Geri Wagner

Slow two-phase flow in artificial fractures: Experiments and simulations

The slow displacement of a wetting fluid by a nonwetting fluid in models of a single fracture was studied experimentally and by computer simulations on identical geometries. The fracture was modeled by the gap between a rough plate and a smooth transparent plate, both oriented horizontally. Two different rough plates were used, a textured glass plate and a polymethyl methacrylate plate with a computer-generated pattern. A nonwetting fluid (air) was injected slowly through an inlet into the model and displaced a wetting fluid (water) initially filling the model. The aperture fields of the artifical fractures were measured using a light absorption technique. The experiments were simulated using modified invasion percolation models, making use of the measured aperture fields. The simulation models captured invasion bursts and fragmentation and redistribution of the invading air. Experiments and simulations were compared step by step, and good qualitative and quantitative agreement was found.

Invasion percolation and secondary migration: experiments and simulations

Abstract Experiments have been carried out to study the displacement of wetting fluids by immiscible non-wetting fluids in quasi-two-dimensional and three-dimensional granular porous media. These experiments included a systematic investigation of the effects of gravity acting on the density difference between the two fluids. The simple invasion percolation model provides a surprisingly realistic simulation of the slow fluid–fluid displacement process in the absence of gravity, and a simple extension of the model can be used to simulate the most important features of gravity stabilized and destabilized fluid–fluid displacement processes. The dimensionless Bond number Bo (the ratio between buoyancy forces and capillary forces on the pore scale) can be used to compare experiments and simulations carried out using different (geometrically similar) porous media, different fluid–fluid interfacial tensions and different fluid densities. The complex patterns generated by gravity stabilized and gravity destabilized fluid–fluid displacement processes can be described in terms of a pattern of blobs of size ξ that have a fractal structure on length scales l in the range ϵ≤l≤ξ, where ϵ is the characteristic porous medium grain size. The blob size ξ is related to the Bond number by the simple scaling relationship ξ∼Bo−ν/(1+ν), which was first derived by Wilkinson, 1984 , Wilkinson, 1996 ) for gravity-stabilized displacement. Here, ν is the percolation theory correlation length exponent (ν=4/3 in two-dimensional systems and ν≈0.88 in three-dimensional systems). The experiments and simulations have been extended to include fluid–fluid displacement in fracture apertures and the effects of flow of the wetting fluid under the influence of a hydraulic potential gradient. These experimental and simulation results have important implications for our understanding of secondary migration. They indicate that the residual hydrocarbon saturation in the enormous volume of porous sedimentary rock (carrier rocks) between the hydrocarbon source and the reservoir can be very low, thus allowing significant quantities of oil and gas to reach the reservoir. Simulations have been carried out to explore the effects of heterogeneities on gravity destabilized fluid–fluid displacement processes and fluid–fluid displacement in fracture apertures. However, the structure of the carrier rocks is highly dynamic on the time scales over which secondary migration takes place (of the order of 108 years, in many cases). A better understanding of the pore structure of the carrier rocks and its dynamics on long time scales is needed to more accurately model secondary migration.

Buoyancy-driven invasion percolation with migration and fragmentation

A modified invasion percolation model that includes migration and fragmentation processes is presented. Fluid clusters were formed using the ordinary gravity invasion percolation model. Migration of fluid clusters was driven by buoyancy forces. The fluid formed a fragmented structure oriented in the direction of the buoyancy forces. The fragments formed temporary “pipelines” through which fluid was transported from the source to the tip of the structure. In the presence of a trapping rule the fragmented structure could bifurcate. The structures obtained in the simulations in two dimensions can be described in terms of standard percolation and invasion percolation.

INVASION PERCOLATION IN FRACTAL FRACTURES

Invasion percolation (IP) with trapping was studied on two-dimensional substrates with a correlated distribution of invasion thresholds. The substrates represented the void spaces of variable-aperture fracture models that were formed by (2+1)-dimensional self-affine rough surfaces and their replicas, displaced relative to each other. Void spaces generated in this manner were self-affine on length scales less than the horizontal displacement rd, and uncorrelated on larger length scales. The geometry of the IP clusters reflected the void space geometry, with different scaling behavior on length scales below and beyond rd.

