When a naturally fractured carbonate formation is treated with acid at pressures below the fracturing pressure, the acidizing process will likely be different from either a matrix treatment or an acid fracturing treatment. Our previous experimental study shows three kinds of acid etching patterns after acidizing naturally fractured carbonates, and it also illustrates their relationships with the fracture properties and acidizing conditions. Based on the experimental observations, a mathematical model of acidizing naturally fractured carbonates has been developed. The model includes bulk solution transport, acid transport and reaction, and the change of fracture width by acid dissolution. A new approach was used to treat leakoff acid and acid-rock reactions at the fracture walls. The new acidizing model for numerically generated rough-surfaced fractures predicts the same kinds of acid etching patterns and the same relationships between acid etching patterns and the fracture width, surface roughness, and leakoff rate as observed in experiments.
Matrix acidizing is a stimulation method to improve the well productivity by pumping acid at a pressure lower than formation fracture pressure, and it usually creates wormholes in an unfractured carbonate. Acid fracturing is another method used to improve well productivity in carbonate reservoirs, in which acid flows through the relatively wide hydraulic fracture and etches the fracture walls. Considerable research has been done on both matrix acidizing and acid fracturing, and many models have been developed. 1–19 However, these models are developed for either matrix flow conditions or flow in a relatively wide hydraulic fracture. When a naturally fractured carbonate formation is treated with acid at pressures below the formation fracture pressure, the acidizing process is neither like matrix acidizing nor like acid fracturing. Our previous experimental studies20 show that in acidizing naturally fractured carbonates with a fracture width on the order of 10–3 cm, most acid flows through and reacts with rock inside the fracture, not the matrix. When the fracture width is smaller than 2×10–3 cm, wormholes are created along the fracture surface from the inlet to the outlet, which is like matrix acidizing. When the fracture width is between 3×10–3 and 8×10–3 cm, a channel that is broad near the inlet and narrower farther away from the inlet is created along the fracture surface. When the fracture width is larger than 1×10–2 cm, acid etches most of the fracture surface, as in acid fracturing. When the fracture surface becomes rougher, wormholes are more easily created. Leakoff has little effect on the etching pattern along the fracture surface but creates wormholes perpendicular to the fracture surfaces.
We developed a mathematical model to describe the acidizing process in naturally fractured carbonates. The model is based on mass conservation for the acid solution, acid transport, and the change of fracture width by acid dissolution. Pressure, fracture width, and acid concentration as functions of position and time can be predicted by the model. The simulation results compared well with the experimental observations.
We initially formulated the model for the laboratory scale in order to make direct comparisons with our experimental results. The model domain is a block of carbonate core sample with a length, l, a thickness, 2wm and a height, h, as shown in Fig. 1. A single fracture is placed in the middle and through the entire core sample. Acid is introduced from one side, flows along the fracture, and flows out of the fracture from the opposite side. The leakoff acid penetrates the matrix perpendicular to the fracture. The coordinate system is defined such that the x direction is the effluent acid flow direction, which is aligned with the length of the fracture, the z direction is aligned with the height of the fracture, and the y direction is perpendicular to the fracture surface. The fracture plane is the x-z plane, and the leakoff is in the y direction. The control volume is a parallelepiped with dimensions of ?x, ?z, and b, where b is the fracture width at the point (x, z) (Fig. 2).
A mass balance for acid solution inside the fracture is
where vx and vz=the average velocities in the fracture at the point (x, z) in the x and z directions respectively, vl=the leakoff velocity, and b=the fracture width at the point (x, z).
To determine whether the flow regime of acid in the fracture is laminar or turbulent, the Reynolds number, NRe, was calculated. For flow between parallel plates, the Reynolds number is
The average velocity, v, =the volumetric flow rate divided by the cross-sectional area of the fracture (bavh), so in terms of volumetric flow rate, the Reynolds number is
where q=the pumping rate, ?=the acid solution density, h=the fracture height, and µ=the acid solution viscosity. For injection of water at 10 ml/min into a 5-cm-high fracture as in our lab experiments, the Reynolds number is about 13.
For such a low Reynolds number, the flow is laminar, and therefore the permeability of the fracture at a certain position is ß2/12. The average velocities of acid flow in the fracture, vx and vz, can be calculated by
Equations 4 and 5
where p=the pressure at the point (x, z).