Joint inversion in coupled quasi-static poroelasticity


  • Marc A. Hesse,

    1. Department of Geological Sciences, University of Texas at Austin, Austin, Texas, USA
    2. Institute for Computational Engineering and Sciences, University of Texas at Austin, Austin, Texas, USA
    Search for more papers by this author
  • Georg Stadler

    1. Institute for Computational Engineering and Sciences, University of Texas at Austin, Austin, Texas, USA
    Search for more papers by this author

Corresponding author: M. A. Hesse, Department of Geological Sciences, University of Texas at Austin, Austin, Texas, USA. (


Geodetic surveys now provide detailed time series maps of anthropogenic land subsidence and uplift due to injection and withdrawal of pore fluids from the subsurface. A coupled poroelastic model allows the integration of geodetic and hydraulic data in a joint inversion and has therefore the potential to improve the characterization of the subsurface and our ability to monitor pore pressure evolution. 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 compute the maximum a posteriori (MAP) estimate of the posterior permeability distribution and a Gaussian approximation of the posterior. Computing the MAP estimate requires the solution of a large-scale minimization problem subject to the poroelastic equations, for which we propose an efficient Newton-conjugate gradient optimization algorithm. The covariance matrix of the Gaussian approximation of the posterior is given by the inverse Hessian of the log posterior, which we construct by exploiting low-rank properties of the data misfit Hessian. First and second derivatives are computed using adjoints of the time-dependent poroelastic equations, allowing us to fully exploit transient data. Using three increasingly complex model problems, we find the following general properties of poroelastic inversions: Augmenting standard hydraulic well data by surface deformation data improves the aquifer characterization. Surface deformation contributes the most in shallow aquifers but provides useful information even for the characterization of aquifers down to 1 km. In general, it is more difficult to infer high-permeability regions, and their characterization requires frequent measurement to resolve the associated short-response timescales. In horizontal aquifers, the vertical component of the surface deformation provides a smoothed image of the pressure distribution in the aquifer. Provided that the mechanical properties are known, coupled poroelastic inversion is therefore a promising approach to detect flow barriers and to monitor pore pressure evolution.

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 km2 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 CO2 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.

Figure 1.

Geometries for the model problems in section 5. (a) Two-dimensional modified Mandel problem from geomechanics. (b) Two-dimensional modified Segall problem from reservoir engineering. The insert shows a detail of the aquifer and is also applicable to the quarter five-spot model problem. (c) The three-dimensional quarter five-spot problem from reservoir engineering. The insert on the top left shows the map view of the five-spot well pattern.

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.

2 The Linear Poroelastic Forward Problem

As a prototype of geomechanical coupling between the flow of pore fluids and the deformation of the solid skeleton, we consider the linear poroelastic theory developed by Biot [1941] in the two- or three-dimensional domain Ω over the time interval [0,T]. Wang [2000] gives the initial boundary value problem for the evolution of the fluid pressure p=p(x,t) and the solid displacement u=u(x,t) as

display math(1a)
display math(1b)

where ΩT:=Ω×(0,T) and (·)t denotes the time derivative. The initial and boundary conditions for the pressure p are given by

display math(1c)
display math(1d)
display math(1e)

where inline image is a disjoint splitting of the boundary Ω, and inline imageand inline image. For a (possibly different) disjoint splitting of the boundary inline image, the boundary conditions for the displacement u are

display math(1f)
display math(1g)

where inline imageand inline image. While this type of boundary conditions is sufficient for the model problems discussed below, the framework presented here is not limited to these boundary conditions.

Equation (1a) arises from the conservation of fluid mass, where Sε is the specific storage at constant strain, κ(x) is the permeability, μ is the fluid viscosity, p(x,t) is a boundary pressure, g(x,t) is a boundary flux, and f(x,t) is a volumetric source or sink term due to fluid injection or production at wells. Equation (1b) arises from the total momentum conservation of the porous medium, where σ is the elastic stress tensor, u is the solid displacement vector, α is the Biot-Willis parameter, ud is a boundary displacement, g(x,t) a boundary force, and f(x,t) is a volumetric body force. The elastic stress tensor in terms of the solid displacements is given by

display math(2)

where G and ν are the drained shear modulus and Poisson's ratio, respectively, and I is the identity matrix.

The Biot-Willis parameter and the specific storage coefficient at constant strain can be written in terms of the mechanical parameters as

display math(3)
display math(4)

where B is Skempton's pore pressure coefficient and νu is the undrained Poisson's ratio.

For the formulation of the inverse problem and the numerical discretization, it is convenient to consider (1a)(1g) in weak form. We introduce the solution spaces

display math

and the homogeneous variation and test function spaces

display math

for the pressure and the displacements, respectively. We assume the variables to be sufficiently regular for the variational forms given below to be well defined. Multiplying (1a) by inline image and (1b) by inline image, integrating over ΩT, using the boundary conditions and integrating by parts, the weak form of the poroelastic equations becomes: Find inline imagesuch that

display math(5a)
display math(5b)

Note that in the weak form (5a)(5b), the Dirichlet boundary conditions (1e) and (1g) have been added to the definition of the trial spaces inline imageand inline image, while the Neumann boundary conditions (1d) and (1f) are incorporated in the weak form (5a)-(5b).

3 The Bayesian Poroelastic Inverse Problem

3.1 Bayesian Inverse Problems

Bayesian inference provides a systematic way to combine different observations, observation error and model error, and to incorporate prior knowledge that is independent of observations. It seeks those model parameters (e.g., the permeability field κ(x)) which render the actual observations likely, taking into account observation and model errors. The solution of a Bayesian inverse problem is the posterior probability density function (pdf), which is defined over the parameter space [Tarantola, 2005; Kaipio and Somersalo, 2005].

If the inversion parameters are the coefficients corresponding to the discretization of a function (such as the permeability field), physically meaningful results should—aside from numerical approximation errors—not depend on the discretization of the parameter space. To properly account for statistical inverse problems where the unknown parameter is a function, an infinite dimensional Bayesian inversion theory has been developed (see the review in Stuart [2010]). This function space approach formulates the statistical inverse problem in infinite dimensions and, when properly discretized, leads to well-defined finite dimensional Bayesian inverse problems that approximate the infinite dimensional Bayesian solution. Below, we summarize the finite dimensional Bayesian approach following the notation in Tarantola [2005] and point out conditions that guarantee that the finite dimensional problem is an appropriate numerical approximation of the underlying infinite dimensional problem.

In the poroelastic problem considered here, the parameter-to-observable map inline imagerequires the solution of a system of coupled partial differential equations for given parameters, inline image, in our case the coefficients of a discretization of the permeability field κ(x). Assuming that the error between the model simulation output, h(κ), and the observations, inline image, is Gaussian, Bayes' theorem states that the posterior pdf is given by

display math(6)

where ∝ denotes that the left and right side coincide up to a constant, and the log likelihood, which measures the acceptability of κ given dobs is

display math(7)

The data covariance matrix inline imagein (7) allows to incorporate the assessment of measurement error and the significance of model data discrepancies. The prior pdf πprior(κ) represents prior knowledge that is independent of the data dobs, for instance, by imposing a correlation between parameters. Here we will restrict ourselves to Gaussian priors, i.e.,

display math(8)

where inline image denotes the prior mean and Γprior the prior covariance matrix. An important requirement for being able to formulate a Bayesian inverse problem in infinite dimensions (and guarantee convergence of the discretizations) is that the prior provides a sufficiently strong correlation between spatially close points, such that the prior variance is pointwise bounded [Stuart, 2010; Bui-Thanh et al., 2013]. Even when, as above, a Gaussian prior is chosen and Gaussian noise for the parameters is assumed, the posterior pdf is not Gaussian unless the parameter-to-observable map h(κ) is linear [Tarantola, 2005]. Since this is in general (and in the problem considered in this paper) not the case, the posterior pdf has to be explored numerically. A full exploration of the posterior pdf has to rely on sampling methods and is often infeasible for high-dimensional parameters and computationally expensive parameter-to-observable maps h(·). The focus of this work is on the computation of the most likely permeability field that maximizes πpost (section 3.2), and on a local Gaussian approximation of the posterior around that point (section 3.3), which—depending on the nonlinearity of the parameter-to-observable map—can be an excellent approximation of πpost.

