# Acceleration of Gas Flow Simulations in Dual-Continuum Porous Media Based on the Mass-Conservation POD Method

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## Abstract

**:**

^{−4}%~3.87% for the matrix and 8.27 × 10

^{−4}%~2.84% for the fracture).

## 1. Introduction

## 2. Problems Arising from the Typical POD Modeling Approach

#### 2.1. Model Derivation via the Typical Approach

_{x}and l

_{y}are the domain length in the x and y directions, respectively. Note that $\phi $ is only the spatial function and $a$ is only the temporal function. Using this property, the above two equations can be transformed to:

- $\begin{array}{ll}H{B}_{m}^{k}& =\frac{2}{{\left(\Delta x\right)}^{2}}{\displaystyle \sum _{j=1}^{ny}\left[{\left({K}_{xxk}{\phi}_{m}^{k}{p}_{k}^{2}\right)}_{nx+1,j}Diri{X}_{nx+1,j}^{k}+{\left({K}_{xxk}{\phi}_{m}^{k}{p}_{k}^{2}\right)}_{0,j}Diri{X}_{0,j}^{k}\right]}\\ & +\frac{2}{{\left(\Delta y\right)}^{2}}{\displaystyle \sum _{i=1}^{nx}\left[{\left({K}_{yyk}{\phi}_{m}^{k}{p}_{k}^{2}\right)}_{i,ny+1}Diri{Y}_{i,ny+1}^{k}+{\left({K}_{yyk}{\phi}_{m}^{k}{p}_{k}^{2}\right)}_{i,0}Diri{Y}_{i,0}^{k}\right]}\end{array}$,
- $\begin{array}{ll}HB{V}_{m,n}^{k}& =\frac{1}{\Delta x}{\displaystyle \sum _{j=1}^{ny}\left[{u}_{0,j}^{k}{\left({\phi}_{m}^{k}{\phi}_{n}^{k}\right)}_{0,j}\left(1-Diri{X}_{0,j}^{k}\right)-{u}_{nx,j}^{k}{\left({\phi}_{m}^{k}{\phi}_{n}^{k}\right)}_{nx+1,j}\left(1-Diri{X}_{nx+1,j}^{k}\right)\right]}\\ & +\frac{1}{\Delta y}{\displaystyle \sum _{i=1}^{nx}\left[{v}_{i,0}^{k}{\left({\phi}_{m}^{k}{\phi}_{n}^{k}\right)}_{i,0}\left(1-Diri{Y}_{i,0}^{k}\right)-{v}_{i,ny}^{k}{\left({\phi}_{m}^{k}{\phi}_{n}^{k}\right)}_{i,ny+1}\left(1-Diri{Y}_{i,ny+1}^{k}\right)\right]}\end{array}$,
- $\begin{array}{ll}HB{P}_{m,n}^{k}& =\frac{2}{{\left(\Delta x\right)}^{2}}{\displaystyle \sum _{j=1}^{ny}\left[{\left({K}_{xxk}{\phi}_{m}^{k}{p}_{k}\right)}_{nx+1,j}{\left({\phi}_{n}^{k}\right)}_{nx,j}Diri{X}_{nx+1,j}^{k}+{\left({K}_{xxk}{\phi}_{m}^{k}{p}_{k}\right)}_{0,j}{\left({\phi}_{n}^{k}\right)}_{1,j}Diri{X}_{0,j}^{k}\right]}\\ & +\frac{2}{{\left(\Delta y\right)}^{2}}{\displaystyle \sum _{i=1}^{nx}\left[{\left({K}_{yyk}{\phi}_{m}^{k}{p}_{k}\right)}_{i,ny+1}{\left({\phi}_{n}^{k}\right)}_{i,ny}Diri{Y}_{i,ny+1}^{k}+{\left({K}_{yyk}{\phi}_{m}^{k}{p}_{k}\right)}_{i,0}{\left({\phi}_{n}^{k}\right)}_{i,1}Diri{Y}_{i,0}^{k}\right]}\end{array}$,
- $H{D}_{m,n,l}^{k}={\displaystyle \sum _{j=1}^{ny}{\displaystyle \sum _{i=1}^{nx}{\left[{\phi}_{l}^{k}\left({K}_{xxk}\frac{\partial {\phi}_{n}^{k}}{\partial x}\frac{\partial {\phi}_{m}^{k}}{\partial x}+{K}_{yyk}\frac{\partial {\phi}_{n}^{k}}{\partial y}\frac{\partial {\phi}_{m}^{k}}{\partial y}\right)\right]}_{i,j}}}$,
- $H{I}_{m,n,l}^{M}={\displaystyle \sum _{j=1}^{ny}{\displaystyle \sum _{i=1}^{nx}{\left[{K}_{M}{\phi}_{m}^{M}{\phi}_{l}^{M}\left({\phi}_{n}^{M}-{\phi}_{n}^{F}\right)\right]}_{i,j}}}$, $H{I}_{m,n,l}^{F}={\displaystyle \sum _{j=1}^{ny}{\displaystyle \sum _{i=1}^{nx}{\left[{K}_{M}{\phi}_{m}^{F}{\phi}_{l}^{M}\left({\phi}_{n}^{M}-{\phi}_{n}^{F}\right)\right]}_{i,j}}}$.

