Alison Ord

Numerical modelling of single-layer folding: clarification of an issue regarding the possible effect of computer codes and the influence of initial irregularities

The influence of initial perturbation geometry and material propel-ties on final fold geometry has been investigated using finite-difference (FLAC) and finite-element (MARC) numerical models. Previous studies using these two different codes reported very different folding behaviour although the material properties, boundary conditions and initial perturbation geometries were similar. The current results establish that the discrepancy was not due to the different computer codes but due to the different strain rates employed in the two previous studies (i.e. 10(-6) s(-1) in the FLAC models and 10(-14) s(-1) in the MARC models). As a result, different parts of the elasto-viscous rheological field were bring investigated. For the same material properties, strain rate and boundary conditions, the present results using the two different codes are consistent. A transition in Folding behaviour, from a situation where the geometry of initial perturbation determines final fold shape to a situation where material properties control the final geometry, is produced using both models. This transition takes place with increasing strain rate, decreasing elastic moduli or increasing viscosity (reflecting in each case the increasing influence of the elastic component in the Maxwell elastoviscous rheology). The transition described here is mechanically feasible but is associated with very high stresses in the competent layer (on the order of GPa), which is improbable under natural conditions

An equivalent algorithm for simulating thermal effects of magma intrusion problems in porous rocks

An equivalent algorithm is proposed to simulate thermal effects of the magma intrusion in geological systems, which are composed of porous rocks. Based on the physical and mathematical equivalence, the original magma solidification problem with a moving boundary between the rock and intruded magma is transformed into a new problem without the moving boundary but with a physically equivalent heat source. From the analysis of an ideal solidification model, the physically equivalent heat source has been determined in this paper. The major advantage in using the proposed equivalent algorithm is that the fixed finite element mesh with a variable integration time step can be employed to simulate the thermal effect of the intruded magma solidification using the conventional finite element method. The related numerical results have demonstrated the correctness and usefulness of the proposed equivalent algorithm for simulating the thermal effect of the intruded magma solidification in geological systems.

Bifurcation in Growth Patterns for Arrays of Parallel Griffith, Edge and Sliding Cracks

This paper presents the recent finding by Muhlhaus et al [1] that bifurcation of crack growth patterns exists for arrays of two-dimensional cracks. This bifurcation is a result of the nonlinear effect due to crack interaction, which is, in the present analysis, approximated by the dipole asymptotic or pseudo-traction method. The nonlinear parameter for the problem is the crack length/ spacing ratio lambda = a/h. For parallel and edge crack arrays under far field tension, uniform crack growth patterns (all cracks having same size) yield to nonuniform crack growth patterns (i.e. bifurcation) if lambda is larger than a critical value lambda(cr) (note that such bifurcation is not found for collinear crack arrays). For parallel and edge crack arrays respectively, the value of lambda(cr) decreases monotonically from (2/9)(1/2) and (2/15.096)(1/2) for arrays of 2 cracks, to (2/3)(1/2)/pi and (2/5.032)(1/2)/pi for infinite arrays of cracks. The critical parameter lambda(cr) is calculated numerically for arrays of up to 100 cracks, whilst discrete Fourier transform is used to obtain the exact solution of lambda(cr) for infinite crack arrays. For geomaterials, bifurcation can also occurs when array of sliding cracks are under compression.

Models for Mineral Phase Nucleation and Growth

This chapter concerns the processes associated with the nucleation and growth of new mineral grains during deformation. Four end-member models are explored: (1) networked mineral reactions; (2) isochoric replacement of old grains involving stress-driven diffusion processes (‘pressure solution’); (3) classical nucleation and growth processes; and (4) the isochoric replacement of old grains by new ones involving dissolution–precipitation processes. Evidently the detailed mechanisms operating at the reaction front become important in that the replacement process, at least for models (2) and (4), are coupled at the scale of the reacting front resulting in self-accelerating (that is, autocatalytic) kinetics. Models for porphyroblast development are explored from two aspects. First, interpretations of grain size distribution are discussed from a stochastic viewpoint in terms of the competition between processes that diffuse the distribution through distribution space and grain growth/reduction processes that tend to localise the distribution. This leads to a consideration of the Fokker–Planck equation. Second, a deterministic model for porphyroblast growth is presented in terms of reaction–diffusion–transport equations. A specific model, the Gray–Scott model, is explored. This model produces microstructures similar to natural microstructures, in particular, multifractal microstructures with zoned grains that resemble natural examples.