3.2 The Maximum A Posteriori Estimate

The maximum a posteriori (MAP) estimate κMAP of πpost is defined as the point in parameter space that maximizes the posterior pdf or, equivalently, minimizes the negative log of πpost. Thus, combining (6), (7), and (8), κMAP is the solution of the following nonlinear least squares optimization problem

display math(9)

We will employ a Newton method to find κMAP. Starting with an initial guess for the permeability field κ, Newton's method iteratively updates these parameters based on successive quadratic approximations of, which requires gradients (i.e., first derivative) and Hessians (i.e., second derivative) of inline image with respect to κ. The parameters are updated as follows:

display math

where κ is the current permeability vector, and the Newton update direction inline image is obtained by solving the linear system

display math(10)

Here inline image is the gradient of the least squares functional inline image in (9), and inline image is the Hessian operator, i.e., the second derivative of evaluated at κ. For a Newton-type method, it is known that a step length of β=1 results in fast convergence close to the solution. To guarantee convergence from any initialization of the parameter, the step length β>0 is reduced such that the functional in (9) is sufficiently decreased in each iteration, e.g., it satisfies the Armijo condition [Nocedal and Wright, 2006].

In the next section, we discuss the Gaussian approximation of the posterior distribution about the MAP point κMAP.

3.3 Low-Rank-Based Gaussian Approximation of πpost

As discussed above, the only source of non-Gaussianity of the posterior pdf πpost is the nonlinearity of the parameter-to-observable map. If this map is only mildly nonlinear over the set of parameters that are consistent with the prior and the data, the posterior can be well approximated by a Gaussian centered at the MAP point given by

display math(11)


display math(12)

where, as above, inline imageis the Hessian of inline image(as defined in (9)) evaluated at the MAP estimate, and we have used the fact that the gradient of inline image vanishes at the MAP point, i.e., inline image. Choosing a scaling factor that makes inline imagea proper pdf, the approximation (11) of the posterior is a Gaussian with mean κMAP and covariance matrix given by inline image, the inverse of the Hessian. The explicit computation of the Hessian requires as many forward PDE solves as there are parameters, which makes this prohibitive for large-scale inverse problems. Thus, we need efficient algorithms to approximate the Hessian matrix using low-rank ideas by exploiting its structure. Following Bui-Thanh et al. [2013], the Hessian can be written as the sum of inline image, the Hessian of the data misfit term (see inline imagein (15)) and the inverse of the prior covariance operator:

display math(13)

For many ill-posed problems, data typically inform only a limited number of modes of the parameter field, and thus the spectrum of the misfit Hessian often decays quickly. Thus, inline image can typically be well approximated by a low-rank matrix, enabling an efficient computation of the Hessian (13) (or a prior preconditioned version of the misfit Hessian Bui-Thanh et al. [2013]). The computation of this low-rank approximation requires applications of the data misfit Hessian inline image to vectors. In general, each application requires the solution of two PDEs, which are specified for the poroelastic system in section 3.4. The construction of the low-rank approximation for the misfit Hessian can then be computed using, for instance, the Lanczos algorithm.

Having computed a low-rank approximation of the data misfit Hessian, the Sherman-Morrison-Woodbury formula allows an efficient inversion of the Hessian (13) to compute the posterior covariance matrix inline imagefor (11). The posterior covariance matrix can then be used to draw samples from inline image, visualize the pointwise variance or compute other statistical quantities. In particular, it provides information on the confidence we can have in the inferred parameters.

Both Newton's method for finding the MAP estimate and the construction of inline image require the computation of derivatives of inline image with respect to κ. This is complicated by the implicit dependence of inline imageon the parameters through the nonlinear parameter-to-observable map. In the next section, we present an effective computation of these derivatives using adjoint equations. For that purpose, we specify the parameter and its prior and the likelihood function, which involves the Biot equations (1a)(1g) as part of the parameter-to-observable map h(κ).

3.4 Adjoint-Based Derivatives for Inversion in Poroelasticity

Adjoint-based methods for the efficient computation of derivatives of functionals whose evaluation involves the solutions of partial differential equations are reviewed in the textbooks Borzì and Schulz [2012]; Tröltzsch [2010]; Gunzburger [2003]; Hinze et al. [2009]. The first application of adjoint methods for joint inversion in poroelasticity can be found in Iglesias and McLaughlin [2012], where the parameter-to-observable map occurring in poroelasticity is analyzed. Moreover, a Newton-conjugate gradient method to solve the deterministic inverse problem is presented, which amounts to minimizing inline image as defined in (15) below. The weak form of the adjoint equations, whose derivation we sketch in this section coincides with Iglesias and McLaughlin [2012]. We also present the strong form of the adjoint equations and explicitly specify the incremental forward and adjoint equations required to apply the Hessian matrix to vectors. Instead of prior knowledge or regularization, Iglesias and McLaughlin [2012] use early truncation in the conjugate gradient algorithm to cope with the ill-posedness of the inverse problem. We next summarize the inference problem.

We target the inversion of the permeability in the poroelastic equations (1a)(1g) in the subset Ω0 of the domain Ω, which corresponds to the aquifer. Figure 1 shows sketches of the geometries used in the model problems studied in section 5. We assume that the permeability is given by

display math(14)

and invert for the log permeability ζ defined on Ω0. This parameterization ensures that κ is positive in Ω0; as an alternative, one can choose to invert for κ directly and use inequality constraints to enforce positivity.

The definition of the parameter-to-observable map is thus that it maps the log permeability ζ=ζ(x) to time series surface deformation and well pressure data. To compute the derivatives of inline imageefficiently, we use adjoint equations. All expressions in this section are given in infinite dimensional form, which provides a natural and “clean” path to computing derivatives. These expressions are then discretized (see section 4) resulting in a finite dimensional inverse problem as discussed in section 3.1.

We assume given time series point data ud of the displacement on the top surface inline image, and time series well data pd for the pressure p. For convenience of the notation, the below formulation assumes uobs and pobs to be defined everywhere in space and we use Dirac delta functions δu,δp and δκ to restrict them to the actual measurement points. The negative log likelihood is thus given by

display math(15)

where the implicit dependence of u and p on ζ through the solution of the poroelastic equations is used in the definition of inline image. Moreover, the negative log of the prior is

display math(16)

where ζobs are point data for the log permeability, ζm is an estimate for the mean log permeability, and γ,γm≥0 are parameters that characterize the prior, where γ controls to prior knowledge on the smoothness of the log permeability field, and γm its distance to the mean. Above,

display math

where δ(·) denotes the Dirac delta function, and inline imagewith Nu,Np≥0 are measurement points in Ω for u and p, and inline image with Nκ≥0 are points where data for the log permeability ζ(x) is available. Here we use diagonal noise covariance matrices, and thus inline image are the variances (i.e., the diagonal entries of the noise covariance matrices)—compare with the first term in (9). The nonlinear least squares functional to be minimized for computing the MAP estimate, therefore, is

display math(17)

In the following, we denote by (·,·)S the integral over a set S⊂Ω; see Appendix A. To compute the derivative of inline imagewith respect to ζ, we use the Lagrangian functional

display math(18)

In this definition, we consider the state variables u and p independent of ζ, and the equations for poroelasticity are enforced through the Lagrange multipliers v and r. We next turn to the computation of the gradient of inline image.

