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
$\begin{array}{l}J_{AQ}\end{array}$	aquifer Productivity Index
$\begin{array}{l}\sigma\end{array}$	aquifer transmissibility
$\begin{array}{l}\displaystyle \tau = \frac{V_{AQ} \, c_t}{J}\end{array}$	aquifer relaxation time
Detailing Inputs
$\begin{array}{l}V_{AQ}\end{array}$	aquifer volume
$\begin{array}{l}A_{AQ}\end{array}$	aquifer area
$\begin{array}{l}A_e\end{array}$	pay area

Assumptions

Pseudo Steady State Flow		Oil field performs as single well with $\begin{array}{l}A_e\end{array}$ cross-sectional area in
$\begin{array}{l}p_{AQ}(t)=p_i = \rm const\end{array}$	Aquifer pressure is constant
$\begin{array}{l}J_{AQ} = \rm const\end{array}$	Aquifer Productivity Index is constant

Equations

(1)	$\begin{array}{l}\displaystyle \frac{d Q_{AQ}}{dt} + \frac{1}{\tau} Q_{AQ} = J \cdot ( p_i - p(t))\end{array}$

$\begin{array}{l}\displaystyle J_{AQ} = \frac{2 \pi \sigma}{\ln \frac{A_{AQ}}{A_e} - \frac{3}{4} }\end{array}$

where

$\begin{array}{l}Q_{AQ}(t)\end{array}$	water influx from aquifer at time moment $\begin{array}{l}t\end{array}$	$\begin{array}{l}\sigma\end{array}$	aquifer transmissibility
$\begin{array}{l}J = \rm const\end{array}$	aquifer productivity index	$\begin{array}{l}\displaystyle \tau = \frac{V_{AQ} \, c_t}{J}\end{array}$	aquifer relaxation time
$\begin{array}{l}p_i\end{array}$	initial formation pressure	$\begin{array}{l}A_{AQ}\end{array}$	aquifer area
$\begin{array}{l}p(t)\end{array}$	field-average formation pressure at time moment $\begin{array}{l}t\end{array}$	$\begin{array}{l}A_e\end{array}$	pay area

Derivation

Const PI expansion:

$\begin{array}{l}\displaystyle q_{AQ} = \frac{d Q_{WAQ}}{dt} = J \cdot ( p_{AQ} - p(t))\end{array}$

Finite-volume reservoir PSS depletion:

$\begin{array}{l}\displaystyle p_{AQ} = p_i - \frac{Q_{WAQ}}{V_{WAQ} c_t}\end{array}$

Reference

1. Fetkovich, M.J. 1971. A Simplified Approach to Water Influx Calculations—Finite Aquifer Systems. J Pet Technol 23 (7): 814–28. SPE-2603-PA. http://dx.doi.org/10.2118/2603-PA

Page tree

Motivation

Inputs & Outputs

Assumptions

Equations

See Also

Reference

Page tree

Fetkovich PSS Aquifer Drive @model

Motivation

Inputs & Outputs

Assumptions

Equations

See Also

Reference