an integrated extended kalman filter-implicit level set

Under consideration for publication in Inverse Problems 1

An Integrated Extended Kalman Filter-Implicit Level SetAlgorithm for monitoring Planar Hydraulic Fractures

A. PEIRCE , F. R O C H I N H A

Anthony Peirce; Department of Mathematics, University of British Columbia, Vancouver, British Columbia, V6T 1Z2, Canada,

Fernando Alves Rochinha; Mechanical Engineering Department Federal University of Rio de Janeiro, Brazil.

(Received 4 April 2011)

We describe a novel approach to the inversion of elasto-static tiltmeter measurements to monitor a planar hydraulic frac-

ture propagating within a three dimensional elastic medium. The technique involves integrating the Extended Kalman

Filter (EKF) for state prediction and updates using tiltmeter measurement time-series combined with a novel Implicit

Level Set Algorithm (ILSA) for the solution of the coupled elasto-hydrodynamic equations in order to locate the unknown

fracture free boundary. By means of a scaling argument a strategy is derived to tune the parameters in the algorithm

to enable measurement information to compensate for significant un-modeled dynamics. A sequence of numerical ex-

periments are considered in which significant changes are introduced to the fracture geometry by altering the confining

geological stress field. Even though there is no confining stress field in the dynamic model used by the new EKF-ILSA

scheme, it is able to use synthetic measurements from tiltmeters in order to arrive at remarkably accurate predictions

of the fracture opening and the fracture footprint. These experiments also explore the robustness of the algorithm to

noise and to placement of tiltmeter arrays operating in the near-field and far-field regimes. In these experiments the

appropriate parameter choices and strategies to improve the robustness of the algorithm to significant measurement

noise are explored.

Key words: Hydraulic Fracture Monitoring, Level Set Method, Kalman Filtering, Tiltmeter Measurements, CrackDetection, Inverse Problems, Free Boundary Problems.

1 Introduction

Hydraulic fractures (HF) are a class of brittle fractures that propagate in pre-stressed solid media due to the injection

of a viscous fluid. These fractures occur naturally when pressurized magma from deep underground chambers form

vertical intrusions driven by buoyancy forces, which can, in turn, form geological structures such as dykes and sills

[44, 38]. HF have also found numerous industrial applications. In the oil and gas industry, HF are deliberately

created in reservoirs to enhance the recovery of hydrocarbons by the creation of permeable pathways [11]. In the

mining industry, HF have been used to weaken the rock surrounding underground excavations to enhance the so-

called block-caving process [20, 48]. In the production of geothermal energy HF, are used create fracture networks

through which water can be pumped to extract the heat energy from the hot rock. HF have also been used for

waste disposal, the protection of environmentally sensitive areas, and are likely to play a key role in the gravitational

trapping of CO2 in low permeability host rock.

There has recently been considerable concern raised about the adverse environmental effects of hydraulic fracturing

[14]. This stems largely from the uncontrolled propagation of hydraulic fractures into subterranean water reservoirs

causing loss of hydrocarbons and fracturing fluid into the agricultural and domestic water supply. Moreover, the

propagation of HF into undesirable locations can have severe safety consequences in the mining industry. It is

therefore of considerable interest to be able to monitor the progress of propagating HF. Unfortunately, other than

2 Anthony Peirce and Fernando Alves Rochinha

the fluid pressure history at the well-bore and the volume of fluid pumped, there is very little information readily

available in the HF treatment process.

Since the late 1960’s tiltmeters [46] have been used to monitor propagating HF. Tiltmeters measure changes in the

inclination of the rock at selected sample points due to the altered displacement gradient field induced by propagating

HF. These tiltmeters can be located in the well-bore itself, in neighboring off-set boreholes, or on the earth’s surface

[12, 51, 25]. Indeed, tiltmeter monitoring is offered as a commercial service for the oil and gas industry [36]. Interest in

this form of monitoring has been mixed as the information that can be extracted from the inversion of the elasto-static

displacement gradient field is limited. Indeed, by means of a moment expansion [27] it is possible to demonstrate

that it is only feasible to invert the first two moments of the crack opening from remote displacement gradient

measurements. This is due to elliptic nature of the elasto-static PDE, for which there is a rapid decay O(1/r3)

of the displacement gradient field with distance from the fracture. Thus inverting the elasto-static displacement

gradient field is a classic ill-posed problem that is exacerbated by a paucity of tiltmeter measurements. Indeed, this

ill-posedness has likely hampered the development and deployment of extensive tiltmeter arrays in practice. In fact,

the more recent development in the petroleum industry has been toward monitoring microseismic images [53] or a

combination of tiltmeter measurements with microseismic images [50]. While microseismic networks, based on the

hyperbolic PDE of elasto-dynamics, provide different information, it is difficult to distinguish the first motions due

to the propagating HF from secondary movements triggered in the jointed rockmass that surrounds the HF. Thus a

cloud of microseismic events are typically observed rather than being located on the HF fracture surface itself. This

problem hampers the clear identification of the propagating failure surface using a microseismic network.

The approach to the tiltmeter inversion problem that we adopt in this paper is somewhat different. Following our

initial study using the Extended Kalman Filter (EKF) for the inversion of tiltmeter data from a one dimensional

HF propagating in a state of plane strain [39], we consider the application of this methodology to monitoring two

dimensional planar HF. In 1D the identification of the free boundary involves the location of only two points, while

the determination of the crack opening involves the identification of a 1D function on the domain defined by these

free boundary points. The planar problem is a lot more complex. Locating the free boundary involves identifying a

closed curve in a plane, while identifying the crack opening involves determining a surface defined on the domain

circumscribed by this boundary curve. Parametrization of the free boundary and discretization can reduce the problem

to one involving parameter identification from remote measurements. For the elasto-static data, which involves the

displacement gradients at the tiltmeter locations at a particular time-step, there is severe ill-posedness resulting in

a loss of uniqueness, so that, at best, only the first two moments of the crack opening can be recovered. This yields

little information about the fracture geometry. However, we formulate a different inverse problem using the same

data by stringing together these elasto-static tilltmeter snapshots into time-series of observations. These time-series

of tilt measurements are then combined with a dynamic model for the fracture propagation to arrive at improved

estimates of the fracture geometry and opening via the EKF. One approach to this identification problem, which was

adopted in the 1D case [39], is to explicitly parameterize the fracture free boundary as well as the fracture widths

as an augmented state vector for direct identification by the EKF. There is, however, strong coupling between the

geometric changes that are introduced by perturbations in the boundary and the fracture widths close to the tips. If

we used a direct parametrization of the free boundary in 2D this would substantially complicate the calculation of the

Frechet derivatives required to formulate the EKF. However, the Implicit Level Set Algorithm (ILSA), developed for

the efficient modeling of the HF free boundary problem, makes it possible to avoid having to explicitly parameterize

An Integrated Extended Kalman Filter Implicit Level Set Algorithm for monitoring Planar Hydraulic Fractures 3

the free boundary. Instead, the ILSA scheme is able to locate the free boundary by using the applicable local tip

asymptote along with state estimates comprising only the fracture width. It is this integrated EKF-ILSA scheme

that we describe in this paper.

Traditional data assimilation methods rely upon the Kalman Filter and extensions of this method. The main ideas,

which are based on the use of Bayesian Inference, are described in detail in [3, 24]. The Extended Kalman Filter has

been used extensively for inverse problems related to a wide range of applications (see for example [28, 41, 18, 31])

and particularly relevant for fracture problems is the work described in [5, 29]. In this latter work, the EKF has been

used to identify parameters from experiments in a cohesive crack model by considering a pseudo time-stepping of the

flow of experimental data and sequential estimation by the EKF until convergence is achieved. Though also dealing

with fracture detection, the problem we consider in this paper is quite different. Indeed, we are using the EKF to

compensate for significant un-modeled dynamics in a complex system of highly nonlinear integro-partial differential

equations. The objective of this inversion problem is not only to identify some parameters in the model, but also to

identify large perturbations in both the crack opening displacement and the fracture geometry.

In section 2, we describe the forward hydraulic fracture model comprising a system of integro-partial differential

equations along with a free-boundary problem and outline the novel ILSA numerical algorithm to locate the free

boundary; In section 3, we provide a brief introduction to the EKF formulation that we use in this paper and present

the details of the integrated EKF-ILSA scheme that we propose. In section 4, we provide the results of numerical

experiments chosen to demonstrate the performance of the EKF-ILSA scheme for three distinct problems in which

significant geometric perturbations are introduced by changes in the confining geological stress field; in section 5 we

provide some concluding remarks.

2 Forward model for a planar Hydraulic Fracture

2.1 Governing equations

Hydraulic fractures tend to propagate in planar regions perpendicular to the minimum principal direction of the

geological confining stress field. The dynamics of the propagating HF are governed by a coupled system of nonlinear,

degenerate, hyper-singular, integro-partial differential equations along with propagation and boundary conditions

that determine the location of the fracture free boundary. Analytic solution of these equations is only possible for

special geometries (e.g. plane strain [6, 1] or radially symmetric [40]) that essentially reduce the problem to one spatial

dimension. The numerical solution of these equations presents considerable challenges, including: (i) the resolution

of the multiscale solution structure close to the fracture tip; (ii) the singular tip behavior, caused by the degeneracy

of the PDEs, which results in an expression for the tip velocity involving an indeterminate form that is difficult

to evaluate numerically. This in turn makes the moving boundary extremely difficult to locate with established

algorithms; (iii) a spatial discretization reduces the problem to a stiff system of ODEs for which implicit backward

difference time-stepping and specialized algorithms for the solution of the coupled equations at each time-step are

necessary [33].

Recently, an Implicit Level Set Algorithm (ILSA) [34] has been developed for the efficient solution of this complex

free boundary problem. This scheme is able to incorporate multiscale tip behavior, obtained via detailed asymptotic

analysis [15, 10, 16, 30], into an algorithm defined on a relatively coarse structured Eulerian mesh. This ILSA solution

scheme forms the basis for the forward model that is used in this paper in combination with the Extended Kalman


Filter to devise a novel HF monitoring technique. Since all the numerical experiments presented are in dimensionless

form, for the sake of brevity, we choose to only present the dimensionless form of the model equations and details of

their discretization. The dimensional equations and their reduction to dimensionless form are detailed in [34].

