Patent Publication Number: US-2020300082-A1

Title: Calculation method for dynamic fluid loss of acid-etched fracture considering wormhole propagation

Description:
BACKGROUND OF THE INVENTION 
     Field of the Invention 
     The invention relates to the technical field of oil and gas development, in particular to a calculation method for dynamic fluid loss of acid-etched fracture considering wormhole propagation in the process of pad acid fracturing. 
     Description of Related Art 
     Carbonate reservoir is an important part of global oil and gas resources. More than half of the world&#39;s oil and gas reservoirs are carbonate reservoirs. The carbonate reservoir in China is also considerable, equivalent to 58.3 billion tons of crude oil. The acid fracturing, as the key technology of carbonate reservoir stimulation, has been widely applied to many oilfields at home and abroad. 
     The acid fracturing technologies used in carbonate reservoir stimulation include common acid fracturing and pad acid fracturing. The common acid fracturing is to directly use acid fluid to fracture the formation to produce fractures and etch the fracture wall. The pad acid fracturing is the technology to fracture the formation by inert fracturing fluid system with high viscosity to make artificial fractures, and then inject acid fluid into the fractures. 
     For reactive acid fluid system, the fluid will continuously etch the rock surface, so it cannot effectively form filter cake; at the same time, the acid fluid from the fracture to the bedrock will produce wormholes, and the wormholes will bring about more fluid loss, which is a self-amplification process, and may lead to excessive acid fluid loss. Therefore, the classical formula for calculating the fluid loss coefficient of inert fracturing fluid system is no longer suitable for calculating the fluid loss coefficient of acid fluid with reactive activity. 
     SUMMARY OF THE INVENTION 
     The purpose of the invention is to provide a calculation method for the dynamic fluid loss of acid-etched fracture considering wormhole propagation in the process of pad acid fracturing, in view of the state of the art that the formula for calculating the fluid loss coefficient of inert fracturing fluid system is no longer suitable for calculating the fluid loss coefficient of acid fluid with reactive activity. 
     The calculation method for the dynamic fluid loss of acid-etched fracture considering wormhole propagation in the process of pad acid fracturing, disclosed in the present invention, comprises the following steps: 
     Step 1: Dividing the construction time T of injecting acid fluid into artificial fracture into m time nodes at equal intervals, then the time step 
     
       
         
           
             
               Δ 
                
               
                   
               
                
               t 
             
             = 
             
               T 
               m 
             
           
         
       
     
     and t n =nΔt, where, n=0, 1, 2, 3, . . . , m, and to is the initial time, i.e., the time to start to inject the acid fluid; 
     Step 2: Calculating the fluid loss velocity v l (0) in the fracture at to; the acid fluid does not react with the rock at to, and the fluid loss velocity v l (0) is determined by the fluid loss velocity of the hydraulic fracture propagating to the last time node; the fluid loss velocity v l (0) is calculated by Formula (1) in Step 2: 
     
       
         
           
             
               
                 
                   
                     
                       
                         v 
                         l 
                       
                        
                       
                         ( 
                         0 
                         ) 
                       
                     
                     = 
                     
                       
                         2 
                          
                         
                           C 
                            
                           
                             ( 
                             
                               x 
                               , 
                               b 
                             
                             ) 
                           
                         
                       
                       
                         
                           b 
                           - 
                           τ 
                         
                       
                     
                   
                   ; 
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     wherein, C (x, t) is the fluid loss coefficient at x place in the fracture at t, in m/min 0.5 ; 
     b is the construction time of artificial fracture, in min; 
     τ is the time when the fluid reaches fracture x, in min. 
     Step 3: Calculating the flowing pressure distribution P(n) in the fracture at t n ; the calculation process is as follows: 
     The average velocity v x  and v y  of acid fluid flowing any point (x, y) in the fracture: 
     
       
         
           
             
               
                 
                   
                     
                       v 
                       x 
                     
                     = 
                     
                       
                         - 
                         
                           
                             w 
                             2 
                           
                           
                             1 
                              
                             2 
                              
                             
                               μ 
                               a 
                             
                           
                         
                       
                        
                       
                         
                           ∂ 
                           P 
                         
                         
                           ∂ 
                           x 
                         
                       
                     
                   
                   ; 
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
             
               
                 
                   
                     v 
                     y 
                   
                   = 
                   
                     
                       - 
                       
                         
                           w 
                           2 
                         
                         
                           1 
                            
                           2 
                            
                           
                             μ 
                             a 
                           
                         
                       
                     
                      
                     
                       
                         
                           ∂ 
                           P 
                         
                         
                           ∂ 
                           y 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     The amount of change in the acid fluid mass in the unit volume within a unit time is equal to the total inflow of acid fluid minus the total outflow, and then the mass conservation equation of the acid fluid in the fracture is worked out: 
     
       
         
           
             
               
                 
                   
                     
                       
                         - 
                         
                           
                             ∂ 
                             
                               ( 
                               
                                 
                                   v 
                                   x 
                                 
                                  
                                 w 
                               
                               ) 
                             
                           
                           
                             ∂ 
                             x 
                           
                         
                       
                       - 
                       
                         
                           ∂ 
                           
                             ( 
                             
                               
                                 v 
                                 y 
                               
                                
                               w 
                             
                             ) 
                           
                         
                         
                           ∂ 
                           y 
                         
                       
                       - 
                       
                         2 
                          
                         
                           v 
                           l 
                         
                       
                     
                     = 
                     
                       
                         ∂ 
                         w 
                       
                       
                         ∂ 
                         t 
                       
                     
                   
