Inverse Theory for Petroleum Reservoir Characterization and by Dean S. Oliver, Albert C. Reynolds, Ning Liu

By Dean S. Oliver, Albert C. Reynolds, Ning Liu

This e-book is a advisor to using inverse conception for estimation and conditional simulation of move and delivery parameters in porous media. It describes the speculation and perform of estimating houses of underground petroleum reservoirs from measurements of move in wells, and it explains how one can signify the uncertainty in such estimates. Early chapters current the reader with the required historical past in inverse thought, chance and spatial records. The publication demonstrates the right way to calculate sensitivity coefficients and the linearized courting among versions and construction facts. It additionally exhibits how you can boost iterative tools for producing estimates and conditional realizations. The textual content is written for researchers and graduates in petroleum engineering and groundwater hydrology and will be used as a textbook for complex classes on inverse thought in petroleum engineering. It contains many labored examples to illustrate the methodologies and a range of routines.

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9) Note that for any M-dimensional vector x   x1 M  x2    Gx = [g1 g2 . . gM ]  .  = xj gj .  .. 10) j =1 xM Then, it is clear that every vector in the range of G is a linear combination of the columns of G. It can be shown that if the columns of G are linearly independent then there are no nontrivial solutions to Gm = 0 and vice versa. If the number of linearly independent columns of G is equal to Nd , then there is at least one vector m which satisfies Gm = dobs . On the other hand, if M < Nd , then there exist data vectors dobs such that the equation Gm = dobs has no solution.

P ··· ··· 0 = Up .. .. .. .. ··· ..

E. the three five-dimensional column vectors, x1 = [1, −1, 0, 0, 0]T , x2 = [0, 0, 1, 0, −1]T and x3 = [0, 0, 0, 1, −1]T , form a basis for the null space of G. If d is in the range of G, then there exist m in S(m) such that d6 = β(m1 + m2 + · · · + m5 ) and di = β(m1 + m2 ) for i = 1, 2, . . e. d1 = d2 = · · · = d5 . Any such d can be written as d = d1 [1, 1, 1, 1, 1, 0]T + d6 [0, 0, 0, 0, 0, 1]T . 26) Thus, the two vectors y1 = [1, 1, 1, 1, 1, 0]T and y2 = [0, 0, 0, 0, 0, 1]T form a basis for the range of G.

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