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-

6

½½

½½½=

S

∂S∈ ∂Sc ⊂ Sc

∈ Sc

∈ St

χ

η

ζ

Figure 1. A schematic of a hydraulic fracture occupying the region S in the χ−η plane. The squares represent the structuredmesh used by the ILSA scheme to discretize the governing equations. The sets St, Sc, and ∂Sc are used to identify differentclasses of elements that are used in the ILSA scheme described in sub-section 2.3

We consider a coordinate system (χ, η, ζ) and a dimensionless time scale τ . The evolving fracture is assumed to

occupy the region S(τ) in the (χ, η)-plane that is circumscribed by the fracture front ∂S(τ) (see figure 1). The

dimensionless quantities Gj that appear in the equations below are defined to be:

Gc =C ′T 1/2

∗W∗

, Ge =E′W∗P∗L∗

, Gm =µ′L2

∗W 2∗P∗T∗

, Gv =Q0T∗W∗L2∗

, Gk =K ′L1/2

∗E′W∗

, (2.1)

Here E′ = E1−ν2 , where E and ν are the rock Young’s modulus and Poisson’s ratio; µ′ = 12µ, where µ is the dynamic

fluid viscosity; C ′ = 2CL where CL is Carter’s leak-off coefficient (see [7]); and K ′ = 4( 2π )

12 KIC is the modified stress

intensity the factor. In addition, T∗ is a characteristic time scale; W∗ is a characteristic fracture aperture; P∗ is a

characteristic net pressure; L∗ is a characteristic length scale, and Q0 is the characteristic volumetric fluid injection

rate.

2.1.1 The Elasticity Equation

The hypersingular integral equation [17] relating the crack opening displacement Ω(χ, η, τ) to the fluid pressure

Πf(χ, η, τ) is given by

Π = Πf(χ, η, τ)− Σoϕ(χ, η) = −Ge

8π

∫

S(τ)

Ω(χ′, η′, τ)dS(χ′, η′)[(χ′ − χ)2 + (η′ − η)2

]3/2(2.2)


Here Σo = σo

P∗is the dimensionless confining stress where σo is the characteristic confining stress, ϕ(χ, η) represents

the spatial variation of the confining stress field, and Π = Πf − Σoϕ(χ, η) is the net pressure.

2.1.2 The Fluid Flow Equation

A combination of the Poiseuille’s law and the conservation of mass yields Reynolds’ lubrication equation, which

assumes the form:∂Ω∂τ

+ GcH(τ − τo)√

τ − τo=

1Gm

∇ · (Ω3∇Πf

)+ Gvψ(τ)δ(χ, η) (2.3)

The term involving Gc represents the fluid leak-off rate according to Carter’s model [7]. Here H is the Heavyside

function and τo(χ, η) is the dimensionless exposure time, defined to be the time at which the point (χ, η) within

the fracture was first exposed to fluid. The leak-off term introduces non-locality into the model equations which is

history dependent.

2.1.3 Boundary and propagation conditions

A zero flux boundary condition is imposed at each point of the fracture boundary ∂S(τ):

limξ→0

Ω3 ∂Πf

∂ξ= 0 (2.4)

where ξ is a local coordinate representing the normal distance to the fracture tip. The propagation of the moving

boundary ∂S(τ) is governed by the following asymptotic condition, derived from Linear Elastic Fracture Mechanics

(LEFM), that the local stress intensity factor be in limit equilibrium with the dimensionless toughness Gk, which

can be expressed [37] as:

limξ→0

Ωξ1/2

= Gk (2.5)

2.1.4 Propagation regimes and multiscale tip behavior

If Gk > 0 then the asymptotic behavior of the fracture opening is always given by (2.5). However, this classic LEFM

square root behavior may not be the appropriate asymptotic expansion that applies at the computational length

scale. In the absence of a fluid lag, there are two competing physical processes namely energy dissipation and fluid

balance that determine the dominant asymptotic behavior. There are, in turn, two competing dissipative processes

namely the viscous dissipation associated with driving the fluid into the fracture cavity and the solid toughness

associated with the energy required to fracture the rock. The two competing fluid balance processes are the fluid

storage within the fracture and the fluid leak-off into the surrounding rockmass. Further details of the multiscale,

multiphysics behavior associated with this four-parameter phase space can be found in [34].

Since the objective of this paper is to explore the performance of a combined EKF-ILSA algorithm rather than

the multiplicity of different solutions associated with this phase space, we choose to restrict the examples presented

in this study to solutions in the neighborhood of the viscous-storage vertex at which Gc = 0 and Gk = 0. There

is considerable practical value to this mode of fracture propagation. Indeed, assuming typical parameter values for

impermeable rock, radial HF have been shown [40] to spend many years in the viscosity dominated regime - way

beyond the typical treatment time which is of the order of a few hours.

For the idealized viscous-storage vertex, which corresponds physically to a HF propagating between two de-bonded


impermeable half-spaces, the tip asymptotic behavior of the fracture aperture Ω has been shown for the plane strain

case [9, 15, 10, 16, 30] to be of the form:

Ωξ→0∼ βm0 v1/3ξ2/3 (2.6)

where βm0 = 21/3 ·35/6 and v is the tip fluid velocity given by v = limξ→0

Ω2 ∂Πf

∂ξ . In [34] it is shown that this asymptotic

behavior also holds for arbitrarily shaped planar fractures with smooth boundaries ∂S(τ). We will see that the tip

asymptotic behavior (2.6) plays a central role in locating the fracture free boundary ∂S(τ) in the ILSA scheme.

2.2 Discrete equations

For the sake of completeness we briefly describe the scheme outlined in [34] omitting the leak-off terms since we will

be only considering numerical experiments involving the viscous propagation regime. Following [34] we assume that

the fracture will grow within a rectangular region that has been tessellated into a fixed uniform rectangular mesh

with spacings ∆χ and ∆η in the two coordinate directions. The fracture footprint S(τ) is then covered by rectangular

elements ∆Sm,n such that S ⊆ ∪∆Sm,n. Constant displacement discontinuity (DD) elements are used for the elasticity

computations [8] along with collocation at element centers, while the lubrication equation is discretized by replacing

spatial derivatives by second order finite difference quotients to yield a five node finite difference stencil [45]. The

resulting, extremely stiff, system of ordinary differential equations that result from this discretization process is solved

using the backward Euler scheme.

2.2.1 The Discrete Elasticity Equation

The elasticity equation (2.2) is discretized by assuming that the fracture opening Ω(χ, η, τ) is piecewise constant over

each rectangular element ∆Sm,n, i.e.

Ω(χ, η, τ) =∑m,n

Ωmn(τ)Hmn(χ, η) (2.7)

in which

Hmn(χ, η) =

1 for (χ, η) ∈ ∆Sm,n

0 for (χ, η) /∈ ∆Sm,n(2.8)

is the characteristic function for element (m,n). Substituting this approximation into the integral equation (2.2)

and evaluating the pressures at the collocation points comprising the element centers, yields a system of algebraic

equations of the form:

Πkl(τ) =∑m,n

Ck−m,l−nΩmn(τ) (2.9)

where

Ck−m,l−n = − 18π

[√(χk − χ)2 + (ηl − η)2

(χk − χ)(ηl − η)

]χ=χm+∆χ, η=ηn+∆η

χ=χm−∆χ, η=ηn−∆η

It is convenient to write (2.9) in the operator form:

Π = CΩ (2.10)


2.2.2 The discrete Fluid Flow Equation

In order to discretize the fluid flow equation (2.3) in a way that is compatible with (2.9) we use the pressures Πkl(τ)

and widths Ωkl(τ) at element centers along with central difference approximations of the partial derivatives to arrive

at the following spatial discretization:

ddτ

Ωkl(τ) =1

∆χ

(Ω3

k+ 12 l

(Πk+1l −Πkl)∆χ

− Ω3k− 1

2 l

(Πkl −Πk−1l)∆χ

)+

1∆η

(Ω3

kl+ 12

(Πkl+1 −Πkl)∆η

− Ω3kl− 1

2

(Πkl −Πkl−1)∆η

)+ skl(τ) (2.11)

where skl(τ) represents the source and sink terms that apply in the element located at (χk, ηl) and Ωk+ 12 l etc

represent the aperture values at element edges that are determined by the following averaging operator Ωk+ 12 l =

12 (Ωk+1l + Ωkl). It is convenient to express this system of differential equations in the following operator form:

dΩdτ

= A(Ω)Π + s(τ) (2.12)

where A(Ω) is the second order difference operator defined on the right side of (2.11).

2.2.3 Coupled Evolution Operator: Backward Euler Scheme

Using (2.10) to eliminate the pressure Π from (2.12), we obtain the following evolution equation for Ω:

dΩdτ

= A(Ω)CΩ + s(τ) (2.13)

where s(τ) represents a vector of sources and sinks. The evolution operator A(Ω)C can be shown [35] to be extremely

stiff resulting in a CFL condition of the form ∆τ ≤ O(min∆χ3, ∆η3). We therefore use the L-stable Backward

Euler scheme to march the solution process forward in time, resulting in the following system of nonlinear equations

for Ω(τ + ∆τ) that need to be solved at each time-step:

Ω(τ + ∆τ)− Ω(τ) = ∆τA(Ω(τ + ∆τ))CΩ(τ + ∆τ) + ∆τs(τ + ∆τ) (2.14)

Introducing the notation ∆Ω := Ω(τ + ∆τ) − Ω(τ) we may express (2.14) in the following form suitable for fixed

point iteration to determine Ω(τ + ∆τ) = limi→∞

Ωi:

(I −A(Ωi)C

)∆Ωi = ∆τA(Ωi)CΩ(τ) + ∆τs(τ + ∆τ)

Ωi+1 = Ω(τ) + ∆Ωi (2.15)

2.3 The Implicit Level Set Algorithm (ILSA)

The fundamental ansatz behind the ILSA scheme is that a given tip asymptote applies at the computational length

scale, which is characterized by the mesh size O(∆χ, ∆η). For the examples considered in this paper we assume

that the viscous tip asymptote (2.6) holds one or two cell lengths from the tip. This asymptotic relation establishes

the fracture aperture at a distance ξ measured normal to the fracture boundary ∂S(τ). This leads naturally to the

following procedure to locate the free boundary. If we assume that an initial estimate of the fracture footprint is

known (as an initial guess, the footprint at the previous time-step is a natural choice), then by solving the coupled

equations (2.14) for Ω(τ + ∆τ) we can use the apertures nearest to the tip to obtain a new estimate for the normal

distance to the fracture tip by inverting (2.6). In order to implement this iterative procedure, we divide the elements


∆Sm,n that cover the region S occupied by the fracture in the current iteration into two disjoint sets, Sc and St.

The elements totally within the boundary ∂S are referred to as the “channel elements” and occupy the region Sc.