                   ; 
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     wherein, μ a  is the viscosity of acid fluid, in mPa·s; 
     P is pressure, in MPa; 
     v l  is the fluid loss velocity, in m/min; 
     w is the fracture width, in m; w is taken as the width w f  of artificial hydraulic fracture at t 0 , and taken as the width w a  of acid etched fracture at t n  (notes: where the subscript n is greater than 0). 
     If the time step Δt is small enough, it can be considered that the fluid loss velocity of acid fluid does not change within Δt, that is, the fluid loss velocity at t n−1  can be used within the period from t to t n . Therefore, in the calculation of the flowing pressure distribution P(n) at to, adopt the fluid loss velocity v l (n−1) at t n−1 , substitute the fluid loss velocity v l (n−1), Formula (2) and Formula (3) into Formula (4), and work out the flowing pressure distribution P(n) in the fracture. 
     Step 4: Calculating the width w a (n) of acid etched fracture at t n  in acid fracturing; the specific process is as follows: 
     The acid fluid concentration at each point in the fracture before acid injection is set as 0. In the process of acid fracturing, the injection rate of acid fluid is constant, there is no acid fluid flowing at the top and bottom of the fracture (i.e., at y=−H and y=H), the pressure at the fracture outlet (i.e., at x=L) is the formation pressure, and the acid fluid concentration at the fracture inlet is C f   0 . 
     The P(n) obtained in Step 3 is taken as the internal boundary condition, and the width w a (n) of acid-etched fracture at to is calculated by combining the initial condition formula (5) and the boundary condition formula (6). 
     Initial condition: 
     
       
         
           
             
               
                 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 P 
                                  
                                 
                                   ( 
                                   
                                     x 
                                     , 
                                     y 
                                   
                                   ) 
                                 
                               
                               = 
                               0 
                             
                             , 
                             
                               ∀ 
                               x 
                             
                             , 
                             y 
                             , 
                             
                               t 
                               = 
                               0 
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 
                                   C 
                                   f 
                                 
                                  
                                 
                                   ( 
                                   
                                     x 
                                     , 
                                     y 
                                   
                                   ) 
                                 
                               
                               = 
                               0 
                             
                             , 
                             
                               ∀ 
                               x 
                             
                             , 
                             y 
                             , 
                             
                               t 
                               = 
                               0 
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 
                                   w 
                                   a 
                                 
                                  
                                 
                                   ( 
                                   
                                     x 
                                     , 
                                     y 
                                   
                                   ) 
                                 
                               
                               = 
                               
                                 
                                   w 
                                   f 
                                 
                                  
                                 
                                   ( 
                                   
                                     x 
                                     , 
                                     y 
                                     , 
                                     end 
                                   
                                   ) 
                                 
                               
                             
                             , 
                             
                               ∀ 
                               x 
                             
                             , 
                             y 
                             , 
                             
                               t 
                               = 
                               0 
                             
                           
                         
                       
                     
                   
                   . 
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     Boundary condition: 
     
       
         
           
             
               
                 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 
                                   
                                     ∫ 
                                     
                                       - 
                                       H 
                                     
                                     H 
                                   
                                    
                                   
                                     
                                       
                                         w 
                                         a 
                                         3 
                                       
                                       
                                         12 
                                          
                                         
                                           μ 
                                           a 
                                         
                                       
                                     
                                      
                                     
                                       
                                         ∂ 
                                         P 
                                       
                                       
                                         ∂ 
                                         x 
                                       
                                     
                                   
                                 
                                  
                                 
                                   | 
                                   
                                     x 
                                     = 
                                     0 
                                   
                                 
                                  
                                 dy 
                               
                               = 
                               
                                 q 
                                 inj 
                               
                             
                             , 
                             
                               t 
                               &gt; 
                               0 
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 P 
                                  
                                 
                                   ( 
                                   
                                     l 
                                     , 
                                     y 
                                   
                                   ) 
                                 
                               
                               = 
                               
                                 P 
                                 e 
                               
                             
                             , 
                             
                               ∀ 
                               y 
                             
                             , 
                             
                               t 
                               &gt; 
                               0 
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 
                                   
                                     ∂ 
                                     P 
                                   
                                   Φ 
                                 
                                  
                                 
                                   | 
                                   
                                     y 
                                     = 
                                     
                                       - 
                                       H 
                                     
                                   
                                 
                               
                               = 
                               0 
                             
                             , 
                             
                               ∀ 
                               x 
                             
                             , 
                             
                               t 
                               &gt; 
                               0 
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 
                                   
                                     ∂ 
                                     P 
                                   
                                   Φ 
                                 
                                  
                                 
                                   | 
                                   
                                     y 
                                     = 
                                     H 
                                   
                                 
                               
                               = 
                               0 
                             
                             , 
                             
                               ∀ 
                               x 
                             
                             , 
                             
                               t 
                               &gt; 
                               0 
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 
                                   C 
                                   f 
                                 
                                  
                                 
                                   ( 
                                   
                                     0 
                                     , 
                                     y 
                                   
                                   ) 
                                 
                               
                               = 
                               
                                 C 
                                 f 
                                 0 
                               
                             
                             , 
                             
                               ∀ 
                               y 
                             
                             , 
                             
                               t 
                               &gt; 
                               0 
                             
                           
                         
                       
                     
                   
                   . 
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     Equation for reaction equilibrium in acid-etched fracture: 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             
                               
                                 ∂ 
                                 
                                   ( 
                                   
                                     
                                       C 
                                       f 
                                     
                                      
                                     
                                       v 
                                       x 
                                     
                                      
                                     
                                       w 
                                       a 
                                     
                                   
                                   ) 
                                 
