Subsurface characterization is important for the detection and development of underground resources. Useful subsurface models can correctly make production forecast and optimize development plans, such as infill well location in an aquifer, geothermal, and hydrocarbon reservoir development. A useful subsurface model should honor geological concepts as well as all the available measurements, such as fluid production history, geophysical measurements. However, common subsurface modeling methods cannot efficiently honor the two concepts. To build subsurface models in a way that it is easy to condition them to both geological concepts and available measurements, we develop a new machine learning method, referred to as the stochastic pix2pix method. In this method, we use convolutional neural networks and adversarial neural networks to stochastically generate new subsurface models matching both geological concepts and static measurements, such as seismic and well data. This method first extracts the depositional patterns from analog training images, such as outcrops, high-resolution seismic images, and depositional process-based reservoir models, and then minimizes the Jensen-Shannon entropy between the training images and new subsurface models, as well as the mismatch of static measurements. The hydraulic inverse problem is solved with a machine learning-based proxy model on the model parameter space defined by stochastic pix2pix. The stochastic pix2pix method helps maintain the match of geological concepts and static measurements during the inversion. To verify and benchmark our procedure, we show the conditional subsurface models generated with stochastic pix2pix reproduce the geological concepts as good as synthetic unconditional process-based models. we successfully build reservoir models for channel and turbidite fan systems, where the depositional patterns of common geobodies are well reproduced. The synthetic well data, seismic interpretation, net-to-gross ratio, and time records of fluid production are well-matched with this new method. Additionally, we generate conditional subsurface models 90% faster than with conventional object-based modeling methods and with more accurate reproductions of the available measurements.