A rich variety of spatiotemporal patterns exists on reactive rings having a nonuniform activity. Simulations of a case with a nonuniformity indicate that when the local phase-plane features change along the ring, novel motions evolve which cannot form on a uniformly active ring. An example is a rotating pulse which can move only in one direction but not in the other. The nonuniformity increases the number of possible patterns which can exist on a specific ring relative to that existing on a uniformly active one. Thus the nonuniformity increases the sensitivity of the observed patterns to changes in the initial conditions or perturbations.

Analytical multi-linear models are popular in the well testing analysis of multi-stage fractured horizontal wells (MFHWs). But such models are challenged now. In high density network of hydraulic fractures, the fluid flow around fracture tips no longer behaves like linear flow. Moreover, the production behaviors of hydraulic fractures are different at different stages in heterogeneous reservoirs. To overcome the two challenges, this study develops a semi-analytical composite model for MFHWs in heterogeneous reservoirs. The stimulated reservoir is regarded as a combination of several closed sub-systems with hydraulic fractures centered or uncentered. Each sub-system is further divided into several regions based on reservoir heterogeneity and flow characteristics. Reservoir and hydraulic fracture properties can be different in those regions. The production from each region is modeled by the solutions of linear flow equations or radial flow equations or source/sink functions. The production from a sub-system is modeled by coupling the solutions of regions in the sub-system. The production from the whole reservoir is then modeled by coupling the solutions of all sub-systems. The composite model was validated by analytical solutions in homogeneous reservoirs and numerical solutions in heterogeneous reservoirs. Calculations of production pressure and rates with this composite model generate type curves for the pressure and rate transient analysis of MFHWs. Two field cases are further analyzed with the type curves. One novelty of this model is to consider the non-linear flow around hydraulic fracture tips. Therefore, our model is better prepared for incorporating pressure-dependent heterogeneous reservoir properties in the future. The other novelty of this study is the ability to model the different production behavior at different fracture stages in complex heterogeneous reservoirs.

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This article presents the intrapair and interpair electron correlation energies of LiF . The 2 total interpair correlation energy predominates. Hence, in building models for correcting post-Hartree᎐Fock calculations to obtain accurate electron correlation energies, the interpair electron correlation energy must be considered. Based on this result and our previous results, a fundamental assumption underlying the so-called higher-level correction in the Pople G1 the G2 theory is flawed.