                               
                               
                                 ∂ 
                                 x 
                               
                             
                             + 
                             
                               
                                 ∂ 
                                 
                                   ( 
                                   
                                     
                                       C 
                                       f 
                                     
                                      
                                     
                                       v 
                                       y 
                                     
                                      
                                     
                                       w 
                                       a 
                                     
                                   
                                   ) 
                                 
                               
                               
                                 ∂ 
                                 y 
                               
                             
                             + 
                           
                         
                       
                       
                         
                           
                             
                               2 
                                
                               
                                 C 
                                 f 
                               
                                
                               
                                 v 
                                 l 
                               
                             
                             + 
                             
                               2 
                                
                               
                                 
                                   k 
                                   g 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       C 
                                       f 
                                     
                                     - 
                                     
                                       C 
                                       w 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     - 
                     
                       
                         
                           ∂ 
                           
                             ( 
                             
                               
                                 C 
                                 f 
                               
                                
                               
                                 w 
                                 a 
                               
                             
                             ) 
                           
                         
                         
                           ∂ 
                           t 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     Equation for local reaction on fracture wall: 
         k   g ( C   f   −C   w )= R ( C   w )  (8).
 
     Equation for width change of acid-etched fracture: 
     
       
         
           
             
               
                 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       2 
                     
                      
                     
                       
                         
                           β 
                           i 
                         
                         
                           
                             ρ 
                             i 
                           
                            
                           
                             ( 
                             
                               1 
                               - 
                               φ 
                             
                             ) 
                           
                         
                       
                        
                       
                         ( 
                         
                           
                             2 
                              
                             η 
                              
                             
                               v 
                               l 
                             
                              
                             
                               C 
                               f 
                             
                           
                           + 
                           
                             2 
                              
                             
                               
                                 R 
                                 i 
                               
                                
                               
                                 ( 
                                 
                                   C 
                                   w 
                                 
                                 ) 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       
                         ∂ 
                         
                           w 
                           a 
                         
                       
                       
                         ∂ 
                         t 
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     Where: C f  is the acid fluid concentration in the center of fracture, in kmol/m 3 ; 
     q inj  is the injection rate of acid fluid, in m 3 /min; 
     P e  is the pressure at the fracture outlet, in MPa; 
     C f   0  is the acid fluid concentration at the fracture inlet, in kmol/m 3 ; 
     k g  is the mass transfer coefficient, in m/s; 
     C w  is the acid fluid concentration at fracture wall, in kmol/m 3 ; 
     R(C w ) is the corrosion rate of irreversible reaction in single step, in m·kmol/(s·m 3 ); 
     β i  is the solubility between acid fluid and limestone or dolomite, in kg/kmol; 
     ρ i  is the density of limestone or dolomite, in kg/m 3 ; 
     i is the subscript, indicating different rock types; 
     ϕ is the porosity, dimensionless; 
     η is the percentage of lost acid fluid that reacts with fracture wall rock; in most cases n≈0. 
     Step 5: Calculating the wormhole propagation and the fluid loss velocity v l (n) at t n ; the calculation process is as follows: 
     Formulas (11) to (13) are used to simulate the dynamic propagation of the wormhole, and then Formula (10) is substituted to calculate v. The fluid loss velocity v l (n) is equal to the fluid loss velocity v(x, y) on the fracture surface. 
     Equation for distribution of acid fluid concentration: 
     
       
         
           
             
               
                 
                   
                     
                       
                         ∂ 
                         
                           ( 
                           
                             φ 
                              
                             
                                 
                             
                              
                             
                               C 
                               f 
                             
                           
                           ) 
                         
                       
                       
                         ∂ 
                         t 
                       
                     
                     + 
                     
                       
                         ∂ 
                         
                           ∂ 
                           x 
                         
                       
                        
                       
                         ( 
                         
                           
                             v 
                             x 
                           
                            
                           
                             C 
                             f 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         ∂ 
                         
                           ∂ 
                           y 
                         
                       
                        
                       
                         ( 
                         
                           
                             v 
                             y 
                           
                            
                           
                             C 
                             f 
                           
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       
                         ∂ 
                         
                           ∂ 
                           x 
                         
                       
                        
                       
                         ( 
                         
                           φ 
                            
                           
                               
                           
                            
                           
                             D 
                             
                               e 
                                
                               x 
                             
                           
                            
                           
                             
                               ∂ 
                               
                                 C 
                                 f 
                               
                             
                             
                               ∂ 
                               x 
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         ∂ 
                         
                           ∂ 
                           y 
                         
                       
                        
                       
                         ( 
                         
                           φ 
                            
                           
                               
                           
                            
                           
                             D 
                             
                               e 
                                
                               y 
                             
                           
                            
                           
                             
                               ∂ 
                               
                                 C 
                                 f 
                               
                             
                             
                               ∂ 
                               y 
                             
                           
                         
                         ) 
                       
                     
                     - 
                     
                       
                         ∑ 
                         i 
                       
                        
                       
                         
                           
                             R 
                             i 
                           
                            
                           
                             ( 
                             
                               C 
                               w 
                             
                             ) 
                           
                         
                          
                         
                           
                             a 
                             vi 
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     Equation for reaction between acid fluid and rock: 
     
       
         
           
             
               
                 
                   
                     
                       R 
                       i 
                     
                      
                     
                       ( 
                       
                         C 
                         w 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           k 
                           ci 
                         
                          
                         
                           k 
                           si 
                         
                          
                         
                           γ 
                           
                             
                               H 
                               + 
                             
                             , 
                             s 
                           
                         
                       
