# Reduced Numerical Model for Methane Hydrate Formation under Conditions of Variable Salinity. Time-Stepping Variants and Sensitivity

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## Abstract

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## 1. Introduction

## 2. Mathematical Model

Symbol | Definition | Units/value |
---|---|---|

Data about reservoir and fluids | ||

$x=({x}_{1},\phantom{\rule{3.33333pt}{0ex}}{x}_{2},\phantom{\rule{3.33333pt}{0ex}}{x}_{3})$ | Spatial coordinate | [m] |

t | Time variable | [yr] |

G | Gravitational acceleration | 9.8 $\mathrm{m}/{\mathrm{s}}^{2}$ |

$D\left(x\right)$ | Depth of point x from sea level | [m] |

${D}_{ref}\left(x\right)$ | Seafloor depth | [m] |

In 1D case $x={x}_{3}$, $H={D}_{ref}$ | ||

$z=D\left({x}_{3}\right)\phantom{\rule{3.33333pt}{0ex}}-\phantom{\rule{3.33333pt}{0ex}}H$ | Depth below seafloor | [m] |

(G)HSZ | (Gas) Hydrate stability zone | |

P | Pressure | [Pa,MPa] |

${G}_{H}$ | Hydrostatic gradient | $\approx {10}^{4}\phantom{\rule{3.33333pt}{0ex}}\mathrm{Pa}/\mathrm{m}$ |

T | Temperature | [K] |

${G}_{T}$ | Geothermal gradient | [K/m] |

q | Darcy volumetric flux of liquid phase | [m/yr] |

${D}_{M}={D}_{S}={D}^{0}{S}_{l}{\varphi}_{0}$ | Diffusivity of component C in the liquid phase | [${\mathrm{m}}^{2}$/yr] |

${D}^{0}={10}^{-9}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{2}/\mathrm{s}=3\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{2}/\mathrm{yr}$ | ||

${\rho}_{l}$ | Seawater density | 1030 $\mathrm{kg}/{\mathrm{m}}^{3}$ |

${\rho}_{h}$ | Hydrate density | 925 $\mathrm{kg}/{\mathrm{m}}^{3}$ |

${\chi}_{hM}$ | Mass fraction of methane in hydrate phase | 0.134 kg/kg |

$R={\chi}_{hM}{\rho}_{h}/{\rho}_{l}$ | Constant used for methane concentration | 0.1203 kg/kg |

${\varphi}_{0},\phantom{\rule{3.33333pt}{0ex}}\varphi ={S}_{l}{\varphi}_{0}$ | Porosity in Ω without/with hydrate present | |

${K}_{0},\phantom{\rule{3.33333pt}{0ex}}K$ | Permeability in Ω without/with hydrate present | |

${\chi}_{lS}^{sw}$ | Seawater salinity | 0.035 [kg/kg] |

${f}_{M}$ | Supply of methane (source/sink term) | [kg/kg/yr] |

α | Parameter of the reduced model | [kg/kg] |

Variables in the model | ||

${S}_{l},\phantom{\rule{3.33333pt}{0ex}}{S}_{h}=1\phantom{\rule{3.33333pt}{0ex}}-\phantom{\rule{3.33333pt}{0ex}}{S}_{l}$ | Void fraction of liquid and hydrate phases | |

${\chi}_{lM}$ | Mass fraction of methane (solubility) in liquid phase | [kg/kg] |

${\chi}_{lS}$ | Mass fraction of salt (salinity) in liquid phase | [kg/kg] |

${N}_{M},{N}_{S}$ | Mass concentration of methane and salt | [kg/kg] |

#### 2.1. Mass Conservation

#### 2.2. Phase Equilibria and [NCC-M] Constraint

#### 2.2.1. Data for ${\chi}_{lM}^{max}$

#### 2.2.2. Other Constraints

#### 2.3. Boundary and Initial Conditions

## 3. Numerical Model

#### 3.1. Implementing Phase Constraint [NCC-M] in Fully Implicit Models

#### 3.2. Implementing Phase Constraints in Non-Implicit Models

**Simple flash.**The simplest situation is when ${N}_{M}$ is known and we know ${\chi}_{lM}^{max}$. To determine ${S}_{l}$ and ${\chi}_{lM}$ we simply use Equations (3e), (3c) to calculate

