Uncertainty is one of the important topics in Natural Resource Economics that is under investigation. The purpose of this dissertation is to develop a Markovian depletion model for petroleum resource where the uncertainty is the geologic uncertainty. The analysis first derives the mathematical formulation for the expected net present value of a petroleum reserve expressed in monetary terms for an exploration in an unknown district. The computation uses a Markov process to quantify the uncertainty and the methodology is the spectral representation of a generator matrix. The price, wage and rental rates are assumed to be constant. It is also assumed that the firm that does the exploration is the firm that will produce the petroleum. The depletion model uses the Markovian exploration model as a constraint and computes the maximum discounted profit relative to the wage rate, using an optimum control method. The state variables are the net reserves and the total cumulative reserve additions per worker while the control variables are the capital labor ratios and the labor per capita ratio. A Cobb Douglas production function with constant returns to scale is assumed for the production of petroleum; the production and development cost is a linear function of capital and labor while the exploration cost is also dependent on the Markov process. The analysis indicates that under optimum production and when the stock of the resource is augmented by exploration, the change in the value of the deposit yields an interest rate on the scarcity rent equal to r, and the price of petroleum is equal to the marginal cost of production plus the scarcity rent. Also, the value of the capital used in exploration is proportional to the value of the petroleum produced. The analysis also shows that in general the Markovian model is a better representation for an exploration decision process than a decision tree process, and that the Markovian exploration model can be used in a three sector growth model. Finally, the Markovian exploration model may be used for any other mineral resource by modifying the exploration network.