Heterogeneity and anisotropy of fluid-bearing rocks remain a challenge of central importance in the quantification of subsurface energy resources. Borehole geophysical measurements are the principal source of data used to quantify in situ rock properties, but they are often riddled with uncertainty resulting from the combined effects of rock heterogeneity/anisotropy and mud-filtrate invasion. This uncertainty often persists in geological formations deemed homogeneous. Accurate interpretation of borehole geophysical measurements requires modeling of multiphase flow resulting from invasion of mud filtrate into porous and permeable rocks. For the specific case of spatially complex rocks, there is a need for experimental and numerical methods that integrate all pertinent information about the interactions between fluids (including mud filtrate) and rocks to develop realistic models of fractional flow, i.e., saturation-dependent relative permeability and capillary pressure in the near-wellbore region. This dissertation combines new laboratory measurements with numerical simulations to estimate fluid-transport properties of spatially complex rocks subjected to two-phase immiscible fluid displacement. At the heart of these experimental procedures is a new high-resolution imaging technique based on X-ray radiography that uses a microfocus computed tomography scanner and thin rectangular rock samples to (a) capture and quantify fluid displacement patterns, (b) provide time-lapse images of fluid distribution, and (c) visualize external and internal mudcake deposition. The work also includes the development of a new method to appraise the quality of nuclear magnetic resonance measurements used to quantify rock-pore structure. Injection experiments were performed to study the impact of connate fluid properties, drilling fluid properties, and rock petrophysical properties on two-phase flow taking place during mud-filtrate invasion, injection, or production. Experimental and numerical results indicate that the spatial distribution of fluids (both connate fluids and mud filtrate), flow patterns, and mudcake deposition resulting from mud-filtrate invasion depend heavily on the nature and degree of rock heterogeneity, bedding plane orientation, and anisotropy during both drainage and imbibition. In rocks considered homogeneous, fluid displacements approach piston-like behavior, as predicted by the Buckley-Leverett theory of fractional flow, while in spatially complex rocks, high-resolution time-lapse images uncover preferential flow paths along high-permeability sections of the rock, hence giving rise to low sweep efficiency. At late experimental times, both the spatial distribution of fluids and sweep efficiency are significantly influenced by variations in capillary pressure and transmissibility across the rock sample. Laboratory experiments also emphasize the impact of viscous and/or capillary forces on two-phase flow behavior during mud-filtrate invasion. It is found that mud properties dominantly control both invasion rate and mudcake thickness growth, independently of rock properties. The new hybrid laboratory-simulation approach is effective for examining the time evolution of fractional flow, as it offers an alternative to the laborious and time-consuming traditional steady-state laboratory methods used for measuring relative permeability and capillary pressure. Furthermore, the new laboratory methods introduced in this dissertation are fast and reliable to simultaneously assess flow-related petrophysical properties of spatially complex rocks and to examine competing fluid displacement mechanisms. Overall, the combination of experimental and numerical results improves our understanding of two-phase immiscible flow in heterogeneous rocks and of the various effects that mud-filtrate invasion can have on borehole geophysical measurements during or after drilling operations. This new approach foreshadows new interpretation methods for production-oriented formation evaluation.