**Two-variable flash.**Given ${N}_{M},\phantom{\rule{3.33333pt}{0ex}}{N}_{S}$ we can solve for the three unknowns ${S}_{l}$, ${\chi}_{lM}$, ${\chi}_{lS}$ using Equations (3c), (3d) and (9). The implementation is especially easy if Equation (4) is used. This flash solver typically takes 2 or 3 iterations to complete, but may fail when ${S}_{h}$ is close to 1.

#### 3.3. Notation in Fully Discrete Model

**Accumulation and source terms.**For each $i,n$ we calculate the approximation of accumulation and source terms as follows

**Advection terms.**It suffices to consider only methane advection, since salt advection si treated the same way. We consider first the case $q>0$. The advective flux

**Diffusion terms.**For the spatially dependent diffusion coefficient ${D}_{M}\left(x\right)$ and the variable ${\chi}_{lM}\left(x\right)$ we have, in a standard way [30,31]

#### 3.4. Advection Step

#### 3.5. Diffusion Step

#### 3.5.1. Variant (I): Fully Implicit

#### 3.5.2. Variant (SI): Semi-Implicit

#### 3.5.3. Variant (SEQ): Sequential

## 4. Comparison of Performance of the Time Stepping Variants

**Figure 2.**Evolution of hydrate saturation and of salinity for the base case. (

**left**) Plot of ${S}_{h}$, (

**right**) Plot of ${\chi}_{lS}$. Variable ${\chi}_{lM}$ equals ${\chi}_{lM}^{max}$ at these times and is not shown.

#### 4.1. Accuracy of the Time-Stepping Variants and Choice of Time Step

**Figure 3.**Plots of ${S}_{h}$ for different time steps τ (denoted on figure by $dt$), and different time-stepping variants fully implicit (I), semi-implicit (SI), and sequential (SEQ). (

**left**) Plots over the full range of depth and ${S}_{h}$ are essentially indistinguishable. (

**right**) The zoom of the left plot shows a small sensitivity to the choice of time step and of the model variant.

**Table 2.**Maximum hydrate saturation ${S}_{h}$ obtained with different model variants and time steps at $T=10\phantom{\rule{3.33333pt}{0ex}}K$ and $T=25\phantom{\rule{3.33333pt}{0ex}}K$, all parameters as in base case.

