MATHEMATICAL

COMPUTRR MODELLING

PERGAMON Mathematical and Computer Modelling 29 (1999) 95-99

A Stochastic Model for Drilling Optimization

YANG LIU+ Qinhuangdac Division, Daqing Petroleum Institute

Qinhuangdao City 066004, P.R. China

G. CHEN+ Department of Electrical and Computer Engineering

University of Houston, Houston, TX 77204-4793, U.S.A.

(Received September 1998; revised and accepted November 1998)

Abstract-Drilling optimization problems in oilfields are usually formulated and solved by using

deterministic mathematical models, in which uncertain (indeterminate) factors or random issues are

not taken into consideration. However, it has been widely experienced that random factors (such

as those from soil layers, drill bits, and surface equipment) greatly affect the drilling performance.

This paper introduces a new stochastic model for describing such random effects. This model, when

used to optimization design, is more practical and provides a better characterization for real oilfield

situations as compared with other deterministic models, and has been demonstrated to be more

efficient in solving real design problems of drilling optimizations. @ 1999 Elsevier Science Ltd. All

rights reserved.

Keywords-Computer modeling, Stochastic modeling, Stochastic optimization, Oil drilling.

1. INTRODUCTION

This paper addresses a practical design problem of oilfield drilling optimization. Traditionally,

drilling optimization problems are formulated and solved by using deterministic mathematical

models, in which uncertain (indeterminate) factors or random issues are not taken into account

[l-3]. However, it has been widely experienced that random factors (such as those from the

soil layers, drill bits, and surface equipment) greatly affect the drilling performance, thereby

leading a deterministic model to fail to achieve realistic optimality. In this paper, we introduce

a new stochastic model for describing such random effects. This optimization design model is

more practical in the sense that it provides a better description for a real oilfield environment

as compared with other deterministic models. This model has been demonstrated to be more

efficient in solving real design problems of drilling optimizations.

2. A STOCHASTIC MODEL FOR DRILLING

The variables that can affect penetration during drilling include the rock-drill ability, drilling

pressure, rotary speed, mud flux, jet diameter on the drill bit, mud properties, jet arrangement,

etc. It is a complex, constrained, and higher-dimensional dynamic problem that demands design

optimization.

tThis research was supported by the Daqing Oilfield Administration Bureau.

tThis research was supported by the Energy Lab at the University of Houston.

0895-7177/99/$ - see front matter. @ 1999 Elsevier Science Ltd. All rights reserved.

PII: SO895-7177(99)00084-9

Typeset by A,++!-‘QX

96 Y. Lru AND G. CHEN

To describe this problem more precisely, let the parameters vector (to be designed) be X, which contains the aforementioned variables:

X=(2ir...&JT. (1)

In order to optimize the drilling performance, we first need to build a good model that describes

the laws of effects of the variables throughout the drilling process. This model consists of three basic parts, which are discussed separately in the following three sections.

2.1. Stochastic Drilling Rate Model

A mathematical model for drilling rate can be established as, for example [3],

R(X) =exp (K+g xi&), (2)

where R(X) denotes the drilling rate and X = (xi,. . . ,x,) T is the design parameters vector, with K and Bi being the uncertain factors, i = 1,. . . , n. Traditionally, the uncertain parameters

are determined by the simple least-squares curve fitting technique regardless of their physical meaning and influence to the results of the model.

We have found, through statistical analysis on real data, that K and Bi (i = 1,. . . , n) are not constants but some mutually independent normal random variables. Thus, a better drilling rate model is proposed to be the following stochastic one:

E(X) =exp (k+$xiEi), (3)

where - denotes the randomness of the variable. With this model, g(X) is a log-normally dis- tributed random variable with mean PR and variance ui (or deviation UR):

(4)

&X> =exP(%~ +4(X>) (exp(&(X)) -l),

PY(X> = PK + f:xillEk i=l

(5)

(6)

i=l

2.2. Stochastic Drill-Bit Life Model

We have observed that the drill-bit life time, T^, is also a random variable [4]. This random

variable is described by G?(X) = G(X) (8)

for a function t(X) and a drill-bit wearing factor, g, which is another log-normally distributed random variable with mean PT and variance a$, satisfying

k@(X) = CLb r(X), (9)

aT(X) = cb t(X). (10)

2.3. Stochastic Footage Model

The footage function in a drilling model describes how much a single drill-bit can drill and is given by

F(X) = Z(X) F(X), (11)

so is also a log-normal random variable.

Stochastic Model 97

3. STOCHASTIC MODEL FOR OPTIMAL DESIGN

In general, a drilling optimization problem can be formulated as a nonlinear programming prob-

lem that takes the drilling cost into account as the objective function subject to some constraints.