Construction of a DLA cluster model

On the occasion of the 50th birthday of a distinguished colleague, a three-dimensional wood model of a computer-generated DLA cluster was built. In this paper the design of the model and the construction process are described. The experiment may be carried out in the framework of a classroom experiment to demonstrate some of the fundamental concepts used in current research on growth phenomena. It is suitable as a first introduction to fractal geometry. Zusammenfassung. Anlaslich des 50sten Geburtstag eines bedeutenden Kollegen wurde ein dreidimensionales Holzmodell eines computererzeugten DLA-Clusters gebaut. Hier beschreiben wir die Entwickelung des Modells und den Konstruktionsprozess. Das Experiment kann im Rahmen eines Klassenzimmerexperiments durchgefuhrt werden um einige der fundamentalen Konzepte innerhalb der aktuellen Forschung auf dem Gebiet der Wachstumsphanomene zu demonstrieren. Es eignet sich als eine erste Einfuhrung in die fraktale Geometrie.

Simulations of migration, fragmentation and coalescence of non-wetting fluids in porous media

A model based on invasion percolation was used to simulate the migration on a non-wetting fluid through a porous medium filled with an immiscible wetting fluid under the influence of a gradient such as that provided by gravity. The migrating fluid clusters undergo both fragmentation and coalescence. The fragment size distribution obtained from two-dimensional simulations in which the gradient g is slowly increased from 0 can be represented by the scaling form Ns(g)∼s-2ƒ(s⧸|g|-z where z=1+(D−1)ν⧸(ν+1). Here D is the fractal dimensionality of invasion percolation, with trapping, and ν is the ordinary percolation correlation length exponent.

Fragmentation of Invasion Percolation Cluster in Two-Dimensional Porous Media

Two-phase fluid displacement in a two-dimensional porous medium has been studied experimentally and by simulations. A random porous medium was saturated with a wetting fluid (water/glycerol). A non-wetting fluid (air) was then slowly injected, forming a fractal invasion percolation-like structure. Shortly before breakthrough, the pressure of the wetting fluid was increased and the non-wetting fluid was forced to withdraw. The structure formed by the non-wetting fluid disintegrated into fragments. The displacement patterns were found to agree well with patterns obtained from simulations based on an invasion percolation model.

Fractal Structures in Secondary Migration

Secondary migration is the process by which hydrocarbons are transported from mature source rocks through water-saturated rocks, faults or fractures and become concentrated as trapped accumulations of oil and gas. The forces governing secondary migration of hydrocarbons are buoyancy and capillarity (Schowalter, 1979). Experimental data suggest that the secondary migration of oil in porous, permeable sediments takes place along restricted pathways or conduits (Dembicki and Anderson, 1989). These conduits are formed after the oil has penetrated far enough into the reservior rock for the bouyancy forces acting on the oil to overcome the capillary pressure in the pore throats. Long-range petroleum migration of the order of 100 km in the horizontal direction and about 2 km in the vertical direction is not uncommon (England et al., 1987).

Slow Two-phase Flow in Single Fractures: Fragmentation, Migration, and Fractal Patterns Simulated Using Invasion Percolation Models

Abstract—The slow displacement of a wetting fluid by an invading non-wetting fluid in single fractures was studied using experiments and simulations. In the experiments, the fracture aperture was modeled by the gap between a rough plate and a smooth transparent plate. The displacement was simulated using invasion percolation models and two types of self-affine fracture aperture models; model A with an infinite in-plane correlation length, and model B with a finite in-plane correlation length. Simulations were also performed on self-affine models that precisely represented the aperture fields of the experiments. At length scales below the in-plane correlation length, the simulated displacement patterns show scaling properties that may be tuned by changing the characteristics of the underlying geometry. In the experiment-matched simulations, we observed closely corresponding displacement patterns.

FRACTALS AND SECONDARY MIGRATION

The process of secondary migration, in which oil and gas are transported from the source rocks, through water saturated sedimentary carrier rocks, to a trap or reservoir can be described in terms of the gravity driven penetration of a low-density non-wetting fluid through a porous medium saturated with a wetting fluid. This process has been modeled in the laboratory and by computer simulations using homogeneous porous media. Under these conditions, the pattern formed by the migrating fluid can be described in terms of a string of fractal blobs. The low density internal structure of the fractal blobs and the concentration of the transport process onto the self-affine strings of blobs (migration channels) both contribute to the small effective hydrocarbon saturation in the carrier rocks. This allows the hydrocarbon fluids to penetrate the enormous volume of carrier rock without all of the hydrocarbon being trapped in immobile isolated bubbles. In practice, heterogeneities in the carrier rocks play an important role. In some cases, these heterogeneities can be represented by fractal models and these fractal heterogeneity models provide a basis for more realistic simulations of secondary migration. Fractures may play a particularly important role and migration along open fractures was simulated using a self-affine fractal model for the fluctuating fracture aperture.