#### 2.2. Numerical Methods and Parameters

- (1)
- Through the numerical computation of Equations (3) and (4), a sample matrix of pressure can be collected at different moments as:$$S=\left[\begin{array}{ccc}{p}^{M}\left({t}_{1}\right)& \cdots & {p}^{M}\left({t}_{Ns}\right)\\ {p}^{F}\left({t}_{1}\right)& \cdots & {p}^{F}\left({t}_{Ns}\right)\end{array}\right]$$$$C=\frac{1}{Ns}{\displaystyle {\int}_{0}^{ly}{\displaystyle {\int}_{0}^{lx}{S}^{T}Sdxdy}}$$
- (2)
- Take the eigenvalue decomposition for the kernel to obtain eigenvalues and eigenvectors:$$CV=\left[\begin{array}{cccc}{\lambda}_{1}& & & \\ & {\lambda}_{2}& & \\ & & \ddots & \\ & & & {\lambda}_{Ns}\end{array}\right]V$$
- (3)
- Calculate the POD modes using the eigenvectors and samples:$$\left[\begin{array}{l}{\phi}_{1}^{M}{\phi}_{2}^{M}\cdots {\phi}_{Ns}^{M}\\ {\phi}_{1}^{F}{\phi}_{2}^{F}\cdots {\phi}_{Ns}^{F}\end{array}\right]=SV/\Vert SV\Vert $$
_{2}norm.

#### 2.3. Problem Analyses

_{1}and t

_{2}. Thus, the typical POD model has quite low robustness and low precision, and it should be improved.

^{2}, −1.98 × 10

^{3}and 4.85 × 10

^{3}for N = 1, 2, 3 respectively in Table 2. This means that the effect of the matrix–fracture interaction is hundreds or thousands of times larger than the effect of the diffusion, so that the behavior is indeed controlled by the matrix–fracture interaction in the typical POD modeling. However, this interaction effect is artificially generated from the projection of the transfer term $-\alpha {K}_{M}{p}_{M}\left({p}_{M}-{p}_{F}\right)$ and $\alpha {K}_{M}{p}_{M}\left({p}_{M}-{p}_{F}\right)$ separately in the matrix equation and fracture equation. Thus, the large unphysical mass transfer term causes the computation of the typical POD model to have very low precision (N = 1) or even to be broken up (N > 1). It is confirmed by the above comparison that the previous theoretical analyses are correct quantitatively. To obtain a reliable and accurate POD model for the dual-porosity, dual-permeability system, the artificial mass transfer should be avoided.

