Estimating threshold-exceeding probability maps of environmental variables with Markov chain random fields

Estimating and mapping spatial uncertainty of environmental variables is crucial for environmental evaluation and decision making. For a continuous spatial variable, estimation of spatial uncertainty may be conducted in the form of estimating the probability of (not) exceeding a threshold value. In this paper, we introduced a Markov chain geostatistical approach for estimating threshold-exceeding probabilities. The differences of this approach compared to the conventional indicator approach lie with its nonlinear estimators—Markov chain random field models and its incorporation of interclass dependencies through transiograms. We estimated threshold-exceeding probability maps of clay layer thickness through simulation (i.e., using a number of realizations simulated by Markov chain sequential simulation) and interpolation (i.e., direct conditional probability estimation using only the indicator values of sample data), respectively. To evaluate the approach, we also estimated those probability maps using sequential indicator simulation and indicator kriging interpolation. Our results show that (i) the Markov chain approach provides an effective alternative for spatial uncertainty assessment of environmental spatial variables and the probability maps from this approach are more reasonable than those from conventional indicator geostatistics, and (ii) the probability maps estimated through sequential simulation are more realistic than those through interpolation because the latter display some uneven transitions caused by spatial structures of the sample data.     

Unstructured Gridding for MODFLOW from Prior Groundwater Flow Models: A New Paradigm

This work introduces a new unstructured gridding approach relying on feedback from a previous groundwater flow model. All cells in a relatively coarse model using a rectilinear grid are recursively subdivided following a cell wise specific discharge-based indicator to generate quadtree, octree or Voronoï grids. This technique leverages the full potential of the latest MODFLOW engines. The suitability of this approach is demonstrated on challenging single and multilayered heterogeneous formations. The proposed method is straightforward to implement in existing software packages. It supports iterative updating of groundwater flow models from the legacy rectilinear to unstructured grids.     

Axisymmetric Modeling Using MODFLOW-USG

Axisymmetric groundwater models are used for simulating radially symmetric conditions. Groundwater simulators built specifically to model axisymmetric conditions are most commonly used for simulating aquifer tests. Although some numerical models capable of simulating flow and solute transport that are developed in the cartesian coordinate system framework offer flexibility to simulate axisymmetric conditions, most of the numerical groundwater models, such as the MODFLOW family of codes, are based on structured grids in which axisymmetric flows cannot be directly simulated. Researchers in the past have provided methods to manipulate aquifer properties to mimic axisymmetric conditions. This study presents a methodology that takes advantage of the unstructured grids of MODFLOW-USG to simulate axisymmetric models within the MODFLOW framework. To develop axisymmetric models, the intercell interface area arrays of MODFLOW-USG were calculated to accurately represent coaxial cylindrical model cells. Three examples are presented to demonstrate the application of MODFLOW-USG for axisymmetric modeling: a pumping well with delayed yield effects, a vadose zone flow model simulating an infiltration basin, and a density-dependent saltwater intrusion problem for a circular island. Results were verified against analytical solutions and published numerical codes.     

Efficient Storage of Large MODFLOW Models

We present a methodology for storing the bulkier portions of a set of MODFLOW input and output files in a compressed binary format using the HDF5 library. This approach results in compression ratios of up to 99% with no significant time penalty. The highly compressed format is particularly beneficial when dealing with large regional models or Monte Carlo simulations. The strategy is focused on the list‐ and array‐based portions of the input files including the cell property and recharge arrays, and is compatible with models containing parameters, including pilot points. The utilities are based on a modified version of the MODFLOW code and are, therefore, compatible with any standard MODFLOW simulation. We present used cases and instructions on how to use the utilities.     

Grid cell distortion and MODFLOW's integrated finite-difference numerical solution

