We present the first quantitative study and complete model of the wormholing phenomenon, leading to a means of predicting and optimizing carbonate acidizing treatments. Laboratory experiments on a gypsum model system and computer simulations show that for a given geometry, wormholes can be quantified by a unique parameter, their equivalent hydraulic length. The behavior of this quantifying parameter vs. all the system parameters is studied and allows the quantitative prediction of the efficiency of an acidizing treatment. This study highlights the fractal nature of the phenomenon, which is accounted for in the equations, and the strong effect of the sample geometry. Three types of etching can be obtained: compact, wormhole type, or homogeneous. The optimum conditions for achieving the best skin decrease correspond to the creation of wormholes and can then be defined in terms of fluid reactivity and injection rate.
Much less is known about matrix acidizing of carbonate reservoirs than sandstone acidizing; models for sandstone acidizing have existed for more than 10 years. The main reason for this lack of knowledge is that in sandstone acidizing the kinetics of dissolution is limited by the surface reaction, which makes, in a first approximation, the global reaction rate insensitive to the flow rate and the displacement stable (the rock is homogeneously etched, in spite of possible large-scale instabilities), thus allowing a macroscopic formulation.n contrast, it is well known that the acidizing of carbonates with highly reactive acids leads to the creation of wormholes-i.e., empty channels that bypass most of the matrix. This seems to be the result of the mass-transfer limitation of the kinetics. 3 This characteristic causes the local reaction rate to be velocity-dependent and therefore the etched pattern to display instabilities. Little quantitative knowledge in this domain has been gained recently. Several more or less qualitative studies have aimed at understanding the key parameters limiting the extension of the wormholes; it is currently believed (more intuitively or by analogy with fracture acidizing than from laboratory works) that decreasing the diffusion constant or increasing the acid flow rate will improve the penetration of the live acid.
The basic factor limiting the physical interpretation of the worm-holing process has been recognized as being its stochastic nature. These patterns have recently been characterized as fractals (i.e., for a radial geometry, the number of branches of the pattern at a given distance r from the center varies according to a power law of r), opening the way to a quantitative study. The common mechanism in dissolution wormholing and viscous fingering is enhancement of local rate of advance initiated by perturbations arising from natural inhomogeneities.
The close analogy between wormhole patterns and structures generated with the model of diffusion-limited aggregation (DLA) has been shown quantitatively. As a result, this model emphasized the preponderant role of the sample geometry on the dissolution pattern. Patterns obtained with linear one-dimensional (ID) samples are fundamentally different from those obtained under three-dimensional (3D) radial or 3D linear condition, geometries that are characteristic of matrix injection from a wellbore and fluid loss from a fracture face, respectively.
In this paper, we present further results obtained with our plaster/water model system and from quantitative computer simulations on a network.
We show that the quantitative description of wormholes for a given geometry requires only one parameter. Its growth law depends on the geometry of the system.
We study the effect of all the treatment and system parameters to allow the derivation of the equations giving the absolute penetration of the reactive fluid (we give these equations for two geometries, linear 1D and radial). Previous experimental results obtained with the system limestone/HCl are reinterpreted with our model and found to agree quantitatively. In addition, we show a good comparison between the skin predicted with our model and that measured during a field treatment.
We discuss a physical interpretation of these equations and their limits of validity. Three domains are defined: compact and stable dissolution at low flow rates, wormhole (unstable) domain, and homogeneous etching at high rates.
We then draw practical conclusions, especially concerning the existence of an optimum injection rate; flow rates typically used in the field appear to be much too high.
Dissolution experiments have been carried out on plaster samples (plaster has long been used as a model system for studying the dissolution of natural limestone by groundwater), made by mixing water and reagent grade CaSO4 -0.5H2O in the ratio of 1/1.1 by weight. After setting in an appropriate mold, the plaster (4) = 50 % and k=60 md was saturated with water equilibrated with plaster. Pure water was then injected at a constant flow rate, q, with the injection pressure, p, recorded. Here we describe results obtained with two geometries: linear ID-cylindrical samples for which we varied the diameter/length ratio (from 2rIL = 50 MM/30 nun to 25 mm/130 mm [2 in./1.18 in. to I in./5.1 in.]), the water being pumped linearly through one end face; and radial-from thin disk (typically 1 mm [0.04 in.] thick and 200 mm [7.9 in.] in diameter) to cylinders (h=60 mm [2.4 in.] and ro=25 mm [1 in.]), water being injected radially from a small hole (2-mm [0.08-in.] diameter) drilled along their axis.
The kinetics of dissolution of plaster by water is known to be limited by mass transfer. We verified this from kinetic measurements performed with a capillary and a rotating-disk apparatus and found a D=9 × 10 - 10 M2-.s-1 at 20 degrees C [97 X 10(-10) ft2 Sec-1 at 68 degrees F]. Using a capillary of fixed length and radius and varying the fluid rate, q, we determined an outlet concentration that depended on q-A, while surface-reaction kinetics would have yielded a dependence on q - 1. With the rotating-disk apparatus, the reaction rate (number of moles per unit of time and per unit of area) was proportional to the square root of the rotational speed as predicted by the theory lo for mass-transfer-limited kinetics.
The modeling of flow through a porous medium by means of a network of ducts and by solving the flow and mass-conservation equations at each node has been applied to many problems, particularly to miscible and immiscible displacement. Hoeftier and Fogleri applied this approach to the case of reactive fluids and obtained qualitative results.