                       
                         
                           k 
                           ci 
                         
                         + 
                         
                           
                             k 
                             si 
                           
                            
                           
                             γ 
                             
                               
                                 H 
                                 + 
                               
                               , 
                               s 
                             
                           
                         
                       
                     
                      
                     
                       
                         C 
                         f 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     Equation for volume change of different minerals: 
     
       
         
           
             
               
                 
                   
                     
                       
                         ∂ 
                         
                           V 
                           i 
                         
                       
                       
                         ∂ 
                         t 
                       
                     
                     = 
                     
                       - 
                       
                         
                           
                             M 
                             
                               a 
                                
                               c 
                                
                               i 
                                
                               d 
                             
                           
                            
                           
                             
                               R 
                               i 
                             
                              
                             
                               ( 
                               
                                 C 
                                 w 
                               
                               ) 
                             
                           
                            
                           
                             a 
                             vi 
                           
                            
                           
                             α 
                             i 
                           
                         
                         
                           ρ 
                           i 
                         
                       
                     
                   
                   . 
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     Equation for porosity change: 
     
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       φ 
                     
                     
                       ∂ 
                       t 
                     
                   
                   = 
                   
                     - 
                     
                       
                         ∑ 
                         i 
                       
                        
                       
                         
                           
                             ∂ 
                             
                               V 
                               i 
                             
                           
                           
                             ∂ 
                             t 
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     Where: D ex  is the effective propagation coefficient tensor in x direction, in m 2 /s; 
     D ey  is the effective propagation coefficient tensor in y direction, in m 2 /s; 
     R i (C w ) is the dissolution reaction rate between acid and different minerals, in kmol/s m 2 ; 
     a vi  is the surface area per unit volume of different minerals, in m 2 /m 3 ; 
     k si  is the reaction rate constant, in m/s; 
     k ci  is mass transfer coefficient, in m/s; 
     γ H+,s  is the activity coefficient of the acid fluid, dimensionless; 
     V i  is the volume fraction of the i th  mineral, dimensionless; 
     M acid  is the molar mass of the acid fluid, in kg/kmol; 
     α i  is the solubility of the acid fluid, in kg/kg; 
     ρ i  is the density of the i th  mineral, in kg/m 3 . 
     Step 6: Substituting the calculated fluid loss velocity v l (n) into Step 3 to calculate the pressure P(n+1) in the fracture at t n+1 , and then repeating Step 3 to Step 6 in turn to work out w a (n+1) and v l (n+1), and ending the repeated calculation when the injected volume of the acid fluid is equal to the set total volume. 
     In the above calculation method, the same symbols involved in all formulas have the same meaning. After being marked once, they are all common for all formulas. 
       FIG. 1  shows the flow chart of calculation of dynamic fluid loss of acid-etched fracture considering wormhole propagation of the present invention. 
     The inventor found that the existing patent CN201810704079.4 discloses a calculation method for dynamic comprehensive fluid loss coefficient of acid fracturing in fractured reservoir. Compared with the calculation methods in the existing patent and the present invention, there are the following differences: (1) The present invention aims at the artificial fractures produced before acid fluid injection and the initial pressure in the fracture is no longer the original formation pressure, but the published patent aims at the natural fracture and the initial pressure in the fracture is the original formation pressure. (2) In the present invention, the acid fluid loss considering the wormhole effect changes dynamically as the injection process continues, while in the published patent, only Steps (3) to (6) are repeated in the calculation process, and the acid fluid loss coefficient considering the wormhole effect is a constant value. (3) The present invention aims at the pad fracturing process, specifically forming artificial fractures first and then injecting acid fluid, while the published patent relates to natural fractures, and no artificial fractures or filter cake is produced, so the calculation method of fluid loss coefficient at the initial time is different. 
     The present invention has the following beneficial effects: 
     The calculation method of the present invention takes into account the influence of the dynamic propagation of the wormholes and the continuous change of the width of acid-etched fracture on the fluid loss velocity, so as to more accurately calculate the fluid loss during acid fracturing. It is of great significance for rational design of acid fluid volume, calculation of effective acting distance of acid fluid, and prediction of the stimulation effect of acid fracturing. 
     Other advantages, objectives and characteristics of the present invention will be partly embodied by the following description, and partly understood by those skilled in the art through research and practice of the present invention. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Aspects of the present invention are best understood from the following detailed description when read with the accompanying figures. It is noted that, in accordance with the standard practice in the industry, various features are not drawn to scale. In fact, the dimensions of the various features may be arbitrarily increased or reduced for clarity of discussion. 
         FIG. 1  is a flow chart of calculation of dynamic fluid loss of acid-etched fracture considering wormhole propagation. 
         FIG. 2  illustrates the shape and geometric dimensions of the artificial fracture before acid fluid injection. 
         FIG. 3  is a width profile of etched fracture. 
         FIG. 4  is a comparison curve between the fracture width after etching and the artificial fracture width before etching. 
         FIG. 5  is a profile of distribution of acid fluid concentration. 
         FIG. 6  is a curve showing the distribution of acid fluid concentration along the fracture length. 
         FIG. 7  is a curve showing the fluid loss velocity along the fracture length. 
     
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     In the following detailed description of the preferred embodiments of the present invention, reference is made to the accompanying drawings. It is to be understood that the preferred embodiments described herein are only used to illustrate and interpret the present invention and are not intended to limit the present invention. 
     Example 1 
     The pad acid fracturing technology was used in a carbonate reservoir, and the basic parameters measured are shown in Table 1. 
     