A Segregated Algorithm for Simulating Chemical Dissolution Front Instabilities in Fluid-Saturated Porous Rocks

When fresh pore-fluid flow enters a solute-saturated porous medium, where the concentration of the solute (i.e. aqueous mineral) reaches its equilibrium concentration, the concentration of the aqueous mineral is diluted so that the solid part of the solute (i.e. solid mineral) is dissolved to maintain the equilibrium state of the solution. This chemical dissolution process can result in the propagation of a dissolution front within the fluid-saturated porous medium. Due to the dissolution of the solid mineral, the porosity of the porous medium is increased behind the dissolution front. Since a change in porosity can cause a remarkable change in permeability, there is a feedback effect of the porosity change on the pore-fluid flow, according to Darcy’s law. It is well known that because pore-fluid flow plays an important role in the process of reactive chemical-species transport, a change in pore-fluid flow can cause a considerable change in the chemical-species concentration within the porous medium (Steefel and Lasaga 1990, 1994, Yeh and Tripathi 1991, Raffensperger and Garven 1995, Shafter et al. 1998a, b, Xu et al. 1999, 2004, Ormond and Ortoleva 2000, Chen and Liu 2002, Zhao et al. 2005a, 2006c). This means that the problem associated with the propagation of a dissolution front is a fully coupled nonlinear problem between porosity, pore-fluid pressure and reactive chemical-species transport within the fluid-saturated porous medium. If the fresh pore-fluid flow is slow, the feedback effect of the porosity change is weak so that the dissolution front is stable. However, if the fresh pore-fluid flow is fast enough, the feedback effect of the porosity change becomes strong so that the dissolution front becomes unstable. In this case, a new morphology (i.e. dissipative structure) of the dissolution front can emerge due to the self-organization of this coupled nonlinear system. This leads to an important scientific problem, known as the reactive infiltration instability problem (Chadam et al. 1986, 1988, Ortoleva et al. 1987), which is closely associated with mineral dissolution in a fluid-saturated porous medium.

The Particle Simulation Method for Dealing with Spontaneous Crack Generation Problems in Large-Scale Geological Systems

Cracking and fracturing are one class of major failure mechanisms in brittle and semi-brittle materials. Crustal materials of the Earth can be largely considered as brittle rocks, and so cracking and fracturing phenomena are ubiquitous. Cracks created within the Earth’s crust often provide a very useful channel for mineral-bearing fluids to flow, particularly from the deep crust into the shallow crust of the Earth. If other conditions such as fluid chemistry, mineralogy, temperature and pressure are appropriate, ore body formation and mineralization can take place as a result of such fluid flow. Because of the ever-increasing demand for mineral resources in the contemporary world, exploration for new mineral resources has become one of the highest priorities for many industrial countries. For this reason, extensive studies (Garven and Freeze 1984, Yeh and Tripathi 1989, 1991, Steefel and Lasaga 1994, Raffensperger and Garven 1995, Zhao et al. 1997a, Schafer et al. 1998a, b, Zhao et al. 1998a, Xu et al. 1999, Zhao et al. 2000b, Schaubs and Zhao 2002, Zhao et al. 2002a, 2003e, 2005a) have been conducted to understand the detailed physical and chemical processes that control ore body formation and mineralization within the upper crust of the Earth. Thus, the numerical simulation of spontaneous crack generation in brittle rocks within the upper crust of the Earth has become an important research topic in the field of computational geoscience.

