Based on the theory of the extended finite element method (XFEM), which was first proposed by Moës for dealing with the problem characterized by discontinuities, an extended finite element model for predicting productivity of multifractured horizontal well has been established. The model couples four main porous flow regimes, including fluid flow in the awayfromwellbore region of reservoir matrix, radial flow in the nearwellbore region of reservoir matrix, linear flow in the awayfromwellbore region of fracture, and radial flow in the nearwellbore region of fracture by considering mass transfer between fracture and matrix. The method to introduce the interior well boundary condition into the XFEM is proposed, and therefore the model can be highly adaptable to the complex and asymmetrical physical conditions. Case studies indicate that this kind of multiflow problems can be solved with high accuracy by the use of the XFEM.
Prediction of productivity of multifractured horizontal wells has always been a research hotspot. Various methods including the analytical method [
In recent decades, the XFEM has been continuously developing and already has become an important and effective technique for the analysis of the problems characterized by discontinuities, singularities, and complex geometries, such as modeling crack growth and complex flow in fractured reservoir [
Naturally, the XFEM was developed for simulation of strong discontinuities in fracture mechanics. It was later extended to solve problems for weak discontinuity such as interface problems and fluid flow in fractured reservoir. Sukumar et al. presented a method that coupled the level set method to the XFEM to predict the weak discontinuity due to material interfaces in composite [
This paper aims to establish an extended finite element model for predicting of productivity of multifractured horizontal well, which couples fluid flow in the awayfromwellbore region of the porous matrix, radial flow in the nearwellbore region of the matrix, linear flow in the awayfromwellbore region of fracture, and radial flow in the nearwellbore region of fracture by considering mass transfer between fracture and matrix. The method to load the interior well boundary condition is also proposed, and therefore the model can be completely adaptable in the complex and asymmetrical physical conditions.
In this section we demonstrate the derivation of the governing equations for fluid flow in multifractured horizontal well. Firstly, we sketch the physical model and the main assumptions.
Consider a twodimensional rectangular domain
Fluid flow in hydraulically multifractured reservoir.
Fluid flow in fractured horizontal well involves four main kinds of flow mechanisms, including fluid porous flow in the awayfromwellbore region of reservoir matrix (main reservoir flow), radial flow in the nearwellbore region of reservoir matrix (reservoir radial flow), linear flow in the awayfromwellbore region of fracture (fracture linear flow), and radial flow in the nearwellbore region of fracture (fracture radial flow). Different flow regimes are coupled by considering mass transfer between fracture and matrix (Figures
The local Cartesian coordinate systems in the fracture.
Flow regimes in the hydraulically fractured reservoir.
The essential boundary conditions are imposed on the external and the internal boundary of the model as
Mass transfer is considered to couple the fluid flow in the fracture and the porous matrix. Regarding this, the complementary boundary condition is written as
(a) Fluid flow in the physical model is in steady state; (b) fracture radial flow exists within a circle in the vertical plane with the center at the wellbore and diameter equal to reservoir thickness; (c) reservoir radial flow region exists in the nearwellbore region surrounding the nearwellbore fracture, and its range is similar to fracture radial flow; (d) the fluid pressure within the horizontal wellbore keeps constant; (e) the fluid pressure within the fracture keeps constant along the inward direction that is perpendicular to the fracture line, while the pressure gradient and the fluid flow are discontinuous.
