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Location and mapping of hydrofractures from
arrival times at wells
Victor Pereyra and Mihai Popovici
Inc., Mt.View, CA
victor@ca. wai. com and mihai@3dgeo. com
Abstract
We combine a fast eikonal solver with two optimization
algorithms to locate and map fractures caused by injection
in the secondary recovery of hydrocarbons. The data used is
arrival times on wells, although the method is applicable to
other problems and acquisition geometries. We describe the
problem and the various processes involved and illustrate the
numerical behavior with a synthetic data example.
1 Introduction
We consider the problem of locating and mapping fractures
produced by forced injection in a reservoir. The initial as-
sumption is that the velocity of propagation of elastic waves
is known on a mesh covering the area under investigation and
that geopliones are placed on wells to passively listen to hy-
dro cracking. The medium will be assumed to be isotropic.
The measured quantities are times of arrival of signals pro-
duced by the fractures. Neither the locations nor the origin
times of the signals are known.
We propose to use differential times in order to eliminate
the origination time from the problem. Then we will calculate
travel time tables from the receivers to every point on a three
dimensional mesh using a fast eikonal solver. We will produce
also calculated differential times by substracting each one of
these tables from a fixed reference one.
The algorithm will then consists of finding the best match-
ing set of differential times in the resulting calculated tables.
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60 Computational Acoustics and Its Environmental Applications
2 The problem and its solution
Let P^ = [zi>yj,Zk],i = l,...,/;j = I,-, J\k = l,...,If,
be an uniform mesh in three dimensions, with mesh spacirigs:
6x,6y,6z. Let Vijk represent the velocity of propagation of
pressure waves at the point P^. Let G\ — [flW, fly, #*/],/ =
0, ...,L be a set of geophone positions. Finally, let T° be a
set of arrival times recorded at the geophone positions and
corresponding to a signal produced by a crack in the vicinity
of the geophones.
Now we create the differential times:
The next step requires calculating travel times from the
geophones to each point in the mesh. This generates L + 1
travel time tables: T/", and by taking the differences with Tfi
we similarly create the calculated difference times: DTf, I =
1,...,L.
This quite expensive step will be done quickly by using a
fast eikonal solver implemented by M. Popovici.
Once these tables are created, the problem is reduced to a
minimization one, namely:
In order to solve this problem we need to extend our mesh
function DT^ to continuous values by interpolation, and then
we can call upon an appropriate derivative free minimization
procedure. We have tested an intelligent search algorithm
due to Nelder and Mead [8] as implemented by Hill [4] , and a
derivative-free scheme called PRAXIS, due to R. Brent [1].
3 Calculation of Travel Times
3.1 Seismic Ray Tracing
Seismic ray tracing in 2D complex media or 3D layered
media is a well understood process. Here we will indicate
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Computational Acoustics and Its Environmental Applications 61
briefly how to handle the additional difficulties that three-
dimensional isotropic blocky media presents.
For wave propagation in isotropic media, rays are the or-
thogonal trajectories to wave fronts. If the media is homoge-
neous, rays are straight lines, while for inhomogeneous media,
ordinary differential equations have to be integrated in order
to accurately calculate the rays. These are the so called ray
equations, which can be derived either from the Eikonal equa-
tion, or by invoking Fermat's principle of minimum time.
A convenient form of the ray equations in 3-D is:
TI = vw
w = V%, (1)
where ?? = (a;(g),?/(g),z(g)), w(a) = w )?,, is arc length
along the ray, and u = l/v is the slowness, with v the velocity
of propagation. Observe that since s is arc length, then ||
17(5) |J2= 1, and therefore w is a vector in the ray direction
with length equal to u. That is why this vector is sometimes
referred to as a slowness vector.
The simplest form of ray tracing is shooting, in which the
initial position, and the initial direction of the ray are pre-
scribed:
77(0) = %, w(0) = wo. (2)
We use shooting only as a vehicle to initialize a source-
receiver, global or bending type, iterative calculation, or to
check a posteriori if a two-point ray has changed signature (a
ray signature is an ordered sequence of reflecting interfaces).
This combination of shooting and bending was first employed
(to the best of our knowledge) in two-dimensions in 1983, and
published in (Pereyra, 1987).
Equations (1, 2) describe an initial value problem that
can be solved numerically by a standard technique. In fact,
for smooth velocity fields, these equations present no special
problems.
