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In this work, we present methodologies for optimization of hydraulic fracturing design under uncertainty specifically with reference to the thick and anisotropic reservoirs in the Lower Tertiary Gulf of Mexico. In this analysis we apply a stochastic programming framework for optimization under uncertainty and apply a utility framework for risk analysis. For a vertical well, we developed a methodology for making the strategic decisions regarding number and dimensions of hydraulic fractures in a high-cost, high-risk offshore development. Uncertainty is associated with the characteristics of the reservoir, the economics of the fracturing cost, and the fracture height growth. The method developed is applicable to vertical wells with multiple, partially penetrating fractures in an anisotropic formation. The method applies the utility framework to account for financial risk. For a horizontal well, we developed a methodology for making the strategic decisions regarding lateral length, number and dimensions of transverse hydraulic fractures in a high-cost, high-risk offshore development, under uncertainty associated with the characteristics of the reservoir. The problem is formulated as a mixed-integer, nonlinear, stochastic program and solved by a tailored Branch and Bound algorithm. The method developed is applicable to partially penetrating horizontal wells with multiple, partially penetrating fractures in an anisotropic formation.

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Geophysical measurements can often be described in terms of multichannel, autoregressive data models from which one can directly derive measures of the harmonic composition of the underlying geophysical process and its inherent self-predictability. / We explore methods for and uses of multichannel autoregressive data modelling in a geophysical context. / Autoregressive data modelling using the least-squares linear prediction method is generalized to multichannel time series. A recursive algorithm is obtained for the formation of the system of multichannel normal equations which determine the least-squares solution of the multichannel linear prediction problem. Solution of these multichannel normal equations is accomplished by the Cholesky factorization method. / The corresponding multichannel Maximum Entropy spectra derived from these least-squares estimates of the autoregressive model parameters are compared to that obtained using those parameters estimated by a multichannel generalization of Burg's algorithm. Numerical experiments have shown that the multichannel spectra obtained using the least-squares method provides for more accurate frequency determination for truncated sinusoids in the presence of additive white noise.

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