Based on the forth main assumption, this paper uses the same pressure field and the same mesh to approximate the fluid pressure in both the fracture and the matrix. This treatment is different from that in [
In order to derive the weak form of the governing equations, we must first constitute the respective continuity equations of the four flow regimes as the strongform equations.
The continuity equation for the porous flow in the awayfromwellbore region of the reservoir matrix is written as
The continuity equation for the linear flow in the awayfromwellbore region of the fracture is given as follows
The continuity equation for the radial flow in the nearwellbore region of the fracture can be written as follows:
Similar to (
The symbol
Based on the strongform equations of the four flow regimes, we can derive the weak form equations and then couple them to get the final two kinds of governing equations. There are four kinds of weakform equations and the derivation processes of the first two are similar to that in [
The weak form of the continuity equation of fluid flow in the awayfromwellbore region of the porous medium is given by
Divergence theorem is applied in the derivation of (
The weak form of the continuity equation of fluid flow in the awayfromwellbore region of the fracture is given by
Equation (
Similar to the derivation of (
In the nearwellbore region of the reservoir matrix surrounding the fracture radial flow region, the weak form equation of fluid flow can be derived by reference to the derivation of (
Couple the reservoir radial flow and the fracture radial flow by substituting (
Here we refer to the element with the wellbore located on its edge as “well element.” Equation (
The fluid flow jump across the fracture line reflects the feature of weak discontinuity which means that the pressure field is continuous, whereas its gradient is discontinuous. We can use the absolute value of the signed distance function to enrich the classical finite element approximation [
The fluid pressure is approximated as the linear combination of the standard and enriched shape function as
The shape enrichment function and its gradient are as follows:
In the XFEM theory, split nodes are defined as nodes with their support intersecting the fracture. According to the number of the split nodes that an element contains, all the elements can be subsumed under three types, including the standard finite element, the split element, and the blending element. The quadrature rule differs by the element type.
To cope with the standard finite element, we directly use the standard secondorder Gauss quadrature rule, and there are 4 Gauss points per element. With regard to the split element, we first divide the element into triangular subdomains, then apply Hammer quadrature rule to every subdomain, and finally add together the integration of all the Gauss points in the element. Each split element generally contains 104 Gauss points.
In order to avoid the decrease in the convergence rate of the blending element, we adopt the method of constructing the shifted enrichment function [
In the context of the XFEM, the mesh no longer needs to match the geometry of the fracture. So if the whole fracture is located in the interior of the domain, we cannot load the internal boundary condition as simply as the outer boundary condition. Here, a solution based on the method of Lagrangian multiplier is put forward [
Substitute the new operator
Go on to substitute the original functional
In the specific implementation process, in order to reduce the difficult of numerical simulation, the wellbore is located on the element edge.
The total productivity of multifractured horizontal well can be predicted by adding together the flow of the different fractures:
Skin factor
In this section, Case
A horizontal well with three vertical hydraulic fractures (Shuiping Well 1) is producing oil on the condition of constant wellbore flowing pressure. The actual flow rate of the well is 6.52 m^{3}/d with a pressure drawdown of 10.14 MPa after 12 months of production. The reservoir properties and well data for case 1 are summarized in Table
Reported reservoir and well data (Case
Name of parameters (unit)  Value 

