We present an efficient adjoint model based on the deep-learning surrogate to address high-dimensional inverse modeling with an application to subsurface transport. The proposed method provides a completely code nonintrusive and computationally feasible way to approximate the model derivatives, which subsequently can be used to derive gradients for inverse modeling. This conceptual deep-learning framework, that is, an architecture of deep convolutional neural network through combining autoencoder and autoregressive structure, efficiently produces an analogously analytical adjoint with the help of auto-differentiation module in the popular deep-learning packages. We intentionally retain training data at the specific time instances where the measurements are taken, the storage of the intermediate states and computation of their adjoint, therefore, are completely avoided. This proposed adjoint state method is tested on a synthetic two-dimensional model for parameter estimations. The preliminary results reveal the feasibility of the proposed adjoint state method in term of computational efficiency and programming flexibility.

Shale gas reservoirs (SGR) have been a central supply of carbon hydrogen energy consumption and hence widely produced with the assistance of advanced hydraulic fracturing technologies. On the one hand, due to the inherent ultralow permeability and porosity, there is stress sensitivity in the reservoirs generally. On the other hand, hydraulic fractures and the stimulated reservoir volume (SRV) generated by the massive hydraulic fracturing operation have contrast properties with the original reservoirs. These two phenomena pose huge challenges in SGR transient pressure analysis. Limited works have been done to take the stress sensitivity and spatially varying permeability of the SRV zone into consideration simultaneously. This paper first idealizes the SGR to be four linear composite regions. What is more, the SRV zone is further divided into subsections on the basis of nonuniform distribution of proppant within the SRV zone which easily yields spatially varying permeability away from the main hydraulic fracture. By means of perturbation transformation and Laplace transformation, an analytical multilinear flow model (MLFM) is obtained and validated as a comparison with the previous models. The flow regimes are identified, and the sensitivity analysis of critical parameters is conducted to further understand the transient pressure behaviors. The research results provided by this work are of significance for an effective recovery of SGR resources.