3.4.1 Gradient Computation

The gradient, inline image, of inline image with respect to ζ can be obtained as derivative of inline imagewith respect to ζ, provided variations with respect to all other variables vanish [Borzì and Schulz, 2012; Tröltzsch, 2010]. The latter condition gives rise to state and adjoint equations and their derivation is given in Appendix B.

The gradient is obtained requiring the variations of the Lagrangian functional with respect to ζ vanish and it is given by

display math(19)

for x∈Ω0, and inline image for xΩ0. Note that in inline image, the time evolution of the pressure p and the adjoint pressure r gets collapsed by the integration over the time interval [0,T]. To compute inline image, the pressure p and the adjoint pressure r are required. The former is found as the solution of the Biot equations, whose weak form is recovered from the Lagrangian functional by requiring that its variations with respect to r and v vanish. In this context, the Biot equations are also referred to as the state equations and given by (1a)(1g).

The adjoint variables are found by solving an adjoint Biot system, which is obtained assuming that variations of the Lagrangian functional with respect to p and u vanish. We obtain the boundary value problem for the evolution of the adjoint fluid pressure r=r(x,t) and the adjoint solid displacement v=v(x,t) as

display math(20a)
display math(20b)

The boundary conditions for the adjoint pressure and the final time condition are given by

display math(20c)
display math(20d)
display math(20e)

Note that the adjoint pressure r is given at the final time t=T. Thus, (20) must be solved backward in time. The boundary conditions for the adjoint displacements are given by

display math(20f)
display math(20g)

The adjoint Biot system given here in strong form is equivalent to the weak expressions obtained by Iglesias and McLaughlin [2012].

The computation of the gradient inline imagefor a given log permeability ζ proceeds as follows. First, the forward Biot problem (1a)(1g) is solved. The resulting displacement u and pressure p is then used to compute the data misfit “source” terms for the adjoint problem (20). This adjoint problem is then solved to yield the adjoint displacement v and pressure r. The forward and adjoint pressures, along with the current log permeability iterate ζ then enter into the evaluation of the gradient inline imagein (19). Next, we discuss the computation of the Hessian operator inline imageoccurring on the left side of the Newton equation (10).

3.4.2 Hessian-Vector Product

When the Hessian operator in the Newton system (10) can be derived and is discretized, it gives rise to a dense square matrix inline imageof dimension equal to the number of inversion parameters, i.e., the number of unknowns in the discretization of the log permeability ζ. For large-scale inverse problems, explicitly forming and storing this matrix is often not an option since it requires the inverse of the linearized forward map [Borzì and Schulz, 2012]. Instead, we solve the Newton system (10) using the linear conjugate gradient (CG) method, which does not require the explicit Hessian, but only the action of the Hessian on a vector at each CG iteration. Next, we present expressions for this Hessian vector product in terms of the solution of a pair of linearized forward and adjoint problems. These expressions are simply stated here; their derivation follows by taking second variations of the Lagrangian functional. In the equations for the second derivatives given below, certain terms are boxed. These terms are often neglected to ensure that inline image is positive definite (see the discussion at the end of this section). For the application of the Hessian inline image to a parameter perturbation inline imageone obtains

display math(21)

The incremental variable inline imageis found by solving the incremental state equation for inline image:

display math(22a)
display math(22b)

with the initial and boundary conditions

display math(22c)
display math(22d)
display math(22e)
display math(22f)
display math(22g)

The incremental adjoint pressure inline image, also required in (21) is found by solving the incremental adjoint equation system

display math(23a)
display math(23b)

with the boundary and initial conditions

display math(23c)
display math(23d)
display math(23e)
display math(23f)
display math(23g)

The Newton update direction computed from (10) is a descent direction only if the Hessian is positive definite, which is only guaranteed close to a minimum [Nocedal and Wright, 2006]. Therefore, we use the Gauss-Newton approximation to guarantee a positive definite system. In this approximation, the terms in the Hessian expressions that involve the adjoint variable—surrounded by boxes in (21) and (23)—are neglected. Since the adjoint system (20) is driven only by the point misfits of the displacement and the pressure, the adjoint velocity is expected to be small when the data misfit is small, which occurs close to the solution of the inverse problem provided the model or observational errors are not large. The Gauss-Newton Hessian is thus often a good approximation of the full Hessian. In such cases, even though one loses the strict quadratic convergence guarantee of Newton's method, one can still obtain fast, superlinear convergence, and independence of the number of iterations from the number of inversion parameters. In the remainder of this article, we employ the Gauss-Newton approximation of the Hessian, but for simplicity we occasionally drop the term “Gauss” when referring to the Hessian matrix. Note first that the (Gauss-Newton) Hessian (21) has the same form as the gradient (19). Second, the incremental forward and incremental adjoint problems (22) and (23) have the same matrix operator as the state and the adjoint systems and differ only in the source terms.

4 Discretization and Solvers

We use linear and quadratic continuous finite elements for the spatial discretization of the poroelastic equations (1a)(1g), and the implicit Euler method for the discretization in time. More sophisticated spatial discretizations are possible and our inversion methods are generally independent of the discretization. For instance, mixed and staggered grid discretizations [Ferronato et al., 2010; Aguilar et al., 2008; Phillips and Wheeler, 2007] can guarantee local conservation and avoid oscillations at interfaces with strong permeability contrast. The advantages these schemes have over standard discretizations will carry over to the adjoint and incremental systems presented here. While Iglesias and McLaughlin [2012] solve the governing equations sequentially with a fixed-stress coupling proposed by Kim et al. [2011], we solve the system of equations fully coupled.

We follow a discretize-then-optimize approach for the adjoint system (20), i.e., the adjoint equation is implemented as the discrete adjoint of the state equation. To give a sketch for this approach, we consider a discretization of (1a)(1g) given by

display math(24)

where the superscript k denotes the time step, K is the number of overall time steps, the block matrices L and N are given by

display math(25)

the finite element coefficients pk of p and uk of u at the kth time step are

display math

and X0 is the initial condition. In (25), M corresponds to the discretization of the pressure mass matrix, K to the discretization of the Poisson operator (which depends on the permeability), D and G to the discrete divergence and gradient operators, E to the (negative) discrete elasticity operator, and Fk to the discretization of volume and boundary forces.

Thus, the overall space-time discretization of the state equation can be written as

display math(26)


display math

and inline image corresponds to the space-time force vector and has LX0 in its first component to enforce the initial condition. Using the finite dimensional analogue of the Lagrangian functional (18), one finds the space-time adjoint system

display math(27)

where Xobs is observation data, inline image is a discretization of the observation operator, and inline imageis the space-time adjoint vector, with time steps

display math

Writing (27) separately for each time step, one obtains Euler integration

display math

for k=K−1,…,0. This is a discretization of the continuous adjoint equations (20a)(20g). In particular, the natural time integration for the adjoint is again the implicit Euler scheme, but backward in time. Analogously to the adjoint equation, the incremental equations used for the computation of second derivatives inherit their discretization from the poroelastic state equation.

For a fixed size of the time step, a matrix system with L as coefficient matrix has to be solved in every time step of the implicit Euler method (see (25)). In our implementation, L is factorized up-front and stored in memory, such that each step in the time integration only requires a forward and backward substitution with triangular matrices. If the physics of the problem does not allow a fixed time step size, we factorize L for different time step sizes up-front. These factorizations are not only used for the forward equation but also for the adjoint equations, as well as for the incremental state and adjoint systems. Only when the log permeability is updated, they are recomputed. This significantly reduces the computational cost for the time integration of state, adjoint and incremental state, and adjoint equations.

