Solution:

Let's assume that the initial quantity of the mixture in the barrel is 4 units (3 units of wine and 1 unit of water).

Step 1: Calculating the initial quantity of wine and water in the mixture
- The ratio of wine to water in the initial mixture is 3:1.
- So, the initial quantity of wine in the mixture = 3/4 * 4 = 3 units.
- The initial quantity of water in the mixture = 1/4 * 4 = 1 unit.

Step 2: Drawing off a fraction of the mixture
Let's assume that a fraction 'f' of the mixture is drawn off from the barrel.

- The quantity of wine drawn off = 3/4 * f.
- The quantity of water drawn off = 1/4 * f.

Step 3: Adding water to the mixture
After drawing off a fraction of the mixture, water is added to the barrel to maintain the volume.

- The quantity of water added = 1/4 * f.

Step 4: Calculating the final quantity of wine and water in the mixture
- The quantity of wine remaining in the mixture = 3 units - 3/4 * f.
- The quantity of water remaining in the mixture = 1 unit - 1/4 * f + 1/4 * f = 1 unit.

Step 5: Calculating the ratio of wine to water in the resultant mixture
- The ratio of wine to water in the resultant mixture is 1:1.
- So, the quantity of wine remaining in the mixture = the quantity of water remaining in the mixture.

Therefore, we can write the equation:
3 - 3/4 * f = 1

Step 6: Solving the equation to find the value of 'f'
- Subtracting 3 from both sides of the equation:
-3/4 * f = -2

- Multiplying both sides of the equation by -4/3:
f = -2 * (-4/3)
f = 8/3

Step 7: Conclusion
To make the ratio of wine and water in the resultant mixture 1:1, a fraction of 8/3 of the initial mixture must be drawn off and substituted by water.

Note: Since the fraction of the mixture cannot be negative, we need to consider only the positive value of 'f'. So, the fraction of the mixture that needs to be drawn off and substituted by water is 8/3.