## 3. A New POD Modeling Approach Based on System Mass Conservation

#### 3.1. Establishment of the New POD Model

- $\begin{array}{cc}H{B}_{m}^{new}& =\frac{2}{{\left(\Delta x\right)}^{2}}{\displaystyle \sum _{j=1}^{ny}\left\{\begin{array}{l}{\left({\phi}_{m}^{*}\right)}_{nx+1,j}\left[{\left({K}_{xxM}{p}_{M}^{2}\right)}_{nx+1,j}Diri{X}_{nx+1,j}^{M}+{\left({K}_{xxF}{p}_{F}^{2}\right)}_{nx+1,j}Diri{X}_{nx+1,j}^{F}\right]\\ +{\left({\phi}_{m}^{*}\right)}_{0,j}\left[{\left({K}_{xxM}{p}_{M}^{2}\right)}_{0,j}Diri{X}_{0,j}^{M}+{\left({K}_{xxF}{p}_{F}^{2}\right)}_{0,j}Diri{X}_{0,j}^{F}\right]\end{array}\right\}}\\ & +\frac{2}{{\left(\Delta y\right)}^{2}}{\displaystyle \sum _{i=1}^{nx}\left\{\begin{array}{l}{\left({\phi}_{m}^{*}\right)}_{i,ny+1}\left[{\left({K}_{yyM}{p}_{M}^{2}\right)}_{i,ny+1}Diri{Y}_{i,ny+1}^{M}+{\left({K}_{yyF}{p}_{F}^{2}\right)}_{i,ny+1}Diri{Y}_{i,ny+1}^{F}\right]\\ +{\left({\phi}_{m}^{*}\right)}_{i,0}\left[{\left({K}_{yyM}{p}_{M}^{2}\right)}_{i,0}Diri{Y}_{i,0}^{M}+{\left({K}_{yyF}{p}_{F}^{2}\right)}_{i,0}Diri{Y}_{i,0}^{F}\right]\end{array}\right\}}\end{array}$,
- $\begin{array}{cc}HB{V}_{m,n}^{new}& =\frac{1}{\Delta x}{\displaystyle \sum _{j=1}^{ny}\left\{\begin{array}{l}{\left({\phi}_{m}^{*}\right)}_{0,j}\left[{u}_{0,j}^{M}{\left({\phi}_{n}^{M}\right)}_{0,j}\left(1-Diri{X}_{0,j}^{M}\right)+{u}_{0,j}^{F}{\left({\phi}_{n}^{F}\right)}_{0,j}\left(1-Diri{X}_{0,j}^{F}\right)\right]\\ -{\left({\phi}_{m}^{*}\right)}_{nx+1,j}\left[{u}_{nx,j}^{M}{\left({\phi}_{n}^{M}\right)}_{nx+1,j}\left(1-Diri{X}_{nx+1,j}^{M}\right)+{u}_{nx,j}^{F}{\left({\phi}_{n}^{F}\right)}_{nx+1,j}\left(1-Diri{X}_{nx+1,j}^{F}\right)\right]\end{array}\right\}}\\ & +\frac{1}{\Delta y}{\displaystyle \sum _{i=1}^{nx}\left\{\begin{array}{l}{\left({\phi}_{m}^{*}\right)}_{i,0}\left[{v}_{i,0}^{M}{\left({\phi}_{n}^{M}\right)}_{i,0}\left(1-Diri{Y}_{i,0}^{M}\right)+{v}_{i,0}^{F}{\left({\phi}_{n}^{F}\right)}_{i,0}\left(1-Diri{Y}_{i,0}^{F}\right)\right]\\ -{\left({\phi}_{m}^{*}\right)}_{i,ny+1}\left[{v}_{i,ny}^{M}{\left({\phi}_{n}^{M}\right)}_{i,ny+1}\left(1-Diri{Y}_{i,ny+1}^{M}\right)+{v}_{i,ny}^{F}{\left({\phi}_{n}^{F}\right)}_{i,ny+1}\left(1-Diri{Y}_{i,ny+1}^{F}\right)\right]\end{array}\right\}}\end{array}$,
- $\begin{array}{cc}H{B}_{m}^{new}& =\frac{2}{({\Delta x}^{2}}{\displaystyle \sum _{j=1}^{ny}\left[\begin{array}{l}{\left({\phi}_{m}^{*}\right)}_{nx+1,j}\left[{\left({K}_{xxM}{p}_{M}\right)}_{nx+1,j}{\left({\phi}_{n}^{M}\right)}_{nx,j}Diri{X}_{nx+1,j}^{M}+{\left({K}_{xxF}{p}_{F}\right)}_{nx+1,j}{\left({\phi}_{n}^{F}\right)}_{nx,j}Diri{X}_{nx+1,j}^{F}\right]\\ +{\left({\phi}_{m}^{*}\right)}_{0,j}\left[{\left({K}_{xxM}{p}_{M}\right)}_{0,j}{\left({\phi}_{n}^{M}\right)}_{1,j}Diri{X}_{0,j}^{M}+{\left({K}_{xxF}pF\right)}_{0,j}{\left({\phi}_{n}^{F}\right)}_{1,j}Diri{X}_{0,j}^{F}\right]\end{array}\right]}\\ & +\frac{2}{{\left(\Delta y\right)}^{2}}{\displaystyle \sum _{i=1}^{nx}\left\{\begin{array}{l}{\left({\phi}_{m}^{*}\right)}_{i,ny+1}\left[{\left({K}_{yyM}{p}_{M}\right)}_{i,ny+1}{\left({\phi}_{n}^{M}\right)}_{i,ny}Diri{Y}_{i,ny+1}^{M}+{\left({K}_{yyF}{p}_{F}\right)}_{i,ny+1}{\left({\phi}_{n}^{F}\right)}_{i,1}Diri{Y}_{i,ny+1}^{F}\right]\\ +{\left({\phi}_{m}^{*}\right)}_{i,0}\left[{\left({K}_{yyM}{p}_{M}\right)}_{i,0}{\left({\phi}_{n}^{M}\right)}_{i,1}Diri{Y}_{i,0}^{M}+{\left({K}_{yyF}{p}_{F}\right)}_{i,0}{\left({\phi}_{n}^{F}\right)}_{i,1}Diri{Y}_{i,0}^{F}\right]\end{array}\right\}}\end{array}$,
- $H{D}_{m,n,l}^{new}={\displaystyle \sum _{j=1}^{ny}{\displaystyle \sum _{i=1}^{nx}{\left[\frac{\partial {\phi}_{m}^{*}}{\partial x}\left({K}_{xxM}{\phi}_{l}^{M}\frac{\partial {\phi}_{n}^{M}}{\partial x}+{K}_{xxF}{\phi}_{l}^{F}\frac{\partial {\phi}_{n}^{F}}{\partial x}\right)+\frac{\partial {\phi}_{m}^{*}}{\partial y}\left({K}_{yyM}{\phi}_{l}^{M}\frac{\partial {\phi}_{n}^{M}}{\partial y}+{K}_{yyF}{\phi}_{l}^{F}\frac{\partial {\phi}_{n}^{F}}{\partial y}\right)\right]}_{i,j}}}$,