For large, three-dimensional problems, direct factorization of L might not be an option and block-preconditioned iterative Krylov solvers [Ferronato et al., 2010] or multigrid methods [Gaspar et al., 2008] can be employed. These methods, devised for the Biot system can also be used to solve the adjoint and incremental equations and the efficiency of a solver for the Biot equations thus carries over to the computation of the MAP estimate in the inverse problem.

The discretization of inline image(as defined in (16)) results in the second term on the right hand side of (9). The pointwise variance as well as the correlation between neighboring permeability parameters are determined by the prior covariance matrix Γprior, which depends on γ, γm and inline image, k=1,…,Nκ. While we use continuous time data in the definition (17) of the negative log likelihood function inline image, in our inversions the time series at each spatial measurement point is projected to piecewise linear interpolation between fixed time instances, which are given in Table 2. This makes the time observations finite dimensional and the data independent of the time discretization.

For the inversion study presented in the next section, we use synthetic observational data. That is, we solve the poroelastic equations using a target log permeability field to create synthetic observations. To mitigate the inverse crime, which occurs when the same numerical model is used to both synthesize the data and drive the inverse numerical solution [Kaipio and Somersalo, 2005], we add Gaussian random noise to this data.

5 Inversion Results for Model Problems

We next study properties of poroelastic joint inversion using three model problems. We present results for two commonly used two-dimensional problems from geomechanics and reservoir engineering, and for a three-dimensional problem from reservoir engineering; see Figure 1. The parameters for the poroelastic model problems presented in this section are summarized in Table 1, the parameters used for the inversion in Table 2, and the boundary and initial conditions in Appendix C.

Table 1. Values of the Parameters in (1a)(1g) for the Numerical Results in Section 5a
  1. a

    Numbers in parentheses refer to the equation defining the permeability field. A parameter that has been varied systematically is indicated by “var.” and values will be given in the text or the figure legend. The parameters are grouped into those that are common to all model problems, those that are only required for a particular problem, and the parameters for the inversion.

κm2(28)(30)Figures 10a and 11a
Tyrs or s1 (s)2010
inline imagem2-1.37·10141.37·1014
Table 2. Parameter Values Used for Joint Inversions Results in Section 5a
  1. a

    The time measurements for the modified Segall and the quarter five-spot points are uniform (once per year). For the modified Mandel problem, the measurement times are 10−6,3·10−6,10−5,3·10−510−43·10−4,10−3,10−2,10−1,and1. We assume the same noise variances for each data point, i.e., the noise covariance matrices are identity matrices weighted by inline image for the displacement data, by inline image for the pressure data, and by inline image for the permeability data. The modified Mandel problem is solved for three different noise levels for the surface displacement data, i.e., for three different values of σu.

t-datayrs or s10−6…1 (s)1,2,…,201,2,…,10

5.1 Modified Mandel Problem

Mandel [1953] considered the constant loading of a poroelastic domain closed on three sides and held at constant pressure on the right boundary, as shown in Figure 1a. The application of the force, F, leads to an initial undrained response and an instantaneous pressure increase to p=B(1+νu)F/3 throughout the domain. Over time, the fluid drains to the right and the domain contracts. At the beginning of the drainage, the pressure at the near left side of the domain rises above the undrained response before it decays to zero. This nonmonotonic pressure response following a constant applied boundary stress is a result of the coupling between (1a) and (1b) through the volumetric strain rate.

In the original problem, the load is applied by a rigid body so that the vertical displacement does not vary along the top of the domain. In this case, the transient solution is one-dimensional and Figures 2a–2ac show the analytic solution for this problem given by Mandel [1953] and Cheng and Detournay [1988]. The solution is shown as a function of dimensionless time, defined as τ=cεt/L2, where L is the domain length and cε the hydraulic diffusivity of the aquifer at constant strain, defined as cε=κ/μfSε. The availability of an analytic solution for the Mandel problem makes it a useful benchmark for poroelasticity, see, for example, Phillips and Wheeler [2007] and Barbeiro and Wheeler[2010].

Figure 2.

Mandel problem: (a–c) Comparison of the numerical solution (circle) to the analytic solution (line) of Mandel [1953] and Cheng and Detournay [1988] at different dimensionless times τ=cεt/L2. The parameters for the analytic solution are ν=0.2, νu=0.5, and B=1, which correspond to Cheng and Detournay [Cheng and Detournay, Figure 6]. (d–f) The solution to the modified Mandel problem, defined by (1a)(1g) and (C1), with a constant permeability field.

The simplicity of the Mandel problem and its widespread use as a benchmark for the forward problem make it an appealing test for the inversion. The original Mandel problem, however, has a rigid upper boundary so that the vertical component of the surface deformation is constant and uniform. If the top boundary is allowed to deform, the resulting nonuniform surface deformation can be used to invert for the unknown permeability structure (Figures 2d–2f). Differently from the original Mandel problem, the pressure evolution in the modified Mandel problem used in the inversions presented below shows a monotonic pressure evolution.

To illustrate the basic characteristics of the poroelastic inverse problem, two model problems with the following simple permeability variation are considered

display math(28)

where κl and κr are the left and right permeability plateaus shown in Figures 3c and 3f. The model parameters are given in Table 1, and the detailed boundary and initial conditions are given in section C1. The only data considered in this inversion are the surface displacements at a finite number of locations, i.e., Np=Nκ=0, at a finite set of times. The first case is a left-facing step in permeability (κl=5.003·10−1 and κr=103) and the second case a right-facing step in permeability (κl=103 and κr=5.003·10−1), where the value 5.003·10−1 is chosen so that the harmonic mean,

display math(29)

in both cases. The characteristic diffusive timescale for the drainage of the layer, given by tc=L2μSε/<κ>, is therefore the same in both cases and all results are given in dimensionless time, τ=t/tc≤1.

Figure 3.

Inversion of the modified Mandel problem. (a–c) Inversion for a left-facing step in permeability. (d–f) Inversion for a right-facing step in permeability. (Figures 3a and 3d) Transient evolution of the forward solution. (Figures 3b and 3e) The surface deformation of the forward model (line) and the data used for the inversion (circle). The data shown contains normally distributed noise with a variance corresponding to 10−2 of the mean response, in Figures 3a and 3c, respectively. (Figures 3c and 3f) The target κ distribution (black) with the MAP estimates corresponding to three values of the noise variance in the surface deformation data.

The initial instantaneous undrained response and the final steady state drained response of the medium are independent of the permeability and therefore the same in both cases. The transient response, however, differs and determines the ability to infer the MAP estimate κMAP.

The first case is shown in Figures 3a–3c. The high-permeability region is next to the draining boundary on the right. This leads to a fast initial drainage of the high-permeability region, 0≤τ<10−3, and a subsequent slow drainage of the low-permeability region, 10−2<τ≤1. At early time the deformation is associated with pressure gradients during the drainage of the high-permeability layer and at late times the deformation is associated with pressure gradients during the drainage of the low-permeability layer. The surface deformation therefore contains distinct signals reflecting the two permeability values and the inferred κMAP captures both permeability plateaus. This however requires that the data resolve the two separate response timescales of the medium.

The second case is shown in Figures 3d–3f. The low-permeability region next to the drained boundary provides a barrier to drainage, so that the drainage occurs slowly over the entire time interval, 0<τ<1. The high-permeability region in the interior of the domain is masked, because there is no corresponding short time dynamic response and no deformation due to the lack of a pressure gradient. The inferred κMAP therefore only captures the low-permeability region that sets the timescale of the response and where the pressure gradient and the deformation are localized. As the noise in the data increases the error of κMAP in the high-permeability region increases rapidly.

