# A numerical method for efficient 3D inversions using Richards equation

## TL;DR

This paper introduces an efficient numerical method for 3D inversion of hydraulic properties using Richards equation, enabling large-scale, deterministic parameter estimation with reduced computational resources.

## Contribution

The paper presents a Jacobian-free, vector product-based inversion algorithm for Richards equation, scalable to large 3D problems and compatible with modest computational resources.

## Key findings

- Successfully applied to 3D saturated hydraulic conductivity inversion
- Allows larger problems than traditional Jacobian-based methods
- Operates efficiently on modest computational hardware

## Abstract

Fluid flow in the vadose zone is governed by Richards equation; it is parameterized by hydraulic conductivity, which is a nonlinear function of pressure head. Investigations in the vadose zone typically require characterizing distributed hydraulic properties. Saturation or pressure head data may include direct measurements made from boreholes. Increasingly, proxy measurements from hydrogeophysics are being used to supply more spatially and temporally dense data sets. Inferring hydraulic parameters from such datasets requires the ability to efficiently solve and deterministically optimize the nonlinear time domain Richards equation. This is particularly important as the number of parameters to be estimated in a vadose zone inversion continues to grow. In this paper, we describe an efficient technique to invert for distributed hydraulic properties in 1D, 2D, and 3D. Our algorithm does not store the Jacobian, but rather computes the product with a vector, which allows the size of the inversion problem to become much larger than methods such as finite difference or automatic differentiation; which are constrained by computation and memory, respectively. We show our algorithm in practice for a 3D inversion of saturated hydraulic conductivity using saturation data through time. The code to run our examples is open source and the algorithm presented allows this inversion process to run on modest computational resources.