Those partially filled elements that intersect the crack front are called “tip elements”, which occupy the region St

(see figure 1). Thus Sc ⊂ S, ∂S ∩Sc = ∅ and S ⊂ Sc∪St = ∪∆Sm,n. As our reference points that we use to estimate

the normal distance to the free boundary, we choose the ribbon of elements ∂Sc ⊂ Sc, each of which share at least

one side with a tip element in St. The current trial solution values of Ω at the centers of the ribbon elements ∂Sc

combined with the asymptotic relation (2.6) provide the boundary conditions for the eikonal equation,

|∇T (χ, η)| = 1 (2.16)

The solution of (2.16) is the signed distance function whose level set T (χ, η) = 0 is the fracture front ∂S.

We adopt the convention that T < 0 for points inside the fracture boundary and T > 0 outside the fracture

boundary. Then assuming that the fracture is evolving under a viscous mode of propagation, we invert (2.6) to

obtain:

T (χ, η) = −ξ ∼ −(

Ωβm0 v1/3

) 32

for all (χ, η) ∈ ∂Sc (2.17)

Rather than using the indeterminate form expression for the front velocity v = limξ→0

Ω2 ∂Πf

∂ξ , which is difficult to evaluate

numerically with any precision, we use the following expression for the front velocity in terms of two successive signed

distance functions

v = −Tτ+∆τ − Tτ

∆τ(2.18)

Eliminating v from (2.17) using (2.18) reduces the problem of determining T (χ, η, τ + ∆τ) for (χ, η) ∈ ∂Sc to that

of solving the cubic equation:

T 3τ+∆τ − TτT 2

τ+∆τ + ∆τ

(Ωτ+∆τ

βm0

)3

= 0 (2.19)

Since Tτ < 0, an application of Descarte’s Rule implies that (2.19) has only one negative real root, which determines

the boundary condition values T (χ, η, τ + ∆τ) for (χ, η) ∈ ∂Sc uniquely. Having determined the boundary values

for T , an upwind differencing scheme, developed for the numerical solution of conservation laws [32, 43], is used to

determine the signed distance function T (χ, η, τ + ∆τ) in the immediate neighborhood of the crack front using the

Fast Marching Method (FMM) (for further details see [32, 43, 34]). The fracture front is then determined by locating

the curve T (χ, η, τ + ∆τ) = 0.

2.4 Tiltmeter observation model

For the observation model to be used in conjunction with the EKF in this paper, we consider the time-series of

displacement gradients due to the evolving fracture surface measured by a distant array of tiltmeters. Tiltmeters

measure the angles between their axes and the gravity vector, which are related to the curl of the local displacement

field [25], i.e., ω = ∇× u, where u = (u1, u2, u3) = (uχ, uη, uζ) is the local displacement vector at each of the tilts.

In order to relate these displacement gradients to the evolving hydraulic fracture, we make use of the following

expression for the displacement field due to a planar crack occupying the region S(τ) in the χ − η plane. For a

crack having an opening displacement Ω in the ζ-direction normal to the fracture plane, the scaled displacement


components uk are given by

uk(χ, η, ζ, τ) =1

8π(1− ν)

∫

S(τ)

Ω(χ′, η′, τ)Uk(χ− χ′, η − η′, ζ)dS(χ′, η′) (2.20)

where Uk(χ, η, ζ) can be expressed [42, 8] in terms of derivatives of the harmonic function Φ(χ, η, ζ) = 1ρ , where

ρ =√

χ2 + η2 + ζ2, as follows:

Uχ = −(1− 2ν)Φ,χ − ζΦ,χζ ; Uη = −(1− 2ν)Φ,η − ζΦ,ηζ ; Uζ = 2(1− ν)Φ,ζ − ζΦ,ζζ (2.21)

For rectangular elements, such as the ∆Sm,n used to discretize the fracture plane S, it is convenient to determine

the integral of Φ, which we represent by Φ, over a typical rectangle and to evaluate the following fundamental

derivatives: Φ,χ = log(ρ + (η − η′)), Φ,η = log(ρ + (χ − χ′)), and Φ,ζ = arctan( ρζ(η−η′)(χ−χ′) ) where the limits of

integration ]χ′=∆χ

2

χ′=−∆χ2

]η′=∆η

2

η′=−∆η2

have been omitted. The remaining higher order derivatives required to produce the

rotation vector ω can be obtained from these expressions for Φ,k by direct differentiation.

3 The Integrated Extended Kalman Filter-Implicit Level Set Algorithm

In this section we describe the integrated EKF-ILSA scheme for monitoring evolving hydraulic fractures. Kalman

Filters have been used extensively to calibrate models by identifying model parameters and/or states by assimilating

often noisy and incomplete data from experiments or field measurements. The Kalman Filter is based on a Bayesian

Probabilistic Inference [3] approach to the a posteriori state estimation at a given time given all the previous

observations up till that time.

A particularly challenging aspect of this problem is capturing the moving fracture boundary, which represents

changes to the geometry of the domain on which the integro-PDE is defined. One strategy to estimate the location of

the fracture boundary would be to allocate additional state variables to parameterize the fracture boundary explicitly,

which can then be estimated by the filter itself. However, there is extremely strong and complex geometric coupling

between the parameters that define the front and the primary state variables comprising the fracture widths Ωmn.

Evaluating the Jacobian of the forward model with this augmented set of state variables would require the expensive

numerical evaluation of the Frechet derivatives of the forward model due to boundary perturbations, which would be

computationally prohibitive. On the other hand the ILSA scheme described above provides an elegant and efficient

method to capture the fracture free boundary by making use of the tip widths in the neighborhood of the free

boundary and the applicable tip asymptotic behavior. We now describe how these two methods can be integrated to

provide an efficient iterative procedure to identify the location of the free boundary without explicit parametrization.

The combined iterative estimation algorithm uses the EKF to provide an improved estimate of the state, comprising

only the fracture widths, while the ILSA scheme uses these updated width estimates to locate the free boundary.

This technique does not require explicit parametrization of the identified free boundary.

3.1 The Extended Kalman Filter(EKF)

The Kalman Filter [3] provides the optimal linear estimator for the underlying state of a linear dynamical system,

at a given time, which takes into account all the observations up till that time. The Extended Kalman Filter [3, 24]

applies this Kalman Filter methodology to nonlinear dynamical systems by linearizing the system about a nominal

state comprising the current state estimate. Naturally, as with all such linearizations, its validity depends on how

close the current estimate is to the actual state (see[3]).


We briefly summarize the EKF formulation [3, 24] that we use in the EKF-ILSA model. We consider a typical

time interval [τk, τk+1] and re-write the discrete dynamical system and observer models defined in subsections 2.2-2.4

as follows:

xk+1 = fk(xk) + wk (3.1)

yk = Hkxk + vk (3.2)

where the state vector xk contains the discrete widths Ωmn(τk) and the observation vector yk contains the rotation

components ωi(τk) observed at the tiltmeter stations at time τk; and wk ∼ N(0, Γwk) and vk ∼ N(0, Γvk

) are

assumed to be independent, normally distributed, random variables with covariance matrices Γwkand Γvk

, respec-

tively. The matrix Γwkrepresents un-modeled dynamics while Γvk

represents noise in the measurements themselves

or uncertainty in the measurement model. Here fk(·) is defined implicitly by the operator equation

(I −∆τA(Ω(τk+1))CΩ(τk+1) = Ω(τk) + ∆τs(τk+1) (3.3)

and Hk is the matrix representation of the discretized linear operator (defined in (2.20) and detailed in the subsequent

discussion) that maps the fracture widths xk to the tiltmeter rotation components contained in the vector yk.

We define xk|j := E[xk|x1:j , y1:j ] to be the state estimate at time τk given all the data up to time τj . Similarly, we

define Γk|j := E[(x− xk|j)(x− xk|j)T |x1:j , y1:j ] to be the state covariance matrix at time τk given all the data up to

the time τj . We also define the following linear operator Fk, which is the Jacobian of the state propagation operator:

Fk =∂fk(xk|k)

∂x

For the coupled system the Jacobian Fk, frequently referred to as the transition matrix, has the form

Fk =[I −∆τA(Ωk|k)C −∆τA′(Ωk|k)CΩk|k

]−1 (3.4)

where A′ represents the derivative of A. We have found that by neglecting the term involving A′, so that we

have essentially the same operator used in the fixed point scheme (2.15), we obtain the following approximation

Fk ≈[I −∆τA(Ωk|k)C

]−1, which proves to be more stable and computationally more efficient than the full Jacobian

given in (3.4).

The extended Kalman filter can now be expressed in the following Predictor-Corrector form

Prediction Step

Given xk|k and Γk|k compute

xk+1|k = fk(xk|k) (3.5)

Γk+1|k = FkΓk|kFTk + Γwk

(3.6)

Corrector Step

Given the predicted values xk+1|k and Γk+1|k compute

Kk+1 = Γk+1|kHTk+1(Hk+1Γk+1|kHT

k+1 + Γvk+1)−1 (3.7)

xk+1|k+1 = xk+1|k + Kk+1(yk+1 − hk+1(xk+1|k)) (3.8)

Γk+1|k+1 = (I −Kk+1Hk+1)Γk+1|k (3.9)

where the matrix Kk+1 is the so-called Kalman Gain. For the initial estimate for the state we assume the radial

An Integrated Extended Kalman Filter Implicit Level Set Algorithm for monitoring Planar Hydraulic Fractures11

solution [40] as a starter crack x1|1 = Ω(τ1). For the model and noise covariances we assume that Γwk= σ2

wI and

Γvk= σ2

vI. As an initial guess for the state covariance matrix Γ1|1 = σ2wI.

The Scaled EKF Equations and Parameter Choices

Because we will be typically dealing with remote measurements, which are typically small due to the rapid decay

O( 1ρ3 ) of the displacement gradient fields, it is useful to re-scale the update equations in order to gain insight into

how the algorithm parameters should be selected. Since the measurements and the values of the Green’s function

operator are going to be small we introduce the following notation

ε = ‖H1‖2 , Hk = εHk, yk = εYk, and Γi|j = σ2wGi|j

The covariance prediction equation (3.6) assumes the form:

Γk+1|k = σ2wGk+1|k

= σ2w

(FkGT

k|kFTk + I

)

While the expression for the Kalman Gain (3.8) can be rewritten in the form

Kk+1 = ε−1Gk+1|kHTk+1

(Hk+1Gk+1|kHTk+1 + γI

)−1

= ε−1Kk+1

where

Kk+1 := Gk+1|kHTk+1

(Hk+1Gk+1|kHTk+1 + γI

)−1

and

γ :=σ2

v

σ2wε2

(3.10)

Finally, in terms of these new scaled variables, the state correction can be expressed as follows:

xk+1|k+1 = xk+1|k + Kk+1(yk+1 −Hk+1xk+1|k)

= xk+1|k +Kk+1(Yk+1 −Hk+1xk+1|k)

The point of introducing this re-scaling is that the parameter γ defined in (3.10) provides insight into the choice

of parameters σv and σw. If γ → 0 then we observe that Kk+1 → H−1k+1, which implies that xk+1|k+1 → H−1

k+1yk+1.