The results of a Bayesian uncertainty analysis using data containing noise with σu=1.7·10−4 are shown in Figure 4. To illustrate our assumptions on the prior probability distribution, πprior, in Figure 4a we show the prior mean, <κ>, samples drawn from the prior distribution, and the pointwise marginal distributions of the prior. These samples illustrate the smoothness and thus the correlation length that is imposed on the inferred permeability fields by the prior. The gray shading visualizes the pointwise variance, i.e., information on the diagonal of the prior covariance operator. Quantities illustrating properties of the Gaussian approximation of the posterior distribution πpost for the left-facing and right-facing steps are shown in Figures 4b and 4c, respectively. We depict samples from the posterior distribution, the pointwise marginals of the distribution, and the MAP point κMAP (which is the mean of the Gaussian approximation of the posterior distribution). The darker narrow regions indicate that the posterior is well constrained, while wide gray regions indicate larger uncertainty in the MAP reconstruction. Observe that the variance of πpost in the high-permeability region is much larger for the left-facing step. This uncertainty in the permeability in this region is also illustrated by the different samples drawn from πpost. The difference between samples drawn from πpost in the high-permeability region is up to 2 orders of magnitude.

Figure 4.

Bayesian inference of the permeability for the modified Mandel problem. Shown are the mean (yellow line), the point marginals (gray) and three samples (in shades of red) from the prior and the Gaussian approximation of the posterior pdf. (a) Prior probability distribution of the log-permeability. (b) Posterior distribution for the log permeability for the left-facing step together with the target (black line). (b) Posterior distribution for the log permeability for the right-facing step together with the target (black line).

5.2 Modified Segall Problem

Segall [1985] studied the subsidence due to fluid extraction from a laterally extensive, horizontal, poroelastic aquifer of thickness, h, located at a mean depth, D, embedded in an elastic half space, as shown in Figure 1b. The Segall problem is a useful benchmark for the poroelastic forward problem, because it is based on a more realistic geological scenario that includes a deformable overburden. The effect of this overburden on poroelastic aquifer characterization and pressure monitoring is one of the main open questions this manuscript aims to address.

Here we consider a slightly modified problem of fluid extraction from a permeable aquifer surrounded by a low-permeability aquitard in a large but finite domain. The fluid is extracted from a single production well competed over the entire aquifer that is located in the center of the domain. The width of the domain, L, must be much larger than the horizontal diffusive distance of the pressure perturbation, inline image, where <κ> is the harmonic average of the aquifer permeability in the horizontal direction. The height of the domain, H, is four times the maximum depth of burial of the aquifer. In the simulations presented below both the aquifer and the aquitard are assumed to be poroelastic rather than assuming a purely elastic aquitard. This will allow some pressure dissipation from the aquifer and may slow down the lateral pressure propagation [Chang et al., 2013].

The permeability of the domain is heterogeneous and given by

display math(30)

where κ and κ0 are the permeability of the aquifer and the ambient aquitard, respectively. The bottom of the permeable layer is given by zb=H−(D+h/2) and the top is given by zt=H−(Dh/2). For simplicity, the permeability of the aquifer is considered to vary periodically,

display math(31)

where inline image is the mean value of log10κ and λ is the wavelength.

The model problem considered here is based on the physical properties considered by Segall [1985], and all parameter values used in the simulations discussed below are given in Table 1, while the boundary condition are stated in section C2. After 20 years of production the final cone of depression around the production well is almost 20 cm deep in the vicinity of the well and the surface subsidence as spread approximately 15 km on either side of the well. This distance is on the same order of magnitude as the diffusive distance of the pressure perturbation in the aquifer, δD≈8 km.

Four observation wells are located within the cone of depression at ±5 km and ±10 km distance from the producer. The permeability and the fluid pressure in the aquifer are only known at these five wells. The wells are located such that they always sample κ0 in all heterogeneous cases considered below, to facilitate the comparison of different wavelengths of heterogeneity. The two components of the surface deformation are measured within ±15 km of the well at 50 locations spaced evenly with 600 m. This ensures at least two measurements for the shortest wavelength considered below, λ=1.25 km.

Note that we only infer the permeability variation inside the aquifer, Ω0, and assume constant permeability in the ambient aquitard. First, the joint inversion with different combinations of the three data types will be considered. Figure 5 compares inversions considering only pressure (P), only surface deformation (S), and joint inversions considering all data (S,K,P). The inferred κMAP in the aquifer shows that the error increases with distance from the production well in all cases. This is due to the diffusive character of pore fluid flow, which leads to a decay of all perturbations over a diffusion distance, δD, so that the signal-to-noise ratio increases with distance from the producer. The error in the high-permeability regions is higher than in low-permeability regions with similar distance to the production well. This is reminiscent of the masking of the high-permeability area in section 5.1 and due to the lack of strong pressure gradients and associated deformation in these regions.

Figure 5.

Numerical results for individual and joint inversions of surface deformation, S, pore fluid pressure, P, and permeability, K. All graphs have the same legend shown in graph c. (a–d) Results for an aquifer with permeability wavelength of 10 km. (e–h) Results for an aquifer with permeability wavelength of 5 km. (i–l) Results for an aquifer with permeability wavelength of 2.5 km. (a, e, and i) The target permeability field and the three inferred permeability fields using the data indicated in the legend and the permeability data. (b, f, and i) u1, (c, g, and k) u2, and (d, h, and l) p for the forward problem using the target permeability after 20 years of production and the data for the inversion that are derived from it. These are compared with the results from three different inversions as indicated by the legend.

For λ larger than the well spacing, the permeability variation near the well can be inferred from pressure data alone, but the error is reduced if surface deformation is taken into account. For λ comparable to the well spacing, the inversion based on pressure alone leads to significant errors, because the phase of the inferred κMAP is shifted. In contrast, the permeability variation inferred only from surface deformation has a relatively small error near the well. As λ becomes small, the elastic overburden damps the small scale deformation and the short wavelength features in the permeability field cannot be resolved. Since the flow is predominantly horizontal and perpendicular to the unresolved features, κMAP approaches the effective permeability for flow across layers, given by the harmonic mean of (31),

display math(32)

In particular, the inversions in Figure 5i that do not prescribe κ at the wells show how κMAP approaches this limit. In the SKP-case, κMAP forms peaks at the outer observation wells, where the data pin it at inline image, but the homogenization of the unresolved permeability features requires a lower effective permeability of <κ>.

Figure 7 shows that joint inversion always reduces the error. For long wavelength permeability variations, the inversions with single data types capture the main features and the error is already small so that joint inversion only leads to small reductions in the error. The noise in the data always introduces significant error for short permeability wavelengths, so that joint inversion only leads to minor improvements. At intermediate wavelengths, however, the error of inversions with single data types can be large, but the joint inversion with different data can significantly reduce the error in the inferred permeability field.

Figure 6.

Error in the inferred permeability field in the aquifer for joint inversions with different combinations of surface deformation data, S, pore fluid pressure data in the wells, P, and aquifer permeability data at the wells, K. Both the S and P measurements are taken in time intervals of 2 months. The detailed results for the joint inversion cases SKP, S, and P are shown in Figure 5.

Figure 7.

Bayesian inference of the permeability for the modified Segall problem with a permeability wavelength of 2.5 km and a depth of 1 km. Shown are the mean (yellow line), the point marginals (gray), and three samples (in shades of red) from the prior and the Gaussian approximation of the posterior pdf. (a) Prior probability distribution of the log permeability. (b) Posterior distribution for the log permeability using only pressure data together with the target (black line). (c) Posterior distribution for the log permeability using only surface deformation data. (d) Posterior distribution for the log permeability using surface deformation, pressure, and permeability data together with the target (black line).

