Bringing the Benefits of Adjoint Sensitivity to MODFLOW 6
MF6-ADJ is a new non-intrusive adjoint sensitivity tool that brings the power and efficiency of adjoint analysis to MODFLOW 6 — without any modification of the source code. Developed by scientists and engineers at INTERA, MF6-ADJ makes it possible to compute model sensitivities for thousands of parameters with just one additional adjoint run, offering increased speeds of hundreds to tens of thousands of times compared to traditional finite-difference perturbation methods. Key benefits include:
- Efficient and scalable sensitivity analysis
- Fully compatible with standard MODFLOW 6 releases
- Designed for maintainability and future flexibility
Why It Matters
Understanding how model outputs respond to uncertain parameters is critical for calibration, uncertainty analysis, and model optimization. Traditional “perturbation” sensitivity methods require one model run per parameter, which quickly becomes impractical for large groundwater systems.
The adjoint state method overcomes this by computing sensitivities for all parameters in a single backward run — dramatically improving efficiency and insight. Most adjoint implementations are intrusive, requiring modification of the forward code –MF6-ADJ removes this barrier by being completely non-intrusive.
MF6-ADJ Solution
MF6-ADJ connects directly to MODFLOW 6 through the MODFLOW Application Programming Interface (API). This enables the tool to extract and process solution data without altering MODFLOW’s internal source code. Core features include:
- Non-intrusive – Works with existing MODFLOW 6 executables
- Broad Compatibility – Supports confined/unconfined flow, structured/unstructured grids
- Flexible Solvers – Compatible with both Standard and Newton-Raphson schemes
- Fine-Grained Results – Computes sensitivities at each model node
- Parameter Coverage – Hydraulic conductivity (horizontal/vertical), storage, recharge, boundary heads and conductance, injection/extraction rates
Performance And Validation
MF6-ADJ has been rigorously validated against both analytical solutions and finite-difference perturbation methods. Results show excellent agreement and speedups ranging from hundreds to tens of thousands of times, depending on model size and grid resolution. These gains enable practical, high-resolution sensitivity analysis even for complex, real-world groundwater systems.
MF6-ADJ is ideal for:
- Model calibration and optimization (e.g., integration with PEST)
- Uncertainty and risk analysis
- Parameter screening and diagnostic sensitivity mapping
- Model structure assessment and conceptual evaluation
The tool efficiently generates sensitivity maps that help identify the most influential parameters and guide calibration priorities.
INTERA’s Legacy
This work builds on nearly 50 years of INTERA’s experience developing and applying adjoint methods in groundwater and environmental modeling.
MF6-ADJ was developed by:
Mohamed Hayek, PhD; Jeremy White, PhD; Katherine Markovich, PhD; Joseph Hughes, PhD; and Marsh Lavenue, PhD (ret.)
Read the Paper
To learn more about MF6-ADJ, read the publication here: MF6‐ADJ: A Non‐Intrusive Adjoint Sensitivity Capability for MODFLOW 6 – Hayek – Groundwater
Access the Code
For the link to our GitHub, please click here: GitHub – INTERA-Inc/mf6adj
Meet INTERA’s MF6-ADJ Developers and Contributors
Mohamed Hayek, PhD, is a Senior Scientist at INTERA with BSc, MSc, and PhD degrees in Mathematics. He has nearly two decades of experience in hydrogeology and environmental modeling, specializing in flow and transport processes in saturated and unsaturated media. He leads the development of MF6-ADJ, an open-source tool enabling adjoint sensitivity analysis within MODFLOW-6 and has applied advanced modeling techniques to safety assessments of radioactive waste repositories, CO₂ sequestration, and contaminant transport. Dr. Hayek has contributed to national waste disposal programs in Switzerland, France, and the U.S., and has authored more than 40 publications in international journals and conference proceedings.
Jeremy White, PhD, is a Principal Hydrogeologist at INTERA with 20 years of experience specializing in MODFLOW-6 and groundwater modeling. With a BS in Computer Science and MS/PhD in Hydrogeology, he leads development of PEST++ and pyEMU, open-source tools enhancing MODFLOW-6 applications in parameter estimation, uncertainty analysis, and optimization. His MODFLOW-6 contributions include the CSUB subsidence package and GPU-based solutions. Key projects include the GULF Groundwater Flow and Subsidence Assessment (2019–2022), using MODFLOW-6 for regulatory support in Texas, and the Lisbon Remedial Design Modeling (2022–Present), leveraging MODFLOW-6 API for reactive transport. Dr. White teaches MODFLOW-6 and PEST++ workshops and contributes to FloPy, advancing reproducible modeling.
Katie Markovich is a former Senior Hydrogeologist at INTERA with a BSc degree in geology, a PhD degree in hydrologic sciences, and 10 years of experience in hydrogeology. Dr. Markovich’s expertise ranges from the development of conceptual and numerical flow models, recharge estimation in groundwater systems, and model uncertainty and optimization analysis for decision support.
Joseph Hughes, PhD, Principal Hydrologist at INTERA, has 30+ years of experience in hydrologic modeling, co-developing MODFLOW-6 and FloPy. With degrees in Geology (BS), Geochemistry (MS), and Hydrogeology (PhD), he led MODFLOW-6 enhancements like the CSUB package and API (2017–2025). Dr. Hughes applied MODFLOW-6 in projects like Long Island Sound Flooding (2022–2025), coupling it with hydrodynamic codes. His publications and workshops advance MODFLOW-6 for water resources.
Marsh Lavenue, PhD, Principal Hydrogeologist at INTERA (retired), has 30+ years of experience in MODFLOW-based modeling and calibration. With advanced hydrogeology degrees, he developed GRASP-INV for MODFLOW calibration at WIPP (1991–1996) and applied MODFLOW in forensic modeling for litigation (1998–2000). Dr. Lavenue peer-reviewed Hanford Site MODFLOW models (2015–Present) and authored works on uncertainty, enhancing MODFLOW-6-compatible workflows for environmental assessments.
Dr. Abhishek Singh, with over 20 years of experience in hydrogeologic modeling, is a Principal Engineer and President of INTERA’s Water Resources and Supply Line of Business. Holding a BE in civil engineering and MS and PhD degrees in civil and environmental engineering, he specializes in modeling complex hydrogeologic systems, including surface-water/groundwater interactions, seawater intrusion, and subsidence. Dr. Singh leads projects in California focused on the Sustainable Groundwater Management Act (SGMA), aquifer storage and recovery, and recycled water recharge. He has authored numerous technical publications and journal articles on groundwater modeling, calibration, and uncertainty/risk analyses.
Dr. RamaRao is a leading expert in hydrogeologic simulations with over 40 years of experience in groundwater flow and transport modeling. He has developed codes for single-phase fluid flow, heat transfer, and contaminant transport in porous and fractured media, as well as stochastic inverse modeling codes integrating adjoint sensitivity analysis, parameter optimization, and geostatistical analysis for heterogeneous aquifers. Dr. RamaRao’s contributions to adjoint sensitivity analysis include comparison of alternative stochastic inverse methodologies and development of programs for multi-phase flow code (TOUGH2) and for contaminant transport (SWIFT II); development of a stochastic inverse modeling code for groundwater flow (GRASP-INV) combining geostatistical analysis, adjoint sensitivity, and parameter optimization with groundwater flow simulation; and development of the adjoint sensitivity capability for STOMP (Subsurface Transport over Multiple Phases). He has published extensively on groundwater modeling and stochastic methods.
Light Speed Sensitivities Using Adjoint States with MODFLOW6
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