Accuracy analysis of solving two-phase filtration problem using neural network modeling methods

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When designing the development of oil and gas fields, the use of mathematical modeling methods is required to select the optimal option for field development. One of the key tasks is to simulate the dynamics of flooding producing wells with an operating reservoir pressure maintenance system. The target modeling parameters include the time of water breakthrough into the producing well and the amount of product flooding at the breakthrough time. Practically, such calculations are performed on 3D hydrodynamic simulators based on the numerical solution of differential filtration equations. The accuracy of numerical modeling in this case largely depends on the quality of constructing the grid calculation area, while the quality significantly depends on the shape of the calculation area. The increase in the size of grid blocks, typical for hydrodynamic modeling, has a negative effect on the calculation accuracy. One of the alternative methods of modeling physical processes is neural network modeling. A recent widespread method is physically informed neural networks capable of approximating exact solutions of differential equations with high accuracy. The key feature of this approach is organizing neural network learning process both on precise initial and boundary values, as well as on the execution of predefined systems of differential and algebraic equations. In this case, a trained neural network can calculate desired values at any point in the definition area, and not only at the nodal points when using finite difference solutions. Thus, this works aims to develop neural network methods for calculating phase saturation in large-scale modeling of two-phase filtration; and to evaluate the accuracy of the solutions obtained. The article provides a comparative analysis of solutions to the one-dimensional Buckley–Leverett problem. Three methods of obtaining solutions are considered: an exact analytical solution; a numerical solution obtained by finite difference methods; and a grid-less neural network approximation based on a multilayer perceptron model. The upwind method was used as a finite difference method, which provides the most accurate reproduction of saturation changes. A multilayer perceptron with two types of activation functions was used as a physically informed neural network. An additional condition, which corresponds to the law of conservation of mass at the jump, needed to be included in the error functional in order to determine the position of the saturation jump. The results showed that after training, the neural network solution can reproduce the evolution of rarefaction waves and saturation jumps with high accuracy. The nodes density of the training sample can be reduced without significantly reducing the accuracy of the neural network approximation. The results obtained can be used in the development of hybrid algorithms for modeling oil displacement processes.