Fluid displacement in porous media can usually be formulated as a Riemann problem. Finding the solution to such a problem helps shed light on the dynamics of flow and consequently optimize operational parameters such as injected fluid composition. We developed an algorithm to find solutions to a class of Riemann problems of multiphase flow in porous media. In general, the solution to a Riemann problem in state space is a curve connecting the left and right states of the problem. The solution curves studied here are composed of classical wave curves. For a given Riemann problem, our procedure to find the solution consists of three steps: (1) guess the initial lengths of the solution's constitutive wave curves; (2) construct each wave curve off the last state of its antecedent wave curve; and (3) iterate over the lengths of the constitutive wave curves, using an iterative solver, until the solution curve ends at the right state of the problem. We used benchmark cases from literature to verify the accuracy of the developed algorithm. Using the developed algorithm, we found solutions to some challenging cases where otherwise numerical simulators would be needed to find the type of the involved waves (i.e., rarefaction, shock or composite waves) and the coordinates of the middle states in the state space. Saturation profiles, oil cut and oil recovery for all the studied cases were computed. This information will assist us to: gain insight about the dynamics of flow, interpret core flooding measurements, assess the accuracy of developed models for foam physical properties, and verify the results of numerical simulators.