The idea to treat fossil fuel as capital is found in the first chapter of the book "Small is Beautiful: (A Study of) Economics as if People Mattered" (1973) by E. F. Schumacher. Is there recent research following this idea?
There is a large literature considering the depletion of natural resources, including fossil fuels, in relation to GDP. Some writers explicitly refer to such resources as a kind of natural capital. Others may refer to natural resource assets, or simply to natural resources. A key question within this literature is whether depletion of natural resources should be recognised via a deduction within net domestic product (NDP), in (broadly) the same way that a deduction is made for depreciation of man-made capital. The range of views is wide. Some examples:
Repetto et al (1989) attempted to calculate Net Domestic Income (NDI) for Indonesia for 1971-84, taking account of resource depletion, and found that the differences from Gross Domestic Income (GDI) were very large (p 8). Various resources were considered including petroleum (pp 37ff). It was argued that, for resource-dependent countries, not applying the depreciation concept to the stock of natural resources was "a major omission and inconsistency" (p 4)
Nature's Numbers (1999), a report sponsored by the US Bureau of Economic Analysis, devoted a large section (pp 59-105) to "Accounting for Subsoil Mineral Resources" (including fossil fuels). It recognised that conventional national accounts treat natural capital in a way which is inconsistent with that of man-made capital (pp 25-6), but stopped short of recommending a deduction for depletion.
Natural Resource and Environmental Economics, a textbook by Perman et al, gives a useful overview of the issues (in the 3rd edition I have it's Chapter 19, especially section 19.2.4 on measuring national income). It identifies several different methods that have been proposed for estimating the value of resource depletion. In their concluding remarks (section 19.5), the authors express scepticism as to whether concepts of "adjusted GDP" are worthwhile, mainly because of the difficulty of determining the appropriate valuation basis.
The World Bank has published estimates of the percentage by which world Gross 'National' Income (GNI) would be reduced if allowance were made for depletion of energy resources. Over the period 1970-2015, this varies between about 0.1% and 3%. Illustrating the valuation problem, the pattern of change with sharp increases around 1973 and 1979 (times at which OPEC substantially raised the price of oil) suggests that these estimates have been heavily influenced by market prices. Physical depletion of fossil fuel reserves certainly did not change as dramatically as that pattern might suggest (see the chart here, fossil fuel consumption being a reasonable proxy for depletion).