## 1 Introduction

### 1.1 Surface Deformation Due to Subsurface Hydrology

Anthropogenic land subsidence associated with withdrawal of fluids from the subsurface due to groundwater extraction and the production of hydrocarbons has been recognized for decades [*Poland and Davis*, 1969], for recent reviews, see *Gambolati et al.* [2006] and *Galloway and Burbey* [2011]. While subsidence is generally slow, the rates of the induced land subsidence can locally reach several centimeters per year and cumulative subsidence over several decades of fluid withdrawal can reach tens of meters in severe cases, such as the rapid compaction of lacustrine sediments beneath Mexico City [*Ortega-Guerrero et al.*, 2009; *Osmanoglu et al.*, 2011]. Anthropogenic land uplift is associated with the injection of fluids for enhanced oil production, natural gas and carbon dioxide storage, aquifer storage and recharge, and the injection of industrial waste [*Teatini et al.*, 2011]. Cumulative land uplift due to fluid injection is on the order of several centimeters and is therefore often not recognized, unless it is systematically monitored. In both cases—subsidence and land uplift—the lateral extent of the deformation is large and we will refer to it below as (regional) surface deformation.

#### 1.1.1 Geodetic Surface Deformation Measurements

The systematic measurement and monitoring of the evolution of regional surface deformation is challenging due to the large spatial scales, the need for stable geodetic reference marks, and repeated surveys [*Galloway and Burbey*, 2011]. In the last decades, ground-based geodetic surveys have been complemented by the global positioning system (GPS) and by remotely sensed geodetic surveys such as airborne light detection and ranging (LiDAR) as well as satellite-based differential interferometric synthetic aperture radar (InSAR) [*Galloway et al.*, 1998; *Burgmann et al.*, 2000] and Persistent/Permanent Scatterer interferometry (PS-InSAR) [*Ferretti et al.*, 2001]. Satellite InSAR and PS-InSAR are ideally suited for the measurement of surface deformation associated with fluid injection or extraction, because it allows the mapping of areas as large as 10,000 km^{2} while a single picture element (pixel) may have a resolution of up to 5 m [*Hanssen*, 2001]. Within a pixel, InSAR can detect vertical displacements on the order of 10 mm or less and PS-InSAR may allow the detection of displacements with an accuracy of 1–2 mm [*Ferretti et al.*, 2001]. In the last decade, these techniques have been used to generate detailed time series images of regional surface deformation that have greatly advanced our understanding of the subsurface. *Galloway and Hoffmann* [2007] show that satellite InSAR has (1) allowed the identification of the structural and stratigraphic boundaries to fluid flow and deformation, (2) placed constraints on the magnitudes of subsurface material properties and their heterogeneity, and (3) allowed calibration of numerical models of subsurface flow and the associated deformation.

### 1.2 Hydrological Inverse Problems

All properties of the subsurface are spatially variable and often anisotropic, but the permeability often has the strongest spatial variation and usually dominates the flow field. The characterization of the permeability field is therefore key to successful prediction of flow and transport processes in the subsurface. In practice, the information available for this characterization is sparse and the uncertainty in the estimated properties is large. To make the inverse problem well posed and to characterize the uncertainty of the estimate, hydrological inversions use Bayesian inference to incorporate prior information [*Carrera and Neuman*, 1986; *McLaughlin and Townley*, 1996; *Woodbury and Ulrych*, 2000; *Castagna and Bellin*, 2009; *Rubin et al.*, 2010].

The characterization of aquifers based purely on hydraulic data, briefly reviewed in section 1.2.1, often has very large uncertainties. The reduction of this uncertainty is the motivation to include additional data into the characterization and has led to increased interest in hydrogeophysical methods and joint inversions of hydraulic and geophysical data. The poroelastic inversion discussed here is a special case of hydrogeophysical inversion that aims to add deformation measurements to hydraulic data. Section 1.2.2 summarizes previous work on poroelastic inversion.

#### 1.2.1 Hydraulic Tomography

In hydrology or hydrogeology, the inversion of multiple hydraulic data sets to infer the spatial variation of petrophysical properties of the aquifer is often referred to as hydraulic tomography (for a recent summary, see *Illman et al.* [2010]). Most commonly, hydraulic tomography aims to invert data from multiple well heads using a steady single-phase flow equation based on optimal control theory [*Chavent et al.*, 1975; *Sykes et al.*, 1985], the quasi-linear geostatistical method developed by *Kitanidis and Vomvoris* [1983] or successive linear estimators [*Yeh and Liu*, 2000]. Transient hydraulic tomography studies have either used optimal control theory [*Chavent et al.*, 1975], or considered temporal moments of the flow equation, which transforms the evolution equations into a set of steady equations for the moments [*Li et al.*, 2005; *Zhu and Yeh*, 2005]. An alternative approach is hydraulic tomography based on pressure arrival times [*Vasco et al.*, 2000; *Brauchler et al.*, 2003], which is commonly employed in geomechanical inversions, see section 1.2.2.