In this limit the Kalman Filter places more emphasis on the measurements than on the dynamics of the model itself

in arriving at an estimate of the state. This would be appropriate, for example, if there was significant uncertainty

in the model, in which case σw would need to be large relative to σv. Alternatively, this situation might occur if the

data was known to be significantly more accurate than the level of confidence in the model dynamics. Conversely,

if σw → 0, which would be the case if there was a great deal more confidence in the model dynamics than in

the measurement process, then γ → ∞ so that Kk+1 → 0 and the change made by the Kalman Filter to the state

predicted by the model dynamics is negligible. In the experiments explored in this paper we are interested in exploring

situations in which there is significant un-modeled dynamics. Thus we would like to be operating in the regime in

which γ << 1. A choice that we have found particularly effective is σv = φσwε, which implies that γ = φ2. Here φ

is a parameter that can be used to adjust the resolution of the monitoring process. Values of φ are typically chosen

in the range 0.01 < φ < 2, and can be used to adjust the weight that the EKF-ILSA scheme places on the measured

data relative to the desired weight on the accuracy of the dynamics of the forward model. Values of φ in the range

0.01 < φ < 1/2 are suitable for situations in which there is significant un-modeled dynamics while the measured


data has little noise. In this case it is possible to achieve a significant resolution of the fracture boundary and state.

If there is a significant amount of noise in the data, then for this range of φ values, the fluctuations in the data can

cause significant perturbations in the estimated state and the location of the fracture front ∂S. In order to immunize

the EKF-ILSA scheme from the effect of significant noise in the data, a choice of φ in the range 1/2 < φ < 2 can

significantly damp the effect of the noise on the estimated state. However, in this case one can expect a concomitant

loss of resolution of the un-modeled dynamics.

3.2 The Integrated Algorithm

Having given a detailed description of both the ILSA and the EKF schemes we now outline the integrated algorithm

that forms the central focus of this paper. This algorithm comprises the following two steps: (I) in the first step,

the discrete fracture widths Ωmn are treated as the primary state variables that are updated using the EKF based

on the coupled equations and observed data while the fracture footprint is frozen; (II) in the second step, the ILSA

algorithm is then used to update the fracture front positions based on the new estimates of the widths. Step (I),

involving the EKF width prediction, and step (II), involving ILSA front correction, are then repeated until the front

iterations have converged. By this stage, both the widths and the front positions have been updated to account for

both the dynamics of the forward model (2.2)-(2.6) and the observed data via (2.20).

In order to keep track of, and to distinguish between, the time-steps and the front iterations, we use subscripts

to denote time-step information and superscripts to denote the front iteration counter. Thus, for example, xjk+1|k

represents the predicted value for the state at time-step k + 1 given all the data at time-step k in the jth front

iteration. If the superscript is absent, we assume that we are referring to the value of that quantity once all the

front iterations have converged. Thus xk|k refers to the updated state at the end of the kth time-step given all the

data up till that step and once the final fracture footprint has converged to Sk. In the convergence check below Ξ(S)

represents the characteristic function of S.

It is possible for the Kalman update to predict width components that are negative, i.e., xj+1k+1|k+1, i < 0, where the

additional subscript refers to a particular component. This physically impossible and unstable situation is prevented

by resetting those particular components to the values predicted by the forward model, i.e., xj+1k+1|k+1, i = xj

k+1|k, i.

Negative widths are likely to occur if the EKF-ILSA footprint has, in some region, gone beyond that of the synthetic

fracture. In this case the EKF-ILSA fracture front would need to recede in this region until it matches the synthetic

fracture front. The way in which the EKF update achieves this is to significantly reduce the fracture widths in this

region - sometimes even becoming negative. There are a number of strategies that the ILSA scheme can use to deal

with negative width components in the EKF correction: (i) the corrected components that have negative widths

can be re-set to their predicted values, which should be positive since they result from a converged solution to the

coupled equations (2.15); (ii) elements in the EKF correction with negative width components can be deleted from

the set of active state variables, which will allow the fracture to recede from this region; (iii) a width constraint can

be imposed, by which those components in the EKF correction with negative widths are set to some minimum level.

Repeated width constraints for a particular element in successive time-steps makes it a likely candidate for deletion.

We have adopted strategy (i) for the experiments presented in this paper as it involves minimal adaptation of the

standard ILSA scheme. Changes in geometry in which there is active recession of the fracture front will require a

strategy for the deletion of elements. Option (ii) is a fairly drastic approach as it can induce significant changes to

the fracture geometry. Option (iii) is a staged approach to the deletion of elements proposed in (ii).


Initialization:

Set the state to the radial exact solution x1|1 = Ωexact(τ1)

Set the domain to the exact circle S1 = (χ, η) :√

(χ− χc)2 + (η − ηc)2 ≤ ρexact(τ1) Set the EKF parameters φ, σw, and the covariance matrix Γ1|1 = σ2

wI.

Time-step loop: k = 1 : Nt

τk+1 = τk + ∆τ

Initialize guesses for current step: Set the front ∂S1k+1 = ∂Sk and the covariance matrix Γ1

k|k = Γk|k

Front iteration loop: j = 1 : Nf

Given xk|k and ∂Sjk+1 solve (2.15) for xj

k+1|k = fk(xk|k)

Evaluate F jk ≈

[I −∆τA(xj

k+1|k)C]−1

and Hjk+1

If k=1, set ε =∥∥H1

2

∥∥2

and then set σv = φσwε

Γjk+1|k = F j

kΓjk|k(F j

k )T + Γwk

Kjk+1 = Γj

k+1|k(Hjk+1)

T[Hj

k+1Γjk+1|k(Hj

k+1)T + Γvk+1

]−1

xj+1k+1|k+1 = xj

k+1|k + Kjk+1(yk+1 −Hj

k+1xjk+1|k)

If xj+1k+1|k+1, i < 0 set xj+1

k+1|k+1, i = xjk+1|k, i (to prevent negative widths).

Γj+1k+1|k+1 = (I −Kj

k+1Hjk+1)Γ

jk+1|k

Use xj+1k+1|k+1 to set BC for T j+1

k+1 along ∂Sc,jk+1 using (2.19) and use FMM to solve

∣∣∣∇T j+1k+1 (χ, η)

∣∣∣ = 1.

Update the front position ∂Sj+1k+1 = (χ, η) : T j+1

k+1 (χ, η) = 0 and the velocity vj+1k+1 field using (2.18).

Check for convergence∥∥∥Ξ(Sj+1

k+1)− Ξ(Sjk+1)

∥∥∥ < tol × Ξ(Sj+1k+1)

end front iteration loop

end time-step loop

4 Numerical Results

4.1 Description of the numerical experiments

4.1.1 Geometric changes due to variations in the confining stress field

We now present results for a sequence of numerical experiments in which synthetic data at an array of tiltmeters

has been created for hydraulic fractures propagating in situations in which significant changes to the geometry of

the evolving fractures are induced by spatial variations in the confining stress field Σoϕ(χ, η). Gaussian white noise

with mean zero of varying amplitude σN between 0% and 5% is added to the tiltmeter data to simulate typical

measurement errors experienced in the field. In the identification of the state and the fracture boundary by the

EKF-ILSA scheme, the forward model used by the EKF assumes no confining stress field, i.e., ϕ(χ, η) = 0. Thus

without the corrections introduced by the EKF component of the algorithm due the measured tiltmeter data, the

solution produced by the dynamic model would correspond to a radially symmetric crack. This amounts to significant

un-modeled dynamics that has to be compensated for by the EKF-ILSA algorithm.


4.1.2 Error measures

In order to be able to present the data in a compact form for ready comparison of the performance of the algorithm

with different choices of parameter φ, we introduce the following error measures:

Ew(τk; φ, σN ) : =

∫Ss

k∪SE−Ik

∣∣Ωs(χ, η, τk)− ΩE−I(χ, η, τk)∣∣ dS

∫Ss

k

|Ωs(χ, η, τk)| dS (4.1)

EF (τk; φ, σN ) : =A [(Ss

k ∪ SE−Ik

) \ (Ssk ∩ SE−I

k

)]

A [Ssk]

(4.2)

where the operator A [·] represents the area of the indicated region, Ssk is the region occupied by the synthetic

fracture at time τk, and SE−Ik represents the corresponding region occupied by the fracture identified by the EKF-

ILSA scheme. Ew(τk) represents the relative width error measured by the integral of absolute value of the difference

between the widths over the combined region Ssk∪SE−I

k over which the two fractures fall divided by the total volume

in the synthetic fracture. In addition, we assume that Ωs = 0 in that part of SE−Ik that does not intersect with Ss

k,

and likewise ΩE−I = 0 in that part of Ssk that does not intersect with SE−I

k . We observe that 0 ≤ Ew(τk) ≤ 2, the

lower bound occurs when Ωs ≡ ΩE−I and Ssk ≡ SE−I

k while the upper bound occurs if we assume that both schemes

preserve volume and that Ssk ∩SE−I

k = ∅. The latter situation is unlikely for the problems considered here since both

fractures are assumed to share the same injection point. The quantity EF (τk) represents the relative footprint error

measured by the ratio of the areas of the two regions Ssk and SE−I

k that lie outside their intersection Ssk ∩ SE−I

k

divided by the area occupied by the synthetic fracture. We note that 0 ≤ EF (τk), where the lower bound corresponds

to Ssk = SE−I

k . There is not necessarily an upper bound on EF (τk) because there is no reason for the areas to be

preserved or even to be finite. If the regions are disjoint, i.e., Ssk ∩ SE−I

k = ∅, then EF (τk) ≥ 1. As the number of

plots that we are able to present is limited, we focus more on EF , since it is the fracture footprint as a function of

time that is the most important desired quantity in the field monitoring of HF.

