Motivation
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| Aquifer Drive |
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| Aquifer Drive |
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Inputs & Outputs
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Physical Model
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Mathematical Model
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| Q^{\downarrow}_{AQ}= B \cdot \int_0^t W_{eD} \left( \frac{(t-\tau)\chi |
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_a \right) \dot p(\tau) d\tau |
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q^{\downarrow}_{AQ}(t)= \frac{dQ^{\downarrow}_{AQ}}{dt} | LaTeX Math Block |
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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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RC1\fracpartial p_1}{\partial t_D} = downarrow}_{AQ}(t)= \frac{ |
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\partial^2 p_1}{\partial r_D^2} + \frac{1}{r_D}\cdot \frac{\partial p_1}{\partial r_DdQ^{\downarrow}_{AQ}}{dt} |
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CT(t_D 0,r_D)= 0 LaTeX Math Block |
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=1 = 11 | \frac{\partial p_1}{\partial |
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r_D}
\bigg|_{(, r_D=r_a/r_e)0...
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Transient flow in Radial Composite Reservoir:
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r_D^2} + \frac{1}{r_D}\cdot \frac{\partial p_ |
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1(t_D, r_D)}{\partial r_D}
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Computational Model
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\dot p(\tau) = \frac{d p}{d \tau} |
One can easily check that
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honors the whole set of equations LaTeX Math Block Reference |
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and as such defines a unique solution of the above problem.Water flowrate within
sector angle at interface with oil reservoir will be:...
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where
is flow velocity at aquifer contact boundary, which is: LaTeX Math Block |
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u(t,r_e) = M_a \cdot \frac{\partial p_a(t,r)}{\partial r} \bigg|_{r=r_e} |
where
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body | M_a = \frac{k_a}{\mu_w} |
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is aquifer mobility.Water flowrate becomes:
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q^{\downarrow}_{AQ}(t)= \theta \cdot r_e \cdot h_a \cdot M_a \cdot \frac{\partial p_a(t,r)}{\partial r} \bigg|_{r=r_e} |
Cumulative water flux:
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Q^{\downarrow}_{AQ}(t) = \int_0^t q^{\downarrow}_{AQ}(t) dt = \theta \cdot r_e \cdot h_a \cdot M_a \cdot \int_0^t \frac{\partial p_a(t,r)}{\partial r} \bigg|_{r=r_e} dt |
Substituting
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into LaTeX Math Block Reference |
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leads to:...
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\sum_\alpha W_{eD}
\left( \frac{ (t-\tau_\alpha) \chi}{r_e^2} \right)\Delta p_\alpha
= B \cdot W_{eD}
\left( \frac{ ( |
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B \cdot W_{eD}
\left( \frac{ ( |
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_2
+ ... + B \cdot W_{eD}
\left( \frac{ ( |
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This computational model is using a discrete convolution (also called superposition in some publications) with time-grid
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body | \{ \tau_1, \, \tau_2, \ ... \ , \tau_N \} |
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.In practical exercises with manual or spreadsheet-assisted calculations the time-grid is usually uniform:
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body | \{ \tau_1 =\Delta \tau, \, \tau_2 = 2 \cdot \Delta \tau, \ ... \ , \tau_N = N \cdot \Delta \tau\} |
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with the time step of a month to ensure the formation pressure does not change much since the previous time step.Moving to annual time step may accumulate a substantial mistake if formation pressure has varied substantially in some years.
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title | Polynomial approximations for WeD |
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Polynomial approximation of are available (http://dx.doi.org/10.2118/15433-PA).
Table 1. Polynomial approximation of for infinite aquifer | LaTeX Math Inline |
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body | W_{eD}=\sqrt{\frac{t_D}{\pi}} |
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body | \displaystyle W_{eD}=\frac {1.2838 \cdot t_D^{1/2} + 1.19328 \cdot t_D +0.269872 \cdot t_D^{3/2} +0.00855294 \cdot t_D^2} {1+0.616599 \cdot t_D^{1/2}+0.0413008 \cdot t_D} |
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body | \displaystyle W_{eD}=\frac{-4.29881+2.02566 \cdot t_D}{\ln t_D} |
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Scope of Applicability
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The benefit of VEH approach is that net pay formation pressure history
is usually known so that water influx calculation based on aquifer properties LaTeX Math Inline |
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body | \{ B, \, r_a, \, \chi \} |
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is rather straightforward.In the past the VEH approach was considered as tedious in calculating superposition during the manual exercises.
In modern computers the convolution is a fast fully-automated procedure and VEH model is considered as a reference in the range of analytical aquifer models.
Although the model is derived for linear and radial flow it also shows a good match for bottom-water drive depletions
The above integral represents the integration over the
area in plane (see Fig. 1): LaTeX Math Block |
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Q^{\downarrow}_{AQ}(t) = \theta \cdot h_a \cdot M_a \cdot \iint_D d\xi \ d\tau \, \dot p(\tau)
\frac{\partial p_1}{\partial r_D} \left( \frac{(\xi-\tau)\chi_a}{r_e^2}, r_D \right) \Bigg|_{r_D=1}
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Fig. 1. Illustration of the integration
area in planeChanging the integration order from
to leads to: LaTeX Math Block |
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Q^{\downarrow}_{AQ}(t) = \theta \cdot h_a \cdot M_a \cdot \int_0^t d\tau \int_\tau^t d\xi \ \dot p(\tau)
\frac{\partial p_1}{\partial r_D} \left( \frac{(\xi-\tau)\chi_a}{r_e^2}, r_D \right) \Bigg|_{r_D=1}
=
\theta \cdot h_a \cdot M_a \cdot \int_0^t \dot p(\tau) d\tau \int_\tau^t d\xi \
\frac{\partial p_1}{\partial r_D} \left( \frac{(\xi-\tau)\chi_a}{r_e^2}, r_D \right) \Bigg|_{r_D=1} |
Replacing the variable:
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\xi = \tau + \frac{r_e^2}{\chi_a} \cdot t_D \rightarrow t_D = \frac{(\xi-\tau)\chi_a}{r_e^2} \rightarrow d\xi = \frac{r_e^2}{\chi_a} \cdot dt_D |
and flux becomes:
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Q^{\downarrow}_{AQ}(t) = \theta \cdot h_a \cdot M_a \cdot \frac{r_e^2}{\chi_a} \cdot \int_0^t \dot p(\tau) d\tau \int_0^{(t-\tau)\chi_a/r_e^2}
\frac{\partial p_1( t_D, r_D)}{\partial r_D} \Bigg|_{r_D=1} dt_D |
and finally:
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Q^{\downarrow}_{AQ}(t) = \theta \cdot h_a \cdot M_a \cdot \frac{r_e^2}{\chi_a} \cdot \int_0^t \dot p(\tau) d\tau \int_0^{(t-\tau)\chi_a/r_e^2}
\frac{\partial p_1( t_D, r_D)}{\partial r_D} \Bigg|_{r_D=1} dt_D |
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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
Reference
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1. van Everdingen, A.F. and Hurst, W. 1949. The Application of the Laplace Transformation to Flow Problems in Reservoirs. Trans., AIME 186, 305.
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