Motivation

The most accurate way to simulate Gas Cap expansion (or shrinkage) is full-field 3D Dynamic Flow Model where Gas Cap 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 Gas Cap is not available.

Besides many practical applications require only knowledge of one element of the Gas Cap expansion process – a pressure support and not the sweep in the invaded zones.

This allows building a Gas Cap Drive @model using analytical methods.

Inputs & Outputs

Inputs		Outputs
$\begin{array}{l}p(t)\end{array}$	field-average formation pressure at time moment $\begin{array}{l}t\end{array}$	$\begin{array}{l}Q^{\downarrow}_{GC}(t)\end{array}$	Cumulative subsurface gas influx from Gas Cap
$\begin{array}{l}p_i\end{array}$	Initial formation pressure	$\begin{array}{l}\displaystyle q^{\downarrow}_{GC}(t) = \frac{dQ^{\downarrow}_{GC}}{dt}\end{array}$	Subsurface gas flowrate from Gas Cap
$\begin{array}{l}V_{gi}\end{array}$	Initial Gas Cap volume
$\begin{array}{l}Z(p)\end{array}$	Gas compressibility factor

Physical Model

Isothermal expansion	Uniform pressure depletion in Gas Cap
$\begin{array}{l}T = \rm const\end{array}$	$\begin{array}{l}p_{GC}(t) = p(t)\end{array}$

Mathematical Model

(1)	$\begin{array}{l}\displaystyle Q^{\downarrow}_{GC}(t) = V_{gi} - V_{GC}(t) = V_{gi} \cdot \left( \frac{Z}{p} \cdot \frac{p_i}{Z_i} - 1 \right)\end{array}$

(2)	$\begin{array}{l}\displaystyle q^{\downarrow}_{GC}(t) = \frac{dQ_{GC}}{dt}\end{array}$

Proxy Models

Low pressure depletion $\begin{array}{l}\displaystyle \frac{\delta p(t)}{p_i} \ll 1\end{array}$

(3)	$\begin{array}{l}\displaystyle Q^{\downarrow}_{GC}(t) = V_{gi} \cdot \frac{\delta p(t)}{p_i}\end{array}$

(4)	$\begin{array}{l}\displaystyle q^{\downarrow}_{GC}(t) = \frac{V_{gi}}{p_i}\cdot \frac{d p}{dt}\end{array}$

Derivation

When pressure depletion is not strong $\begin{array}{l}\displaystyle \frac{\delta p(t)}{p_i} \ll 1\end{array}$ then compressibility factor maybe considered as relatively constant $\begin{array}{l}Z = \rm const\end{array}$ which leads to:

(5)

$\begin{array}{l}\displaystyle Q^{\downarrow}_{GC}(t) = V_{gi} \cdot \left( \frac{Z}{p} \cdot \frac{p_i}{Z_i} - 1 \right) = V_{gi} \cdot \left( \frac{p_i}{p} - 1 \right) = V_{gi} \cdot \left( \frac{p_i}{p_i - \delta p} - 1 \right) = V_{gi} \cdot \left( \frac{1}{1 - \frac{\delta p}{p_i}} - 1 \right) \approx V_{gi} \cdot \left( 1 + \frac{\delta p}{p_i} - 1 \right) = V_{gi} \cdot \frac{\delta p(t)}{p_i)}\end{array}$

Page tree

Motivation

Physical Model

Mathematical Model

Proxy Models

See Also