A multiscale method for subsurface inverse modeling: Single-phase transient flow

High-resolution geologic models that incorporate observed state data are expected to effectively enhance the reliability of reservoir performance prediction. One of the major challenges faced is how to solve the large-scale inverse modeling problem, i.e., to infer high-resolution models from the given observations of state variables that are related to the model parameters according to some known physical rules, e.g., the flow and transport partial differential equations. There are typically two difficulties, one is the high-dimensional problem and the other is the inverse problem. A multiscale inverse method is presented in this work to attack these problems with the aid of a gradient-based optimization algorithm. In this method, the model responses (i.e., the simulated state data) can be efficiently computed from the high-resolution model using the multiscale finite-volume method. The mismatch between the observations and the multiscale solutions is then used to define a proper objective function, and the fine-scale sensitivity coefficients (i.e., the derivatives of the objective function with respect to each node’s attribute) are computed by a multiscale adjoint method for subsequent optimization. The difficult high-dimensional optimization problem is reduced to a one-dimensional one using the gradient-based gradual deformation method. A synthetic single-phase transient flow example problem is employed to illustrate the proposed method. Results demonstrate that the multiscale framework presented is not only computationally efficient but also can generate geologically consistent models. By preserving spatial structure for inverse modeling, the method presented overcomes the artifacts introduced by the multiscale simulation and may enhance the prediction ability of the inverse-conditional realizations generated.