The direct way fracking can reduce the budget is by stimulating economic activity and thus tax revenues. This is obvious.
This piece is about another, less obvious, less intuitive, indirect way fracking can reduce the budget deficit. It is based on the fact that the sum of the budget deficit, the capital inflow to finance the trade deficit, and the difference between domestic saving and domestic investment equals zero. If you expand or shrink any of these three imbalances, it puts pressure on the others to expand or shrink to maintain the net zero balance.
As fracking expands domestic oil and gas production, it likely will reduce U.S. demand for energy imports and shrink our trade deficit. This reduces the net capital inflow required to finance the trade deficit. The reduced capital inflow will tend to reduce the gaps between domestic investment and saving and government expenditures and tax revenue—the deficit in question.
Let me back up and elaborate. Income minus consumption gives us saving, by definition. Income minus consumption also gives us investment, since investment represents output not consumed. Therefore, taking consumption out of the equation, total saving must equal total investment.
National saving is composed of personal saving, business saving, and government saving, i.e. an excess of tax revenue over expenditures. Personal saving, as we know, is low but positive these days. Business saving is moderately positive. However, net negative government saving (the budget deficit) overwhelms the others and make total national saving negative. Since we invest more than we save domestically, the saving deficit must be made up by importing foreign saving in the form of the capital inflow that finances the trade deficit. (See the postscript for a further explanation of this.
Therefore, I repeat, these three variables—the investment saving imbalance, the government spending-taxing imbalance and the inverse of the export-import imbalance are linked together (they total zero) and are mutually determined. Other things equal, the reduction in the trade deficit due to fracking will reduce imported capital and put pressure on investment relative to saving and government spending relative to taxing. At least some of the correction is likely to lead to a smaller budget deficit.
P.S. In a closed economy with no government, income will adjust to make saving and investment equal in equilibrium. Introducing, government spending and taxing, the two injections into the income stream (other than consumption) will be investment and government spending while the two leakages will be saving and taxing. Therefore, the sum of the injections will equal the sum of the leakages in equilibrium, although there is no requirement for a separate balance of taxing and spending and saving and investment. Introducing foreign trade, exports become a third injection while imports become a third leakage. In equilibrium, investment plus government spending plus exports will equal saving plus taxes plus imports. In our recent past, the excess of government spending over taxes requires a net capital inflow (to finance the excess of imports over exports) to finance the excess of domestic investment over saving. If fracking reduces the excess of imports over exports the other two imbalances must adjust, thus putting downward pressure on the budget deficit.