By Dean S. Oliver, Albert C. Reynolds, Ning Liu
This ebook is a advisor to using inverse concept for estimation and conditional simulation of move and delivery parameters in porous media. It describes the idea and perform of estimating homes of underground petroleum reservoirs from measurements of stream in wells, and it explains tips on how to signify the uncertainty in such estimates. Early chapters current the reader with the required history in inverse idea, chance and spatial information. The booklet demonstrates how one can calculate sensitivity coefficients and the linearized dating among types and construction information. It additionally exhibits how one 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 conception in petroleum engineering. It comprises many labored examples to illustrate the methodologies and a range of routines.
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Extra resources for Inverse Theory for Petroleum Reservoir Characterization and History Matching
Example text
Since there is no information available to distinguish between the individual values of the first three components of m, it is reasonable that the first three components of mest are equal. Subtracting the “first equation" of Eq. 35 and the estimate of Eq. 175. 1. The result suggests that the choice of m0 had no effect on the estimate, which is true for the specific example considered. 66) in which case, Eq. e. the m closest to m0 that satisfies Eq. 50) does not depend on the specific values of the components of m0 .
The initial condition is imposed on the finite-difference problem by requiring that pi0 = pin . 60) In general, the solution p(x, t) of the initial boundary-value problem specified by Eqs. 54) will not satisfy the finite-difference system, Eqs. 59), exactly because of the approximations we have used in deriving the finite-difference equations, for example in approximating partial derivatives by difference quotients. The expectation is that the solution, pin (i = 1, 2, . . , N, n = 1, 2, . ), of Eqs.
2 Solutions of discrete linear inverse problems Inserting this result into Eq. 54) gives m = m0 + GT (GGT )−1 (dobs − Gm0 ). 57) Note that the estimate of m in Eq. 57) is a linear combination of the columns of GT , which suggests that the columns of GT provide useful basis vectors for construction of estimates. 3. 3]T . 50, respectively. 30. 58) where G= 1 1 1 0 0 . 2 T . 63) and so Eq. 1. The inverse of permeability estimated from two inaccurate pressure measurements at x = 3 and 5. The estimate is required to honor the data exactly.