4.1.3 Observation tiltmeter array and measurement noise

For our observation model, we use a planar array of tiltmeters located at a grid of nodal coordinates defined as

follows:

χ = a : h : b, η = a : h : b, ζ = d. (4.3)

We assume that tilt components with respect to the ζ axis are measured at each tiltmeter station on this grid, so that

a time-series comprising the following components of the rotation vector: ωχ = uζ,η − uη,ζ and ωη = uχ,ζ − uζ,χ, are

generated by the synthetic model. We will model noise and measurement errors by adding to each tilt measurement

ω(χj , ηj , ζj , τk) white noise that is normally distributed, having a mean of zero, and a variance of 1, i.e., N(0, 1), and

which has an amplitude of σNωmax,j and is independent of all the other noise components. Thus the measurements

that are fed to the EKF-ILSA scheme are as follows:

ω(χj , ηj , ζj , τ ; σN ) = ω(χj , ηj , ζj , τ) + σN × ωmax,j ×N(0, 1) (4.4)

For the amplitude components we assume that ωmax,j = maxkω(χj , ηj , ζj , τk) and that σN × 100 is a parameter

that represents the percentage noise introduced into the measurements.


4.1.4 Near-field and Far-field Regimes

Lecampion et al [25] have identified a criterion to distinguish far-field from near-field regimes for tiltmeter measure-

ments. By means of far-field asymptotic expansions of the displacement gradients, this criterion was derived based

on the ability to distinguish a geometric change represented by the difference between the tilt field of a square DD

of side-length 2c and that due to a point DD of equivalent volume using tilts in a plane a distance d away. The

criterion is as follows: if the dimensionless distance δ = d2c < 1.5 = δc, then the tilt measurements are considered to

be near-field, while if δ > δc the tilt measurements are considered to be far-field. In the experiments we will vary d in

order to be able to test the EKF-ILSA scheme both in the near field and the far field regimes. To use this criterion,

we take the critical dimension to be the side-length 2c =√A [S] of a hypothetical square having an equivalent area

to that of S. Here operator A [·] is the same as that defined in (4.2). For a fixed d, this ratio will decrease as the

fracture enlarges, so a fracture can move from one regime to another as it evolves.

4.2 Monitoring an HF in a medium with a Stress Gradient

In the first experiment the synthetic tiltmeter data was generated by considering an HF propagating in the viscous

regime in a linearly varying in situ confining stress field. The confining stress field Σoϕ(χ, η), which was introduced

in (2.2), is assumed to have the following form

ϕ(χ, η) =ηM − η

ηM − ηm

where ηm = 0.25 and ηM = 32.25 are the minimum and maximum η coordinates of the rectangular mesh. In all

the experiments considered in this subsection we use a mesh size ∆χ = ∆η = 0.5 and a time-step ∆t = 3.18. As a

starter solution we considered the solution of a small penny-shaped fracture propagating in the viscosity dominated

regime [40, 34], which has an initial radius ρ = 2.25 at a time τ = 13.94. This problem represents a buoyancy-driven

hydraulic fracture (see for example [38, 47]) as can easily be seen by re-writing the fluid-flow equation (2.3) in terms

of the net pressure Π = Πf − Σoϕ(χ, η). Eliminating Πf in favor of Π, we obtain the following additional term

Σo

Gm∇ · (Ω3∇ϕ(χ, η)

)= − Σo

Gm(ηM − ηm)∂Ω3

∂η(4.5)

on the right side of (2.3). This first-order, convective, buoyancy term represents the tendency of the fracture to follow

the line of least resistance, which is to grow preferentially in the vertical η direction. For the forward model used

with the EKF-ILSA scheme, we assume no confining stress field so that ϕ(χ, η) ≡ 0. Thus the buoyancy term (4.5)

is no longer present in the fluid flow equation (2.3) of the forward model. This introduces significant un-modeled

dynamics into the equations governing the evolving synthetic hydraulic fracture that are not seen by the EKF-ILSA

scheme except via the feedback from the tiltmeter measurements.

4.2.1 Solution used to generate the synthetic tilts

In figure 2(a) we plot every tenth fracture front in the sequence of elliptic-shaped buoyancy-driven fractures that

correspond to the times τk100k=1, which were used to generate the time-series of synthetic tiltmeter measurements.

The effective side-length of the equivalent DD, for use in the near/far-field criterion, ranges from 2c = 3.99 initially

to 2c = 17.52 at the final time horizon τ100 = 333.65. The rectangular computational mesh used to generate this

solution is shown in the background. In figure 2(b) we plot the vertical cross sections of the width profiles that


0 5 10 15 20 25 30

5

10

15

20

25

χ

η

(a) ∂Sk

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

η Ω

(χb,η,τk)

(b) Ω(χb, η, τk) vs η

Figure 2. Left figure: A sub-sequence of the fracture fronts ∂Sk (shown red online) along with the rectangular computationalmesh used to generate the synthetic tiltmeter measurements. Right figure: Vertical cross-sections of the corresponding widthprofiles Ω(χb, η, τk) through the well-bore located at (χb, ηb) = (16.25, 8.25). The η coordinate of the well-bore is indicated bythe solid circle.

correspond to the fracture fronts shown in figure 2(a). The clear emergence of the bulbous fracture fronts reported

in [38, 47] can be seen.

4.2.2 Near-Field monitoring without noise

For this group of results we select the following values for the tilt array parameters that are defined in (4.3):

a = 0, b = 32, h = 9, and d = 10. For the range of front positions shown in figure 2(a), the dimensionless distance

parameter δ falls in the interval 2.51 ≥ δ ≥ 0.57, which implies that the fracture transitions regimes as it grows and

passes the critical value δc = 1.5. Indeed, for a brief initial period 13.94 ≤ τ . 45 the evolving fracture operates in

the far-field regime for this tiltmeter array, after which it spends the majority of the time in the near-field regime.

In figures 3 to 6 we plot the synthetic and EKF-ILSA fracture fronts and corresponding fracture widths at the

time-steps τ25 = 94.97, τ50 = 174.53, τ75 = 254.09, and τ50 = 333.65. In all of these plots we associate the following

symbols with the EKF-ILSA scaling parameter values • : φ = 0.1, ¥ : φ = 0.5, J: φ = 1.0, N : φ = 1.5, and I: φ = 2.0.

In parts (a) of each of these figures, the starter fractures are represented by dashed lines and the synthetic fronts,

which we are trying to identify using the EKF-ILSA scheme, are represented by the solid lines without symbols (shown

in red in the online version). In parts (b) of these figures the solid (red online) curves represent the synthetic fracture

width profiles that we are trying to identify using the EKF-ILSA scheme. We observe that the best identification

of the front positions and widths are achieved for the choice φ = 0.1 and that these estimates degrade as φ and

therefore γ is increased. Thus small values of φ, which correspond to small γ values, means that the EKF will place

relatively more weight on the measurements than on the dynamics of the forward model. The slight asymmetry in

the estimated front positions is due to the fact that the tilt array is not completely symmetrically positioned relative

to the axes of symmetry of the synthetic fracture. We observe that as the time period over which the EKF-ILSA

estimation scheme has been operating increases, the front and width estimates improve due to the improved estimates

of the covariance matrices Γk+1|k+1 accumulated over the duration of the HF process.


8 10 12 14 16 18 20 22 24

4

6

8

10

12

14

16

χ

η

(a) ∂S25

4 6 8 10 12 14 160

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

η Ω

(χb,η,τ

25)

(b) Ω(χb, η, τ25) vs η

Figure 3. Left figure: The synthetic front ∂S25 indicated by the solid line (red online) and the and the EKF-ILSA frontestimates using φ = 0.1 (•) φ = 0.5 (¥), φ = 1.0 (J), φ = 1.5 (N), and φ = 2.0 (I). The starter fracture is representedby the dashed circle (shown red online). Right figure: Vertical cross-sections of the corresponding synthetic width profile andEKF-ILSA estimates Ω(χb, η, τ25) through the well-bore located at (χb, ηb) = (16.25, 8.25). The same symbols have been usedas those in part (a) to represent the different φ values.

10 15 20 25

4

6

8

10

12

14

16

18

20

χ

η

(a) ∂S50

4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

η

Ω(χb,η,τ

50)


Figure 4. Left figure: The synthetic front ∂S50 indicated by the solid line (red online) and the and the EKF-ILSA frontestimates using φ = 0.1 (•) φ = 0.5 (¥), φ = 1.0 (J), φ = 1.5 (N), and φ = 2.0 (I). Right figure: Vertical cross-sections ofthe corresponding synthetic width profile and EKF-ILSA estimates Ω(χb, η, τ50) through the well-bore located at (χb, ηb) =(16.25, 8.25). The same symbols have been used as those in part (a) to represent the different φ values.

To see how the estimates of ∂S and Ω evolve with time, we plot the error measures EF and Ew in figure 7. In these

plots the times, τk : k = 25, 50, 75, and 100, at which the fronts and widths in figures 3 to 6 are sampled, are

represented by the solid circles along the τ axis. We observe that both these errors reduce monotonically as the value

of φ is decreased. In addition, as time evolves we observe that the errors typically peak relatively early and then

ultimately asymptote as time advances. We note that the peaks in the errors occur close to the far-field to near-field


5 10 15 20 25 30

4

6

8

10

12

14

16

18

20

22

24

χ

η

(a) ∂S75

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

η Ω

(χb,η,τ

75)



0 5 10 15 20 25 30

5

10

15

20

25

χ

η

(a) ∂S100

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

η

Ω(χb,η,τ

10

0)

(b) Ω(χb, η, τ100) vs η


transition τ ≈ 45 so that the subsequent reduction in errors could be due to the fractures moving further into the

near-field regime of the tilt array as they enlarge. These reductions in error as time evolves could also, in part, be

due to the improved estimates of the covariance matrix Γk+1|k+1 accumulated as time advances. The solid curves

without symbols in each of these figures (shown red online) represent the difference between the synthetic fracture

solution with the stress gradient and the radially symmetric viscous solution that would develop in the absence of a


0 50 100 150 200 250 300 3500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

τ

E F(τ

;φ,σN

=0)

φ = 0.1 φ = 0.5 φ = 1.0 φ = 1.5 φ = 2.0 Radial

(a) EF vs τ

0 50 100 150 200 250 300 3500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

τ E w

(τ;φ,σN

=0)

φ = 0.1 φ = 0.5 φ = 1.0 φ = 1.5 φ = 2.0 Radial

(b) Ew vs τ

Figure 7. Left figure: Time evolution of the footprint error measure EF for different values of φ. The solid line representsthe footprint error measure used to probe the difference between the radially symmetric and the synthetic fracture footprints.Right figure: Time evolution of the width error measure Ew for different values of φ. The solid line represents the width errormeasure used to test the difference between the radially symmetric and the synthetic solution widths.

stress gradient. The latter solution is that which would be produced by the forward model without corrections from

the measurements introduced by the EKF-ILSA scheme.

