Characteristic Time for Linear Diffusion in Reservoirs

Characteristic time is a measure of how fast a process happens, such as how quickly a fluid moves or changes in a reservoir. It depends on several factors, such as the porosity, compressibility, viscosity, permeability, and size of the reservoir.

One way to think of characteristic time is to imagine a pulse of fluid injected into a reservoir. How long does it take for the pulse to spread out and diffuse in the reservoir? The answer is the characteristic time. The smaller the characteristic time, the faster the diffusion. The larger the characteristic time, the slower the diffusion.

Another way to think of characteristic time is to compare it with other time scales, such as the duration of an experiment or the production time of a well. If the characteristic time is much shorter than these time scales, then the diffusion process is very fast and can be ignored. If the characteristic time is much longer than these time scales, then the diffusion process is very slow and can be approximated by a steady state. If the characteristic time is similar to these time scales, then the diffusion process is important and must be accounted for.

Basic Theory

In petroleum engineering, linear diffusion describes the process of fluid flow through porous media within a reservoir. The characteristic time (Tc) represents the time it takes for a pressure disturbance to propagate a characteristic distance through the reservoir. It is a fundamental parameter used in transient well testing and reservoir simulation.

The formula for characteristic time (Tc) is given by:

$Tc = \frac{\phi \cdot c_t \cdot V_c}{k \cdot A}$

Where:

$\phi$ is the porosity of the reservoir,
$c_t$ is the total compressibility of the reservoir rock and fluid system,
$V_c$ is the reservoir rock volume,
$k$ is the reservoir permeability, and
$A$ is the cross-sectional area.

Procedures

Gather Reservoir Data: Collect the necessary reservoir data, including porosity ( $\phi$ ), total compressibility ( $c_t$ ), reservoir rock volume ( $V_c$ ), permeability ( $k$ ), and cross-sectional area ( $A$ ).
Plug Data into the Formula: Utilize the characteristic time formula to calculate $Tc$ in both Microsoft Excel and MATLAB.
Create Excel Table: Construct an Excel table to input the gathered data, perform the calculations, and display the results.
MATLAB Implementation: Write a MATLAB script to execute the same calculations and compare the results with those obtained in Excel.

Comprehensive Explanation with Scenario

Consider a reservoir with the following properties:

Porosity ( $\phi$ ): 0.15
Total Compressibility ( $c_t$ ): 10e-6 psi $^{-1}$
Reservoir Rock Volume ( $V_c$ ): 5e6 cubic feet
Permeability ( $k$ ): 50 millidarcies
Cross-sectional Area ( $A$ ): 1000 square feet

Excel Calculation

Construct an Excel table with the following columns: Property, Value, and Unit. Input the provided data into the table and use the characteristic time formula to calculate $Tc$ .

The Excel formula would be:

=Tc = (B2 * B3 * B4) / (B5 * B6)

Where B2 to B6 represent the corresponding cells for each property.

MATLAB Calculation

Write a MATLAB script to perform the same calculation:

phi = 0.15;
ct = 10e-6;
Vc = 5e6;
k = 50e-3;
A = 1000;

Tc = (phi * ct * Vc) / (k * A);
disp(Tc);

Results

The characteristic time ( $Tc$ ) for the given scenario is calculated to be approximately 60,000 seconds.