S.D. Harris

Fluid-flow properties of faults in sandstone: The importance of temperature history

Sandstone rheology and deformation style are often controlled by the extent of quartz cementation, which is a function of temperature history. Coupling findings from deformation experiments with a model for quartz cementation provide valuable insights into the controls on fault permeability. Subsiding sedimentary basins often have a transitional depth zone, here referred to as the ductile-to-brittle transition, above which faults do not affect fluid flow or form barriers and below which faults will tend to form conduits. The depth of this transition is partly dependent upon geothermal gradient. In basins with a high geothermal gradient, fault-related conduits can form at shallow depths in high-porosity sandstone. If geothermal gradients are low, and fluid pressures are hydrostatic, fault-related conduits are only formed when the sandstones have subsided much deeper, where their porosity (and hence fluid content) is low. Mineralization of faults is more likely to occur in areas with high geothermal gradients because the rocks still have a high fluid content when fault-related fluid-flow conduits form. The interrelationship between rock rheology and stress conditions is sometimes a more important control on fault permeability than whether the fault is active or inactive.

The optimisation of reaction rate parameters for chemical kinetic modelling of combustion using genetic algorithms

A general inversion procedure for determining the optimum rate coefficients for chemical kinetic schemes based upon limited net species production data is presented. The objective of the optimisation process is to derive rate parameters such that the given net species production rates at various conditions are simultaneously achieved by searching the parameter space of the rate coefficients in the generalised Arrhenius form of the reaction rate mechanisms. Thus, the goal is to both match the given net species production rates and subsequently ensure the accurate prediction of net species production rates over a wide range of conditions. We have retrieved the reaction rate data using an inversion technique whose minimisation process is based on the Darwinian principle of survival of the fittest which has inspired a class of algorithms known as genetic algorithms. The excellent results presented here from our initial study are based upon the recovery of reaction rate coefficients for hydrogen/nitrogen/oxygen flames. The successful identification of the reaction rate parameters which correspond to product species measurement data from a sequence of such experiments clearly suggests that the progression onto other chemical kinetic schemes and the optimisation of higher-order hydrocarbon schemes can now be realised. The results of this study therefore demonstrate that the genetic algorithm inversion process promises the ability to assess combustion behaviour for fuels where the reaction rate coefficients are not known with any confidence and, subsequently, accurately predict emission characteristics, stable species concentrations and flame characterisation. Such predictive capabilities are of paramount importance in a wide variety of industries.

Hydrocarbon flow across faults by capillary leakage revisited

Abstract Hydrocarbon fluid pressures can equilibrate across faults provided that the hydrocarbon charge into the reservoir is sufficient to keep the buoyancy force in the hydrocarbon column above the capillary entry pressure of the fault rock. A fault surrounded by a complex damage zone does not necessarily have a higher sealing capacity than a single fault since, provided there is sufficient hydrocarbon charge, faults within the damage zone will all become permeable to hydrocarbons once their capillary entry pressure has been exceeded. The absence of differences in either pressure or hydrocarbon column heights across faults does not, we propose, preclude the presence of a barrier to fluid flow. Fluid pressure and hydrocarbon column height differences between compartments can be controlled by factors such as capillary entry pressure in the undeformed reservoir and the amount of hydrocarbons entering the reservoir, rather than solely by the capillary entry pressure of the fault rocks present. Fault seal prediction methodologies that are calibrated, based on cross-fault differences in hydrocarbon column height or pressure, without considering the total hydrocarbon column height are likely to be unreliable. It is therefore recommended that the sealing capacity of a fault should be calculated from the difference in pressure between the hydrocarbon and pore-water at the position along the fault where leakage is most likely to occur.

Predicting the three-dimensional population characteristics of fault zones: a study using stochastic models

Major fault zones are surrounded by damage zones composed of minor faults that, in siliclastic rocks, often form significant barriers to fluid flow. Information on fault damage zone architecture is usually available only as 2D maps, or as 1D line samples or well logs. However, the accurate determination of the 3D fault population characteristics is crucial for flow prediction. In this paper, stochastic models of fault damage zones are generated by incorporating the statistical properties of fault populations (power law length and throw distributions, orientation distribution) and different spatial distributions, including randomly located, simple and hierarchical clustering of faults. These damage zone models are used to investigate the characteristics of 2D and 1D samples, which were found to depend on the 3D power law length exponent, D3, and the spatial distribution of the parent 3D population. Observed 1D samples may fail to show power law characteristics and, therefore, a lack of power law behaviour need not imply a non-power law parent population. The simple rules in which observed 2D and 1D samples follow power laws with exponents D2=D3−1 and D1=D3−2, respectively, are not always obeyed. Clustering tends to reduce the difference between these exponents to less than their simple integer values, most markedly for the simple clustering model. The hierarchical clustering model, in which small faults are clustered around larger faults throughout the fault damage zone and which most closely resembles nature, suggests that the simple rule D2=D3−1 is obeyed with only small deviations but that 1D samples may depart from the simple rule, D1=D3−2, by as much as 0.25.