The reduction of the uncertainty in κMAP due to joint inversions of different data can also be seen from the Gaussian approximation of the posterior about the MAP estimate, shown in Figure 8. Generally, the variance increases with distance from the production well as the data is less informative about the permeability. The variance for the inversion considering only the pressure data (Figure 8b) is significantly larger than for the inversion considering all three data types (Figure 8d). For the intermediate wavelength example shown here, the uncertainty in the permeability estimate based solely on surface deformation data (Figure 8c) is comparable to the inversion with all three data types. This illustrates the reduction of the uncertainty in the aquifer characterization due to the incorporation of surface deformation data.

Figure 8.

Numerical results for joint inversions using surface deformation, pore fluid pressure, and permeability with increasing depth of the aquifer, D, indicated in the legend. (a–c) The effect of aquifer depth on the surface deformation response and the aquifer pressure after 20 years for a permeability wavelength of 2.5 km. (d–g) Comparison of the inferred permeability fields for aquifers with increasing depth with the target permeability field. Results are shown for aquifers with target permeability wavelength of 1.25 km, 2.5 km, 5 km, and 10 km. Figure 6e corresponds to the results shown in the left column.

The effect of the overburden on the resolution of different permeability wavelengths in joint inversions of all three data types are considered. Figures 6a–6c show that the observable surface deformation is reduced as the depth of the aquifer increases, but that the pore fluid pressure in the aquifer remains essentially unchanged. This leads to increased errors in the inferred κMAP for all wavelength as illustrated in Figures 6d–6g. The error increases faster for short permeability wavelength, because the corresponding short wavelength deformation is preferentially smoothed by the increasing overburden. Figure 9 shows the error in the inferred permeability as a function of aquifer depth D and permeability wavelength λ. The error increases with decreasing λ and increasing D. The changing slopes of the error contours show that the inversion is most sensitive to D at intermediate wavelengths.

Figure 9.

The error between the inferred and the target permeability field ||κMAPκ||/||κ|| in percent as a function of both aquifer depth D, and the permeability wavelength λ. The location of the data shown in Figure 6 are shown as small circles (∘) for comparison.

5.3 Quarter Five-Spot Pattern

Oil reservoirs are commonly produced by placing injectors and producers in a regular pattern to ensure uniform oil recovery [Lake, 1989]. The most well-known pattern is the five-spot, where each production well is at the center of a square defined by four injection wells and vice versa. This leads to a regular pattern shown in Figure 1c that has a square cell as its symmetry unit, with a single pair of injection and production wells in opposite corners. The symmetry unit is referred to as a quarter five-spot and all its lateral boundaries are symmetry boundaries. The quarter five-spot is a common model problem in reservoir engineering, and it is chosen as a three-dimensional model problem, because the lateral symmetry boundaries allow a small computational domain.

Similar to the Segall problem discussed above, we consider a horizontal aquifer of thickness h buried at depth D below the surface and surrounded by a low-permeability aquitard. The physical parameter values are given in Table 1, and are generally similar to the Segall problem considered in section 5.2. The geometry of the domain is shown in Figure 1c, the initial condition is p0=0. The boundary conditions are specified in section C3.

A well in the northeast corner injects fluid into the aquifer at a constant rate Q and a well in the southwest corner produces fluid at the same rate. The source term for (1a) is therefore given by

display math

where δ(x,y,z) is the delta function, and the bottom and top of the aquifer are given by zb=H−(D+h/2) and zt=H−(Dh/2), respectively. The permeability in the domain is given by

display math(33)

where the spatial variation of the permeability field inline image in the aquifer is generated using the geostatistical tool box mGstat [Hansen et al., 2013] and an isotropic spherical semivariogram with a range of 0.8L and eight points of known permeability. Two permeability fields have been generated, one with a high-permeability channel connecting the two wells (Figure 10a) and the other with a low-permeability barrier between the two wells (Figure 11a).

Figure 10.

Quarter five-spot problem with a channel. (a) The target permeability field with a high-permeability channel. (b) Estimated permeability field in a hydrological inversion, using only the observed permeability k and the pressure evolution p at the wells. (c) Estimated permeability field in a poroelastic inversion using well data and surface deformation data. The (d–f) vertical surface deformation and the (g–i) pore pressure in the aquifer are shown after 1, 2, and 10 years.

Figure 11.

Quarter five-spot problem with a barrier. (a) The target permeability field with a low-permeability barrier. (b) Estimated permeability field in a hydrological inversion, using only the observed permeability k and the pressure evolution p at the wells. (c) Estimated permeability field in a poroelastic inversion using well data and surface deformation data. The (d–f) vertical surface deformation and the (g–i) pore pressure in the aquifer are shown after 1, 2, and 10 years.

The evolution of the vertical surface deformation for the channel and the barrier are shown in Figures 10d–10f and Figures 11d–11f, respectively. The evolution of the pore pressure in the aquifer is shown in Figures 10d–10f for the channel and Figures 11d–11f for the barrier. The two types of heterogeneity in the aquifer leads to two distinct patterns in surface deformation and pore pressure. This difference contains information that can be utilized to characterize the aquifer. Comparing the pore pressure fields with the associated surface deformation shows that the surface deformation is simply a smoothed version of the pore pressure field in the aquifer. It is therefore possible to measure the propagation of a pressure disturbance in the aquifer by visual inspection of the surface deformation. This is particularly interesting in geological CO2 storage, where the outer radius of the pressure disturbance needs to be monitored to avoid the contamination of potable aquifers. Unlike the CO2 plume, which can be detected by seismic surveys, the much larger pressure plume can only be monitored with costly observation wells or potentially through surface deformation.

For the inverse problem for the permeability distribution in the aquifer, Ω0={x∈Ω|zb<z<zt}, we consider surface displacements measured on a 15 by 15 grid and pressure and permeability data from four observation wells located at (L/2,L/5), (L/2,4L/5), (L/5,L/2), and (4L/5,L/2), as shown in Figures 10a and 11a. Deformation and pressure data are sampled once a year for 10 years. Note that the injection and production wells are not sampled in this case.

We study two types of joint inversions: (1) hydraulic inversion based on the permeability and the pressure evolution at the four observation wells and (2) poroelastic inversion that additionally includes all three components of the surface deformation. The inversion parameters for both cases are given in Table 2, and the resulting MAP estimates for the channel are shown in Figures 10b and 10c, and for the barrier in Figures 11b and 11c. The inversion for the permeability in this model problem is relatively difficult, because the overburden of 1 km leads to significant smoothing of the deformation, and the symmetry of the injection pattern leads to negligible vertical deformation along the diagonal between the injector and producer.

Comparing the results for the hydraulic inversion, given in Figures 10b and 11b, with the results of the poroelastic inversion, given in Figures 10c and 11c, it is clear that the MAP estimate from poroelastic inversion is closer to the target permeability field than the MAP estimate obtained from hydraulic inversion. Therefore, surface deformation improves the aquifer characterization, even if the aquifer is buried at a significant depth of 1 km, as in this case. Similar to the other two model problems discussed above, the MAP estimate for the flow barrier is closer to the target permeability field than the MAP estimate for the high permeability channel. This confirms that it is generally easier to infer flow barriers, even if the flow can bypass them as in this quasi two-dimensional flow field.

6 Discussion

In this paper, we have formulated a Bayesian inverse problem to infer aquifer permeability from geodetic and hydraulic data. Moreover, we have developed numerical algorithms to infer the MAP estimate and to further explore the range of parameters consistent with the data and our prior assumptions. Three model problems of increasing complexity have been used to test the formulation, the algorithms, and to study the behavior of poroelastic inversions. Next, in section 6.1, we will illustrate and discuss the performance of the optimization algorithm for finding the MAP estimate. Then, in section 6.2, we discuss some general qualitative properties of poroelastic inversion, which can already improve our interpretation of field observations of surface deformation. In section 6.3, we discuss some of the limitations of the work presented here, with particular emphasis on the effect of uncertainty in the mechanical parameters and the challenge this poses for field application of poroelastic inversion. Finally, we conclude with a brief outline of promising directions for further research in section 6.4.

