tabulated via numerical solution or approximated by closed-form expression and | LaTeX Math Block Reference |
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.
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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