The correct answer is $\boxed{\frac{X}{2}}$.
The moisture content of a substance is the ratio of the mass of water in the substance to the mass of the dry solid matter in the substance. It is expressed as a percentage.
The solid content of a substance is the ratio of the mass of the dry solid matter in the substance to the total mass of the substance. It is also expressed as a percentage.
If the quantity of sludge with moisture content of 98% is X, then the mass of the dry solid matter in the sludge is 0.02X. The mass of the water in the sludge is 0.98X.
If the moisture content of the sludge is reduced to 96%, then the mass of the water in the sludge will be 0.96X. The mass of the dry solid matter in the sludge will be 0.04X.
The quantity of sludge with moisture content of 96% is $\frac{0.04X}{0.02X} = \frac{1}{2}X$.
Option A is incorrect because the mass of the water in the sludge is not $\frac{1}{4}X$.
Option B is incorrect because the mass of the dry solid matter in the sludge is not $\frac{1}{2}X$.
Option C is incorrect because the quantity of sludge with moisture content of 96% is not X.
Option D is incorrect because the quantity of sludge with moisture content of 96% is not 2X.