Location and mapping of hydrofractures from Victor …...Location and mapping of hydrofractures from...

$: Location and mapping of hydrofractures from Victor …...Location and mapping of hydrofractures from arrival times at wells Victor Pereyra and Mihai Popovici Inc., Mt.View, CA victor@ca.$
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.

Transactions on the Built Environment vol 25, © 1997 WIT Press, www.witpress.com, ISSN 1743-3509

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


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



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



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-



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



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)



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.



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)



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



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



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:



+ 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.



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



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



References

[1] Brent, R.P., Algorithms for finding zeros and extrema

•of functions without calculating derivatives, Dissertation,

Stanford University, 1971.

[2] Engquist, B., arid Osher, S., Stable and entropy satisfy-

ing approximations for transonic flow calculations, Math.

Comp., 1980, 34:45-75.

[3] Gill, P.E., Murray, W. and Wright, M.H., "PmchW Op-

timization". Academic Press, San Francisco, 1981.

[4] Hill, ylpp/. 3W ., 1978, 27, 380-382.

[5] Koshy, M., Pereyra, V., and Meza, J.C., Distributed com-

puting applications in forward and inverse geophysical

modeling, SEG Gist Annual Meeting, Houston, TX. Ex-

fcWcd AWracfs, 349-352, 1991.

[6] Koshy, M., Meza J., and Pereyra, V., Asynchronous

global optimization techniques for medium and large in-

version problems, SEG 65 Annual Meeting, Houston. Ex-

ZcWed j4Wmck, 1091-1094, 1995.

[7] Lentini, M., and Pereyra, V., PASVA4: An ordinary

boundary solver for problems with discontinuous inter-

faces and algebraic parameters, Matematica Aplicada e

Comp%(acmW, 1983, 2, 103-118.

[8] Nelcler, J.A., and Mead, R., A simplex method for func-

tion minimization, Computer Journal, 1965, 7, 308-313.

[9] Pereyra, V.,PASVA3: An adaptive finite difference FOR-

TRAN program for first order nonlinear, ordinary bound-

ary problems, in Lect. Notes Comp. Sc., 76, 67-88, (eds.

Pereyra, V. and Reynoza, A.), Springer-Verlag, 1979.

[10] Pereyra, V., Two-point ray tracing in heterogeneous me-

dia and the inversion of travel time data, in Computa-

tional Methods in Applied Science and Engineering (eds.



Glowinski R, and Lions, J.L.), 553-570, North-Holland,

Amsterdam, 1980.

[11] Pereyra V., Keller, H.B., and Lee, W.H.K., Computa-

tional methods for inverse problems in geophysics, inver-

sion of travel time observations, Phys. Earth Plan. Int.,

1980,21,120-125.

[12] Pereyra, V., Modeling with ray tracing in two-

dimensional curved homogeneous layered media, in

Micro-Computers in Large Scale Scientific Computation,

39-67, (ed. Wouk, A.), SIAM Pub., Philadelphia, 1987.

[13] Pereyra, V., Numerical methods for inverse problems in

three-dimensional geophysical modeling, Applied Numer-

%m/ M&f/tema^ca, 1988, 4, 97-139.

[14] Pereyra, V., and Wright, S.J., Three-dimensional inver-

sion of travel time data for structurally complex geol-

ogy, in Proceedings of Workshop on Geophysical Inver-

Mon, 137-157, (eds. Bednar, J. Bee, et al), SIAM Pub.,

Philadelphia, 1991.

[15] Pereyra, V., Two point ray tracing in general 3D media,

GeopAygzm/ ProspecZmg, 1992, 40, 267-287.

[16] Pereyra, V., Modeling, ray tracing, and block nonlinear

travel time inversion in 3D, to appear in Pure and Applied

Geophysics, 1995.

[17] Powell, M.J.D., An efficient method for finding the mini-

mum of a function of several variables without calculating

derivatives, Comp. J., 1964, 7: 155-162.

[18] VanTrier, J. and Symes, W., Upwind finite-difference cal-

culation of travel times, Geophysics, 1991, 56:812-821.


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