We introduce an inversion approach for determining the water-based mud-filtrate invasion profile, as well as the formation porosity and horizontal permeability, from the induction logging data. The inversion is constrained by a multiphase fluid flow simulator that simulates the mud-filtrate invasion process to obtain the spatial distributions of the water saturation and the salt concentration, which are in turn transformed into the formation resistivity using a resistivity-saturation formula. By ignoring the diffusion effect, we assume that the mud-filtrate invasion process is mainly convective so that it can be equivalently simulated by providing an average invasion rate and the duration of invasion. The average invasion rate can be directly inverted from the fluid-flow-constrained inversion of induction logging data. We also obtain the mud-filtrate invasion profile, which is consistent with the fluid flow physics. The reconstructed mud-filtrate invasion profile benefits the interpretation of the formation test. When the pressure transient data are available, this approach can be also used to jointly invert both induction logging data and pressure transient data to obtain the mud-filtrate invasion profile, as well as a parametric distribution of the TI-anisotropic formation permeability and porosity. Assuming a vertical well penetrating horizontal formations, the fluid flow problem is solved using an implicit black oil finite-difference simulator with brine tracking option based on a cylindrical, axially symmetric grid, whereas the response of the induction logging tool is simulated using a frequency-domain finite-difference solver based on a Cartesian grid. A Gauss-Newton inversion scheme using the multiplicative regularization technique is used for either the fluid-flow-constrained inversion or the joint inversion. The reliability of the inversion results depends on the accuracy of a priori knowledge of the reservoir, which needs to be confirmed via sensitivity analysis.