6.1 Algorithm Performance

To illustrate the typical performance of the Newton-conjugate gradient method to compute the MAP estimate, we show the convergence history for the modified Segall problem (see section 5.2) in Figure 12. Note that the norm of the gradient is decreased by 8 orders of magnitude in only 7 Newton iterations. The number of conjugate gradient iterations increases in later Newton steps, which is a consequence of choosing a tighter tolerance in the conjugate gradient method; an accurate solution of the Newton system becomes more important close to the solution [Nocedal and Wright, 2006]. While this “inexactness” may increase the number of Newton iterations, it often decreases the overall number of CG iterations, making the method computationally more efficient. The overall efficiency of the method is partly due to the use of the discrete adjoints (i.e., we follow a discretize-then-optimize approach), which guarantees accurately computed gradients.

Figure 12.

Convergence to MAP estimate in modified Segall problem. The domain is discretized using 2068 quadrilateral elements, and the mesh is locally refined around the well. Linear basis functions for the pressure and quadratic basis functions for the displacement are used, which amounts to overall 19,195 spatial unknowns for the coupled poroelastic system. The time interval is discretized using 120 implicit Euler time steps. Shown is the norm of the gradient in each Newton step normalized by the norm of the initial gradient and, in red, the number of conjugate gradient steps to approximately solve the Newton system. Since a constant time step is used, the same matrix system has to be solved in every time step of the forward (state and incremental) equations, and the transpose system is used in every time step for the adjoint and incremental adjoint equation. Thus, we only have to compute a single matrix factorization in each Newton step.

The computation of the low-rank data misfit Hessian for the Gaussian approximation of the posterior only requires products of the Hessian with vectors, each of which amounts to solving an incremental state equation and an incremental adjoint equation. The number of required Hessian vector products depends on the numerical rank of the data misfit Hessian, which, in turn, depends on how many modes of the parameters are informed by the data; for the modified Segall problem used for Figure 12, we found that 25 applications of the (data misfit) Hessian were sufficient to obtain an accurate approximation of the inverse Hessian, i.e., the covariance matrix for the local Gaussian approximation. Due to the use of adjoints, this number does not only depend on the discretization of the parameter function (i.e., the number of parameters) but also on the physics of the problem.

Thus, our algorithms for computing the MAP estimate and the Gaussian approximation of the posterior pdf are scalable, i.e., their computational cost is a fixed number (which only depends on the problem physics, but not on the discretization) times the cost of a forward solve. As a consequence, the methods inherit the efficiency of the forward solver, and hence have the potential to be used for large-scale inference problems. This opens the door to realistic geological applications and will help to fully capitalize on the potential of geodetic data in aquifer characterization.

6.2 General Properties of Poroelastic Inversion

The large range of the permeability in natural rocks can lead to a dynamic response that spans a large range of timescales, as illustrated by the modified Mandel model problem, discussed in section 5.1. The timescales over which data are required are determined by the magnitude of the permeability variation. In this model problem, three orders of magnitude of variation in permeability require data spanning at least five orders of magnitude in time in order to find a satisfactory MAP estimate. In natural samples, the variations in permeability are commonly larger and data covering an even larger range of temporal scales is required for the inversion. Collecting data spanning such disparate timescales may be challenging in itself. Time series data from satellite geodesy are available over timescales from a few months to a few decades. The overlap of these geodetic timescales with the response timescales of aquifers to typical fluid injection and production operations makes poroelastic inversion a promising tool for aquifer characterization. Nonetheless, geodetic data are limited both in the very fast and the very slow timescales, which make joint inversion with other data attractive.

In our model problems, the poroelastic deformation contains little information about the vertical permeability variation in the aquifer so that the MAP estimate of the permeability is an effective vertically averaged permeability. Sections 5.2 and 5.3 illustrate that poroelastic inversions are sensitive to permeability variations on much larger spatial scales (namely, on the order of several kilometers) compared to other hydrogeophysical inverse methods, e.g., geoelectrical methods. The availability of relatively cheap and regional data from satellite geodesy that match the wavelength of the poroelastic response enables the characterization of the large-scale structure of regional aquifers. Although this has been recognized previously—for a review, see Galloway and Hoffmann [2007]—this study presents the first systematic analysis of the resolution limits of poroelastic aquifer characterization. Figure 9 summarizes this analysis and shows clearly that our ability to infer the permeability structure is more limited by short wavelength then by aquifer depth, at least for the modified Segall model problem studied here. In other words, variations in permeability shorter than a few kilometers are almost impossible to infer even in shallow aquifers, but our ability to infer larger wavelength only deteriorates slowly with depth. If this behavior translates to more complex three-dimensional examples, it is encouraging for the poroelastic characterization and monitoring of deeper aquifers encountered in reservoir engineering and in geological CO2 storage. On the other hand, it requires that poroelastic characterization is complemented by other hydrogeophysical data that is sensitive to shorter wavelength.

In all three model problems considered here, the error and the uncertainty of the MAP estimate in low-permeability regions is smaller than in high-permeability regions, at least in the subhorizontal aquifers considered here. It therefore appears to be a general feature of poroelastic inversion that flow barriers are easier to detect than flow channels, unless the latter are directly connected to one of the wells. In this case, very high temporal resolution will be required to detect the fast response timescales. This general behavior is due to the large pressure gradients associated with flow barriers that lead to observable deformation. This qualitative knowledge can already improve our interpretation of field observations of surface deformation, and it provides guidance as to what can be expected from the integration of surface deformation observations into aquifer characterization.

6.3 Limitations

The constitutive mechanical model used here does not include a number of potentially significant physical mechanisms such as hysteresis in the storativity, nonelastic or large deformations, and stress-dependent permeability. In general, however, the linear poroelastic model and even simpler models such as the aquitard drainage model are successfully used to model regional uplift and subsidence due to subsurface fluid injection and extraction.

If the actual mechanical parameters are different from those assumed in the poroelastic model, errors are introduced into the MAP estimate of the permeability field, as illustrated in Figure 13. In this example, the model used for the inversion assumes a constant shear modulus, G, but the actual system has a perturbed shear modulus, G+δG, that is 10% higher to the left of the producer, x<0, and 10% lower to the right of the producer, x>0. Figure 13c shows that this discrepancy in the model increases the error in the MAP estimate. The surface deformation is increased in regions of high G and decreased in regions of low G and consequently the permeability is overestimated in the left half where G is increased and underestimated in the right half where G is decreased.

Figure 13.

Illustration of the effect of unaccounted spatial variations in the mechanical properties on the estimated permeability field in the modified Segall problem. Comparison of the (a) horizontal and (b) vertical components of the surface deformation after 20 years for the constant, G, and a perturbed, G+δG, drained shear modulus. (c) Comparison of the target κ with the κMAP with constant, G, and a perturbed, G+δG, drained shear modulus.

If the actual mechanical parameters are different from those assumed in the poroelastic model, errors are introduced into the MAP estimate of the permeability field, as illustrated for the modified Segall problem in Figure 13. In this example, the model used for the inversion assumes a constant shear modulus, G, but the actual system has a perturbed shear modulus, G+δG, that is 10% higher to the left of the producer and 10% lower to the right of the producer. Figure 13c shows that this discrepancy in the model increases the error in the MAP estimate. The surface deformation is increased in regions of high G and decreased in regions of low G and consequently the permeability is overestimated in regions of increased G and underestimated in regions of decreased G.