Reservoir thickness (m)  14 
Length in the 
600 
Length in the 
400 
Matrix permeability (mD)  1.3 
Fracture permeability (D)  30 
Oil viscosity (mPa·s)  3.5 
Oil volume factor  1.13 
Wellbore radius (m)  0.068 
Skin factor  0 
Half fracture lengths  
Fracture 1 (m)  127.5 
Fracture 2 (m)  105.0 
Fracture 3 (m)  107.5 
Fracture width (m)  0.005 
Average distance between fractures (m)  95.4 
Pressure on the external boundary (MPa)  16.41 
Bottomhole flowing pressure (MPa)  6.2742 
The details of the different meshes and the corresponding solutions are shown in Table
Comparison between the numerical results for different meshes.
Key parameters and results  Plan A  Plan B  Plan C  Plan D 

Mesh strategy  Structured, uniform  Structured, uniform  Structured, uniform  Unstructured 
Number of elements 




Average side length of well element (m) 




Calculated bottomhole flowing pressure (MPa) 




Error of bottomhole flowing pressure ( 




Calculated production rate (m^{3}/d) 




The pressure distribution in the steady state (under Plan D) is shown in Figure
Matrix pore pressure distribution.
The distribution of the xcomponent of pressure gradient.
A horizontal well with four vertical hydraulic fractures (Maoping Well 1) is producing oil on the condition of constant wellbore flowing pressure. The actual flow rate of the well is 20.35 m^{3}/d when the pressure drawdown is 9.1 MPa. The reservoir properties and well data for case 2 are summarized in Table
Reported reservoir and well data (Case
Name of parameters (unit)  Value 

Reservoir thickness (m)  11.9 
Length in the 
700 
Length in the 
260 
Matrix permeability (mD)  7.5 
Fracture permeability (D)  30 
Oil viscosity (mPa·s)  4.8 
Oil volume factor  1.084 
Wellbore radius (m)  0.058 
Skin factor  0 
Half fracture length (m)  75 
Fracture width (m)  0.005 
Average distance between fractures (m)  110.95 
Pressure on the external boundary (MPa)  17.94 
Bottomhole flowing pressure (MPa)  8.82 
The calculated production rate of Maoping well 1 based on unstructured mesh is 20.90 m^{3}/d at the same pressure drawdown, and the relative error is 2.7%. In Case
In this coupled model, the accuracy is closely related to the size of the elements surrounding wellbore. Due to the relatively large size of the well element, the numerical results under Plan A are completely wrong. With the mesh from coarse to fine (from left to right in Table
Comparison between the numerical results for different meshes.
Key parameters and results  Plan A  Plan B  Plan C  Plan D 

Mesh strategy  Structured, uniform  Structured, uniform  Structured, locally refined  Unstructured 
Number of elements 




Average side length of well element (m) 




Calculated bottomhole flowing pressure (MPa) 




Error of bottomhole flowing pressure ( 




Calculated production rate (m^{3}/d) 




The pressure distribution (under Plan D) of Case
Matrix pore pressure distribution.
The distribution of the
An asymmetrical model is established to analyze the flow regime and the pressure distribution in the hydraulically fractured reservoir. The computation parameters are basically the same as those reported by Li et al. [
Reservoir and well data (Case
Fracture parameter  Fracture 1  Fracture 2 

Location of wellbore  (−100 m, 52 m)  (50 m, 0 m) 
Length of the upper wing  80 m  80 m 
Length of the lower wing  100 m  80 m 
Conductivity  infinite  infinite 
The mesh of the extended finite element model.
The asymmetrical pressure distribution.
Based on the above, the following conclusions can be drawn.
An effective extended finite element model has been formulated for prediction of productivity of multifractured horizontal well, and it shows favorable prospect in future engineering application.
Within the frame of the XFEM, the model couples four kinds of the flow regimes, including fluid flow in the awayfromwellbore region of porous matrix (main reservoir flow), radial flow in the nearwellbore region of porous matrix surrounding the fracture (reservoir radial flow), linear flow in the awayfromwellbore region of fracture (fracture linear flow), and radial flow in the nearwellbore region of fracture (fracture radial flow).
Cases
Furthermore the simulation results show that the extended finite element model constructed in this paper is more appropriate for the complicated asymmetrical physical condition.
Radial velocity in the nearwellbore region of fracture, m/s
Fluid velocity in the direction normal to the fracture line, m/s
Oil volume factor, rb/stb
Reservoir thickness, m
Velocity vector of the pore fluid, m/s
Fracture permeability, m^{−2}
Reservoir permeability, m^{−2}
Reservoir pressure, Pa
Oil viscosity, Pa
Test function of fluid pressure, Pa
Fracture width along the direction parallel to the fracture line, m
Half of fracture width, m
Flowing bottomhole pressure, Pa
Production rate, m^{3}/s
Radius of wellbore, m
Number of fractures.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The research is supported by the National Natural Sciences Foundation of China (51344005), National Major Special Science and Technology Project of China (2011ZX05014006), and Program for Innovative Research Team of Southwest Petroleum University (2013XJT001).