Simulating Fluid Leak-Off and Flow-Back in a Fractured Subterranean Region

Abstract

In some aspects, a set of governing flow equations can be defined for a one-dimensional flow model representing well system fluid in a fractured subterranean region. A first subset of the governing flow equations can represent flow within a fracture, and a second subset of the governing flow equations can represent flow within a reservoir medium adjacent to the fracture. A reduced set of governing flow equations can be generated by eliminating the second subset from the set of governing flow equations based on fluid coupling between the fracture and the reservoir medium. Well system fluid flow in the fractured subterranean region can be simulated based on the reduced set of governing flow equations.

1 . A fluid flow modeling method comprising:
 defining a set of governing flow equations for a one-dimensional flow model representing well system fluid flow in a fractured subterranean region, a first subset of the governing flow equations representing flow within a fracture, a second subset of the governing flow equations representing flow within a reservoir medium adjacent to the fracture;   generating a reduced set of governing flow equations by eliminating the second subset from the set of governing flow equations based on fluid coupling between the fracture and the reservoir medium; and   simulating, by operation of a computer system, well system fluid flow in the fractured subterranean region based on the reduced set of governing flow equations.   
     
     
         2 . The method of  claim 1 , comprising modeling flow within the fracture by a fracture flow model that includes the first subset of the governing flow equations, and modeling flow within the reservoir medium by a flow-back model that includes the second subset of the governing flow equations, wherein boundary conditions of the flow-back model couple the fracture flow model with the flow-back model. 
     
     
         3 . The method of  claim 1 , wherein simulating well system fluid flow includes:
 computing values of a flow variable for locations within the fracture by solving the reduced set of governing flow equations; and   computing values of the flow variable for locations within the reservoir medium based on the values of the flow variable within the fracture and the coupling between the fracture and the reservoir medium.   
     
     
         4 . The method of  claim 1 , wherein generating the reduced set of governing flow equations reduces the number of unknown values of a solution variable in the one-dimensional flow model. 
     
     
         5 . The method of  claim 1 , comprising generating the reduced set of governing flow equations according to a matrix decomposition of the second subset of the governing flow equations. 
     
     
         6 . The method of  claim 5 , wherein the matrix decomposition represents the second subset of the governing flow equations by a product of a lower triangular matrix and a transpose of the lower triangular matrix. 
     
     
         7 . The method of  claim 5 , wherein the well system fluid includes multiple constituent fluids, and the matrix decomposition represents the second subset of the governing flow equations by a product of a lower triangular matrix and an upper triangular matrix. 
     
     
         8 . The method of  claim 1 , wherein:
 the one-dimensional flow model includes fracture nodes representing locations of fluid flow within the fracture and flow-back nodes representing locations of fluid flow within the reservoir medium;   the first subset of the governing flow equations are discretized at the fracture nodes; and   the second subset of the governing flow equations are discretized at the flow-back nodes.   
     
     
         9 . The method of  claim 8 , wherein the fracture nodes represent a fracture flow path, and the flow-back nodes represent branches from the fracture flow path. 
     
     
         10 . A non-transitory computer-readable medium storing instructions that, when executed by data processing apparatus, perform operations comprising:
 defining a set of governing flow equations for a one-dimensional flow model representing well system fluid flow in a fractured subterranean region, a first subset of the governing flow equations representing flow within a fracture, a second subset of the governing flow equations representing flow within a reservoir medium adjacent to the fracture;   generating a reduced set of governing flow equations by eliminating the second subset from the set of governing flow equations based on fluid coupling between the fracture and the reservoir medium; and   simulating well system fluid flow in the fractured subterranean region based on the reduced set of governing flow equations   
     
     
         11 . The computer-readable medium of  claim 10 , wherein the second subset of governing flow equations are based on the expression:
   α    p     t   −β  p     yy =0,
   
       where  p  represents fluid pressure in a flow-back direction within the reservoir medium, α and β represent flow parameters of a flow-back model,  p   t  represents a partial derivative of  p  with respect to time, and  p   yy  represents a second partial derivative of  p  with respect to distance along a simulated flow path in the reservoir medium. 
     
     
         12 . The computer-readable medium of  claim 10 , wherein the first subset of the governing flow equations are based on the expression: 
       
         
           
             
               
                 
                   
                     A 
                     t 
                   
                   - 
                   
                     
                       ( 
                       
                         
                           
                             A 
                             3 
                           
                           
                             12 
                              
                             
                                 
                             
                              
                             μ 
                           
                         
                          
                         
                           p 
                           x 
                         
                       
                       ) 
                     
                     x 
                   
                   + 
                   
                     β 
                      
                     
                         
                     
                      
                     
                       
                         p 
                         _ 
                       
                       y 
                     
                   
                 
                 = 
                 0 
               
               , 
             
           
         
       
       where A represents a cross-sectional area of the fracture, A t  represents a partial derivative of A with respect to time, p represents fluid pressure in the fracture, the x subscript represents a partial derivative with respect to distance along a simulated flow path in the fracture, μ represents a fluid viscosity coefficient,  p  represents fluid pressure in a flow-back direction within the reservoir medium, and  p   y  represents a partial derivative of  p  with respect to distance along a simulated flow path in the reservoir medium. 
     
     
         13 . The computer-readable medium of  claim 10 , wherein generating the reduced set of governing flow equations reduces the number of unknown values of a solution variable in the one-dimensional flow model. 
     
     
         14 . The computer-readable medium of  claim 10 , the operations comprising generating the reduced set of governing flow equations according to a matrix decomposition of the second subset of the governing flow equations. 
     
     
         15 . The computer-readable medium of  claim 14 , wherein the matrix decomposition represents the second subset of the governing flow equations by a product of a lower triangular matrix and a transpose of the lower triangular matrix. 
     
     
         16 . The computer-readable medium of  claim 14 , wherein the well system fluid includes multiple constituent fluids, and the matrix decomposition represents the second subset of the governing flow equations by a product of a lower triangular matrix and an upper triangular matrix. 
     
     
         17 . A fluid flow modeling system comprising one or more computers that include:
 memory operable to store flow model data associated with a one-dimensional flow model for well system fluid, the one-dimensional flow model including a set of governing flow equations, a first subset of the governing flow equations representing flow within a fracture in a subterranean region, a second subset of the governing flow equations representing flow within a medium adjacent to the fracture; and   data processing apparatus operable to:
 generate a reduced set of governing flow equations by eliminating the second subset from the set of governing flow equations based on fluid coupling between the fracture and the reservoir medium; and 
 simulate well system fluid flow in the fractured subterranean region based on the reduced set of governing flow equations. 
   
     
     
         18 . The fluid flow modeling system of  claim 17 , wherein the reduced set of governing flow equations is generated according to a matrix decomposition of the second subset of the governing flow equations. 
     
     
         19 . The fluid flow modeling system of  claim 18 , wherein the matrix decomposition represents the second subset of the governing flow equations by a product of a lower triangular matrix and a transpose of the lower triangular matrix. 
     
     
         20 . The fluid flow modeling system of  claim 18 , wherein the well system fluid includes multiple constituent fluids, and the matrix decomposition represents the second subset of the governing flow equations by a product of a lower triangular matrix and an upper triangular matrix.