Since we are not interested in shot rays per se, but only as
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62 Computational Acoustics and Its Environmental Applications
a means to initialize a two-point iteration for solving source-
receiver problems, it is not necessary to calculate them to high
precision.
Given %, WQ, a model description, and a ray signature, the
shooting algorithm attempts to produce a ray that starts at
?7Q with direction WQ, travels through the structure, honoring
ray bending in inhomogeneous regions. If successful, we will
have obtained a discrete ray:
(s,-,?7,.,w,-), 2 = 1,...,AT.
The shooting algorithm produces discrete rays with the
same format as the ones required by the two-point solver de-
scribed below. Thus, when a shot ray lands near a receiver
it can be used directly to start a two-point iteration. A de-
tailed ray signature is also produced; this is now an ordered
sequence of regions R^ and patches P? traversed by the rays
that is needed by the two-point solver. By this procedure,
the two-point solver is made essentially independent of the
structural complexity of the model.
A general two-point boundary value approach for source-
receiver ray tracing in inhomogeneous layered media has been
reported earlier in detail in (Pereyra, 1988, 1992). A mul-
tipoint boundary value finite difference solver for nonlinear
systems of first order ordinary differential equations is used
(Pereyra, 1979; Lentini and Pereyra, 1983). This solver has
variable order, variable mesh, and global error estimation ca-
pabilities, combined to provide an accurate, efficient arid ro-
bust algorithm, well suited for high resolution work. Versions
adequate for solving two point boundary value problems in
smooth inhomogeneous media are available in the public do-
main (IMSL, Harwell, NAG libraries, or through the elec-
tronic Numerical Analysis Network na.net).
The ray equations (1) are discretized on a mesh (not nec-
essarily uniform) by the second order trapezoidal rule. We
make sure that discontinuities, i.e., patch crossings, occur at
mesh points where appropriate discontinuity conditions are
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Computational Acoustics and Its Environmental Applications 63
enforced. Global error estimates, adaptive meshes and adap-
tive order through deferred corrections are used to obtain a
solution with a prescribed accuracy in an efficient manner.
The finite difference process used is of global type and it
does not suffer from the instabilities associated with shooting
schemes. An efficient Newton type nonlinear equation solver,
which includes a carefully crafted, sparse linear solver is used
on the resulting nonlinear difference system. The sparse struc-
ture is such that a perfect elimination stable algorithm can be
devised; i.e., no fill-in is produced in the Gaussian elimination
process.
Discontinuities, additional algebraic conditions and unknown
parameters are also handled by our current version. The lin-
ear equation solver produces a sparse triangular decomposi-
tion (LU), which is a discrete version of the linearized ray
equations. This is quite useful for performing economically a
number of additional tasks, like calculation of 3D geometrical
spreading, sensibility studies, and nonlinear travel time inver-
sion or geophysical tomography with bent rays, as we show
below; see also (Pereyra, 1980, Pereyra, Keller, and Lee, 1980,
Pereyra, 1988, 1991).
In summary, this algorithm has the necessary generality to
solve the ray equations (1) subject to the end conditions:
K, (3)
(where S is the (unknown) total arc-length). It can also han-
dle the additional interface conditions arising in piece-wise
discontinuous media and of course many other similar prob-
lems.
Of course, the seismic ray tracing task in geophysics never
consists of calculating an isolated ray, but rather, for a given
shot location (in the case of non-zero offset ray tracing), one
needs to calculate all possible arrivals with a prescribed sig-
nature for a given array of receivers. This array may consist
of just one line of equally spaced receivers, as in the case of
2D surveys, or of a number of lines, either in a regular ar-
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64 Computational Acoustics and Its Environmental Applications
ray or in more general positions, or they can be underground
on wells. In any case, our algorithm takes into account this
fact to accelerate the calculation by using a so-called receiver
continuation strategy.
Receiver continuation is a technique that exploits the fact
that if we have calculated a ray path joining a source with a
receiver, then this ray can be used to initialize the two-point
or bending calculation for a neighboring receiver position. In
this way, the shooting exploratory phase is limited to finding
the first ray that arrives near the receiver array, which is then
used to initiate a sequence of two-point ray calculations by
receiver continuation.