The ground water flow model MODFLOW inherently implements a nongeneralized integrated finite-difference (IFD) numerical scheme. The IFD numerical scheme allows for construction of finite-difference model grids with curvilinear (piecewise linear) rows. The resulting grid comprises model cells in the shape of trapezoids and is distorted in comparison to a traditional MODFLOW finite-difference grid. A version of MODFLOW-88 (herein referred to as MODFLOW IFD) with the code adapted to make the one-dimensional DELR and DELC arrays two dimensional, so that equivalent conductance between distorted grid cells can be calculated, is described. MODFLOW IFD is used to inspect the sensitivity of the numerical head and velocity solutions to the level of distortion in trapezoidal grid cells within a converging radial flow domain. A test problem designed for the analysis implements a grid oriented such that flow is parallel to columns with converging widths. The sensitivity analysis demonstrates MODFLOW IFD's capacity to numerically derive a head solution and resulting intercell volumetric flow when the internal calculation of equivalent conductance accounts for the distortion of the grid cells. The sensitivity of the velocity solution to grid cell distortion indicates criteria for distorted grid design. In the radial flow test problem described, the numerical head solution is not sensitive to grid cell distortion. The accuracy of the velocity solution is sensitive to cell distortion with error <1% if the angle between the nonparallel sides of trapezoidal cells is <12.5 degrees. The error of the velocity solution is related to the degree to which the spatial discretization of a curve is approximated with piecewise linear segments. Curvilinear finite-difference grid construction adds versatility to spatial discretization of the flow domain. MODFLOW-88's inherent IFD numerical scheme and the test problem results imply that more recent versions of MODFLOW 2000, with minor modifications, have the potential to make use of a curvilinear grid.     

Effect of elevation resolution on evapotranspiration simulations using MODFLOW

Surface elevations represented in MODFLOW head-dependent packages are usually derived from digital elevation models (DEMs) that are available at much high resolution. Conventional grid refinement techniques to simulate the model at DEM resolution increases computational time, input file size, and in many cases are not feasible for regional applications. This research aims at utilizing the increasingly available high resolution DEMs for effective simulation of evapotranspiration (ET) in MODFLOW as an alternative to grid refinement techniques. The source code of the evapotranspiration package is modified by considering for a fixed MODFLOW grid resolution and for different DEM resolutions, the effect of variability in elevation data on ET estimates. Piezometric head at each DEM cell location is corrected by considering the gradient along row and column directions. Applicability of the research is tested for the lower Rio Grande (LRG) Basin in southern New Mexico. The DEM at 10 m resolution is aggregated to resampled DEM grid resolutions which are integer multiples of MODFLOW grid resolution. Cumulative outflows and ET rates are compared at different coarse resolution grids. Results of the analysis conclude that variability in depth-to-groundwater within the MODFLOW cell is a major contributing parameter to ET outflows in shallow groundwater regions. DEM aggregation methods for the LRG Basin have resulted in decreased volumetric outflow due to the formation of a smoothing error, which lowered the position of water table to a level below the extinction depth.     

Generating MODFLOW grids from boundary representation solid models

Complex stratigraphy can be difficult to simulate in MODFLOW models. MODFLOW uses a structured grid that requires that each grid layer be continuous throughout the model domain. This makes it difficult to explicitly represent common features such as pinchouts and embedded seams in a MODFLOW model. In this paper, we describe a method for automatically generating MODFLOW-compatible grids from boundary-representation solid models. Solid models are data structures developed originally for computer-aided design applications that define the geometry of three-dimensional objects. Solid models can be used to represent arbitrarily complex stratigraphy. The elevations defined by the solids are then extracted from the solids in a manner that preserves the continuous-layer requirement imposed by MODFLOW. Two basic approaches are described: The first method adjusts the MODFLOW grid dimensions (layer elevations) to fit the solid model boundaries, and the second method creates a regular MODFLOW grid and adjusts the material properties to match the changes in stratigraphy. One of the main benefits of using solid models to define stratigraphy for MODFLOW models is that it provides a grid-independent definition of the layer elevations that can be used to immediately re-create the MODFLOW grid geometry after any change to the grid resolution.     

Spatial prediction of categorical variables: the Bayesian maximum entropy approach

 Being a non-linear method based on a rigorous formalism and an efficient processing of various information sources, the Bayesian maximum entropy (BME) approach has proven to be a very powerful method in the context of continuous spatial random fields, providing much more satisfactory estimates than those obtained from traditional linear geostatistics (i.e., the various kriging techniques). This paper aims at presenting an extension of the BME formalism in the context of categorical spatial random fields. In the first part of the paper, the indicator kriging and cokriging methods are briefly presented and discussed. A special emphasis is put on their inherent limitations, both from the theoretical and practical point of view. The second part aims at presenting the theoretical developments of the BME approach for the case of categorical variables. The three-stage procedure is explained and the formulations for obtaining prior joint distributions and computing posterior conditional distributions are given for various typical cases. The last part of the paper consists in a simulation study for assessing the performance of BME over the traditional indicator (co)kriging techniques. The results of these simulations highlight the theoretical limitations of the indicator approach (negative probability estimates, probability distributions that do not sum up to one, etc.) as well as the much better performance of the BME approach. Estimates are very close to the theoretical conditional probabilities, that can be computed according to the stated simulation hypotheses.     

