Inverse hydrologic modeling using stochastic growth algorithms

Abstract

Abstract. We present a method for inverse modeling in hydrology that incorporates a physical understanding of the geological processes that form a hydrologic system. The method is based on constructing a stochastic model that is approximately representative of these geologic processes. This model provides a prior probability distribution for possible solutions to the inverse problem,. The uncertainty in the inverse solution is characterized by a conditional (posterior) probability distribution. A new stochastic simulation method, called conditional coding, approximately samples from this posterior distribution and allows study of solution uncertainty through Monte Carlo techniques. We examine a fracture-dominated flow system, but the method can potentially be applied to any system where formation processes are modeled with a stochastic simulation algorithm. based on geologic theory. We solve the inverse modeling problem by using this stochastic model to define a broad suite of possible spatial structures, and we then select samples from this suite that are consistent with (i.e., conditioned upon) the available hydraulic response data. The resulting solution is a model that is consistent with flow data and the geologic theory represented in the stochastic model. Our inverse modeling method is Bayesian, so we give a brief description of Bayesian theory and introduce some notation. In the Bayesian approach one first assumes that the unknown hydraulic geometry is randomly chosen from a given probability distribution of possible hydraulic geometries. This distribution is called the prior distribution or just the prior. Probabilities given by the prior can be represented with the notation P(X = X), where X represents the random hydraulic geometry and X is a particular geometry that X could equal. Before data are collected, the prior contains the only information about possible values for X, but after the data are taken some Xs will be more compatible with the data and so become more probable while other Xs are less compatible and less probable. Our Bayesian approach has two main components. First we define X using a physically based stochastic model that represents the geologic processes that form a hydraulic geometry; 3335

Synthetic fracture network characterization with transdimensional inversion

“…Discrete fracture networks (DFN) are realistic mappings of fracture geometries based on the field observation via a stochastic representation of fracture properties. Individual fracture geometry and fracture network connectivity are among the key parameters for such a realistic representation [Hestir et al, 1998[Hestir et al, , 2001Jang et al, 2008;Frampton and Cvetkovic, 2010;Niven and Deutsch, 2012;Dorn et al, 2013;Li et al, 2014]. DFN models could model flow and transport in fractured media, although the strong reliance on the cubic law makes these simulations very sensitive to aperture estimations [Neuman, 1988[Neuman, , 2005.…”

“…For instance, Mauldon et al [1993] and Datta-Gupta et al [1995] simulated the fracture network via partially connected conductors with equal or variable apertures, aligned to a predefined lattice as a finite element approximation of the fractured system. Other concepts keep the number of fractures fixed while adjusting their hydraulic properties [Le Borgne et al, 2007;Le Goc et al, 2010;Dorn et al, 2013;Klepikova et al, 2014], activating-deactivating fractures [Hestir et al, 1998;Niven and Deutsch, 2012], or solving the problem over the statistical parameters of the DFN but not on the exact geometries [Jang et al, 2008]. [Dorn et al, 2013] suggested that using variable dimension DFN inversion would be a good choice to improve the capabilities of the existing DFN-based inversion techniques.…”

Hydraulic Tomography for Detecting Fracture Zone Connectivity

“…Day‐Lewis et al (2006) further conducted an integrated interpretation of field experimental cross‐hole radar, tracer, and hydraulic data at a fractured rock aquifer at the same site and found that combining time‐lapse geophysical monitoring with conventional hydrologic measurements improved the characterization of a fractured rock aquifer. Hestir et al (1998) and Datta‐Gupta et al (1995) used classical hydrologic inverse modeling approach to characterize fractured rocks.…”

Geometry and mechanics of secondary fracturing around small three‐dimensional faults in granitic rock

“…Estimation of statistical parameters describing the distributions of fault and fracture sizes can be done by conditioning the statistical model to match field data using a method called conditional coding [Hestir et al, 1998]. The estimation results can be used to determine connectivity of the fault zone and as a basis for hydrologic inverse modeling [e.g., Martel and Evans, 1996;Hestir et al, 1998]. In the geometry of Figure A1, the derivative is the length of line segment PQ divided by the length of line segment QR, or the slope of line segment PR.…”

“…For example, distributions of secondary fracture locations and sizes can be selected stochastically using the most tensile stress or the strain energy density. Estimation of statistical parameters describing the distributions of fault and fracture sizes can be done by conditioning the statistical model to match field data using a method called conditional coding [Hestir et al, 1998]. The estimation results can be used to determine connectivity of the fault zone and as a basis for hydrologic inverse modeling [e.g., The spline technique has a key advantage over difference methods in the context of polygonal elements: data points need not be evenly spaced or be on a rectangular grid.…”