Motivation

The most accurate way to simulate Aquifer Expansion (or shrinkage) is full-field 3D Dynamic Flow Model where Aquifer Expansion is treated as one of the fluid phases and accounts of geological heterogeneities, gas fluid properties, relperm properties and heat exchange with surrounding rocks.

Unfortunately, in many practical cases the detailed information on the aquifer is not available which does not allow a proper modelling of aquifer expansion using a geological framework.

Besides many practical applications require only knowledge of cumulative water influx from aquifer under pressure depletion.

This allows building an Aquifer Drive Models using analytical methods.

Inputs & Outputs

Inputs

Outputs

$\begin{array}{l}p(t)\end{array}$

field-average formation pressure at time moment $\begin{array}{l}t\end{array}$

$\begin{array}{l}Q^{\downarrow}_{AQ}(t)\end{array}$

Cumulative subsurface water influx from aquifer

$\begin{array}{l}p_i\end{array}$

initial formation pressure

$\begin{array}{l}q^{\downarrow}_{AQ}(t) = \frac{dQ^{\downarrow}_{AQ}}{dt}\end{array}$

Subsurface water flowrate from aquifer

$\begin{array}{l}B\end{array}$

water influx constant

$\begin{array}{l}\chi\end{array}$

aquifer diffusivity

$\begin{array}{l}A_e = \pi \, r_e^2\end{array}$

net pay area

Detailing

Detailing Inputs
$\begin{array}{l}B = \frac{\theta}{\pi} \cdot A_e \cdot h_a \cdot \phi_a \cdot c_t\end{array}$	water influx constant
$\begin{array}{l}\theta\end{array}$	central angle of net pay area ↔ aquifer contact
$\begin{array}{l}h_a\end{array}$	aquifer effective thickness
$\begin{array}{l}\phi_a\end{array}$	aquifer porosity
$\begin{array}{l}c_t=c_\phi +c_w\end{array}$	aquifer total compressibility
$\begin{array}{l}c_\phi\end{array}$	aquifer pore compressibility
$\begin{array}{l}c_w\end{array}$	aquifer water compressibility

Physical Model

Radial Composite Reservoir
Transient flow
Computational approximation to van Everdingen-Hurst (VEH)

	Fig. 1. Carter-Tracy aquifer drive schematic

Mathematical Model

(1)	$\begin{array}{l}\displaystyle \frac{d Q^{\downarrow}_{AQ}}{dt_D} = \frac{ B \cdot (p_i - p(t_D)) - Q^{\downarrow}_{AQ} \cdot p'_D(t_D)}{p_D(t_D) - t_D \cdot p'_D(t_D)}\end{array}$

(2)	$\begin{array}{l}\displaystyle p_D= \frac{370.529 \, \sqrt{t_D} +137.528 \, t_D + 5.69549 \, t_D^{1.5}} {328.834 +265.488 \, \sqrt{t_D} + 45.2157 \, t_D + t_D^{1.5} }\end{array}$

(3)	$\begin{array}{l}\displaystyle q^{\downarrow}_{AQ}(t)=\frac{d Q^{\downarrow}_{AQ}}{dt}\end{array}$

(4)	$\begin{array}{l}\displaystyle p'_D= \frac {716.441 + 46.7984 \, \sqrt{t_D} + 270.038 \, t_D + 71.0098 \, t_D^{1.5} } { 1296.86 \, \sqrt{t_D} + 1204.73 \, t_D + 618.618 \, t_D^{1.5} + 538.072 \, t_D^2 + 142.41 \, t_D^{2.5} }\end{array}$

(5)	$\begin{array}{l}\displaystyle t_D= \frac{\pi \, \chi \, t}{A_e}\end{array}$

Derivation

Equation (1) is not a solution of diffusion equation but an approximation of VEH model which does not involve convolution integral.

Computational Model

(6)	$\begin{array}{l}\displaystyle Q^{\downarrow}_{AQ}(t_{D,n}) = Q^{\downarrow}_{AQ}(t_{D, \, n-1}) + (t_{D,n}-t_{D,\, n-1}) \cdot \frac{ B \cdot (p_i - p(t_{D,n})) - Q^{\downarrow}_{AQ}(t_{D, \, n-1}) \cdot p'_D(t_{D,n})}{p_D(t_{D,n}) - t_{D,\, n-1} \cdot p'_D(t_{D, n})}\end{array}$

Reference

1. Carter, R.D. and Tracy, G.W. 1960. An Improved Method for Calculating Water Influx. Trans., AIME 219: 415.

2. Tarek Ahmed, Paul McKinney, Advanced Reservoir Engineering (eBook ISBN: 9780080498836)

Page tree

Motivation

Inputs & Outputs

Mathematical Model

Computational Model

See Also

Reference

Page tree

Carter-Tracy Aquifer Drive @model

Motivation

Inputs & Outputs

Mathematical Model

Computational Model

See Also

Reference