4.2.3 Near-Field monitoring including noise

We repeat the estimation process with the same tilt array and parameters as described in the previous sub-section

4.2.2, but this time we include the noise defined in (4.4) in which σN > 0. In figure 8(a) we plot a typical near-field tilt

measurement and the 5% noise fluctuations about this curve. In figure 8(b) we represent five synthetic fracture fronts

developing in an elastic medium with a linearly varying stress gradient field by solid lines without markers (shown

red online), while the corresponding EKF-ILSA estimates assuming that φ = 0.1 and σN = 0.01 are represented

by the solid lines with • markers. The extreme perturbations evident in the estimated fracture fronts shown in this

figure are due to the large weighting that the EKF-ILSA scheme places on the measurements for φ = 0.1. While this

parameter choice provided the best estimates of the front positions in the previous subsection, it proves not to be

robust to the perturbations introduced by even a relatively small amount of noise and does not remain stable much

beyond the 50 time-steps represented here.

In order to further explore the performance of the EKF-ILSA scheme in the presence noise, we consider the

following sequence of identifications in which the resolution parameter φ and the noise parameter σN take on the

following values

φ = 0.1 : 0.1 : 2.0; σN = 0 : 0.01 : 0.05

The results of these numerical experiments are summarized in figure 9. Similar to figure 7(a), in figure 9(a) we

plot the time evolution of the footprint error measure EF associated with the EKF-ILSA scheme, but this time we

assume a resolution parameter φ = 1.2 and noisy measurements in which σN = 0.01, 0.02, 0.04, and 0.05. We

observe that the error trajectories for all the noisy measurements σ > 0 closely follow the trajectory of the error in


0 50 100 150 200 250 3000.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

−3

τ

ω(χ

9,η

9,ζ

9,τ

)+

0.05×N

(0,1

)×ω

ma

x,9

(a) ω(χ9, η9, ζ9, τ ; σN = 0.05) vs τ

10 15 20 25

4

6

8

10

12

14

16

18

20

χ

η

(b) ∂Sk

Figure 8. Left figure: A typical tilt measurement ω(χ9, η9, ζ9, τ) without noise shown by the solid (red online) curve andthe corresponding measurement ω(χ9, η9, ζ9, τ ; σN = 0.05) with 5% noise added represented by the • symbols. Right figure:Synthetic fracture fronts (solid - red online) for τk : k = 10 : 10 : 50 developing in a stress gradient and the correspondingEKF-ILSA estimates assuming that φ = 0.1 and σN = 0.01.

0 50 100 150 200 250 300 3500

0.02

0.04

0.06

0.08

0.1

0.12

0.14

τ

E F(τ

;φ=

1.2,σN

)

σN = 0.00

σN = 0.01

σN = 0.02

σN = 0.04

σN = 0.05

(a) EF vs τ

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

φ

E F

σN = 0.00

σN = 0.01

σN = 0.02

σN = 0.04

σN = 0.05

(b) EF vs φ

Figure 9. Left figure: Time evolution of the footprint error measure EF for φ = 1.2, a near-field tiltmeter array (d = 10), anddifferent values of σN . Right figure: Variation of the footprint error measure maxima Emax,F (-.-) and asymptotes EF (τ100)(solid lines) vs φ for different values of σN .


which the measurements are not noisy σN = 0. Thus if the identification process is stable for a given choice of the

resolution parameter φ, one can expect a similar error damping characteristic from the EKF-ILSA scheme for noisy

measurements as one does for the EKF-ILSA scheme in which there is no noise present. To explore the robustness

of this error damping characteristic to noise fluctuations, we summarize the data such as the curves shown in figure

9(a) by sampling their maxima and their asymptotic values, which we assume to be the value at τ100. We use these

special values of EF to characterize the identification performance of the EKF-ILSA scheme over time. In figure 9(b)

we plot Emax,F (-.- lines) and EF (τ100) (solid lines) as a function of φ for a number of different values of the noise

parameter σN . A number of interesting characteristics of the EKF-ILSA scheme become evident from this figure. (i)

The asymptotic error EF (τ100) exhibits roughly a linear increase with φ that is largely independent of the noise level

σN . The error maxima Emax,F exhibit a similar linear increase with φ, but the fluctuations about this straight line are

more marked. (ii) The EKF-ILSA scheme is not always stable for the smaller values of φ when the noise parameter

is increased - this is the phenomenon demonstrated in figure 8(b). To represent this on figure 9(b), we have, for each

σN , colored (red online) and enlarged the symbol representing the smallest φ value for which the EKF-ILSA scheme

is still stable. As the noise level σN increases, so too does the minimal φ value required stabilize the EKF-ILSA

scheme need to increase. Thus a compromise in resolution is required in order to increase the robustness needed to

deal with additional noise.

10 15 20 25

4

6

8

10

12

14

16

18

20

χ

η

(a) ∂S25 and ∂S50

0 5 10 15 20 25 30

5

10

15

20

25

χ

η

(b) ∂S75 and ∂S100

Figure 10. The solid lines (colored red online) represent the synthetic fronts while the symbols represent the φ = 1.2EKF-ILSA fracture fronts for near-field tilts (d = 10) and measurements with noise levels σN = 0 (•); 0.01 (¥); 0.02 (J); 0.04 (N); 0.05 (I).

To demonstrate how the values of the footprint error measures are reflected in discrepancies in the evolving front

positions, in figure 10 we plot the synthetic and φ = 1.2 EKF-ILSA front positions that correspond to the errors at

the times indicated by the solid circles shown along the τ axis in figure 9(a). We observe that for this value of φ

there is very little spread due to the increased noise in the measurements, while the fracture fronts are still identified

relatively accurately.


4.2.4 Far-Field monitoring including noise

For the results presented in this sub-section we select the following values for the tilt array parameters defined in

(4.3): a = 0, b = 32, h = 7 & 9, and d = 50. For the range of front positions shown in figure 2(a), the dimensionless

distance parameter δ falls in the interval δc < 2.85 ≤ δ ≤ 11.51, which implies that the fracture remains in the

far-field regime throughout its evolution.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.05

0.1

0.15

0.2

0.25

0.3

0.35

φ

E F

σN = 0.00

σN = 0.01

σN = 0.02

(a) EF vs φ

0 50 100 150 200 250 300 3500

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

τ

E F(τ

;φ=

0.6,σN

)

σN = 0.00

σN = 0.01

σN = 0.02

σN = 0.03

σN = 0.04

(b) EF vs τ

Figure 11. Left figure: Variation of the footprint error measure maxima Emax,F (-.-) and asymptotes EF (τ100) (solid lines)vs φ for different values of σN . A 4× 4 tilt array with h = 9 was used for all these experiments. Right figure: Time evolutionof the footprint error measure EF for φ = 0.6, different values of σN , and a more dense 5 tiltmeter array with h = 7 operatingin the far-field regime (d = 50).

In figure 11 we use the same format as that used above to summarize the results for this series of numerical

experiments. In figure 11(a) the values of the footprint error measure maxima Emax,F and asymptotes EF (τ100) are

plotted as a function of φ. In this figure we use the same 4 × 4 tilt array with h = 9 as that used above. As with

the near-field case, these parameters increase almost linearly with increasing φ. However, the error level is noticeably

larger for the far-field case d = 50 than for the near-field case d = 10. For example, the asymptotic error EF (τ100) for

the near-field case (d = 10) lies in the range 1.48−5.50 (with a gradient of 0.021) compared to a range of 6.70−18.65

(with a gradient of 0.059) for the far-field case d = 50. From figure 11(a) it can be seen that it is possible to obtain

stable EKF-ILSA identifications for all φ in the range 0.1 ≤ φ ≤ 2.0, provided the amplitude of measurement noise

is restricted to the range 0 ≤ σN ≤ 0.02. In order to achieve a stable identification for larger noise amplitudes

σN > 0.02, it is necessary to use larger values of the resolution parameter φ ≥ 1.3. The footprint error measure

in this case will range between 13.13 % and 24.4 %, or larger. Thus for a far-field tilt array one is presented with

a difficult choice. In order to achieve an identification of the fracture footprint within 10% it is necessary to use a

resolution parameter in the range 0 < φ ≤ 0.6, but then the EKF-ILSA scheme becomes more susceptible to noise

resulting in significant perturbations in the identified front and instabilities in the algorithm. Indeed, using the 4× 4

tilt array with h = 9 on the far-field measurements d = 50, these perturbations and instabilities are observed for

σN > 0.02 when φ is in the range 0 ≤ φ ≤ 1.3.

However, it is possible to make the tilt array significantly more robust by increasing the density of tiltmeters in


a given region. Although we expect that increasing the density of tilts can degrade the overall conditioning of the

system due to redundancy in the measurements, it is this redundancy in measurements that enables the EKF-ILSA

algorithm to distinguish noise from measurement by essentially canceling the random fluctuations - provided the

noise in the different tilt locations is independent.

To illustrate this increase in robustness with tilt array density, we increased the density of the tilt array by

decreasing h to 7 resulting in a 5× 5 tilt array. This significantly improves the robustness of the EKF-ILSA scheme

for smaller values of φ. To illustrate this, in figure 11(b) we present the time evolution of the footprint error measure

EF for φ = 0.6 and noisy measurements having amplitudes σN = 0, 0.01, 0.02, 0.03, 0.04. This error evolution

curve exhibits similar characteristics to those shown for the near-field case in figure 9(a), such as a maximum value

that occurs relatively early in the identification and an asymptote (more of a slant asymptote in this case). Since

there is no transition between far and near field regimes in this case, the appearance of the early maximum for this

tilt array indicates that this turning point characteristic is caused by improved state estimates due to accumulated

information about the covariance matrix Γk+1|k+1 as time evolves. Recall that for the case d = 10, these turning

points happened to occur close to the far to near-field transition time, so in that case it was not possible to select

between these two mechanisms for the formation of the turning points early in the identification process.

10 15 20 25

4

6

8

10

12

14

16

18

20

χ

η

(a) ∂S25 and ∂S50

0 5 10 15 20 25 30

5

10

15

20

25

χ

η

(b) ∂S75 and ∂S100

Figure 12. Synthetic footprints represented by the solid lines (shown red online) and the φ = 0.6 far-field tilt EKF-ILSA frac-ture fronts represented by the following symbols corresponding to the measurements noise levels σN = 0 (•); 0.01 (¥); 0.02 (J); 0.03 (N); 0.04 (I). A more dense 5× 5 tiltmeter array with h = 7 is used to arrive at an EKF-ILSA identification that ismore robust to noise perturbations

In figure 12 we plot the EFK-ILSA identifications of the front locations for φ = 0.6, measurement noise amplitudes

σN = 0, 0.01, 0.02, 0.03, 0.04, and using a 5× 5 tiltmeter array with h = 7. These fronts correspond to the times

indicated by the solid circles shown along the τ axis in figure 11(b). While the performance of the EFK-ILSA scheme

is clearly worse for this far-field case than it is for the near-field case, the identifications are still fairly accurate and

show significant robustness to perturbations introduced by noise in the measurements.