A naive implementation of this simple idea would stop here
and would generally fail to calculate all possible arrivals, since
the two-point continuation will not be feasible if it tries to
move through caustics or into shadow zones, and may also
fail for other physical or computational reasons. What makes
our procedure robust is that as soon as the two-point con-
tinuation fails, the algorithm switches to shooting in order to
find another starting ray, and this search can be made as fine
and extensive as desired by choosing an appropriate control
screen, which also aids us in keeping track of the work done.
Normal incidence or zero offset ray tracing is also easily im-
plemented within this framework. This type of ray tracing is
used to simulate stacked sections. Our implementation calcu-
lates only one half of the trajectory, say from the coincident
source/receiver position to the reflector, since the return ray
must retrace the same path. The normal incidence on the re-
flector is enforced as a new type of boundary condition. Both
zero and non-zero offset diffracted ray paths from designated
edges can be also calculated.
Many of these tasks are amenable to coarse grain paral-
lelization on a network or multi-CPU setting, as we have
demonstrated in (Koshy, Pereyra, and Meza, 1991).
We will use this procedure to generate the synthetic data
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Computational Acoustics and Its Environmental Applications 65
used in the examples below.
3.2 Eikonal Solver
The scalar wave equation in a 3-D medium of constant den-
sity can be written as
c p c p 1
where p = p(t,x,y,z] is the pressure field, and v(x,y,z) is
the earth velocity. The pressure field p(t,x,y,z) is a finite
function and can be therefore expressed as a Fourier time
series
p(t, x, y, 2) = £ P(u>, .r, j/, z)e-' . (5)LJ
Substituting equation (5) into equation (4), we obtain
y]\ 1 '— - —- -j- —'\ —--f-
2 r>(, . ™ „. _\1_ —zw/ n //?\
Equation (6) should hold for any values of w. This is possi-
ble only if the sum of the terms inside the square brackets is
zero for each w. Equation (6) can also be obtained by Fourier
transforming in time the original wave equation (4). There-
fore we have
, , ,
aa-2
c< r) / ^ \ r\ / r*j \
equation which is valid for all values w and is called the reduced
wave equation or Helmholtz equation. In compact notation it
is written as:
(8)
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66 Computational Acoustics and Its Environmental Applications
By analogy with the constant velocity case, for the variable
velocity case we seek solutions to equation (8) in the form:
X%,Z/,z,w) = A(z,y,z,w)e- ). (9)
Introducing the trial solution (9) into the Helmholtz equation
(8) we obtain:
^ J^ = 0.
(10)
The phase term is always non zero so we can rewrite the
equation as
= 0. (11)
To solve equation (11) for large values of w we assume that
A(x, y,z,w) can be expanded asymptotically in inverse powers
of w. We expand the amplitude term as follows:oo
A(,;, y, z, w) - ^ A,,(2\ y, z)(?;w)—. (12)m=0
The sign means equation (12) is an asymptotic equality, the
series is assumed to be an asymptotic expansions of A(.x, y, z, w
as w — > oo.
Grouping the terms according to the powers of w and set-
ting each to zero we obtain the equation:
= 0 for (a;") (13)
which for AQ 0 is called the eikonal equation.
3.3 The frequency-dependent eikonal
A different form for the solution of the reduced wave equa-
tion can be sought in the form
p(x,y,z,u) = A(x,y,z,u)eWw\ (14)
Introducing equation (14) in equation (8) we obtain again
equation (11):
= 0.
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Computational Acoustics and Its Environmental Applications 67
In this case the phase term <Xz,?/,z,w) is frequency depen-
dent. We can employ a different strategy for solving equa-
tion (11) by equating the real and the imaginary part to zero.
Equating the real part to zero we get the frequency-dependent
eikonal equation:
AA = 0, (15)' \ /
while equating to zero the imaginary part we get the transport
equation:
2VA- V<^> + AA<^ = 0. (16)
Equation (15) can be written as
which becomes the eikonal equation if we drop the term con-
taining L>J~^.
4 Solving the eikonal
The 2-D algorithm described by Van Trier and Symes (1989)
is based on the eikonal equation
«' + «' = 8* (17)
where
^= &/
and s(x, z) is the 2-dimensional slowness model and t(x, z] is
the traveltime field. The second equation used is the equality
of the partial derivatives of the fields u(x, z} and v(x, z),
In cylindrical coordinates the eikonal equation becomes
+ = s2 (19)
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68 Computational Acoustics and Its Environmental Applications
where
and the mixed partial derivatives are
du &*t dv(20)
The finite difference implementation of equations (17), (18)
and in cylindrical coordinates (19), (20) is based on advancing
the computational front for the functions u(x, z) and v(x, z).