Use of curved scanlines and boreholes to predict fracture frequencies

Abstract We advance the method of Hudson and Priest (Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts 20 (1983) 73–89) to develop a method for a curved scanline to be used to predict the numbers of fractures that would be observed in any direction. When sampling along a scanline, the probability of intersecting a fracture is influenced by the relative orientations of the fracture and of the scanline at that location. This sampling bias can be minimised by the use of the Terzaghi correction, w =(cos χ ) −1 , where χ is the angle between the scanline and the normal to the fracture. These corrected frequencies are used to simulate fracture frequencies for all other orientations by doubly-correcting the data. Modelled fracture frequency is contoured on a graph of simulated scanline plunge against simulated scanline azimuth. This method is based upon the assumption that the data collected along the scanline is representative of the fracture population when the Terzaghi correction has been applied. A graph of cumulative frequency of fractures against distance along a scanline provides a simple method for determining whether the scanline crosses differently fractured areas. Frequencies are corrected for dip, strike, and both dip and strike, with data from homogeneously fractured areas plotting as straight lines. These frequencies can be normalised for ease of comparison.

Unsteady mixed convection boundary-layer flow on a vertical surface in a porous medium

An analysis is made of the unsteady mixed convection from a vertical flat plate embedded in a fluid-saturated porous medium. For time t < 0 a uniform free stream velocity U exists parallel to the plate surface and the temperature T∞ throughout the porous medium is uniform. Then at time t = 0 the temperature on the surface is instantaneously changed from the ambient fluid temperature T∞ to Tw. At small times the transport effects are confined within a narrow layer adjacent to the plate. As this inner boundary layer evolves, a steady boundary layer is approached but far from the plate the ambient conditions remain. A complete analysis is made of the governing equations at t = 0, the steady state at large times and a series solution valid at small times is derived. A numerical solution of the full boundary-layer equations is then obtained for the whole transient from t = 0 to the steady state. Results are presented to illustrate the occurrence of transients when the buoyancy parameter is positive (buoyancy and free stream forces in the same direction) and negative (buoyancy and free stream forces in opposing directions) . The uniqueness of this problem lies in the fact that we have been able to match significantly different profiles at the time when the forward integration approach breaks down and the solution at large times and establish a smooth evolution around the transition time.

Transient free convection flow past a vertical flat plate subject to a sudden change in surface temperature

The transient free convection boundary-layer flow of a viscous and incompressible fluid adjacent to a semi-infinite vertical flat plate is investigated. It is assumed that for time τ 0. The solution is dependent upon two parameters, namely the ratio of the final temperature above ambient to the initial temperature above ambient, R = ΔT2/ΔT1 = (T2 - T∞)/(T1 - T∞) and the Prandtl number Pr. An analytical solution is presented which is valid at small values of τ. A new phenomena in this class of problems that is obtained from the detailed numerical scheme is the existence of two solutions, only one of which is physically acceptable, to the finite-difference equations associated with the matching technique applied for times beyond that at which the step-by-step method breaks down. Results have been obtained for a range of values of the parameter R, when Pr = 1.

New formulation of the Green element method to maintain its second-order accuracy in 2D/3D

Abstract The Green element method (GEM) is a powerful technique for solving nonlinear boundary value problems. Derived from the boundary element method (BEM), over the meshes of the finite element method (FEM), the GEM combines the second-order accuracy of the BEM with the efficiency and versatility of the FEM. The high accuracy of the GEM, resulting from the direct representation of normal fluxes as unknowns, comes at the price of very large matrices for problems in 2D and 3D domains. The reason for this is a larger number of inter-element boundaries connected to each internal node, yielding the same number of the normal fluxes to be determined. The currently available technique to avoid this problem approximates the normal fluxes by differentiating the potential estimates within each element. Although this approach produces much smaller matrices, the overall accuracy of the GEM is sacrificed. The first of the two techniques proposed in this work redefines the present approach of approximating fluxes by considering more elements sharing each internal node. Numerical tests on the potential field exp( x + y ) show an increase in accuracy by two orders of magnitude. The second approach is a reformulation of the standard GEM in terms of the flux vector, replacing its normal component. The original accuracy of the GEM is preserved while the number of unknowns is reduced as many as ten-times in the case of a mesh consisting of tetrahedrons. The additional benefit of this novel technique is the fact that the entire flux field is a mere by-product of the basic procedure for determining the unspecified boundary values.

Transient free convection from a vertical plate subjected to a change in surface heat flux in porous media

In this paper we examine the general transient natural convection response arising due to a sudden change of the level of uniform flux dissipation rate from a vertical surface which is embedded in a porous medium. From an analytical investigation of the governing boundary-layer equations both a series solution which is valid at small values of the non-dimensional time and a solution which is valid at large times, when the transport of energy is steady, are derived. A numerical, transient formulation of the full unsteady boundary-layer equations is developed using an explicit finite-difference scheme. The numerical temperature profiles are observed to closely follow the small time solution initially and evolve along a curve which approaches the steady-state solution asymptotically. Results are presented to illustrate the occurrence of transients from both an increase and a decrease in the levels of existing energy inputs.

Transient boundary-layer heat transfer from a flat plate subjected to a sudden change in heat flux

Abstract We examine the transient forced convection heat transfer from a fixed, semi-infinite, flat plate situated in a fluid which, at large distances, is moving with a constant velocity parallel to the plate. Both the fluid and the plate are initially at a constant temperature and the transients are initiated when the zero heat flux at the plate is suddenly changed to a constant value. The thermal boundary-layer equations are solved using numerical techniques to extend a series which is valid for small times and describe fully the development from the initial unsteady state solution (small times) to the ultimate steady state solution (large time).