Enhanced Visualization of Acid Carbonate Rock Interaction
Authors: Economides M.J. , Frick T.P. , Nittman J. 
Published: December 2007

Research Note
Visualization of the reaction patterns between acid and carbonate rocks is the subject of this note.
Carbonate rocks (chalks, limestones, and dolomites) account for a sizableportion of all petroleum reservoirs. Wells drilled in these reservoirs usually need stimulation, as all types of wells do. The acid/rock reaction pattern shave often been referred to as wormholes, a term that reflects their actual shape.
The reaction of acid with reservoir rocks is controlled primarily by the diffusive flux, which depends on the concentration gradient from the live acid to the rock surface and, once the acid is in contact, by the reaction rate at the rock surface.
Clearly, the slower mechanism would control the reaction kinetics. In limestones at any temperature and dolomites above 50 C, the activation energy is low; every acid/rock collision results in a reaction, and the reaction rate depends on how fast the acid can be brought in contact with the rock. These systems are controlled by mass-transfer (or diffusion)-limited kinetics.
For sandstones and very low-temperature dolomites, the activation energy is high, and only a small number of acid/rock collisions result in a reaction. These systems are controlled by surface-reaction-limited kinetics.
The phenomenological behavior of the two extreme cases of acid/rock interactions is well understood and is described in a number of publications. Thus, we do not address that topic. Our work focuses on the diffusion-limited case-i.e., acid/carbonate rock interactions.
Several attempts have been made to quantify and thus to predict reaction patterns. Hoefner and Fogler considered simultaneous momentum, mass transfer, and reaction kinetics. They used network models to simulate and predict wormhole patterns.
Wormhole growth is an unstable process, with a number of possible phases whose durations (or even their presence) depend on several variables and ranges of their values. For example, the magnitude of the injection rate in a given formation may cause compact dissolution, wormholes with multiple branches(dendritic shape), or a single dominant wormhole. Thus, appropriately conditioned stochastic modeling can be used to visualize reaction patterns, to discern differences, to reproduce laboratory results, and ultimately to reproduce and predict field results.
One of the first models to simulate the process is Witten and Sander'sdiffusion-limited aggregation (DLA) model. The DLA model is a random simulator, requiring (and implying) field isotropy and a constant-pressure surface at the cluster. In this environment, it would be difficult to bias the randomness.
To circumvent the problem, the dielectric breakdown model (DBM) was introduced. In the DBM, the cluster propagates with a probability to grow at a site conditioned (or weighted) by the local pressure gradient. This feature makes the DBM particularly attractive in describing viscous fingering and wormholes resulting from acidizing because the pressure gradient and flow characteristics of the already-created patterns greatly influence subsequent growth.
Pichler et al. presented a modification to the DBM, the permeability-driven fingering (PDF) model, where the growing cluster is treated as an interface between two regions of distinctly different properties: mobility in viscous fingering and permeability in acidizing. The PDF model lends itself to a number of interesting studies and visualizations.
We have simulated the reaction patterns in linear cores, and showed that one dominant wormhole is created (Fig. 1). Other 2D results demonstrate the screening effect; some branches grow preferentially while others stop. The longest fingers control the flow distribution, and the shape of the cluster depends on the shape of the previous stage. This suggests that experimental results from linear cores may be of questionable value when extrapolated in a radial environment unless appropriate dimensional analysis is undertaken.
For 3D simulation, Zirngast constructed an extension of the 2D dynamic grid. A grid cube was positioned within a sphere of zero potential, representing the outer boundary. A cylindrical outer boundary can be used for a closer approximation to field stimulation patterns that may emanate for a well.
Stochastic models (2D PDF, 3D PDF, and their derivatives) can simulate the growth of reaction patterns (wormholes) between acids and carbonate rocks, allowing their visualization and study. The impact of variables on the wormhole character can readily assessed. The models can and must be conditioned by experimentally determined quantities such as diffusion constants, the relationship between injection rate and the type of stimulation, and the relative impact of the individual mechanisms.

References
Daccord, G., Lenormand, R., and Touboul, E.: "Carbonate Acidizing: Toward a Quantitive Study of the Wormholing Phenomenon," SPEPE (Feb. 1989)63; Trans., AIME, 287. Levich, V.G.: Physicochemical Hydrodynamics, Prentice-Hall Inc., Englewood Cliffs, NJ (1962). Hoefner, M.L. and Fogler, H.S.: "Fluid-Velocity and Reaction-Rate Effects During Carbonate Acidizing: Application of Network Model," SPEPE(Feb. 1989) 56. Witten, T.A. and Sander, L.M.: "Diffusion-Limited Aggregation. A Kinetic Critical Phenomenon," Phys. Rev. Let. (1981) 47/19, 1400. Niemyer, L., Pietronero, L., and Weismann, H.J.: "Fractal Dimension of Dielectric Breakdown," Phys. Rev. Let. (1984) 52/12, 1033. Pichler, T. et al.: "Stochastic Modeling of Wormhole Growth in Carbonate Acidizing With Biased Randomness," paper SPE 25004 presented at the 1992 SPE European Offshore Petroleum Conference, Cannes, Nov. 9–11. Zirngast, E.: "Stochastic Modeling of Wormhole Growth in Three Dimensions," diploma thesis, Mining U. Leoben, Leoben, Austria (1993).
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