Fractional Flow Curve in Excel

The fractional flow curve is a plot of the fraction of water flowing in a porous medium, as a function of the water saturation. It is used to estimate the advance of a water front that displaces oil in an immiscible displacement process.

To calculate the fractional flow curve, we need to know the relative permeabilities of water and oil, and their viscosities. The relative permeability is a measure of how easily a fluid can flow through a porous medium, when another fluid is present. The viscosity is a measure of how resistant a fluid is to flow.

Basic Theory

The fractional flow curve is typically used to model the displacement of one fluid by another in a porous medium. It is derived from the immiscible displacement process, such as water displacing oil in an oil reservoir. The fractional flow (F) is defined as the ratio of the Darcy velocity of the displaced fluid to the total Darcy velocity of the fluid phase.

$F = \frac{u_o}{u_t}$

Where:

$F$ is the fractional flow,
$u_o$ is the Darcy velocity of the oil phase,
$u_t$ is the total Darcy velocity of the fluid phase.

Procedures

Step 1: Define Parameters

Corey Curve Equation: The Corey curve is often used to model fractional flow. The equation is given by $F = S_w^n$ , where $S_w$ is the water saturation, and $n$ is the Corey exponent.
Define Saturation Range: Determine the range of water saturation ( $S_w$ ) for which you want to calculate the fractional flow.

Step 2: Excel Formulas

1. Create a Saturation Table: Generate a table with saturation values from 0 to 1.
2. Calculate Fractional Flow: Use the Corey curve equation to calculate fractional flow for each saturation point.

$F = S_w^n$

Graph the Results: Create a graph of fractional flow against water saturation using Excel’s charting tools.

Example Scenario

Let’s consider a reservoir with a Corey exponent ( $n$ ) of 2. The saturation range is 0 to 1 in increments of 0.1.

Excel Table

Water Saturation (Sw)	Corey Fractional Flow (F)
0.0	=A2^2
0.1	=A3^2
1.0	=A12^2

Excel Graph

Create a scatter plot with the saturation on the x-axis and fractional flow on the y-axis.

Calculation and Results

Using the provided Excel formulas and the given scenario, the fractional flow curve is calculated and graphed. The resulting graph visually represents how the fractional flow changes with water saturation.

MATLAB Comparison

To compare the results, perform the same calculations in MATLAB using the Corey curve equation. MATLAB code snippet:

% Define parameters
n = 2;
Sw = 0:0.1:1;

% Calculate fractional flow using Corey curve equation
F = Sw.^n;

% Plot the results
plot(Sw, F, '-o')
xlabel('Water Saturation (Sw)')
ylabel('Fractional Flow (F)')
title('Fractional Flow Curve')

Compare the MATLAB plot with the Excel graph to ensure consistency.