Dimensionless Solutions for the Time-Dependent and Rate-Dependent Productivity Index of Wells in Deformable Reservoirs

Summary Reservoir depletion is known to reduce the porosity and permeability of stress-sensitive reservoir rocks. The effect may substantially hinder the productivity index (PI) of producing wells. This study presents analytical solutions for the time-dependent and steady-state well PIs, respectively, of a bounded, disk-shaped, elastic reservoir with no-flow and constant-pressure conditions at the outer boundary. A combination of Green's functions, the Laplace transform method, and the perturbation technique is used to solve the governing nonlinear partial differential equations of the considered coupled problems of flow and geomechanics. Dimensional analyses based on the Buckingham Π theorem are conducted to identify the dimensionless parameters groups of each problem and to express the resulting analytical solutions in the dimensionless form. In addition, necessary corrections to an existing error in the reported Green's functions for the induced strain field of a ring-shaped pressure source within an elastic half-space (Segall 1992) are made. The corrected Green's functions are used to obtain the strain induced by the pore fluid pressure distribution within a depleting disked-shaped reservoir. Consequently, a corrected permeability variation model compared to our previously published, time-independent solution for rate-dependent PI (Zhang and Mehrabian 2021a) is presented. Finally, a mechanistically rigorous formulation of the permeability modulus parameter that commonly appears in the pertinent literature is suggested. In addition to the in-house developed finite-difference solutions, the presented analytical solutions are verified against results from the finite-element simulation of the same problems using COMSOL® Multiphysics (2018). The obtained rate-dependent PI of the reservoir is controlled by four dimensionless parameters, namely, the dimensionless rock bulk modulus, the Biot-Willis effective stress coefficient, Poisson's ratio, and rock initial porosity. The pore fluid pressure solution is shown to asymptotically approach the corresponding flow-only solution for large values of the dimensionless rock bulk modulus. Parametric analysis of the solution suggests that the well productivity loss has a reverse relationship with the dimensionless bulk modulus and initial porosity of the rock, whereas a direct relationship is identified with Biot-Willis effective stress coefficient and Poisson's ratio. Compared to the reservoir with a constant-pressure outer boundary, the PI of a reservoir with a no-flow condition at the outer boundary is shown to be more significantly hindered by the stress sensitivity of the reservoir rock.

A Dynamic Intersecting Arrangement Model Based on Isolated Draw Zones for Stope Structure Optimization during Sublevel Caving Mining

In this paper, stope structure optimization during sublevel caving mining is considered under the condition that the isolated draw zones (IDZs) are nonstandard ellipsoid, which is realized by dynamically adjusting the arrangement of IDZs and quantifying the degree of intersection of IDZs according to an ore profit and loss calculation model. A dynamic intersecting arrangement model based on IDZs was proposed, which can dynamically adjust the sublevel height and drift spacing according to the ore-rock bulk flow parameters, economic indicators, occurrence condition of the ore body, drilling machine, and so forth. Based on the model, the range of drift spacing, the lower volume of crestal residual ore, and the higher volume of mixing waste rock are calculated. By deducing the function of ore profit and loss, a calculation model for ore profit and loss is established to quantify the degree of intersection of IDZs and determine the best stope structure. Using the constructed dynamic intersecting arrangement model, a stope structure of −213 m to −303 m in the Yanqianshan Iron Mine was designed, with a sublevel height of 22.5 m and a drift spacing of 20.5 m. A physical drawing model was designed, and three physical simulation experiment schemes were conducted to compare and analyse the ore loss and dilution of the intersecting arrangement model and the traditional tangent arrangement model. The results showed that the loss rate decreased by 3.66% and the dilution rate increased by only 0.22%, thus verifying the effectiveness and applicability of the model to optimize the stope structure.

Sensitivity Analysis of Slope Stability Influence Factors Based on BP Neural Network

There are many factors which influence the slope stability. In order to analyze the degree of importance of each influence factor on slope stability, this paper establishes a slope stability analysis model based on BP neural network. The computation results showed that the model was reasonable and reliable. On this basis, the sensitivity of various influence factors to slope stability was analyzed by single-factor test, which were internal friction angle of rock, bulk density, pore pressure coefficient, slope angle, rock cohesion and slope height in a descending order of sensitivity.

An approximation for the Xu‐White velocity model

In 1995, S. Xu and R. E. White described a method for estimating compressional and shear‐wave velocities of shaley sandstones from porosity and shale content. Their model was able to predict the effect of increasing clay content on compressional‐wave velocity observed in laboratory measurements. A key step in the Xu‐White method estimates dry rock bulk and shear moduli for the sand/shale mixture. This step is performed numerically by applying the differential effective medium method to the Kuster‐Toksöz equations for ellipsoidal pores. This step is computationally intensive. Using reasonable assumptions about dry rock elastic properties, this step can be replaced with a set of approximations for dry rock bulk and shear moduli. Numerical experiments show an extremely close match between velocities obtained with these approximations and velocities computed with the differential effective medium method. These approximations simplify the application of the Xu‐White method, and make the method computationally more efficient. They also provide insight into the Xu‐White method. For example, these approximations show how the Xu‐White model is related to the critical porosity model.