The traveltime field t(x,z) is found subsequently by integrat-
ing the function v(r,0) with respect to r. In cylindrical coor-
dinates the scheme is based on using equation (20) to compute
the values of u(r, 9} on a new computational front of constant
radius, by using the values of u(r, 9) arid v(r, 9) from the pre-
vious computational front. Equation (20) becomes
*'• • •). (2D
Starting with a constant velocity condition in the immediate
vicinity of the source location (u(r,0) = 0, v(r, 0} = s(r, 0)),
we can advance the computational front using the finite-difference
equation (21). Once the values of w(r, 9) are known, the values
of v(r,9] can be computed using the eikonal equation:
(22)
(23)
9 / /i\sJ(r,0) -it2(?
r
",#)2v(r,8) =
In 3-D the eikonal equation is
V? + V^ + %/ =
where
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Computational Acoustics and Its Environmental Applications 69
For a spherical-coordinates system
9 V^ U? 9> + + o . .0 = * (24)
where
dr
90
The cross derivative equation (20) is transformed into the
spherical coordinates system
du dwOr — #0
dv dw(25)
The finite-difference equivalent of equation (21) is the system
u(r + Ar, 9, 0) = u(r, 6», 0) + |f Au;(r, 9, 0)
(26)
v(r + Ar, 9, <t>) = v(r, 0, 4>] + | Aiw(r, 9, </>)
which is used to advance the stencil for a new radial incre-
ment. Once the values of the functions u(r, 9, </>) and v(r, 0, < >)
are known on the spherical front with constant radius (r +
Ar), the third function w(r,0,4>) can be calculated using the
eikonal equation
w(r + Ar, 0, </>) = sqrt{s\r + Ar, 6, 0) -
(r + Ar] (r + Ar) siif
(27)
-}•
The value of the traveltime is found by integration:
In equation (26), the Engquist-Osher scheme [2] is applied
twice, once for calculating the values of Aw across three points
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70 Computational Acoustics and Its Environmental Applications
of consecutive values of 0, and second for calculating the val-
ues of Aw across three consecutive values of <p. The compu-
tational front advances in spherical shells, and on each shell
the computations advance a circle at a time. The angle 0 is
the horizontal angle while the angle </> is the vertical angle.
The Engquist-Osher scheme is applied along each three con-
secutive points on the circle with constant vertical angle (/> to
determine Aw from the equation A% = Atu. For each cir-
cle of constant vertical angle 0 the Engquist-Osher scheme is
applied for three points (<f> — A</>), </> and (</> + Ac/>), which are
perpendicular on the circle in the (r, #,(/>) coordinates. The
scheme is completely vectorizable.
5 Optimization
5.1 Nelder-Mead
The Nelder-Mead algorithm is an implementation of a so-
called polytope or simplex method. We follow the discussion
in [3] to give an introduction to it.
At each stage of the algorithm, n + I points Xi,...,Xn+i,
and their corresponding function values are retained. It is
assumed that the function values are in ascending order:
/n+l > fn > .- > /I-
These points can be considered the vertices of a polytope in
n dimensions. At each iteration, a new polytope is generated
by producing a new point that replaces the "worst" point
Let c denote the centroid of the first n points:
1 Ac = -E
At the begining of each iteration, a trial point is generated
by a single reflection step:
Xj. = c + a(c — Xn+i), with a > 0.
There are three possible cases to consider:
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Computational Acoustics and Its Environmental Applications 71
+ If/i< fr < /n, then Xr replaces XH+I.
• If fr < fi we assume that the reflection direction is "good"
and try to expand the polytope in this direction by defin-
ng:
with /3 > 1. If /g < /,,, then x« replaces XH+I. Otherwise,
Xr replaces
• If /,. > fa then the polytope is assumed to be too large
and a contraction step is carried out:
Xr =c + T/(xn+i - c), z/ /, >
with 0 < 7 < 1. If /, < 772m(/r,A+i) then x^ replaces
, otherwise a further contraction is carried out.
A number of modifications can be made to this basic pro-
cedure in order to improve its performance and increase its
robustness.
