The ability to design effective experiments is crucial for obtaining data that can substantially reduce the uncertainty in the predictions made using computational models. An optimal experimental design (OED) refers to the choice of a particular experiment that optimizes a particular design criteria, e.g., maximizing a utility function, which measures the information content of the data. However, traditional approaches for optimal experimental design typically require solving a large number of computationally intensive inverse problems to find the data that maximizes the utility function. Here, we introduce two novel OED criteria that are specifically crafted for the data consistent inversion (DCI) framework, but do not require solving inverse problems. DCI is a specific approach for solving a class of stochastic inverse problems by constructing a pullback measure on uncertain parameters from an observed probability measure on the outputs of a quantity of interest (QoI) map. While expected information gain (EIG) has been used for both DCI and Bayesian based OED, the characteristics and properties of DCI solutions differ from those of solutions to Bayesian inverse problems which should be reflected in the OED criteria. The new design criteria developed in this study, called the expected scaling effect and the expected skewness effect, leverage the geometric structure of pre-images associated with observable data sets, allowing for an intuitive and computationally efficient approach to OED. These criteria utilize singular value computations derived from sampled and approximated Jacobians of the experimental designs. We present both simultaneous and sequential (greedy) formulations of OED based on these innovative criteria. Numerical results demonstrate the effectiveness in our approach for solving stochastic inverse problems.