#### 3.2. Model Verification

## 4. Conclusions

- (1)
- 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. - (2)
- 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.
- (3)
- 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: ${\epsilon}_{\mathrm{max}}^{F}=2.84\%$, ${\epsilon}_{\mathrm{max}}^{M}=3.87\%$; ${\epsilon}_{\mathrm{min}}^{F}=8.27\times {10}^{-4}\%$, ${\epsilon}_{\mathrm{min}}^{M}=7.69\times {10}^{-4}\%$; ${\epsilon}_{\mathrm{stable}}^{F}=0.83\%$, ${\epsilon}_{\mathrm{stable}}^{M}=1.02\%$.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**Permeability fields: white region-100 md, black region-1 md (1 md = 9.8692327 × 10

^{−16}m

^{2}). (

**a**) Permeability of fracture; (

**b**) Permeability of matrix.

**Figure 4.**Pressure fields comparison at t

_{1}: black solid line-FDM; red dashed line-POD. (

**a**) Fracture (ε

_{F}= 18.89%); (

**b**) Matrix (ε

_{M}= 16.26%).

**Figure 5.**Pressure fields comparison for typical POD modeling at t

_{2}: black solid line-FDM; red dashed line-POD. (

**a**) Fracture (ε

_{F}= 38.67%); (

**b**) Matrix (ε

_{M}= 38.98%).