联合多点地质统计学与序贯高斯模拟的随机反演方法

岩相和储层物性参数是油藏表征的重要参数,地震反演是储层表征和油气藏勘探开发的重要手段.随机地震反演通常基于地质统计学理论,能够对不同类型的信息源进行综合,建立具有较高分辨率的储层模型,因而得到广泛关注.其中,概率扰动方法是一种高效的迭代随机反演策略,它能综合考虑多种约束信息,且只需要较少的迭代次数即可获得反演结果.在概率扰动的优化反演策略中,本文有效的联合多点地质统计学与序贯高斯模拟,并结合统计岩石物理理论实现随机反演.首先,通过多点地质统计学随机模拟,获得一系列等可能的岩相模型,扰动更新初始岩相模型后利用相控序贯高斯模拟建立多个储层物性参数模型;然后通过统计岩石物理理论,计算相应的弹性参数;最后,正演得到合成地震记录并与实际地震数据对比,通过概率扰动方法进行迭代,直到获得满足给定误差要求的反演结果.利用多点地质统计学,能够更好地表征储层空间特征.相控序贯高斯模拟的应用,能够有效反映不同岩相中储层物性参数的分布.提出的方法可在较少的迭代次数内同时获得具有较高分辨率的岩相和物性参数反演结果,模型测试和实际数据应用验证了方法的可行性和有效性.     

Extending the MODPATH Algorithm to Rectangular Unstructured Grids

The recent release of MODFLOW‐USG, which allows model grids to have irregular, unstructured connections, requires a modification of the particle‐tracking algorithm used by MODPATH. This paper describes a modification of the semi‐analytical particle‐tracking algorithm used by MODPATH that allows it to be extended to rectangular‐based unstructured grids by dividing grid cells with multi‐cell face connections into sub‐cells. The new method will be incorporated in the next version of MODPATH which is currently under development.     

MODFLOW as a Configurable Multi-Model Hydrologic Simulator

MODFLOW 6 is the latest in a line of six “core” versions of MODFLOW released by the U.S. Geological Survey. The MODFLOW 6 architecture supports incorporation of additional hydrologic processes, in addition to groundwater flow, and allows interaction between processes. The architecture supports multiple model instances and multiple types of models within a single simulation, a flexible approach to formulating and solving the equations that represent hydrologic processes, and recent advances in interoperability, which allow MODFLOW to be accessed and controlled by external programs. The present version of MODFLOW 6 consolidates popular capabilities available in MODFLOW variants, such as the unstructured grid support in MODFLOW-USG, the Newton-Raphson formulation in MODFLOW-NWT, and the support for partitioned stress boundaries in MODFLOW-CDSS. The flexible multi-model capability allows users to configure MODFLOW 6 simulations to represent the local-grid refinement (LGR) capabilities available in MODFLOW-LGR, the multi-species transport capabilities in MT3DMS, and the coupled variable-density capabilities available in SEAWAT. This paper provides a new, holistic and integrated overview of simulation capabilities made possible by the MODFLOW 6 architecture, and describes how ongoing and future development can take advantage of the program architecture to integrate new capabilities in a way that is minimally invasive and automatically compatible with the existing MODFLOW 6 code.     

Nonlinear Flow Process: A New Package to Compute Nonlinear Flow in MODFLOW

A new MODFLOW package (Nonlinear Flow Process; NLFP) simulating nonlinear flow following the Forchheimer equation was developed and implemented in MODLFOW‐2005. The method is based on an iterative modification of the conductance calculated and used by MODFLOW to obtain an effective Forchheimer conductance. The package is compatible with the different layer types, boundary conditions, and solvers as well as the wetting capability of MODFLOW. The correct implementation is demonstrated using four different benchmark scenarios for which analytical solutions are available. A scenario considering transient flow in a more realistic setting and a larger model domain with a higher number of cells demonstrates that NLFP performs well under more complex conditions, although it converges moderately slower than the standard MODFLOW depending on the nonlinearity of flow. Thus, this new tool opens a field of opportunities to groundwater flow simulation with MODFLOW, especially for core sample simulation or vuggy karstified aquifers as well as for nonlinear flow in the vicinity of pumping wells.     