       
         
           
               
             
               
                 TABLE 1 
               
             
            
               
                   
               
               
                 List of Basic Parameters for Calculation in Example 1 
               
            
           
           
               
               
               
               
            
               
                 Object 
                 Parameters 
                 Unit 
                 Values 
               
               
                   
               
            
           
           
               
               
               
               
            
               
                 Target layer 
                 Stress 
                 MPa 
                 95 
               
               
                   
                 Fracture toughness 
                 MPa · m 0.5   
                 0.5 
               
               
                   
                 Elastic modulus 
                 MPa 
                 71660 
               
               
                   
                 Poisson&#39;s ratio 
                 — 
                 0.23 
               
               
                   
                 Permeability 
                 mD 
                 0.662 
               
               
                   
                 Temperature 
                 ° C. 
                 157 
               
               
                 Caprock/ 
                 Stress 
                 MPa 
                 105 
               
               
                 bottom layer 
                 Elastic modulus 
                 MPa 
                 50264 
               
               
                   
                 Poisson&#39;s ratio 
                 — 
                 0.25 
               
               
                 Acid fluid 
                 Injection rate 
                 m 3 /min 
                 4.5~5 
               
               
                   
                 Volume 
                 m 3   
                 220 
               
               
                   
                 Concentration 
                 wt % 
                 20 
               
               
                   
                 Type 
                 / 
                 Gelled acid 
               
               
                   
                 Reaction order 
                 — 
                 1.2289 
               
               
                   
                 Solubility 
                 kg/kg 
                 1.37 
               
               
                   
                 Activation energy 
                 J/mol 
                 5277 
               
               
                   
                 Mass transfer 
                 m 2 /s 
                 3.6 × 10 −10   
               
               
                   
                 coefficient 
               
               
                   
                 Frequency factor 
                 (mol/L) 1−m /s 
                 3.6985 × 10 −6       
               
               
                   
               
            
           
         
       
     
     Based on the data in Table 1, the artificial fractures produced by the fracturing with non-reactive inert pad fluid are shown in  FIG. 2 . The present invention is performed based on the artificial fracture. The fluid loss velocity v l (0) of the fracture at to in Step 2 in the present invention is calculated as follows: 
     The fracturing time (construction time b) of the pad fluid is 48 min. At the beginning of acid fluid injection, the time step is short and the fluid is mainly near the fracture, so the fluid loss coefficient C(x, b) is considered as the fluid loss coefficient at the fracture, i.e., 0.76×10 −3  m/min 0.5 ; τ is far less than b, and can be ignored and taken as 0. The fluid loss velocity is calculated by Formula (1) to be 0.22×10 −3  m/min. 
     The results calculated from Step 3 to Step 6 are reflected in form of cloud chart or irregular curve, and cannot be represented by a specific value or algebraic expression, so that Step 3 to Step 6 are directly repeated until the end of acid fluid injection. The final calculation result is shown in  FIGS. 3 to 7 . 
     The calculation method for the dynamic fluid loss of acid-etched fracture considering wormhole propagation in the process of pad acid fracturing, disclosed in the present invention, comprises the following steps: 
     Step 1: Dividing the construction time T of injecting acid fluid into artificial fracture into m time nodes at equal intervals, then the time step 
     
       
         
           
             
               Δ 
                
               
                   
               
                
               t 
             
             = 
             
               T 
               m 
             
           
         
       
     
     and t n =nΔt, where, n=0, 1, 2, 3, . . . , m, and to is the initial time, i.e., the time to start to inject the acid fluid; 
     Step 2: Calculating the fluid loss velocity v l (0) in the fracture at to; the acid fluid does not react with the rock at to, and the fluid loss velocity v l (0) is determined by the fluid loss velocity of the hydraulic fracture propagating to the last time node; the fluid loss velocity v l (0) is calculated by Formula (1) in Step 2: 
     
       
         
           
             
               
                 
                   
                     
                       
                         v 
                         l 
                       
                        
                       
                         ( 
                         0 
                         ) 
                       
                     
                     = 
                     
                       
                         2 
                          
                         
                           C 
                            
                           
                             ( 
                             
                               x 
                               , 
                               b 
                             
                             ) 
                           
                         
                       
                       
                         
                           b 
                           - 
                           τ 
                         
                       
                     
                   
                   ; 
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     wherein, C (x, t) is the fluid loss coefficient at x place in the fracture at t, in m/min 0.5 ; 
     b is the construction time of artificial fracture, in min; 
     τ is the time when the fluid reaches fracture x, in min. 
     Step 3: Calculating the flowing pressure distribution P(n) in the fracture at to; the calculation process is as follows: 
     The average velocity v x  and v y  of acid fluid flowing any point (x, y) in the fracture: 
     
       
         
           
             
               
                 
                   
                     
                       v 
                       x 
                     
                     = 
                     
                       
                         - 
                         
                           
                             w 
                             2 
                           
                           
                             1 
                              
                             2 
                              
                             
                               μ 
                               a 
                             
                           
                         
                       
                        
                       
                         
                           ∂ 
                           P 
                         
                         
                           ∂ 
                           x 
                         
                       
                     
                   
                   ; 
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
             
               
                 
                   
                     v 
                     y 
                   
                   = 
                   
                     
                       - 
                       
                         
                           w 
                           2 
                         
                         
                           1 
                            
                           2 
                            
                           
                             μ 
                             a 
                           
                         
                       
                     
                      
                     
                       
                         
                           ∂ 
                           P 
                         
                         
                           ∂ 
                           y 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     The amount of change in the acid fluid mass in the unit volume within a unit time is equal to the total inflow of acid fluid minus the total outflow, and then the mass conservation equation of the acid fluid in the fracture is worked out: 
     
