To increase oil recovery of weakly permeable carbonate rock, acid treatments with a weakly concentrated aqueous solution of hydrochloric acid are used. There can be various modes of dissolution of rocks depending on the injection rate of the acid solution in the breed from complete dissolution of the skeleton of the rock with a solid front at low speeds the injection to the emergence of long, single wormhole at high speeds of injection. A one-dimensional unsteady model of the flow of an aqueous hydrochloric acid solution in a porous carbonate rock is developed taking into account the movement of the front of the carbonate rock dissolution reaction. The boundary conditions under which the obtained system of equations is reduced to a self-similar system of equations of the fifth order are found. In contrast to self-similar filtration of a Newtonian fluid in the half space, the degree of dependence of self-similar independent variable is equal to − 3/2, and the dependent variables are the porosity and rate of filtration depend on the exponent of the dependence of rock permeability from changes in porosity after acid treatment. A one-dimensional non-stationary model of the flow of an aqueous solution of hydrochloric acid in a porous carbonate rock was developed taking into account the movement of the front of the dissolution reaction of carbonate rock. Boundary conditions are found under which the resulting system of equations reduces to a self-similar system of fifth-order equations. Unlike self-similar filtering of Newtonian fluid in half-space, the degree of dependence of the self-similar independent variable is −3/2, and the dependent variables (porosity and speed) of filtration depend on the degree of dependence of rock permeability on the change in porosity after acid treatment.
filtration of an aqueous acid solution,
self-similar system of equations
To increase the productivity of a producing well in carbonate reservoirs, hydrochloric acid treatments or acid hydraulic fracturing of the reservoir are used, using an aqueous solution of hydrochloric acid with a concentration of 10-25 % by eliminating the contamination of the bottom-hole zone or creating main cracks. Carbonate rocks consist of the main minerals: calcite (CaCO3), dolomite (CaMg(CO3)2), magnesite (MgCO3) and other carbonates. When conducting acid treatments, the main process occurring in the carbonate rock is its dissolution. During the flow of an aqueous solution of acid in carbonate reservoirs, chemical reactions of the acid with the indicated minerals occur. In reality, reactions with these minerals and reactions with other possible minerals occur simultaneously. As a result, the order of the dissolution reaction of carbonate rock can be fractional. As a result of these reactions of carbonates with hydrochloric acid, water-soluble calcium and magnesium salts, water and carbon dioxide are formed. Thus, we can assume that under reservoir conditions, salts and carbon dioxide are in dissolved form, that is, they belong to the liquid phase. Depending on the feed rate of the aqueous acid solution, various modes of dissolution of the rock are realized from a flat front with almost complete dissolution of the rock material to the appearance of main wormholes. In the general case, the effects of diffusion can be neglected when filtering acid into a carbonate oil-containing formation. Then the main dimensionless parameter controlling the front of the chemical reaction is the Damkeller number — the ratio of the characteristic convection time to the characteristic reaction time. In this work, we construct a mathematical model for filtering an aqueous solution of hydrochloric acid in a porous carbonate reservoir taking into account changes in the porosity and permeability of the rock due to its dissolution with hydrochloric acid. The mathematical model is based on the equations of mechanics of two-phase media and the theory of equilibrium chemical reactions and includes the equation of continuity of the rock skeleton, the equation of continuity for each of the components of the binary liquid phase, Darcy's law of filtration taking into account the change in permeability with a change in porosity according to the power law due to the dissolution of the rock. Exponents in the power law should be determined from experiments on acid treatment of carbonate porous cores.The constructed mathematical model is a system of fifth-order equations. In the general case, the system of equations obtained is rather complicated even for a numerical solution. Of great interest is the possibility of constructing a self-similar representation of the proposed system of equations. In this paper, we managed to find the boundar conditions at the entrance to the half-space, as well as at the front of the chemical reaction, under which the considered system of equations is presented in a self-similar form. As boundary conditions satisfying the self-similarity condition at the entrance to the porous half-space, we should take the conditions of the first kind for the pressure and concentration of acid, that is, the condition of their constancy. On the front of the chemical reaction of carbonate rock with an aqueous solution of hydrochloric acid should also take the conditions of the first kind-the equality of pressure to reservoir pressure and equality to zero concentration of hydrochloric acid. The resulting self-similar system of equations is a system of three ordinary differential equations of the fifth order. The required functions (porosity and filtration rate) turn out to depend on the index of the degree of permeability of the rock from its changing porosity, and the self-similar variable is inversely proportional to time in the degree 3/2. For comparison, when filtering a Newtonian fluid in a porous half-space, the self-similar variable is inversely proportional to time to the power of 1/2. The obtained self-similar representations do not allow the passage to the limit to the flow of a Newtonian fluid without chemical reactions. The distribution of acid concentration and pressure in the porous carbonate half-space do not explicitly depend on time, but rather depend on time indirectly through a self-similar variable.