4.3 Monitoring an HF in a medium with a Symmetric Stress Jump

The numerical experiments considered in this sub-section are motivated by the symmetric stress jump experiments

reported in [19], in which an HF is initiated within a channel having a low confining stress, which is sandwiched

between two regions in which there is a high confining stress. The extent to which the HF penetrates the high stress

zones is of great practical interest in the oil industry. To simulate data from this type of situation, we generate

synthetic tiltmeter data by considering a viscosity driven HF propagating in an elastic medium in which the well-

bore is located midway between two jump discontinuities in the confining stress field Σoϕ(χ, η). In particular, the

well-bore is located at (χb, ηb) = (7.1667, 7.1667), while the confining stress has the following jump discontinuities

across the planes η = 6.6667 and η = 7.6667:

Σo = 3/4; ϕ(χ, η) =

1 if |η − ηb| ≥ 1/20 if |η − ηb| < 1/2

(4.6)

To generate the synthetic tilt measurements we use a rectangular mesh comprising square elements whose edges are

defined by the Cartesian product of the following partitions χk = 0 : ∆χ : 14.3333 and ηk = 0 : ∆η : 14.3333, where

∆χ = ∆η = 1/9. A radially symmetric starter crack is assumed in which the initial radius is set to ρ = 0.2778 at time

τ1 = 0.1260 and the pressure and width profiles initialized to the radially symmetric viscous solution [40]. The time

duration of the simulation is assumed to be τ1 ≤ τ ≤ 21.5521, which is subdivided into 300 uniform time-steps of size

∆τ = 0.0713. The EKF-ILSA scheme assumes a forward model for which ϕ(χ, η) = 0, so that a radially symmetric

fracture would develop without an update from the measurements via the EKF component of the algorithm.

The tilt array in this case is defined by the parameters a = 0, b = 16, h = 3 & 1, and d = 10. The effective-square

side-length, having an area equivalent to the evolving fracture footprint, is initially 2c = 0.49235 and is increased

to 2c = 4.5165 by the end of the simulation. This corresponds to a dimensionless distance parameter δ in the range

20.3108 ≥ δ ≥ 2.2141 > δc. Thus the tilt array is operating in the far-field regime throughout the evolution of the

fracture.

In figure 13 we compare horizontal and vertical cross-sections of the fracture widths for the synthetic profile

indicated by solid lines without symbols (shown red online) and those of the EKF-ILSA identification with φ = 0.01

and no noise σN = 0, represented by the solid lines with • symbols (shown black online). These widths are sampled at

the times τk : k ∈ 10, 20, 30, 50, 75, 100 : 50 : 300. We observe that the EKF-ILSA scheme does not capture the

inflection points in the synthetic vertical width profile, which is associated with the stress jump interfaces η = 6.6667

and η = 7.6667, but approximates these widths by convex profiles of equivalent volume. We note that the EKF-ILSA

scheme closely captures synthetic the fracture width profiles close to the fracture boundaries ∂Sk, particularly in the

horizontal direction in which the fracture footprint dimension is the largest. By virtue of the tip asymptotic relation

(2.6) used to locate the fracture boundary, this enables the EKF-ILSA approximation to capture of the fracture

geometry relatively accurately in spite of the mismatch between the estimated and synthetic widths in the interior

of the domain.

In figure 14(a) we plot the evolving synthetic fracture fronts shown by solid line segments (red online) and the

corresponding fracture fronts identified by the EKF-ILSA scheme with φ = 0.01 and no noise σN = 0, which are

represented by solid lines with • symbols. As before the forward model used in the EKF-ILSA scheme assumes no

confining stress field, which is consistent with a growing sequence of circular fracture fronts. Thus the stress-jumps

represent significant un-modeled dynamics not represented in the forward model that cause substantial changes in


2 4 6 8 10 120

0.5

1

1.5

2

2.5

χ

Ω(χ,ηb,τk)

(a) Ω(χ, ηb, τk) vs χ

5.5 6 6.5 7 7.5 8 8.5 90

0.5

1

1.5

2

2.5

η Ω

(χb,η,τk)


Figure 13. In both plots the solid lines without symbols (colored red online) represent the synthetic fracture widths fordifferent time-steps and the solid lines with • symbols (shown black online) represent the corresponding EKF-ILSA fracturewidths in which φ = 0.01 and there is no noise present in the measurements. Left figure: Horizontal cross-sections of the widthprofiles Ω(χ, ηb, τk) through the well-bore located at (χb, ηb) = (7.1667, 7.1667). Right figure: Vertical cross-sections of thecorresponding width profiles Ω(χb, η, τk) through the well-bore located at (χb, ηb) = (7.1667, 7.1667).

4 6 8 10 12

4

5

6

7

8

9

10

11

χ

η

(a) ∂Sk with φ = 0.01 and σN = 0

4 6 8 10 12

4

5

6

7

8

9

10

11

χ

η

(b) ∂Sk with φ = 0.4 and σN = 0

Figure 14. Both figures show a sub-sequence of the fracture fronts ∂Sk in which the synthetic solution is represented by solidline segments (shown red online) and the EKF-ILSA solution by solid lines with the symbols • (shown black online). No noisewas added to the measurements in either figure. The thick horizontal lines indicate the location of the interfaces across whichthe confining stress field jumps. Left figure: In this case φ = 0.01 for the EKF-ILSA scheme. The fracture fronts shown in thisfigure correspond to the widths shown in figure 13. Right figure: In this case φ = 0.4 for the EKF-ILSA scheme.

the fracture geometry. We have chosen a particularly small value of the resolution parameter to demonstrate that the

EKF-ILSA scheme can faithfully resolve these extreme changes in fracture geometry by making taking into account

the tilt measurements. For this run a 6 × 6 tiltmeter array with spacing h = 3 is used. Unfortunately, this small

a value of the resolution parameter φ is not at all robust to noise in the measurements. In the previous subsection

we found that robustness to noise could be traded for resolution by increasing the parameter φ. However, for this


problem, with extreme changes in the geometry, even a moderate increase in the resolution parameter φ results in a

significant degradation in the quality of the identification of the fracture footprint. To illustrate this degradation we

plot the same sequence of fracture fronts corresponding to φ = 0.4 in figure 14(b).

0 5 10 15 20 250

0.05

0.1

0.15

0.2

0.25

0.3

0.35

τ

E F(τ

;φ=

0.2,σN

)

σN = 0.00

σN = 0.01

σN = 0.02

(a) EF vs τ

4 6 8 10 12

4

5

6

7

8

9

10

11

χ

η

(b) ∂Sk with φ = 0.1 and σN = 0.02

Figure 15. Left figure: Time evolution of the footprint error measure EF for the EKF-ILSA identification of the symmetricstress jump problem using a tiltmeter array operating in the far-field regime. The resolution parameter in this case is φ = 0.2and the results for different values of the noise amplitude parameter σN are shown. Right figure: Using the same symbolconvention as in figure 14 we plot the EKF-ILSA solution for φ = 0.2 and noise σN = 0.02.

From the results presented in figure 14 it is clear that if we want to achieve a reasonably accurate EKF-ILSA

identification of the fracture footprints for this problem, we will have to use a relatively small value of the resolution

parameter, φ = 0.2 for example. In order to achieve robustness to noise for such a small value of φ it will be necessary

to make use of a tilt array with a greater density to exploit the noise cancelation effect achieved by the additional

redundancy in measurements. To this end we reduce the array spacing parameter from h = 3 to h = 1. In figure 15 we

plot results for the EKF-ILSA identification of the symmetric stress jump footprints using the more dense tiltmeter

array h = 1, φ = 0.2 and assuming a noise amplitude σN = 0.02. From 15(a) we observe that the identification

results with this array and parameter choice are robust to noise in the range 0 ≤ σN ≤ 0.02. In figure 15(b) we plot

the front positions sampled at the at the times indicated by the solid circles along the τ axis in figure 15(a). These

front identifications are clearly an improvement on those shown in figure 14(b) while still being robust to moderate

amplitude noise in the measurements.

4.4 Monitoring an HF in medium with an Asymmetric Stress Jump

The numerical experiments considered in this sub-section are motivated by the asymmetric stress jump experiments

reported in [52], in which an HF is initiated within a channel, having an intermediate confining stress, which is

sandwiched between a high stress region and a low stress region. As in the previous sub-section, we generate synthetic

tiltmeter data by considering a viscosity driven HF propagating in an elastic medium in which the well-bore is located

midway between two jump discontinuities in the confining stress field Σoϕ(χ, η). However, in this case we assume

asymmetric jumps in the confining stress field. In particular, the well-bore is located at (χb, ηb) = (7.1667, 7.1667),


while the confining stress has the following asymmetric jump discontinuities across the planes η = 6.6667 and

η = 7.6667:

Σo = 1; ϕ(χ, η) =

1 if η − ηb ≥ 1/21/2 if |η − ηb| < 1/20 if η − ηb ≤ −1/2

(4.7)

The asymmetric, positive and negative, stress jumps about the well-bore confining stress level, result in a completely

different geometry to that observed in the case of the symmetric stress jump. For the case of the symmetric stress

jump the HF is largely trapped between the jump interfaces, with small zones of penetration into the high stress

regions. For the asymmetric stress jump defined in (4.7), the fracture penetrates very little into the higher stress

region, propagates only a moderate distance in the intermediate stress region, and herniates into the low stress region

in which the majority of the fracture footprint and volume ultimately reside.

To generate the synthetic tilt measurements we use precisely the same a rectangular mesh, starter fracture, and

time-step as that used for the symmetric stress jump described in sub-section 4.3. As before the EKF-ILSA scheme

assumes a forward model for which ϕ(χ, η) = 0, so that a radially symmetric fracture would develop without an

update from the measurements via the EKF component of the algorithm.

The tilt array in this case is defined by the parameters a = 0, b = 16, h = 3, and d = 50. The effective-square

side-length, having an area equivalent to the evolving fracture footprint, is initially 2c = 0.49235 and is increased

to 2c = 4.8403 by the end of the simulation. This corresponds to a dimensionless distance parameter δ in the range

101.5538 ≥ δ ≥ 10.3299 > δc. Thus the tilt array is operating well into the far-field regime throughout the evolution

of the fracture.