       
         
           
             
               
                 
                   
                     
                       
                         - 
                         
                           
                             ∂ 
                             
                               ( 
                               
                                 
                                   v 
                                   x 
                                 
                                  
                                 w 
                               
                               ) 
                             
                           
                           
                             ∂ 
                             x 
                           
                         
                       
                       - 
                       
                         
                           ∂ 
                           
                             ( 
                             
                               
                                 v 
                                 y 
                               
                                
                               w 
                             
                             ) 
                           
                         
                         
                           ∂ 
                           y 
                         
                       
                       - 
                       
                         2 
                          
                         
                           v 
                           l 
                         
                       
                     
                     = 
                     
                       
                         ∂ 
                         w 
                       
                       
                         ∂ 
                         t 
                       
                     
                   
                   ; 
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     wherein, μ a  is the viscosity of acid fluid, in mPa·s; 
     P is pressure, in MPa; 
     v l  is the fluid loss velocity, in m/min; 
     w is the fracture width, in m; w is taken as the width w f  of artificial hydraulic fracture at t 0 , and taken as the width w a  of acid etched fracture at to (notes: where the subscript n is greater than 0). 
     If the time step Δt is small enough, it can be considered that the fluid loss velocity of acid fluid does not change within Δt, that is, the fluid loss velocity at t n−1  can be used within the period from t n−1  to t n . Therefore, in the calculation of the flowing pressure distribution P(n) at to, adopt the fluid loss velocity v l (n−1) at t n−1 , substitute the fluid loss velocity v l (n−1), Formula (2) and Formula (3) into Formula (4), and work out the flowing pressure distribution P(n) in the fracture. 
     Step 4: Calculating the width w a (n) of acid etched fracture at T n  in acid fracturing; the specific process is as follows: 
     The acid fluid concentration at each point in the fracture before acid injection is set as 0. In the process of acid fracturing, the injection rate of acid fluid is constant, there is no acid fluid flowing at the top and bottom of the fracture (i.e., at y=−H and y=H), the pressure at the fracture outlet (i.e., at x=L) is the formation pressure, and the acid fluid concentration at the fracture inlet is C f   0 . 
     The P(n) obtained in Step 3 is taken as the internal boundary condition, and the width w a (n) of acid-etched fracture at t n  is calculated by combining the initial condition formula (5) and the boundary condition formula (6). 
     Initial condition: 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               
                                 P 
                                  
                                 
                                   ( 
                                   
                                     x 
                                     , 
                                     y 
                                   
                                   ) 
                                 
                               
                               = 
                               0 
                             
                             , 
                             
                               ∀ 
                               x 
                             
                             , 
                             y 
                             , 
                             
                               t 
                               = 
                               0 
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 
                                   C 
                                   f 
                                 
                                  
                                 
                                   ( 
                                   
                                     x 
                                     , 
                                     y 
                                   
                                   ) 
                                 
                               
                               = 
                               0 
                             
                             , 
                             
                               ∀ 
                               x 
                             
                             , 
                             y 
                             , 
                             
                               t 
                               = 
                               0 
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 
                                   w 
                                   a 
                                 
                                  
                                 
                                   ( 
                                   
                                     x 
                                     , 
                                     y 
                                   
                                   ) 
                                 
                               
                               = 
                               
                                 
                                   w 
                                   f 
                                 
                                  
                                 
                                   ( 
                                   
                                     x 
                                     , 
                                     y 
                                     , 
                                     end 
                                   
                                   ) 
                                 
                               
                             
                             , 
                             
                                 
                             
                              
                             
                               ∀ 
                               x 
                             
                             , 
                             y 
                             , 
                             
                               t 
                               = 
                               0 
                             
                           
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     Boundary condition: 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               
                                 
                                   
                                     ∫ 
                                     
                                       - 
                                       H 
                                     
                                     H 
                                   
                                    
                                   
                                     
                                       
                                         w 
                                         a 
                                         3 
                                       
                                       
                                         12 
                                          
                                         
                                           μ 
                                           a 
                                         
                                       
                                     
                                      
                                     
                                       
                                         ∂ 
                                         P 
                                       
                                       
                                         ∂ 
                                         x 
                                       
                                     
                                   
                                 
                                  
                                 
                                   | 
                                   
                                     x 
                                     = 
                                     0 
                                   
                                 
                                  
                                 dy 
                               
                               = 
                               
                                 q 
                                 inj 
                               
                             
                             , 
                             
                               t 
                               &gt; 
                               0 
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 P 
                                  
                                 
                                   ( 
                                   
                                     l 
                                     , 
                                     y 
                                   
                                   ) 
                                 
                               
                               = 
                               
                                 P 
                                 e 
                               
                             
                             , 
                             
                               ∀ 
                               y 
                             
                             , 
                             
                               t 
                               &gt; 
                               0 
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 
                                   
                                     ∂ 
                                     P 
                                   
                                   
                                     ∂ 
                                     y 
                                   
                                 
                                  
                                 
                                   | 
                                   
                                     y 
                                     = 
                                     
                                       - 
                                       H 
                                     
                                   
                                 
                               
                               = 
                               0 
                             
                             , 
                             
                               ∀ 
                               x 
                             
                             , 
                             
                               t 
                               &gt; 
                               0 
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 
                                   
                                     ∂ 
                                     P 
                                   
                                   
                                     ∂ 
                                     y 
                                   
                                 
                                  
                                 
                                   | 
                                   
                                     y 
                                     = 
                                     H 
                                   
                                 
                               