τ | SEQ | SI | I |
---|---|---|---|

$T=10\phantom{\rule{3.33333pt}{0ex}}K$ | |||

78 | 0.177208 | 0.182844 | 0.182844 |

70 | 0.176441 | 0.181803 | 0.181803 |

50 | 0.176834 | 0.181267 | 0.181267 |

25 | 0.177841 | 0.180908 | 0.180908 |

10 | 0.178834 | 0.180736 | 0.180736 |

5 | 0.179238 | 0.180688 | 0.180688 |

1 | 0.180183 | 0.180651 | 0.180651 |

$T=25\phantom{\rule{3.33333pt}{0ex}}K$ | |||

78 | 0.456162 | 0.463925 | 0.463925 |

70 | 0.456803 | 0.464271 | 0.464271 |

50 | 0.45644 | 0.462797 | 0.462797 |

25 | 0.457708 | 0.462438 | 0.462438 |

10 | 0.458886 | 0.462266 | 0.462266 |

5 | 0.459731 | 0.462218 | 0.462218 |

1 | 0.460878 | 0.462181 | 0.462181 |

#### 4.2. Robustness and Efficiency of the Variants

**Table 3.**Robustness of nonlinear solvers depending on the variant and the time step for the simulations of the base case between $T=25\phantom{\rule{3.33333pt}{0ex}}K$ and $T=50\phantom{\rule{3.33333pt}{0ex}}K$. The robustness is assessed by checking which solver variant is more prone or more robust to failing in the difficult modeling circumstances close to unphysical. We report the critical value ${S}_{h}^{crit}$ obtained before the solver fails, and on the number ${N}_{it}$ of iterations. When ${N}_{it}$ is denoted by “-”, this means the solver did not complete. For SEQ model, ${N}_{it}$ denotes the number of flash iterations. For the SI and I models, ${N}_{it}$ denotes the number of global Newton iterations.

τ | SEQ | SI | I | |||
---|---|---|---|---|---|---|

${S}_{h}^{crit}$ | ${N}_{it}$ | ${S}_{h}^{crit}$ | ${N}_{it}$ | ${S}_{h}^{crit}$ | ${N}_{it}$ | |

78 | 0.75833 | - | 0.767473 | - | 0.773341 | - |

70 | 0.772449 | - | 0.782752 | - | 0.781435 | - |

50 | 0.806955 | - | 0.817198 | - | 0.817198 | - |

25 | 0.873396 | - | 0.880766 | - | 0.880766 | - |

10 | 0.925712 | 2 | 0.932267 | 2 | 0.932267 | 3 |

5 | 0.926744 | 2 | 0.93222 | 2 | 0.93222 | 3 |

**Dependence of the results on q.**Next, it is known that the advective fluxes are the hardest physically to handle for hydrate systems, since they provide the source for the fastest hydrate formation.

**Computational time and the choice of time step.**Finally, we evaluate the computational complexity of the variants, and this is done by comparing the wall clock times for our MATLAB implementation. In order to compare the solvers on equal footing, no additional code vectorization is implemented, but the code takes advantage of the natural MATALB vector data types. In Table 5 we report the wall clock time.

**Figure 4.**Hydrate saturation at $T=31\phantom{\rule{3.33333pt}{0ex}}K$ when different advective fluxes are assumed. For $q=0.1$ for which high saturation is attained already at $T=25\phantom{\rule{3.33333pt}{0ex}}K$ we do not show the plot at $T=31\phantom{\rule{3.33333pt}{0ex}}K$.

**Table 4.**The time T when $max{S}_{h}\approx 0.5$ depending on q, for the base case for each time-stepping variant, respectively, ${T}_{I},{T}_{SI},{T}_{SEQ}$. Here we use $\tau =1$.

q | ${T}_{I}$ | ${T}_{SI}$ | ${T}_{SEQ}$ |
---|---|---|---|

0.1 | 13917 | 13917 | 13972 |

0.01 | 27014 | 27014 | 27091 |

0.005 | 28629 | 28629 | 28691 |

0.0001 | 30568 | 30568 | 30587 |

1e−08 | 30614 | 30614 | 30624 |

**Table 5.**Comparison of computational wall clock time ${T}^{w}\left[s\right]$ for the three model variants and different time steps, for the base case and $T=25\phantom{\rule{3.33333pt}{0ex}}K$.

τ | ${T}_{\mathbf{\text{SEQ}}}^{w}$ | ${T}_{\mathbf{\text{SI}}}^{w}$ | ${T}_{\mathbf{I}}^{w}$ |
---|---|---|---|