Rucci et al. [2010] have already discussed the errors introduced by variations of the mechanical parameters, and Iglesias and McLaughlin [2012] have demonstrated that multiparameter estimation for the permeability and one mechanical parameter is possible. The poroelastic model, however, has five parameters, all of which can vary spatially and are uncertain. The apparent advantage of including surface deformation data may be reduced if several of these parameters have to be estimated simultaneously.

Our implementation currently uses a factorization-based solver for the poroelastic system arising in each time step, limiting it to problems with no more than a few 10,000 unknowns. Additionally, it uses standard first- or second-order finite elements for the displacement and pressure. This choice of elements does not guarantee mass conservation, and the solution can develop oscillation in regions with strong permeability contrasts.

The solution of a Bayesian inverse problem is the posterior probability density function defined over the parameter space. In this paper, we only discuss the computation of the maximum a posteriori (MAP) estimate and a Gaussian approximation of the posterior pdf about the MAP estimate. The Gaussian nature of the posterior pdf is not guaranteed due to the nonlinearity of the parameter-to-observable map and, as a consequence, a full (statistical) exploration of the posterior pdf would have to rely, for instance, on sampling methods.

6.4 Future Work

Most of the above limitations are not fundamental to the approach taken in this paper and can be overcome. We are planning to develop a computationally more efficient solver for the Biot system, which uses an advanced mixed finite element discretization. This will allow the application of the presented inversion approach to field examples of regional subsidence with preexisting surface topography and more complex geological structure.

The demonstration that poroelastic inversion can improve the aquifer characterization in a realistic problem with field data is the most important medium-term goal. To achieve this, we need to determine the uncertainty in the mechanical parameters and hence the number of parameters that need to be estimated simultaneously. In general, this can only be determined for specific cases, and it will depend on the available data and the hydrogeological setting. Most likely, more than one parameter will have to be estimated simultaneously.

In this case, careful quantification of the uncertainty in the estimated parameters will be required to demonstrate that poroelastic inversion has improved the aquifer characterization. This will require that the current Gaussian approximation of the posterior about the MAP estimate is replaced by a more complete characterization of the posterior pdf. This is particularly challenging for problems with high-dimensional parameter space, due to the so-called curse of dimensionality. We will employ a Markov chain Monte Carlo (MCMC) sampling method to further characterize statistical quantities of the posterior pdf. Unfortunately, the use of black-box MCMC for large-scale inverse problems with expensive parameter-to-observable map (which, as in our case, involves the integration of the Biot system) is prohibitive due to the large number of samples required. The key to overcoming this lies in effectively exploiting the problem structure [see, e.g., Oliver et al., 2008; N. Petra, A computational framework for infinite-dimensional Bayesian inverse problems: Part II. Stochastic Newton MCMC with application to ice sheet inverse problems, submitted to Journal of Scientific Computing, Martin et al., 2012]. We are planning on studying and applying these methods and comparing them to other statistical inversion approaches.

7 Conclusions

A Bayesian framework to infer the permeability field in the quasi-static linear poroelastic equations from geodetic and hydraulic observations and from prior knowledge is presented. We demonstrate the efficiency and scalability of the proposed algorithms to compute the MAP estimate and a Gaussian approximation of the posterior distribution.

Three model problems are used to test our implementation and to study the properties of poroelastic inversion. We observe that:

  1. the inversion becomes more difficult with decreasing horizontal heterogeneity wavelength, and the MAP estimate approaches the harmonic average of the unresolved heterogeneity in the short wavelength limit;

  2. the inversion is also impeded with increasing aquifer depth, but a short heterogeneity wavelength is more limiting;

  3. provided the geomechanical properties are known, joint inversions of hydraulic and surface deformation data reduce the error in the inferred permeability field and, thus, have the potential to improve aquifer characterization and monitoring;

  4. high-permeability regions are difficult to infer, unless they are directly connected to a well, in which case high-frequency time series data are required to resolve the short response timescales;

  5. low-permeability regions, in contrast, act as flow barriers which result in detectable surface deformation and are thus well constrained by surface deformation;

  6. the vertical component of the surface deformation provides a smoothed image of the pore pressure in horizontal aquifers, which may allow monitoring of the pressure plume generated by geological CO2 storage.

Appendix A: Definition of Bilinear Forms

The bilinear form (·,·) is a scalar quantity given by the integrals

display math

for scalar value function a,b, vector-valued function a,b, and tensor-valued functions σ,τ, respectively.

Appendix B: Derivatives of the Lagrangian Functional

In section 3.4.1 we state the gradient and the adjoint Biot system. Here we derive these expression using variational calculus.

The gradient is found as variation of the Lagrangian functional inline image defined in (18) with respect to ζ. For an arbitrary parameter direction inline image, we obtain

display math

Integration by parts yields the strong form of the gradient as follows:

display math(B1)

for x∈Ω0, and inline image for xΩ0.

The adjoint poroelastic equations are found requiring that variations of the Lagrangian functional with respect to u and p vanish in all directions inline image, i.e.,

display math(B2)

To compute the strong form of (B2), we isolate the perturbation inline image and use the identity

display math(B3)

Using (B3), the symmetry of σ, and inline image on inline image, we obtain from (B2) that

display math

for all inline image. From the arbitrariness of inline image, we obtain the strong form (20a) and the boundary conditions given in the main text. Next, we require that variations of the Lagrangian functional with respect to p in arbitrary directions inline image vanish, i.e.,

display math

Integration by parts results in the weak form of the evolution equations for the adjoint pressure p:

display math

The corresponding strong form (20b) and the boundary and final time conditions given in the main text are found by using the arbitrariness of the variation inline image.

Appendix C: Boundary Conditions for Model Problems

C1 Modified Mandel Problem

The boundary and initial conditions for the results presented in section 5.1 are as follows:

display math(C1a)
display math(C1b)
display math(C1c)
display math(C1d)
display math(C1e)
display math(C1f)

where t denotes the unit tangent vector at the boundary. The problem geometry is shown in Figure 1a, and the boundary segments are given by Γt={x∈Ω|z=H}, Γb={x∈Ω|z=0}, Γl={x∈Ω|x=0}, and Γr={x∈Ω|x=L}. As before, the subscript T in (C1) denotes that the boundary conditions hold for all t∈(0,T).

C2 Modified Segall Problem

The boundary and initial conditions for the modified Segall problem used in section 5.2 are

display math(C2a)
display math(C2b)
display math(C2c)
display math(C2d)
display math(C2e)

The boundary segments are given by Γt={x∈Ω|z=H}, Γb={x∈Ω|z=0}, Γl={x∈Ω|x=−L}, and Γr={x∈Ω|x=L}. The fluid extraction from the layer is represented by a sink-term given by f=−Q, where Q is the volumetric flow rate of the well.

C3 Quarter Five-Spot Problem

The boundary conditions for the quarter five-spot problem in section 5.3 are as follows:

display math(C3a)
display math(C3b)
display math(C3c)
display math(C3d)
display math(C3e)

where Γ=Ω, and Γt and Γb denote the top and bottom boundaries of Ω, respectively.


The authors would like to thank Silvia Barbeiro for providing us with an implementation of the analytical solution for the Mandel problem. We would like to thank Noemi Petra for helping with the implementation of the Gaussian approximation of the posterior pdf, and for helpful comments on our manuscript. We also appreciate helpful comments and suggestions from Carsten Burstedde. Work by Marc Hesse was supported as part of the Center for Frontiers of Subsurface Energy Security (CFSES), an Energy Frontier Research Center, funded by the U.S. Department of Energy (DOE), Office of Science, Office of Basic Energy Sciences (BES), Materials Sciences and Engineering Division under award DE-SC0001114. Work by Georg Stadler was supported by the US Department of Energy's MMICCs program, under award DE-SC0009286.