4 5 6 7 8 9 10

3

3.5

4

4.5

5

5.5

6

6.5

7

7.5

8

χ

η

(a) ∂Sk with φ = 0.1 and σN = 0

3 4 5 6 7 8 90

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

η

Ω(χb,η,τk)


Figure 16. Left figure: Using the same symbol convention as in figure 14 we plot the EKF-ILSA solution for φ = 0.1 andno noise. Right figure: Vertical cross-sections of the corresponding width profiles Ω(χb, η, τk) through the well-bore located at(χb, ηb) = (7.1667, 7.1667).

In figure 16 we compare the fracture footprints and widths for the synthetic and EKF-ILSA identification assuming

φ = 0.1 and no noise. The EKF-ILSA scheme provides a very good identification of the widths and the fracture front

positions in the low stress region where the fracture has herniated and the widths are large. However, in the high

stress region, where the fracture opening is relatively small, the EKF-ILSA is not able to identify all the details of the


fracture footprints as the fracture widths are much smaller than in other parts of the fracture footprint. The relatively

poor match of the fracture footprint in the high stress region affects the way in which the footprint is approximated

in the intermediate stress zone between the stress jump interfaces. The EKF-ILSA fronts show a moderate inflection

point in this region and is only able to approximate the synthetic fracture front in an average sense.

Given that the EKF-ILSA scheme estimates the front location via the state comprising the fracture widths, it is

not surprising that the best resolution of the fracture footprint is found in the regions where the fracture width is

much larger than in the remainder of the fracture. Conversely, in regions where the fracture widths are much smaller

than in the rest of the fracture, the resolution can be expected to be relatively poor. In the light of this characteristic,

it is interesting to contrast the performance of the EKF-ILSA scheme for this asymmetric fracture with that of the

symmetric fracture discussed in sub-section 4.3. Though both tilt arrays in these two examples are operating in the

far-field regime, the array for the asymmetric stress jump is five times farther from the fracture than in the case

of the symmetric stress jump. In spite of this, the tilt array in the asymmetric case is able to capture the fracture

footprint much more accurately than in the symmetric case as can be seen clearly by comparing the error evolution

plot shown in figure 17(a) with that shown in figure 15(a). This discrepancy can be explained by observing that, in

the symmetric case, the majority of the front identification takes place in the high stress regions where the fracture

widths are relatively small compared to the widths in the well-bore channel. Indeed, the identification of the fracture

fronts in the channel region is particularly accurate. In contrast, for the asymmetric case, the majority of the fracture

front is being identified in the herniated region where the fracture widths are much larger than in the remainder of

the fracture.

0 5 10 15 20 250

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

τ

E F(τ

;φ=

0.6,σN

)

σN = 0.00

σN = 0.01

σN = 0.02

σN = 0.03

σN = 0.04

(a) EF vs τ

4 5 6 7 8 9 10

3

4

5

6

7

8

χ

η

(b) ∂Sk with φ = 0.6 and σN = 0.04

Figure 17. Left figure: Time evolution of the footprint error measure EF for the EKF-ILSA identification of the asymmetricstress jump problem using a tiltmeter array operating in the far-field regime. The resolution parameter in this case is φ = 0.6and the results with the noise amplitude σN = 0.01 and without noise are shown. Right figure: Using the same symbolconvention as in figure 14 we plot the EKF-ILSA solution for φ = 0.6 and noise σN = 0.01.

In figure 17 we demonstrate the performance of the EKF-ILSA scheme with φ = 0.6 in the presence of noise

having the following amplitudes σN = 0.01 : 0.01 : 0.04. Stable inversion results were possible with noise amplitudes

σN = 0 and 0.01 when a 6 × 6 tiltmeter array with h = 3 is used. However, for larger noise amplitudes σN = 0.02


and 0.03 it is necessary to decrease the sample spacing of the array to h = 2, while for σN = 0.04 it is necessary

to decrease the sample spacing of the array to h = 1. We observe that increasing the density of the sample points

does not change the magnitude of the errors but it does enable the EKF-ILSA scheme to cancel independent noise

fluctuations by exploiting the redundancy introduced by the additional sample points in the array. Comparing figure

17(b) with figure 16(a) we observe some degradation in the identification of the front positions with the increase of

the resolution parameter. However, even in the presence of significant noise, the identification of the fracture fronts

is remarkably accurate. The slight asymmetry in the results, is due to the fact that the tilt array is not distributed

symmetrically with respect to the well-bore in the χ direction.

5 Conclusions

In this paper we propose a novel algorithm to invert time-series of tiltmeter measurements sampled at selected points

within an elastic medium in order do identify the fracture footprints and fracture opening as an HF propagates.

The algorithm is based integrating the Extended Kalman filter with the stable and efficient Implicit Level Set

Algorithm, recently developed to solve the highly nonlinear degenerate integro-partial differential equations governing

the dynamics of the moving fracture front. The unique property of the ILSA scheme is that it uses predicted state

values close to fracture tip along with the tip asymptotic relation, for a given propagation regime, to locate the fracture

free boundary. This feature enables integration with the EKF without having to resort to explicit parametrization

and identification of the fracture free boundary, which would introduce complex geometric coupling between the state

variables and make the evaluation of the state Jacobian extremely complex and costly. The proposed scheme uses the

EKF state predictions and measurement updates, assuming a given fracture footprint, to determine corresponding

ILSA front updates to arrive at iterative algorithm that has both state and front updates that are consistent with

the measurements.

In such an estimation algorithm it is typical to have to prescribe the appropriate algorithm parameters to achieve

the desired identification regime. In EKF-ILSA scheme it is necessary to determine the model and noise covariances,

as well as the initial state covariance matrix. In this context of state identification from remote measurements in which

we expect significant un-modeled dynamics, we need to find a rational way to choose these parameters. By means

of the appropriate scaling we are able to identify the appropriate parameter choices for the EKF component of the

algorithm that are suitable for the identification of remote measurements. In particular, we identified the parameter

γ = σ2v

σ2wε2 , which determines the relative weight the Kalman filter places on the observed measurements compared

to the state predicted by the dynamic model. If γ → 0 then the EKF places most weight on the measurements,

while if γ →∞ the EKF places most weight on the state predictions. For measurements of order ε, it is possible to

ensure γ << 1, suitable for dealing with significant un-modeled dynamics, by coupling the model and noise variance

parameters σw and σv according to the relation:

σv = φσwε, so that γ = φ2, where 0.01 < φ < 2

In this regime of significant un-modeled dynamics, the parameter φ can be used to increase the resolution of the

identification, i.e., throwing more weight onto the measurements, by choosing φ ≈ 0.01. Conversely, by increasing φ

the identification can be used to trade resolution, moving the emphasis away from the measurements, for increased

robustness. This strategy for defining the algorithm parameters is by no means unique. Indeed, if a better estimate of


the noise variance parameter σv is available, then it would be more appropriate to choose σw to be a scaled function

of σv so that the algorithm is placed in the desired identification regime as characterized by γ.

We consider three numerical experiments comprising three different physical situations each of which have real-

world realizations: buoyancy driven fracture propagation; breakthrough by an HF propagating in a channel formed by

a symmetric stress jump; and simultaneous herniation of an HF into a lower stress region and restricted penetration

into a high stress region in an asymmetric stress jump. Some interesting patterns emerge from the numerous numerical

results presented: (i) The EKF-ILSA scheme provides successful identifications in both the near and far-field regimes.

As is to be expected, the resolution is better in the near-field regime than in the far-field regime. (ii) The error in of

the EKF-ILSA scheme, as measured by the footprint error measure, increases initially to a maximum level after which

it decreases and eventually tends to an asymptote. This characteristic can be explained by the improved estimates

of the state covariance matrix Γk|k as the data is accumulated. (iii) The resolution of the identification degrades

roughly linearly with increasing φ - as more weight is taken away from the measurements and placed on the state

evolution model, which does not contain information about the un-modeled dynamics. (iv) The EKF-ILSA scheme

can tolerate noise of up to 5% of the maximum measurement amplitude. The robustness to noise in the measurements

typically increases as φ is increased. Another way in which an array of tiltmeters can be made more robust to noise

in the measurements is to increase to density of tilts in a given region. In this way, the EKF-ILSA scheme can exploit

the redundancy in the additional measurements by canceling the independent fluctuations in measurements from

neighbouring stations. Increasing the density of tilts in a region typically does not improve the error characteristic

but does improve the robustness.

Though the deployment of extremely dense arrays to improve robustness to noise may not be practical at this

time, this noise cancelation feature of the EKF-ILSA scheme might be an important consideration in deciding the

placement of tiltmeters or provide a stimulus for the development of new technology. For example, it might be possible

to deliberately place two tilts in close proximity. An array comprising such twinned tilts might be much more robust

to independent noise fluctuations. Indeed, before deploying a tiltmeter array, it would be profitable to perform a

number of simulations to see if the resolution of the EKF-ILSA scheme in a particular region can be optimized by

improving on the tilt locations. Given the susceptibility of the EKF-ILSA scheme to noise in the data, which is likely

fluctuating at a different frequency from that expected of the actual measurements, it would be advantageous to

filter noise from the measurements before feeding the data to the EKF-ILSA scheme.

Having established the basic methodology a number of developments and extensions of the EKF-ILSA scheme can

be considered. The EKF is based on a linearization of the state equation (3.1) about a nominal state comprising

the estimate at the previous update. However, there is a limit to the validity of this form of linear approximation

especially in the light of the fact that the EKF update equations are derived by taking conditional expectations of

the linearized system. However, for a nonlinear function f(x) and random variable x it is, in general, definitely not

true that

E[f(x)|y] 6= f(E[x|y]) (5.1)

This error is compounded even further in the calculation of the covariance matrices. Different approaches to the

evaluating these expectations have been developed such as the so-called Unscented Kalman Filter (UKF) [21, 22, 49].

The UKF approximates the desired expectations by taking appropriate weighted sums evaluated at so-called sigma

points and has been shown to be much more stable than the EKF in the context of highly nonlinear systems. The


UKF is typically computationally equivalent to, or even more efficient than, the EKF, however, for the HF problem

each new sample point will involve a completely new solution of the system of coupled equations (2.14). Thus the

UKF will clearly require more development in this context, which is beyond the scope of this paper.

We believe that this novel EKF-ILSA scheme brings a new approach to the classic elasto-static inverse problem.

Having established the potential of the method in silico via numerical experiments with synthetic measurements,

our next objective is to test the procedure on real field data. If this is shown to be promising then there will be a

great deal of potential for the development more efficient algorithms to make the real-time monitoring of propagating

hydraulic fractures a reality.

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