1 | 591.801 | 439.806 | 441.394 |

10 | 60.2528 | 44.0688 | 47.6352 |

50 | 11.8322 | 8.81442 | 9.63327 |

78 | 7.55206 | 5.655 | 6.08011 |

## 5. Sensitivity to Physical and Coputational Parameters

**Discretization parameters.**As the discretization parameters $h,\tau \to 0$ and the numbers of cells ${N}_{x}=\frac{L}{h}$ and time steps increase, it is expected that the numerical solutions of a PDE model converge to the analytical ones in an appropriate sense dictated by the theoretical numerical analysis. The convergence studies for the purely diffusive one component case of Equation (3) in [17] suggest to vary τ wit h either linearly or faster, and to consider various metrics of convergence in appropriate functional spaces. For the present case with significant advection q and variable salinity, we expect the rates to be inferior of the approximate $O(h\phantom{\rule{3.33333pt}{0ex}}+\phantom{\rule{3.33333pt}{0ex}}\tau )$ rates observed in [17]. The theoretical analysis is underway and will be presented elsewhere.

**Figure 5.**Hydrate saturation for different ${N}_{x}$ and h denoted by dx. See Table 6 for the related quantitative information extracted from the simulations.

**Table 6.**Accuracy and complexity of the computational model depending on ${N}_{x}$, with the time step τ adjusted to vary linearly with h. As the quantity of interest depending on ${N}_{x}$ we show the saturation values at $T=25\phantom{\rule{3.33333pt}{0ex}}K$. This table complements the plots in Figure 5.

${N}_{x}$ | h | τ | $max{S}_{h}$ | Wall-Clock Time |
---|---|---|---|---|

10 | 15.9 | 10 | 0.453079 | 5.6533 |

25 | 6.36 | 4 | 0.455525 | 32.644 |

50 | 3.18 | 2 | 0.459280 | 121.411 |

100 | 1.59 | 1 | 0.462181 | 489.101 |

200 | 0.795 | 0.5 | 0.465253 | 2301.53 |

**Sensitivity to the parameters of the reduced model Equation (4).**There is large uncertainty as to what ${\chi}_{lM}^{max}$ one should use. In particular, there may be an error associated with the look-up table process of finding α described in [3] and due to the lack of information on salinity. More broadly, in a comprehensive model ${\chi}_{lM}^{max}$ depends on the unknown pressure and temperature values, and possibly rock type, thus further variability and uncertainty of $\alpha \left(x\right)$ should be expected.

**Figure 6.**Parameter $\alpha \left(x\right)$ as a function of depth used in Section 5 (

**left**) and the corresponding ${\chi}_{lM}^{max,0}\left(x\right)$ computed from Equation (4) and assuming ${\chi}_{LS}\approx {\chi}_{lS}^{sw}$ (

**right**). On right the plot of ${\chi}_{lM}^{max,0}\left(x\right)$ is also shown. The base case from Ulleung Basin [3] in both plots is denoted with circles. The other cases correspond to $c=-1$, $c=10$, the average of $\alpha \left(x\right)$, and to a randomly perturbed $\alpha \left(x\right)$. The plots for $c=10$ are out of range and are not fully included.

**Figure 7.**Hydrate saturation for different coefficients α. The figure on the (

**right**) is a zoomed in version of that on the (

**left**).

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Peszynska, M.; Medina, F.P.; Hong, W.-L.; Torres, M.E.
Reduced Numerical Model for Methane Hydrate Formation under Conditions of Variable Salinity. Time-Stepping Variants and Sensitivity. *Computation* **2016**, *4*, 1.
https://doi.org/10.3390/computation4010001

**AMA Style**

Peszynska M, Medina FP, Hong W-L, Torres ME.
Reduced Numerical Model for Methane Hydrate Formation under Conditions of Variable Salinity. Time-Stepping Variants and Sensitivity. *Computation*. 2016; 4(1):1.
https://doi.org/10.3390/computation4010001

**Chicago/Turabian Style**

Peszynska, Malgorzata, Francis Patricia Medina, Wei-Li Hong, and Marta E. Torres.
2016. "Reduced Numerical Model for Methane Hydrate Formation under Conditions of Variable Salinity. Time-Stepping Variants and Sensitivity" *Computation* 4, no. 1: 1.
https://doi.org/10.3390/computation4010001