Hydraulic tomography is increasingly applied to laboratory and field cases [*Jiang et al.*, 2004; *Illman et al.*, 2009, 2010; *Liu and Kitanidis*, 2011; *Cardiff and Barrash*, 2011], but *Bohling and Butler* [2010] have demonstrated the large uncertainty in the estimates of the hydraulic parameter fields, even in the simultaneous analysis of a number of carefully controlled hydraulic tests.

#### 1.2.2 Poroelastic Inversion

*Galloway and Hoffmann* [2007] review several case studies that have successfully used time series surface deformation measurements to inform hydrological models and to constrain aquifer properties. These studies assume purely vertical deformation, i.e., they only use equation (1a) in section 2, although increasingly accurate horizontal displacement measurements are available. Here we propose to integrate all components of the observed displacement with hydraulic data through a fully coupled poroelastic inversion of surface deformation and hydraulic data for aquifer characterization.

Unlike, for instance, geoelectrical measurements, surface deformation is not directly related to a physical property of the pore fluid and therefore the fluid distribution, but rather to the force exerted onto the porous matrix by the pore fluid pressure gradients, which are due to pore fluid flow. Poroelastic inversion is therefore a potentially powerful method to monitor the dynamics of flow in the subsurface. In a sequence of papers, Vasco and coworkers show that surface deformation measurements either from satellite geodesy or tilt measurements can be used to infer information about aquifer volume changes and pressure distribution [*Vasco et al.*, 2000, 2001; *Vasco*, 2004]. They show that the permeability distribution can be inferred from pressure fronts at multiple times [*Vasco and Ferretti*, 2005] or from pressure travel time tomography [*Vasco*, 2004; *Vasco et al.*, 2008a, 2008b, 2010]. Both methods are based on Green's functions for efficient and accurate solution of the poroelastic equations. *Rucci et al.* [2010] showed that the pressure travel times are insensitive to variations in mechanical properties. A similar Green's function based approach was also presented by *Du and Olson* [2001] for a joint inversion of pressure observations and surface deformation.

*Iglesias and McLaughlin* [2012] have developed a mathematically rigorous framework for the deterministic poroelastic inversion. Similar to the work presented here, they use adjoints to compute derivatives of the least squares optimization problem arising in deterministic inversion. They present a joint inversion of surface deformation and pressure data for permeability and one of the elastic parameters. For the assimilation of surface deformation data into complex stand-alone aquifer simulation models, *Chang et al.* [2010] have proposed an Ensemble Kalman filter and *Wick et al.* [2013] have proposed an iterative stochastic ensemble method.

Finally, there is related work on the inversion of the dynamic poroelastic wave propagation problem. *Morency et al.* [2009] have developed finite-frequency kernels for wave propagation in porous media based on adjoint methods and applied them to crosswell monitoring of geological CO_{2} storage [*Morency et al.*, 2011]. However, asymptotic analysis by *Vasco* [2009] shows that at the low frequencies of interest here the nature of the dynamic wave propagation problem, which is domination by the propagation of modified elastic waves, is qualitatively very different from the diffusive nature of the quasi-static problem discussed below.

### 1.3 Overview and Contributions

This paper presents an efficient algorithm for the joint coupled inversion for the permeability field in the time-dependent quasi-static poroelastic equations and uses benchmark problems, shown in Figure 1 to systematically study properties, prospects, and limitations of this inference problem.

In section 3, we formulate a Bayesian inverse problem to infer the lateral permeability variation in an aquifer from geodetic and hydraulic data and from prior information. We characterize the solution of the Bayesian problem—the posterior probability density function (pdf)—by computing the maximum a posteriori (MAP) estimate and the Gaussian approximation centered at the MAP estimate. These computations require gradient and Hessian information, which we compute through adjoint methods. Similar to the recent paper by *Iglesias and McLaughlin* [2012], we propose a Newton-conjugate gradient algorithm for the computation of the MAP estimate. The Gaussian approximation of the posterior pdf requires the inverse of the Hessian of the log posterior pdf. To efficiently compute the Hessian matrix, we exploit the low-rank structure of the data misfit part of the Hessian, which is typical for ill-posed inverse problems. Due to the use of adjoint methods, the computational work to compute the MAP estimate and the local Gaussian approximation of the posterior pdf does not depend on the discretization of the permeability field, i.e., on the number of parameters.

In section 5, the proposed inversion algorithm is demonstrated on three synthetic model problems from geomechanics and reservoir engineering. Despite the simple specification of these problems, the inference of the permeability fields is challenging. These model problems are designed to illustrate general properties of joint poroelastic inversions. In particular, we use them to study the influence of the overburden on the inferred permeability fields. These problems also provide benchmarks for the comparison of different poroelastic inversion algorithms. In section 6.1, we demonstrate the efficiency of the algorithm. The current limitations of our algorithm and implementation and our plans to address these limitations are discussed in sections 6.3 and 6.4.