Reduced-order modeling approaches for gas flow in dual-porosity dual-permeability porous media are studied based on the proper orthogonal decomposition (POD) method combined with Galerkin projection. The typical modeling approach for non-porous-medium liquid flow problems is not appropriate for this compressible gas flow in a dual-continuum porous media. The reason is that non-zero mass transfer for the dual-continuum system can be generated artificially via the typical POD projection, violating the mass-conservation nature and causing the failure of the POD modeling. A new POD modeling approach is proposed considering the mass conservation of the whole matrix fracture system. Computation can be accelerated as much as 720 times with high precision (reconstruction errors as slow as 7.69 × 10^{−4}%~3.87% for the matrix and 8.27 × 10^{−4}%~2.84% for the fracture).
A dual-porosity, dual-permeability (dual-continuum) model is an important conceptual model for unconventional oil/gas flow in fractured reservoirs [
Proper orthogonal decomposition (POD) is an efficient numerical method to largely accelerate the computational speed via project governing equations onto low dimensional eigen spaces to establish a reduced-order model. It has been widely utilized for many non-porous-medium flow problems [
For statement convenience, we only consider ideal gas flow in two-dimensional Cartesian coordinates in this paper. Darcy’s law is as follows:
Governed by the Darcy’s law, mass conservation equations for gas flow in dual-porosity dual-permeability porous media become:
To establish the POD model for this flow system, pressure is assumed to be the following linear combination:
According to the typical procedure of POD modeling, one should project the matrix equation, Equation (7), onto
The first integrations on the right hand side of Equations (11) and (12) can be transformed using integration by part:
By substituting these two expressions back to the first term of right hand side of Equations (11) and (12), one can obtain the following expressions:
For the Dirichlet boundary condition, the boundary pressure gradients in Equations (15) and (16) can be expressed as:
For the Neumann boundary condition, boundary pressure gradients can be expressed via Darcy’s law (Equations (1) and (2)). Substitute Equations (15) and (16) and the boundary conditions to Equations (11) and (12), and the projection equations can be expressed as follows:
Projection equation of the matrix:
Projection equation of the fracture:
Equation (20) is the POD model for the gas flow in dual-porosity, dual-permeability porous media using the typical modeling approach. The temporal advancement of coefficients
The symbol “·” means inner product. Equation (22) exists because of an important property of POD modes, namely that they are unitary vectors orthogonal to each other, i.e.,
Through the numerical computation of Equations (3) and (4), a sample matrix of pressure can be collected at different moments as:
Take the eigenvalue decomposition for the kernel to obtain eigenvalues and eigenvectors:
Calculate the POD modes using the eigenvectors and samples:
A numerical case is designed for the gas flow in the dual-porosity, dual-permeability porous media. The computational domain and boundary conditions are shown in
The typical POD model for gas flow in dual-porosity, dual-permeability porous media (Equations (6) and (21)) is computed using the above parameters. To examine the deviation quantitatively, the relative error for pressures in the matrix and the fracture are defined as follows:
It is unusual that the typical POD model can only be computed when the top one POD modes of matrix and fracture (
Before the improvement of the above POD model, the reason for the low robustness and low precision should be revealed firstly to clarify the direction of improvement. We revisited the governing equations (Equations (3) and (4)) of the gas flow in dual-porosity, dual-permeability porous media and realized that these two equations actually reflect the mass transfer of gas between the matrix and fracture. The term
To verify the above theoretical analyses, we examine the value of
The ratio is −3.62 × 10^{2}, −1.98 × 10^{3} and 4.85 × 10^{3} for
According to the theoretical analysis and numerical comparison in
From the analyses in
This equation can be projected onto either
Equation (31) does not contain the projection of interaction terms, such as
A new POD model based on the mass conservation of the whole system is established in
After the determination of the projection onto
The temporal evolutions of relative errors using different POD modes are shown in
To further examine the local precision of the new POD model, it is necessary to compare the local flow fields at two typical moments (0.15 days and 365 days). At 0.15 days, the reconstruction errors of the POD are maximum. At 365 days, the reconstruction errors of the POD stay low and stable. As shown in
The aim of POD modeling is not only high-precision reconstruction, but also a large-acceleration of computation. Hence, the verification of computational speed is important. The comparison of code running time, i.e., CPU time, is listed in
POD modeling for gas flow in dual-porosity, dual-permeability porous media was studied for the purpose of the acceleration of computation with high-precision for potential engineering applications. Some conclusions related to the principle of POD modeling for this type of flow can be made as follows:
For dual-porosity, dual-permeability porous media, the typical method should be avoided to project the matrix equation and fracture equation separately. Otherwise, an artificial mass transfer term, which is 10^{3}~10^{2} times larger than the diffusion term, will be generated to cause the failure of the POD modeling, because it violates the mass conservation of the whole system.
