Mathematical model of drilling mud movement along a shale shaker screen

Introduction. When drilling oil or gas wells, rock from the bottomhole is brought to the surface by drilling mud, which is cleaned of sludge by shale shakers at the first stage. Shale shakers are equipped with such a screen and create such trajectories of vibration of the frame in order to solve a dual problem: on the one hand, to provide the cleaning of the drilling mud coming from the well mouth, and on the other hand - to maintain the proper degree of cleaning. To correctly select the nomenclature of screens, it is necessary to reliably determine the throughput capacity of a shale shaker in the real-time setting. This will allow you to pre-order the required number of screens with the required cell size. Previously, studies were conducted by sieving a mixture of calibrated granules on a standard shaker or by straining mineral oil through a fixed screen. However, this does not fully correspond to the actual conditions of drilling mud screening. The objective of this article was to simulate the throughput of a drilling shale shaker under real conditions by calculating the movement of drilling mud along a vibrating screen using its previously obtained specific throughput capacity.Materials and Methods. When creating a mathematical model of the full throughput capacity of a shale shaker, a previously known mathematical model of fluid flow in an open channel and a finite-dimensional approximation in the form of a sequence of sections of concentrated containers connected by hydraulic conductivities (by analogy with the approximation of long electrical lines and extended gas pipelines by the finite element method) were used. The fluid flow rate over the screen was determined according to Chézy law. In this case, the cross-sectional area of the flow above the screen for a specific section was specified as the arithmetic mean between the initial and final values of the drilling mud height in the corresponding section.Results. A chain calculation scheme for sieving drilling mud was compiled. Based on the material balance, mathematical models were developed for determining the throughput and height of drilling mud above the screen: for the first section of the screen and the first nodal point of the screen; for subsequent screen sections and subsequent screen nodal points; for the last section of the screen and the last nodal point of the screen.A mathematical model of the drilling mud movement along the shale shaker screen has been obtained, which, due to the finite-dimensional representation of this flow along n sections of the screen length, is a system of n integral equations describing concentrated tanks, and n - 1 algebraic equations describing the flow of drilling mud between tanks.Discussion and Conclusion. The movement of drilling mud along the length of the shale shaker screen corresponds to the movement of a steady uneven flow in an open channel. However, in the first case, the volume flow along the length is variable (decreasing to zero), and in the second case, it is constant along the length. Therefore, the use of only one differential equation for steady uneven flow in an open channel is not sufficient in this case. The resulting mathematical model of the drilling mud movement along the shale shaker screen is a mathematical basis for the development of a modeling program to determine the throughput of the shale shaker for real operating conditions, i.e., for a specific drilling mud, a given screen or a given drive system of the shale shaker.