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
$\begin{array}{l}A_e\end{array}$	net pay area

Detailing

Detailing Inputs
$\begin{array}{l}B = \frac{\theta}{2\pi} \cdot A_e \cdot h_a \cdot \phi_a \cdot c_t\end{array}$	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
$\begin{array}{l}c_t=c_r +c_w\end{array}$	aquifer total compressibility
$\begin{array}{l}c_r\end{array}$	aquifer pore compressibility
$\begin{array}{l}c_w\end{array}$	aquifer water compressibility

Assumptions

Transient flow in Radial Composite Reservoir

Equations

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

(2)	$\begin{array}{l}\displaystyle q^{\downarrow}_{AQ}(t)= C_a \cdot \frac{\partial p_a(t,r)}{\partial r} \bigg\|_{r=r_e}\end{array}$

(3)	$\begin{array}{l}\displaystyle p_a(t, r)= p(0) + \int_0^t p_1 \left(\frac{(t-\tau) \cdot \chi}{r_e^2}, \frac{r}{r_e} \right) \dot p(\tau) d\tau\end{array}$

(4)	$\begin{array}{l}\displaystyle \frac{\partial p_1}{\partial t_D} = \frac{\partial^2 p_1}{\partial r_D^2} + \frac{1}{r_D}\cdot \frac{\partial p_1}{\partial r_D}\end{array}$

(5)	$\begin{array}{l}\displaystyle p_1(t = 0, r_D)= 0\end{array}$

(6)	$\begin{array}{l}\displaystyle p_1(t, r_D=1) = 1\end{array}$

(7)	$\begin{array}{l}\displaystyle \frac{\partial p_1}{\partial r_D} \bigg\|_ {r_D=r_a/r_e} = 0\end{array}$

where $\begin{array}{l}\displaystyle \dot p(\tau) = \frac{d p}{d \tau}\end{array}$

Derivation

Transient flow in Radial Composite Reservoir:

(8)	$\begin{array}{l}\displaystyle \frac{\partial p_a}{\partial t} = \chi \cdot \left[ \frac{\partial^2 p_a}{\partial r^2} + \frac{1}{r}\cdot \frac{\partial p_a}{\partial r} \right]\end{array}$

(9)	$\begin{array}{l}\displaystyle p_a(t = 0, r)= p(0)\end{array}$

(10)	$\begin{array}{l}\displaystyle p_a(t, r=r_e) = p(t)\end{array}$

(11)	$\begin{array}{l}\displaystyle \frac{\partial p_a}{\partial r} \bigg\|_ {r=r_a} = 0\end{array}$

One can easily check that pressure from (3) honors the whole set of equations (4)– (11) and as such defines a unique solution of the above problem.

Reference

1. van Everdingen, A.F. and Hurst, W. 1949. The Application of the Laplace Transformation to Flow Problems in Reservoirs. Trans., AIME 186, 305.

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

3. Klins, M.A., Bouchard, A.J., and Cable, C.L. 1988. A Polynomial Approach to the van Everdingen-Hurst Dimensionless Variables for Water Encroachment. SPE Res Eng 3 (1): 320-326. SPE-15433-PA. http://dx.doi.org/10.2118/15433-PA

Page tree

Motivation

Inputs & Outputs

Assumptions

Equations

See Also

Reference

Page tree

van Everdingen-Hurst (VEH) Aquifer Drive @model

Motivation

Inputs & Outputs

Assumptions

Equations

See Also

Reference