                               = 
                               0 
                             
                             , 
                             
                               ∀ 
                               x 
                             
                             , 
                             
                               t 
                               &gt; 
                               0 
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 
                                   C 
                                   f 
                                 
                                  
                                 
                                   ( 
                                   
                                     0 
                                     , 
                                     y 
                                   
                                   ) 
                                 
                               
                               = 
                               
                                 C 
                                 f 
                                 0 
                               
                             
                             , 
                             
                               ∀ 
                               y 
                             
                             , 
                             
                               t 
                               &gt; 
                               0 
                             
                           
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     Equation for reaction equilibrium in acid-etched fracture: 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             
                               
                                 ∂ 
                                 
                                   ( 
                                   
                                     
                                       C 
                                       f 
                                     
                                      
                                     
                                       v 
                                       x 
                                     
                                      
                                     
                                       w 
                                       a 
                                     
                                   
                                   ) 
                                 
                               
                               
                                 ∂ 
                                 x 
                               
                             
                             + 
                             
                               
                                 ∂ 
                                 
                                   ( 
                                   
                                     
                                       C 
                                       f 
                                     
                                      
                                     
                                       v 
                                       y 
                                     
                                      
                                     
                                       w 
                                       a 
                                     
                                   
                                   ) 
                                 
                               
                               
                                 ∂ 
                                 y 
                               
                             
                             + 
                           
                         
                       
                       
                         
                           
                             
                               2 
                                
                               
                                 C 
                                 f 
                               
                                
                               
                                 v 
                                 l 
                               
                             
                             + 
                             
                               2 
                                
                               
                                 
                                   k 
                                   g 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       C 
                                       f 
                                     
                                     - 
                                     
                                       C 
                                       w 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     - 
                     
                       
                         
                           ∂ 
                           
                             ( 
                             
                               
                                 C 
                                 f 
                               
                                
                               
                                 w 
                                 a 
                               
                             
                             ) 
                           
                         
                         
                           ∂ 
                           t 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     Equation for local reaction on fracture wall: 
         k   g ( C   f   −C   w )= R ( C   w )  (8).
 
     Equation for width change of acid-etched fracture: 
     
       
         
           
             
               
                 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       2 
                     
                      
                     
                       
                         
                           β 
                           i 
                         
                         
                           
                             ρ 
                             i 
                           
                            
                           
                             ( 
                             
                               1 
                               - 
                               φ 
                             
                             ) 
                           
                         
                       
                        
                       
                         ( 
                         
                           
                             2 
                              
                             η 
                              
                             
                               v 
                               l 
                             
                              
                             
                               C 
                               f 
                             
                           
                           + 
                           
                             2 
                              
                             
                               
                                 R 
                                 i 
                               
                                
                               
                                   
                               
                               ( 
                               
                                 C 
                                 w 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       
                         ∂ 
                         
                           w 
                           a 
                         
                       
                       
                         ∂ 
                         t 
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     Where: C f  is the acid fluid concentration in the center of fracture, in kmol/m 3 ; 
     q inj  is the injection rate of acid fluid, in m 3 /min; 
     P e  is the pressure at the fracture outlet, in MPa; 
     C f   0  is the acid fluid concentration at the fracture inlet, in kmol/m 3 ; 
     k g  is the mass transfer coefficient, in m/s; 
     C w  is the acid fluid concentration at fracture wall, in kmol/m 3 ; 
     R(C w ) is the corrosion rate of irreversible reaction in single step, in m·kmol/(s·m 3 ); 
     β i  is the solubility between acid fluid and limestone or dolomite, in kg/kmol; 
     ρ i  is the density of limestone or dolomite, in kg/m 3 ; 
     i is the subscript, indicating different rock types; 
     ϕ is the porosity, dimensionless; 
     η is the percentage of lost acid fluid that reacts with fracture wall rock; in most cases η≈0. 
     Step 5: Calculating the wormhole propagation and the fluid loss velocity v l (n) at t n ; the calculation process is as follows: 
     Formulas (11) to (13) are used to simulate the dynamic propagation of the wormhole, and then Formula (10) is substituted to calculate v. The fluid loss velocity v l (n) is equal to the fluid loss velocity v(x, y) on the fracture surface. 
     Equation for distribution of acid fluid concentration: 
     
       
         
           
             
               
                 
                   
                     
                       
                         ∂ 
                         
                           ( 
                           
                             φ 
                              
                             
                                 
                             
                              
                             
                               C 
                               f 
                             
                           
                           ) 
                         
                       
                       
                         ∂ 
                         t 
                       
                     
                     + 
                     
                       
                         ∂ 
                         
                           ∂ 
                           x 
                         
                       
                        
                       
                         ( 
                         
                           
                             v 
                             x 
                           
                            
                           
                             C 
                             f 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         ∂ 
                         
                           ∂ 
                           y 
                         
                       
                        
                       
                         ( 
                         
                           
                             v 
                             y 
                           
                            
                           
                             C 
                             f 
                           
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       
                         ∂ 
                         
                           ∂ 
                           x 
                         
                       
                        
                       
                         ( 
                         
                           φ 
                            
                           
                               
                           
                            
                           
                             D 
                             
                               e 
                                
                               x 
                             
                           
                            
                           
                             
                               ∂ 
                               
                                 C 
                                 f 
                               
                             
                             
                               ∂ 
                               x 
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         ∂ 
                         
                           ∂ 
                           y 
                         
                       
                        
                       
                         ( 
                         
                           φ 
                            
                           
                               
                           
                            
                           
                             D 
                             
                               e 
                                
                               y 
                             
                           
                            
                           
                             
                               ∂ 
                               
                                 C 
                                 f 
                               
                             
                             