Visco-Plastic Flow

Visco-plastic flow is the rate- and temperature-sensitive deformation of solids. The behaviour is characterised by a yield or flow surface which marks the change over from elastic behaviour for stress states inside the surface to visco-plastic behaviour for stress states on the yield surface. The shape and size of the yield surface depends on both the deformation rate and the temperature. The mechanisms of visco-plastic flow involve the motion of point defects, and line defects such as dislocations, disclinations and disconnections, the latter being especially important for multiphase mineral aggregates. Two end-member modes of plastic deformation are (1) that dominated by crystal slip which is essentially characterised by translational deformations and (2) that dominated by the motion of disclinations and disconnections characterised by rotational deformations. The rotational mode commonly involves the coupled motion of surfaces such as twin, subgrain and grain boundaries and results in both grain size reduction and grain growth. The development of mineral lineations and foliations derives directly from the migration of disconnections. Five independent slip systems are necessary to accommodate a general imposed isochoric affine strain but eight deformation mechanisms are necessary to accommodate a general isochoric, affine deformation. These eight mechanisms are made up of a combination of dislocations, disclinations and disconnections. More general mechanisms of visco-plastic deformation include the migration of cellular dislocations. We explore the extrapolation of laboratory determined flow laws to geological conditions.

Geometry: The Concept of Deformation

This chapter introduces the geometrical concept of deformation and the associated concepts of strain and rotation. The geometrical description of deformation is independent of the applied forces, velocities and histories of these quantities. We consider the concept of the deformation gradient and how that quantity describes the changes in the positions of material points, the lengths and orientations of lines, the distortion of arbitrary surfaces and of volume elements. The concept of a phase portrait is introduced as a precursor to its use in dynamical systems. Also considered are special deformations associated with buckling, localisation and plastic slip within crystals. Conditions for compatibility of deformations across a surface are defined. Worked examples for simple shear and inhomogeneous deformations are presented.

Kinematics – Deformation Histories

The subject concerned with movements during a deformation history is called kinematics . In this chapter, we examine some common deformation histories in deformed rocks and discuss the advection of quantities through the deforming material in terms of the material time derivative . This highlights the need to distinguish between spatial and material descriptions of the deformation history. We then define the velocity gradient tensor and the useful measures of deformation history that follow from the use of that tensor, namely, the stretching tensor and the spin tensor . Various forms of the velocity gradient are explored and the eigenvectors of the flow gradient are derived. The nature of these eigenvectors controls the patterns of flow defined by streamlines expressed as phase portraits . This gives us the tools (together with the deformation measures discussed in Chapter 2) to derive expressions for the rates of change of strain, volume and oriented area. We follow with a consideration of spin and vorticity in deforming rocks with an emphasis on unsteady and inhomogeneous deformation histories (including pulsating histories). The use of the kinematic vorticity number is discussed particularly with respect to the concept of objectivity . We complete the chapter with an examination of the relation of kinematic indicators to the eigenvectors of the flow. This introduces the concepts of strong and weak flows .

A Consistent Point-Searching Interpolation Algorithm for Simulating Coupled Problems between Deformation, Pore-Fluid Flow, Heat Transfer and Mass Transport Processes in Hydrothemal Systems

Over the past decade or so, many commercial computational codes have become available for solving a great number of practical problems in both scientific and engineering fields. Primary advantages of using commercial computational codes are: (1) built-in pre-processing and post-processing tools make it very easy and attractive to prepare, input and output data which are essential in a numerical analysis; (2) provision of movie/animation functions enables numerical results, the treatment of which is often a cumbersome and tedious task, to be visualised via clear and colourful images; (3) detailed benchmark solutions and documentation as well as many embedded robust solution algorithms allow the codes to be used more easily, correctly, effectively and efficiently for solving a wide range of practical problems. However, the main disadvantage of using commercial computational codes is that each code is often designed, within a certain limit, for solving some particular kinds of practical problems. This disadvantage becomes more and more obvious because the ever-increasing competitiveness in the world economy requires us to deal with more and more complicated and complex geoscience problems, which are encountered and not solved in the field of contemporary computational geoscience. There are three basic ways to overcome the above difficulties. The first is to develop some new commercial computational codes, which is time consuming and often not cost-effective for numerical analysts and consultants. The second is to extend an existing commercial computational code, which is usually impossible because the source code is often not available for the code users. The third is to use several existing commercial computational codes in combination. This requires development of a data translation tool to transfer data necessary between each of the codes to be used. Compared with the difficulties encountered in the first two approaches, the third one is more competitive for most numerical analysts and consultants.