Monte Carlo simulation for variable-density groundwater flow through reduced-order model coupled with Gaussian process
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摘要:
变密度地下水流系统受水力梯度和密度梯度共同驱动，非线性强，数值模型计算量大，尤其在开展不确定性分析时需要的计算成本很高。常规的数据驱动机器学习方法只能对点监测信息进行模拟分析，不能模拟整个地下水流系统。本研究发展了变密度地下水流降阶模型，利用高斯过程模型对降阶模型的数值误差进行修正组成耦合模型。耦合模型既能克服高斯过程只能模拟有限监测点信息的缺陷，又能提高降阶模型对监测点信息的模拟精度。考虑二维剖面变密度地下水流案例，将渗透系数场设定为空间随机变量，采用基于全阶模型（FSMC）、降阶模型（ROMC）和耦合模型（GP-ROMC）3种蒙特卡罗模拟方法进行不确定分析。研究结果表明：（1）ROMC能替代FSMC开展不确定性分析；（2）水头和盐度的平均相对二范误差与降阶模型维度的关系可用指数函数描述（决定性系数R2≥0.99）；（3）GP-ROMC对监测点信息的模拟精度比ROMC高，GP-ROMC可有效修正降阶模型误差提高蒙特卡罗模拟的精度。研究成果可为地下水建模、不确定性分析、风险评估及参数反演等工作提供重要技术支撑。
Abstract:Variable-density groundwater flow (VDGF) is jointly driven by hydraulic and density gradient, leading to strong nonlinearity, large computational burden of numerical models, and therefore huge computational cost of Monte Carlo simulation for uncertainty analysis. This study developed the reduced-order model (ROM) for VDGF and built the Gaussian process (GP) for simulating the numerical error of the ROM. The coupled model can obtain solutions of head and salinity across the study domain while GP simulates observation information at limited locations. Moreover, the coupled model can provide higher solution accuracies of head and salinity at the observation locations than the ROM. A two-dimensional (cross-section) VDGF test case was considered, where hydraulic conductivity was taken as a spatially random field. MC simulations were performed using three models, including the full-system model, the ROM, and the coupled model, with corresponding MC strategies denoted as FSMC, ROMC, and GP-ROMC, respectively. The results show that ROMC can be an alternative to FSMC for conducting uncertainty quantification. The relationship between head (or salinity) and the dimensional of ROM can be characterized using power functions with determinate coefficients larger than 0.99. GP-ROMC has higher solution accuracy than ROMC, which indicates that GP is capable for simulating the numerical error of ROM. The results in this study are significant for performing simulation, uncertainty quantification, risk assessment, and parameter estimate in the context of groundwater.
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图 3 hf特征向量
${\boldsymbol{p}}_{\boldsymbol{1}}^{\bf{h}}$ 、${\boldsymbol{p}}_{\boldsymbol{2}}^{\bf{h}}$ 和${\boldsymbol{p}}_{\boldsymbol{3}}^{\bf{h}}$ Figure 3.
图 4 c特征向量
${\boldsymbol{p}}_{\boldsymbol{1}}^{\bf{c}}$ 、$ {\boldsymbol{p}}_{\boldsymbol{2}}^{\bf{c}} $ 和${\boldsymbol{p}}_{\boldsymbol{3}}^{\bf{c}}$ Figure 4.
图 5 最后时刻ROMC获得的
${{\boldsymbol{\eta}} _{\bf{h}}}$ 、${\boldsymbol{\sigma}} _{\bf{h}}^{\boldsymbol{2}}$ 、${{\boldsymbol{\kappa}} _{\bf{h}}}$ 及${{\boldsymbol{\zeta}} _{\bf{h}}}$ 的绝对误差分布图Figure 5.
图 6 最后时刻ROMC获得的
${{\boldsymbol{\eta}} _{\bf{c}}}$ 、${\boldsymbol{\sigma}} _{\bf{c}}^{\boldsymbol{2}}$ 、${{\boldsymbol{\kappa}} _{\bf{c}}}$ 及${{\boldsymbol{\zeta}} _{\bf{c}}}$ 的绝对误差分布图Figure 6.
图 7 t = 20，100，200 min时
${{\boldsymbol{\mu}} _{\bf{h}}}$ 与${{\boldsymbol{\mu}} _{\bf{c}}}$ 随m的变化曲线及其指数函数的回归结果Figure 7.
图 12 观测点I、II和III，GP-ROMC与ROMC获得水头与盐度的平均绝对误差和GP-ROMC平均绝对误差相对于ROMC的减少百分比随m的变化曲线
Figure 12.
表 1 不同时刻
${{\boldsymbol{\mu}} _{\bf{h}}}$ 与${{\boldsymbol{\mu}} _{\bf{c}}}$ 关于m的指数函数回归结果Table 1. Regression results of power functions for characterizing the relationship between m and
${{\boldsymbol{\mu}} _{\bf{h}}}$ (or${{\boldsymbol{\mu}} _{\bf{c}}}$ ) obtained at each time stept/min y = αxβ hf c α β R2 α β R2 20 −0.92（−1.03，−0.81） −0.49（−0.53，−0.46） 0.995 −1.67（−1.88，−1.46） −0.93（−1.00，−0.86） 0.994 40 −0.89（−0.99，−0.80） −0.50（−0.53，−0.47） 0.996 −1.35（−1.63，−1.08） −0.96（−1.05，−0.87） 0.991 60 −0.90（−1.00，0.81） −0.50（−0.53，−0.47） 0.996 −1.06（−1.19，−0.93） −0.98（−1.02，−0.94） 0.998 80 −0.94（−1.04，−0.83） −0.49（−0.53，−0.46） 0.994 −0.92（−1.05，−0.79） −0.97（−1.01，−0.93） 0.998 100 −0.99（−1.12，−0.86） −0.48（−0.53，−0.44） 0.992 −0.80（−0.98，−0.62） −0.97（1.03，−0.91） 0.996 120 −1.04（−1.20，−0.89） −0.47（−0.53，−0.42） 0.989 −0.68（−0.88，−0.50） −0.98（−1.05，0.92） 0.996 140 −1.08（−1.25，−0.91） −0.47（−0.52，−0.41） 0.986 −0.59（−0.76，−0.41） −0.99（−1.05，−0.94） 0.997 160 −1.08（−1.25，−0.91） −0.47（−0.52，−0.41） 0.986 −0.47（−0.62，−0.32） −1.01（−1.05，−0.95） 0.998 180 −1.05（−1.20，−0.89） −0.47（−0.52，−0.42） 0.998 −0.33（−0.46，−0.20） −1.01（−1.06，0.97） 0.998 200 −1.00（−1.13，−0.86） −0.48（−0.53，−0.44） 0.991 −0.15（−0.26，−0.03） −1.03（−1.07，−0.99） 0.999 注：系数α和收敛指数β对应的单元数值为：回归值（95%置信区间下边界，95%置信区间上边界）。 
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