ABSTRACT
CONTENTS Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 688 17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689 17.2 Fluid Flow Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693
17.2.1 Multiphase Fluid Systems in Porous Media . . . . . . . . . . . . . . . . . . . . 693 17.2.2 Single-Phase Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694 17.2.3 Multiphase Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695
17.3 Introduction to Genetic Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 699 17.3.1 What are Genetic Algorithms? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 699 17.3.2 Advantages of Using Genetic Algorithms . . . . . . . . . . . . . . . . . . . . . . 700 17.3.3 Components of a Genetic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702
17.3.3.1 Binary encoding of individuals . . . . . . . . . . . . . . . . . . . . . . . . 702 17.3.3.2 Selection operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702 17.3.3.3 Crossover operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703 17.3.3.4 Mutation operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704 17.3.3.5 Further genetic algorithm techniques . . . . . . . . . . . . . . . . . 704
17.3.4 A Theoretical Basis for the Genetic AlgorithmApproach. . . . . 705 17.3.5 Details and Example of the Genetic Algorithm Technique
Employed in this Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706 17.4 Parameter Identification within Rock Samples . . . . . . . . . . . . . . . . . . . . . . . . . 710
17.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 710 17.4.2 Experimental Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713 17.4.3 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714
17.4.3.1 The governing equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714 17.4.3.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715 17.4.3.3 Direct solution for the homogeneous case. . . . . . . . . . . . 716
17.4.4 Inverse Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716 17.4.5 Sensitivity Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717
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17.4.6 The Homogeneous Hydraulic Conductivity Case, k(x) = k0 = constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 718
17.4.7 Linear Hydraulic Conductivity, k(x) = k0 + k1x . . . . . . . . . . . . . . . . 719 17.4.8 Quadratic Hydraulic Conductivity, k(x) = k0 + k1x + k2x2 . . . 719 17.4.9 Simultaneous Retrieval of Constant Hydraulic
Conductivity and Constant Specific Storage. . . . . . . . . . . . . . . . . . . . 719 17.5 Hydraulic Conductivity Measurements in Composite
Homogeneous Rocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 720 17.5.1 Experimental Setup and Limitations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 721 17.5.2 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 721 17.5.3 Steady-State Mathematical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 722 17.5.4 The Composite Homogeneous Situation . . . . . . . . . . . . . . . . . . . . . . . . 723 17.5.5 Retrieval of the Hydraulic Conductivities . . . . . . . . . . . . . . . . . . . . . . 723 17.5.6 Simultaneous Retrieval of the Compressive Storage
of the Upstream Reservoir and the Specific Storages of the Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724
17.5.7 Comparison with Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725 17.6 Hydraulic Conductivity Measurements in
Anisotropic Rocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726 17.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726 17.6.2 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727 17.6.3 Inverse Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 729 17.6.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 730
17.7 Hydraulic Conductivity Measurements in Composite Anisotropic Rocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 730 17.7.1 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 730 17.7.2 Inverse Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 731 17.7.3 Numerical Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 731
17.7.3.1 The case when the fault is perpendicular to the x-axis of the sample, xft = xfb . . . . . . . . . . . . . . . . . . . . 731
17.7.3.2 The case when the fault is inclined to the x-axis of the sample, xft = xfb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734
17.8 Comparison of Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734 17.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737
This chapter is concerned with the inverse problem of the identification of the hydraulic properties of porous materials in the context of petroleum, civil, and mining engineering. The applicability of the novel technique of genetic algorithms (GAs), which attempt to imitate the principles of biological evolution in the construction of optimization strategies and have led to
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the development of a powerful and efficient optimization tool, is studied for such purposes. First, we model a one-dimensional hydraulic pump-flow permeability test
and formulate an inversion technique in order to retrieve homogeneous or spacewise dependent material property coefficients, with the extension in the homogeneous situation to the case of layered materials. The direct problem is solved using the finite-difference method (FDM) while the recovery of the material parameters is achieved through a GA approach. Both exact and simulated noisy data are incorporated at optimally selected instants in time through the test, the data measurements used being consistent with a sensitivity analysis of each problem. Second, both steady-state and transient experiments are modeled for the case of anisotropic materials. The direct solution procedures are based on the boundary element method for the steady-state situation while the FDM is chosen for the transient case. Surface measurements, by means of simulated ports along the sealed boundaries of the materials, serve as information to the GA-based optimization procedure, hence enabling a modified least-squares functional to minimize the difference between the observed and the numerically predicted boundary pressure and/or average hydraulic flux measurements under current hydraulic conductivity tensor and specific storage estimates. Composite anisotropicmaterials, that is, with the incorporation of faults, are also studied. Parameter identifiability in inverse problems is numerically investigated
and the results are found to provide an accurate means of recovering the required material properties. A comparison on the performance of the inversion highlights the advantages that a GA-based optimization approach offers in comparison to a traditional gradient-based optimization procedure.