**Figure 7.**Relative errors of matrix and fracture pressures using the new POD modeling. (

**a**) top 1 mode; (

**b**) top 2 modes; (

**c**) top 3 modes; (

**d**) top 4 modes; (

**e**) top 5 modes; (

**f**) top 6 modes.

**Figure 8.**Flow field comparison for matrix and fracture at 0.15 days: black solid line-FDM; red dashed line-POD (${\epsilon}^{F}=2.84\%$, ${\epsilon}^{M}=3.87\%$).

**Figure 9.**Flow field comparison for matrix and fracture at 365 days: black solid line-FDM; red dashed line-POD (${\epsilon}^{F}=0.83\%$, ${\epsilon}^{M}=1.02\%$).

Parameter | Value | Unit |
---|---|---|

${\varphi}_{M}$ | 0.5 | / |

${\varphi}_{F}$ | 0.02 | / |

${p}_{M}\left({t}_{0}\right)$ | 1,013,250 | Pa |

${p}_{F}\left({t}_{0}\right)$ | 1,013,250 | Pa |

${p}_{1}$ | 2,026,500 | Pa |

${p}_{2}$ | 101,325 | Pa |

${q}_{M}$ | 0 | Kg/(m^{3}·s) |

${q}_{F}$ | 0 | Kg/(m^{3}·s) |

${K}_{M}$ | 8.9177127 × 10^{−11} | m^{2}/(Pa·s) |

W | 16 × 10^{−3} | Kg/mol |

R | 8.3147295 | J/(mol·K) |

T | 298 | K |

$\mu $ | 11.067 × 10^{−6} | Pa·s |

nx | 100 | / |

ny | 100 | / |

Ns | 2433 | / |

l_{x} | 100 | m |

l_{y} | 100 | m |

L_{x} | 0.2 | m |

L_{y} | 0.2 | m |

$\Delta x$ | 1 | m |

$\Delta y$ | 1 | m |

$\Delta t$ | 1296 | s |

Simulation time scope | 365 | days |

N | 1 | 2 | 3 |
---|---|---|---|

$r$ | −3.62 × 10^{2} | −1.98 × 10^{3} | 4.85 × 10^{3} |

$\overline{{\mathit{\epsilon}}_{\mathit{M}}}$ (%) | N = 1 | N = 2 | N = 3 | N = 4 | N = 5 | N = 6 |
---|---|---|---|---|---|---|

Project onto ${\phi}_{m}^{M}$ | 2.7527 | 1.3319 | 2.5884 | 0.9123 | 0.8826 | 0.9110 |

Project onto ${\phi}_{m}^{F}$ | 2.7469 | 1.8863 | 1.9840 | 1.1248 | N/A | N/A |

$\overline{{\mathit{\epsilon}}_{\mathit{F}}}$ (%) | N = 1 | N = 2 | N = 3 | N = 4 | N = 5 | N = 6 |
---|---|---|---|---|---|---|

Project onto ${\phi}_{m}^{M}$ | 1.9049 | 0.9131 | 1.8079 | 0.6702 | 0.7182 | 0.7062 |

Project onto ${\phi}_{m}^{F}$ | 1.8990 | 1.3435 | 1.3795 | 0.7912 | N/A | N/A |

FDM | New POD Model | |
---|---|---|

CPU time | 3600 s | 5 s |

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**MDPI and ACS Style**

Wang, Y.; Sun, S.; Yu, B.
Acceleration of Gas Flow Simulations in Dual-Continuum Porous Media Based on the Mass-Conservation POD Method. *Energies* **2017**, *10*, 1380.
https://doi.org/10.3390/en10091380

**AMA Style**

Wang Y, Sun S, Yu B.
Acceleration of Gas Flow Simulations in Dual-Continuum Porous Media Based on the Mass-Conservation POD Method. *Energies*. 2017; 10(9):1380.
https://doi.org/10.3390/en10091380

**Chicago/Turabian Style**

Wang, Yi, Shuyu Sun, and Bo Yu.
2017. "Acceleration of Gas Flow Simulations in Dual-Continuum Porous Media Based on the Mass-Conservation POD Method" *Energies* 10, no. 9: 1380.
https://doi.org/10.3390/en10091380