5.2 PRAXIS
PRAXIS [1] is an implementation of a modified version of
Powell's [17] method for minimization of /(x) without using
derivatives. The basic idea of Powell's method is as follows.
Let XQ be an initial approximation to the minimum, and
let {ui}i=i,_^ be the columns of the identity matrix. One
iteration of the basic procedure consists of the following steps:
+ For i = 1, ..., n, calculate fa that minimizes /(x; + A-UI),
and define X; = x;_i + / u;.
• For i = 1, ..., n, replace u; by Ui+i.
• Replace UH by x^ - XQ.
• Compute (3 that minimizes /(XQ + /3u,J and replace XQ
by XQ + 0Un.
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72 Computational Acoustics and Its Environmental Applications
These steps are repeated until a stopping criteriurri is sat-
isfied.
If / is quadratic, then it can be shown by induction that
the vectors iin-k+i> , Un are conjugate, and after n steps we
would reach the minimum, provided that no u; vanishes. This
will be true if at each iteration (3\ 0.
A number of modifications and safeguards have been intro-
duced by Brent into this basic procedure as explained in the
reference mentioned at the begining.
6 Testing
In order to test the algorithms we generate an INTEGRA
model by the name of cracks. It consists of a layer over a half
space with a velocity given by a gradient in z plus a 2D lateral
correction in the form of a 31 x 31 tensor product B-spline.
Four vertical wells are located at [4, 5], [5,4], [6, 5], and [5,6],
surrounding a crack located at x = 5, y = 5, z = 5. Eleven
receivers are placed in each well at 1.0 intervals, starting at
depth 0.5.
Rays are traced from the crack to each receiver and the
travel and differential times with respect to the first receiver
are calculated. This is the synthetic data. For this problem,
this step (which will not be needed for real data) took only
27" on a SUN 10 workstation.
We also run the eikonal solver to generate travel time ta-
bles for shots placed at the receivers (reciprocity principle) to
a box containing the crack; from them we generate the cor-
responding arrival time differences. The mesh has origin at
[3.5,3.5,0.0], and [61,51,133] mesh points in the [x,y,z] di-
rections, covering the box [3.5,6.5] x [3.5,6.0] x [0.0,11] with
grid spacings 0.05,0.05,0.0833333.
This is the most time consuming step, taking 72' on a SUN
10 workstation. Of course, we have to put the task in per-
spective. We have generated 44 travel time tables on a mesh
with 413,763 points, so the calculation has proceeded at a
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Computational Acoustics and Its Environmental Applications 73
rate of 4,214 travel times per second. Also this step is easily
parallelizable on a distributed network of computers.
In order to avoid flat spots in the goal functional we use
linear interpolation between the eight mesh points nearer to a
desired target point (z, ?/, z). This is the function provided to
the Nelder-Mead and PRAXIS algorithms for minimization.
#Wells
1
2
3
4
#iter
N-M
172*
254
435
232
#iter
Prax.
102
59
78
70
4
4
4
4
X
.924
.912
.865
.895
5
4
4
5
y
.000
.964
.939
.046
5
5
5
5
z
027
022
005
009
ei
0.
0.
0.
0.
:ror
069
097
148
115
Despite the disparity in the number of iterations, both algo-
rithms take about the same time, and come up with the same
solution (most of the time), so it is hard to choose among
them. Nelder-Mead is a bit slower in some of the cases. It is
also not clear that increasing the number of sensors buy us
much in terms of accuracy.
Fortunately, for one well, Nelder-Mead gave a completely
different result: [4.650, 5.600, 5.029], which makes it unreli-
able. Still, it can be used (most of the time) to check the
results of PRAXIS by doubling the cost of the calculation.
By the way, the longest computing time for the minimization
was for four wells, and it amounted to five minutes on a slow
SUN 10 workstation. For one well, it takes only one minute.
In the case that we record during a time period where the
crack is breaking, it would be possible to apply this algorithm
with continuation. That is, once we locate the first signal in
space, we can then use that value to start the next calcula-
tion. That will reduce the computing time radically and it
will provide a mechanism to map the complete crack event,
including the length and orientation of the crack.
Key Words: Hydrofracture location; eikonal solvers; ray
tracing
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74 Computational Acoustics and Its Environmental Applications
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Transactions on the Built Environment vol 25, © 1997 WIT Press, www.witpress.com, ISSN 1743-3509