Dimensionless time is a way of expressing the time elapsed in a reservoir simulation in terms of the reservoir properties and the injection rate. It is useful for comparing different scenarios and analyzing the performance of different recovery methods.
Myhill and Stegemeier’s method is a technique for calculating dimensionless time for steam injection processes, such as steam flooding or cyclic steam stimulation. It accounts for the effects of heat transfer, phase change, and gravity segregation on the steam zone expansion and the oil recovery. It uses a dimensionless radius and a dimensionless pressure to define the steam zone boundary and the oil displacement front. It also uses a dimensionless steam quality to describe the steam composition along the steam zone.
The method involves solving a set of equations that relate the dimensionless time, radius, pressure, and steam quality. These equations are derived from the mass and energy balances of the steam zone and the liquid zone, as well as the boundary conditions at the injection well and the steam zone front. The method requires some assumptions, such as radial symmetry, constant injection rate, constant reservoir thickness, and negligible capillary pressure.
Basic Theory
Dimensionless time, denoted as td, is a non-dimensional parameter that represents the elapsed time in a reservoir. Myhill and Stegemeier’s method expresses dimensionless time as a function of reservoir properties and operating conditions. The formula for dimensionless time is given by:
td = (0.000263 * φ * ct * t) / (k / (μ * Vt))
where:
- φ is the porosity of the reservoir,
- ct is the total compressibility,
- t is the time in days,
- k is the reservoir permeability,
- μ is the fluid viscosity,
- Vt is the total reservoir volume.
Procedures
- Collect Reservoir Data: Gather data on porosity (φ), total compressibility (ct), reservoir permeability (k), fluid viscosity (μ), and total reservoir volume (Vt).
- Input Data into Excel: Create an Excel table with the collected data. Use proper labels and units for clarity.
- Calculate Dimensionless Time: Utilize the Myhill and Stegemeier formula to calculate dimensionless time in Excel. Set up a formula cell that references the input data.
Example
Consider a reservoir with the following properties:
- Porosity (φ): 0.15
- Total Compressibility (ct): 10.0 x 10-6 psi-1
- Reservoir Permeability (k): 50 Darcy
- Fluid Viscosity (μ): 2 cp
- Total Reservoir Volume (Vt): 1.5 x 106 bbl
- Time (t): 365 days
Excel Table
| Property | Value |
|---|---|
| Porosity (φ) | 0.15 |
| Total Compressibility (ct) | 10.0 x 10-6 psi-1 |
| Reservoir Permeability (k) | 50 Darcy |
| Fluid Viscosity (μ) | 2 cp |
| Total Reservoir Volume (Vt) | 1.5 x 106 bbl |
| Time (t) | 365 days |
Excel Formula
td = (0.000263 * B2 * B3 * B7) / (B4 / (B5 * B6))
Excel Calculation
=0.000263 * B2 * B3 * B7 / (B4 / (B5 * B6))
Result
The calculated dimensionless time (td) for the given scenario is approximately 0.201.
MATLAB Comparison
To cross-verify the Excel results, the same calculation can be performed in MATLAB using the following script:
phi = 0.15;
ct = 10.0e-6;
k = 50;
mu = 2;
Vt = 1.5e6;
t = 365;
td = 0.000263 * phi * ct * t / (k / (mu * Vt));
disp(['Dimensionless Time (td) calculated using MATLAB: ', num2str(td)]);
The MATLAB result should closely match the Excel result, providing confidence in the accuracy of the calculations.
