Flux balance analysis production rate

The Geilikman et al. model relates yield front velocity to the volumetric rate of solids production by mass balance. A front between yielded and intact zones moves radially away from the wellbore as solids production continues. Mass balance analysis allows determination of the position of the moving front. Fluid production enhancement is dependent on the instantaneous solids flux, as well as the radius of the yielded zone around the wellbore (i.e., the solids production history). Viscous resistance to oil flow is reduced when the granular matrix is also flowing. The volumetric fluid production, Qf(t), is given as... 

The reverse of reaction (3.44) has no effect until the system has equilibrated, at which point the two coefficients d In Yco/d In and d In Yco/d In f44b are equal in magnitude and opposite in sense. At equilibrium, these reactions are microscopically balanced, and therefore the net effect of perturbing both rate constants simultaneously and equally is zero. However, a perturbation of the ratio (A 44f/A 44b = K44) has the largest effect of any parameter on the CO equilibrium concentration. A similar analysis shows reactions (3.17) and (3.20) to become balanced shortly after the induction period. A reaction flux (rate-of-production) analysis would reveal the same trends. 

The starting point in this analysis is the construction of a list of steady-state material balance equations to describe the conversion of substrates to metabolic products for the biochemical system of interest. For example, if one considers a simplified scheme of amino acid metabolism in liver, one can write a set of steady-state material balance equations that represent the flow of metabolites through the network (Fig. 9). The equations contain measurable quantities (these are marked with an asterisk) which are the rates of consumption/production of extracellular metabolites. The concentrations of strictly intracellular metabolites (e.g., argininosuccinate) are assumed to be constant. In this particular case, we have eight fluxes to be determined, five of which are measurable (F[, F, F, F, F ). The five equations listed here, which relate these fluxes to each other, can be reduced to four independent equations. Thus, the system can be solved to yield the three unknown intracellular fluxes (Fi, F2, F4). Because the system is overdetermined, it provides an internal check for consistency of the data with each other and the assumed biochemistry. 

In spite of the wide variety of interfacial media encountered, it is possible in many cases to obtain surface balance equations, which are representative and unrestrictive in comparison to actual phenomena. The constitutive laws can be established by dimensional analysis. They relate to surface variables, to flux and rates of production. The phenomenological coefficients can be determined experimentally, or by detailed analysis over the interfacial thickness. The solution implies a coupling between these equations and those for volumes in contact. In the examples presented here, the results are in perfect agreement with those of simple classical theory. 

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