                               ∂ 
                               y 
                             
                           
                         
                         ) 
                       
                     
                     - 
                     
                       
                         ∑ 
                         i 
                       
                        
                       
                         
                           
                             R 
                             i 
                           
                            
                           
                             ( 
                             
                               C 
                               w 
                             
                             ) 
                           
                         
                          
                         
                           
                             a 
                             vi 
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     Equation for reaction between acid fluid and rock: 
     
       
         
           
             
               
                 
                   
                     
                       R 
                       i 
                     
                      
                     
                       ( 
                       
                         C 
                         w 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           
                             k 
                             ci 
                           
                            
                           
                             k 
                             si 
                           
                            
                           
                             γ 
                             
                               
                                 H 
                                 + 
                               
                               , 
                               s 
                             
                           
                         
                         
                           
                             k 
                             ci 
                           
                           + 
                           
                             
                               k 
                               si 
                             
                              
                             
                               γ 
                               
                                 
                                   H 
                                   + 
                                 
                                 , 
                                 s 
                               
                             
                           
                         
                       
                        
                       
                         C 
                         f 
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     Equation for volume change of different minerals: 
     
       
         
           
             
               
                 
                   
                     
                       
                         ∂ 
                         
                           V 
                           i 
                         
                       
                       
                         ∂ 
                         t 
                       
                     
                     = 
                     
                       - 
                       
                         
                           
                             M 
                             
                               a 
                                
                               c 
                                
                               i 
                                
                               d 
                             
                           
                            
                           
                             
                               R 
                               i 
                             
                              
                             
                               ( 
                               
                                 C 
                                 w 
                               
                               ) 
                             
                           
                            
                           
                             a 
                             vi 
                           
                            
                           
                             α 
                             i 
                           
                         
                         
                           ρ 
                           i 
                         
                       
                     
                   
                   . 
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     Equation for porosity change: 
     
       
         
           
             
               
                 
                   
                     
                       
                         ∂ 
                         φ 
                       
                       
                         ∂ 
                         t 
                       
                     
                     = 
                     
                       - 
                       
                         
                           ∑ 
                           i 
                         
                          
                         
                           
                             ∂ 
                             
                               V 
                               i 
                             
                           
                           
                             ∂ 
                             t 
                           
                         
                       
                     
                   
                   . 
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     Where: D ex  is the effective propagation coefficient tensor in x direction, in m 2 /s; 
     D ey  is the effective propagation coefficient tensor in y direction, in m 2 /s; 
     R i (C w ) is the dissolution reaction rate between acid and different minerals, in kmol/s m 2 ; 
     a vi  is the surface area per unit volume of different minerals, in m 2 /m 3 ; 
     k si  is the reaction rate constant, in m/s; 
     k ci  is mass transfer coefficient, in m/s; 
     γ H+,s  is the activity coefficient of the acid fluid, dimensionless; 
     V i  is the volume fraction of the i th  mineral, dimensionless; 
     M acid  is the molar mass of the acid fluid, in kg/kmol; 
     α i  is the solubility of the acid fluid, in kg/kg; 
     ρ i  is the density of the i th  mineral, in kg/m 3 . 
     Step 6: Substituting the calculated fluid loss velocity v l (n) into Step 3 to calculate the pressure P(n+1) in the fracture at t n+1 , and then repeating Step 3 to Step 6 in turn to work out w a (n+1) and v l (n+1), and ending the repeated calculation when the injected volume of the acid fluid is equal to the set total volume. 
     In the above calculation method, the same symbols involved in all formulas have the same meaning. After being marked once, they are all common for all formulas. 
       FIG. 1  shows the flow chart of calculation of dynamic fluid loss of acid-etched fracture considering wormhole propagation of the present invention. 
     The inventor found that the existing patent CN201810704079.4 discloses a calculation method for dynamic comprehensive fluid loss coefficient of acid fracturing in fractured reservoir. Compared with the calculation methods in the existing patent and the present invention, there are the following differences: (1) The present invention aims at the artificial fractures produced before acid fluid injection and the initial pressure in the fracture is no longer the original formation pressure, but the published patent aims at the natural fracture and the initial pressure in the fracture is the original formation pressure. (2) In the present invention, the acid fluid loss considering the wormhole effect changes dynamically as the injection process continues, while in the published patent, only Steps (3) to (6) are repeated in the calculation process, and the acid fluid loss coefficient considering the wormhole effect is a constant value. (3) The present invention aims at the pad fracturing process, specifically forming artificial fractures first and then injecting acid fluid, while the published patent relates to natural fractures, and no artificial fractures or filter cake is produced, so the calculation method of fluid loss coefficient at the initial time is different. 
     The present invention has the following beneficial effects: 
     The calculation method of the present invention takes into account the influence of the dynamic propagation of the wormholes and the continuous change of the width of acid-etched fracture on the fluid loss velocity, so as to more accurately calculate the fluid loss during acid fracturing. It is of great significance for rational design of acid fluid volume, calculation of effective acting distance of acid fluid, and prediction of the stimulation effect of acid fracturing. 
     The above are only the preferred embodiments of the present invention, and are not intended to limit the present invention in any form. Although the present invention has been disclosed as above with the preferred embodiments, it is not intended to limit the present invention. Those skilled in the art, within the scope of the technical solution of the present invention, can use the disclosed technical content to make a few changes or modify the equivalent embodiment with equivalent changes. Within the scope of the technical solution of the present invention, any simple modification, equivalent change and modification made to the above embodiments according to the technical essence of the present invention, are still regarded as a part of the technical solution of the present invention.