A mass conservation POD modeling method is proposed to ensure that no artificial mass transfer is generated by the POD projection process. Original governing equations should be projected onto the POD modes of matrix pressure to maintain a robust POD model.
The new POD model obeying the mass-conservation nature of the whole system can promote computational speed as much as 720 times under high precision:
The work presented in this paper has been supported by National Natural Science Foundation of China (NSFC) (No. 51576210, No. 51325603), Science Foundation of China University of Petroleum-Beijing (No. 2462015BJB03, No. 2462015YQ0409, No. C201602) and supported in part by funding from King Abdullah University of Science and Technology (KAUST) through the grant BAS/1/1351-01-01. This work is also supported by the Foundation of Key Laboratory of Thermo-Fluid Science and Engineering (Xi’an Jiaotong University), Ministry of Education, Xi’an 710049, P. R. China (KLTFSE2015KF01).
Y.W. has contributed to the model derivation, coding, simulation, and preparation of the article; S.S. has contributed to the concept design and language polishing; B.Y. has contributed to the numerical methods.
The authors declare no conflict of interest.
Computational domain and boundary condition.
Permeability fields: white region-100 md, black region-1 md (1 md = 9.8692327 × 10^{−16} m^{2}). (
Numerical errors of pressures in the matrix and fracture for typical POD modeling.
Pressure fields comparison at
Pressure fields comparison for typical POD modeling at
Cumulative energy contribution of the top
Relative errors of matrix and fracture pressures using the new POD modeling. (
Flow field comparison for matrix and fracture at 0.15 days: black solid line-FDM; red dashed line-POD (
Flow field comparison for matrix and fracture at 365 days: black solid line-FDM; red dashed line-POD (
Additional parameters for computation.
Parameter | Value | Unit |
---|---|---|
0.5 | / | |
0.02 | / | |
1,013,250 | Pa | |
1,013,250 | Pa | |
2,026,500 | Pa | |
101,325 | Pa | |
0 | Kg/(m^{3}·s) | |
0 | Kg/(m^{3}·s) | |
8.9177127 × 10^{−11} | m^{2}/(Pa·s) | |
16 × 10^{−3} | Kg/mol | |
8.3147295 | J/(mol·K) | |
298 | K | |
11.067 × 10^{−6} | Pa·s | |
100 | / | |
100 | / | |
2433 | / | |
100 | m | |
100 | m | |
0.2 | m | |
0.2 | m | |
1 | m | |
1 | m | |
1296 | s | |
Simulation time scope | 365 | days |
Magnitude comparison between the net mass transfer and the diffusion term.
1 | 2 | 3 | |
---|---|---|---|
−3.62 × 10^{2} | −1.98 × 10^{3} | 4.85 × 10^{3} |
Time-averaged error of matrix pressure for different projection operators.
Project onto |
2.7527 | 1.3319 | 2.5884 | 0.9123 | 0.8826 | 0.9110 |
Project onto |
2.7469 | 1.8863 | 1.9840 | 1.1248 | N/A | N/A |
Time-averaged error of fracture pressure for different projection operators.
Project onto |
1.9049 | 0.9131 | 1.8079 | 0.6702 | 0.7182 | 0.7062 |
Project onto |
1.8990 | 1.3435 | 1.3795 | 0.7912 | N/A | N/A |
Acceleration ability of the new POD model.
FDM | New POD Model | |
---|---|---|
CPU time | 3600 s | 5 s |