A Generic Closed-Loop Contaminant Treatment System Package for MODFLOW 6

We present a contaminant treatment system (CTS) package for MODFLOW 6 that facilitates the simulation of pump-and-treat systems for groundwater remediation. Using the “nonintrusive” MODFLOW 6 application programming interface (API) capability, the CTS package can balance flows between extraction and injection wells within the outer flow solution loop and applies blended concentration/mass treatment efficiency within the outer transport solution loop. The former can be important when the requested extraction rate cannot be satisfied by the current simulated groundwater system conditions, while the latter can be important for simulating incomplete/imperfect treatment schemes. Furthermore, the CTS package allows users to temporally vary all aspects of a simulated CTS system, including the configuration and location of injection and extraction wells, and the CTS efficiency. This flexibility combined with the API-based implementation provide a generic and general CTS package that can be applied across the wide range of MODFLOW 6 simulation options and that evolves in step with MODFLOW 6 code modifications and advancements without needing to update the CTS package itself.     

Robust representation of dry cells in single-layer MODFLOW models

Dewatered or "dry" grid cells in the USGS ground water modeling software MODFLOW may cause nonphysical artifacts, trigger convergence failures, or interfere with parameter estimation. These difficulties can be avoided in two dimensions by modifying the spatial differencing scheme and the iterative procedure used to resolve nonlinearities. Specifically, the spatial differencing scheme is modified to use the water level on the upstream side of a pair of adjacent cells to calculate the saturated thickness and hence intercell conductance for the pair. This makes it possible to explicitly constrain the water level in a cell to be at or above the cell bottom elevation without introducing nonphysical artifacts. Thus constrained, all initially active cells will remain active throughout the simulation. It was necessary to replace MODFLOW's Picard iteration method with the Newton-Raphson method to achieve convergence in demanding applications involving many dry cells. Tests using a MODFLOW variant based on the new method produced results nearly identical to conventional MODFLOW in situations where conventional MODFLOW converges. The new method is extremely robust and converged in scenarios where conventional MODFLOW failed to converge, such as when almost all cells dewatered. An example application to the Edwards Aquifer in south-central Texas further demonstrates the utility of the new method.     

MODFLOW/MT3DMS-based simulation of variable-density ground water flow and transport

This paper presents an approach for coupling MODFLOW and MT3DMS for the simulation of variable-density ground water flow. MODFLOW routines were modified to solve a variable-density form of the ground water flow equation in which the density terms are calculated using an equation of state and the simulated MT3DMS solute concentrations. Changes to the MODFLOW and MT3DMS input files were kept to a minimum, and thus existing data files and data files created with most pre- and postprocessors can be used directly with the SEAWAT code. The approach was tested by simulating the Henry problem and two of the saltpool laboratory experiments (low- and high-density cases). For the Henry problem, the simulated results compared well with the steady-state semianalytic solution and also the transient isochlor movement as simulated by a finite-element model. For the saltpool problem, the simulated breakthrough curves compared better with the laboratory measurements for the low-density case than for the high-density case but showed good agreement with the measured salinity isosurfaces for both cases. Results from the test cases presented here indicate that the MODFLOW/MT3DMS approach provides accurate solutions for problems involving variable-density ground water flow and solute transport.     

A Semi‐Structured MODFLOW‐USG Model to Evaluate Local Water Sources to Wells for Decision Support

In order to better represent the configuration of the stream network and simulate local groundwater‐surface water interactions, a version of MODFLOW with refined spacing in the topmost layer was applied to a Lake Michigan Basin (LMB) regional groundwater‐flow model developed by the U.S. Geological. Regional MODFLOW models commonly use coarse grids over large areas; this coarse spacing precludes model application to local management issues (e.g., surface‐water depletion by wells) without recourse to labor‐intensive inset models. Implementation of an unstructured formulation within the MODFLOW framework (MODFLOW‐USG) allows application of regional models to address local problems. A “semi‐structured” approach (uniform lateral spacing within layers, different lateral spacing among layers) was tested using the LMB regional model. The parent 20‐layer model with uniform 5000‐foot (1524‐m) lateral spacing was converted to 4 layers with 500‐foot (152‐m) spacing in the top glacial (Quaternary) layer, where surface water features are located, overlying coarser resolution layers representing deeper deposits. This semi‐structured version of the LMB model reproduces regional flow conditions, whereas the finer resolution in the top layer improves the accuracy of the simulated response of surface water to shallow wells. One application of the semi‐structured LMB model is to provide statistical measures of the correlation between modeled inputs and the simulated amount of water that wells derive from local surface water. The relations identified in this paper serve as the basis for metamodels to predict (with uncertainty) surface‐water depletion in response to shallow pumping within and potentially beyond the modeled area, see Fienen et al. (2015a).     

A reservoir skeleton-based multiple point geostatistics method

Traditional stochastic reservoir modeling, including object-based and pixel-based methods, cannot solve the problem of reproducing continuous and curvilinear reservoir objects. The paper first dives into the various stochastic modeling methods and extracts their merits, then proposes the skeleton-based multiple point geostatistics (SMPS) for the fluvial reservoir. The core idea is using the skeletons of reservoir objects to restrict the selection of data patterns. The skeleton-based multiple point geostatistics consists of two steps. First, predicting the channel skeleton (namely, channel centerline) by using the method in object-based modeling. The paper proposes a new method of search window to predict the skeleton. Then forecasting the distributions of reservoir objects using multiple point geostatistics with the restriction of channel skeleton. By the restriction of channel centerline, the selection of data events will be more reasonable and the realization will be achieved more really. The checks by the conceptual model and the real reservoir show that SMPS is much better than Sisim (sequential indicator simulation), Snesim (Single Normal Equation Simulation) and Simpat (simulation with patterns) in building the fluvial reservoir model. This new method will contribute to both the theoretical research of stochastic modeling and the oilfield developments of constructing highly precise reservoir geological models.

     

Approaches to the simulation of unconfined flow and perched groundwater flow in MODFLOW

Various approaches have been proposed to manage the nonlinearities associated with the unconfined flow equation and to simulate perched groundwater conditions using the MODFLOW family of codes. The approaches comprise a variety of numerical techniques to prevent dry cells from becoming inactive and to achieve a stable solution focused on formulations of the unconfined, partially-saturated, groundwater flow equation. Keeping dry cells active avoids a discontinuous head solution which in turn improves the effectiveness of parameter estimation software that relies on continuous derivatives. Most approaches implement an upstream weighting of intercell conductance and Newton-Raphson linearization to obtain robust convergence. In this study, several published approaches were implemented in a stepwise manner into MODFLOW for comparative analysis. First, a comparative analysis of the methods is presented using synthetic examples that create convergence issues or difficulty in handling perched conditions with the more common dry-cell simulation capabilities of MODFLOW. Next, a field-scale three-dimensional simulation is presented to examine the stability and performance of the discussed approaches in larger, practical, simulation settings.     

基于随机地震反演的Russell流体因子直接估算方法

本文研究了一种基于随机地震反演的Russell流体因子直接估算方法,该方法是一种基于蒙特卡罗的非线性反演,能够有效地融合测井资料中的高频信息,提高反演结果的分辨率.本文应用贝叶斯理论框架,首先通过测井数据计算井位置处的Russell流体因子,利用序贯高斯模拟方法(sequential Gaussian simulation,SGS)得到流体因子的先验信息;然后构建似然函数;最后利用Metropolis抽样算法对后验概率密度进行抽样,得到反演的Russell流体因子.其中对每道数据进行序贯高斯模拟时,采用一种新的逐点模拟方式,具有较高的计算速度.数值试验表明:反演结果与理论模型和实际测井数据吻合较好,具有较高的分辨率,对于判识储层含流体特征具有较好的指示作用.     

MODFLOW 2000 head uncertainty,a first-order second moment method

A computationally efficient method to estimate the variance and covariance in piezometric head results computed through MODFLOW 2000 using a first-order second moment (FOSM) approach is presented. This methodology employs a first-order Taylor series expansion to combine model sensitivity with uncertainty in geologic data. MODFLOW 2000 is used to calculate both the ground water head and the sensitivity of head to changes in input data. From a limited number of samples, geologic data are extrapolated and their associated uncertainties are computed through a conditional probability calculation. Combining the spatially related sensitivity and input uncertainty produces the variance-covariance matrix, the diagonal of which is used to yield the standard deviation in MODFLOW 2000 head. The variance in piezometric head can be used for calibrating the model, estimating confidence intervals, directing exploration, and evaluating the reliability of a design. A case study illustrates the approach, where aquifer transmissivity is the spatially related uncertain geologic input data. The FOSM methodology is shown to be applicable for calculating output uncertainty for (1) spatially related input and output data, and (2) multiple input parameters (transmissivity and recharge).