In recent years, the drilling rate has become another main concern in this design [4,5].

The consideration of the drilling rate in a design leads to the following criterion:

mp P (g(X) 2 &) , (12)

where RL denotes the lower bound for the drilling rate R(X), and P is the probability function.

It can be verified, based on the stochastic model given in the last section, that the density

function f~ of E is given by

It follows that

Let

Then

where

P (f?(X) 2 RL) = lrn fR(r) dr. RI.

s = ln (4 - P

UY

P (k(X) 2 RI,) = cx, -QY& exp (-f) dS := Q(-P(X)),

Since Q(.) is a monotone decreasing function, the above optimization is equivalent to

Similarly, the constraints on T^(X) and p(X), namely,

PT (p(x) 2 TL) 2 p;,

PF (p(x) 2 FL) > pi,

are transformed to the following equivalent conditions:

pT(x) 2 Q-l (p;) , PFW) 2 Q-l (Pi),

(13)

04)

(15)

(16)

(17) (18)

where

pT(x) = /tC - lncTL) C’T ’

pFcxj = PF - In cFL)

gF ’

98 Y. LIU AND G. CHEN

Fr=E(ln(T^(X))),

PF = E (In (@(X))) ,

,,=/Var(ln(F(X))),

gF = /Var (In (F(X))).

To this end, the optimal drilling parameters design problem can be formulated as

where Gi(X), i = 1, . . . , m, are some other drilling constraint functions, if exist, which are to be

specified in a design. It is clear from this optimization formulation that both the objective and constraints are

probabilistic, showing the maximum possibility of realization of optimal drilling variables. This is significantly different from the traditional drilling optimization problems.

Here, it should be noted that this model is only for a single bit drilling process. The whole well

section’s optimal drilling design is completed via a multistage decision-making procedure based

on all such single drill-bit optimization results [4].

4. A DESIGN EXAMPLE WITH COMPARISON

A specific sequential quadratic programming algorithm [6] has been developed and used to solve the above single-bit drilling optimization model. The numerical results indicate that drilling performance is greatly improved as compared to the traditional design via a deterministic opti-

mization. A typical example is given here for comparison. The characteristic values of the random

factors in different soil layers for the commonly used PDC drilling bits are listed in Table 1. The developed sequential quadratic programming algorithm was run on a Pentium PC computer and the optimal solution was obtained with the results listen in Table 2.

In these two tables, the layer sections (NEN2, etc.) are some test oilfields in China, where this stochastic drilling optimization model was used to complete a design for guiding the drilling activities. As a result of the comparison on 300 drills of a one-day oilfield testing, the average drilling rate was 12 percent higher than that designed by using the traditional deterministic model [7].

In Table 2, H (m) is the depth, & (l/s) is the flux, P (Mpa) is the pump pressure, hrd (kw) is the drilling bit hydraulic power, p (g/cm3) is the mud weight, W (kn) is the impact force on the bit, and N (r/min) is the drill-bit rotary speed.

5. CONCLUSIONS

A new stochastic model is developed in this paper, taking into account random effects during the drilling process such as those from the soil layers, drill bits, and surface equipment. Drilling

Stochastic Model 99

Table 1. Characteristic values of random factors in different soil layers.

Layers Depth

SEC. (m)

NENs

NENi

YAO

QIzs

&II

QU4

QU3

1200

1400

1500

1700

1800

1900

2200

PK PBl

(“k) ffB1

23.24 0.0031

(0.66) (0.001)

5.84 0.0035

(0.18) (0.0006)

9.06 0.0053

(0.28) (0.005)

22.48 0.0066

(0.72) (0.0012)

2.73 0.0065

(0.06) (0.004)

4.37 0.0123

(0.09) (0.002)

1.64 0.0148

(0.038) (0.003)

PB2

OBl

0.0036

(0.0065)

0.0034

(0.0010)

0.0046

(0.007)

0.0034

(0.0004)

0.0064

(0.0002)

0.0051

(0.0005)

0.0037

(0.0003)

PB3 PB‘l

OBl WBl

0.0013 0.0012

(0.0003) (0.0006)

0.0012 0.0001

(0.0002) (0.0001)

0.0012 0.0007

(0.0001) (0.0001)

0.0009 0.0012

(0.0003) (0.0005)

0.0004 0.0001

(0.0003) (0.0005)

0.0004 0.0007

(0.0002) (0.0001)

0.0003 0.0003

(0.0001) (0.0005)

Table 2. Optimal values for drilling with PDC bits.

Q p Nd P w 32 14.0 341.2 1.20 62

31 14.0 326.2 1.21 72

31 14.0 314.0 1.23 77

30 14.1 305.3 1.24 86

30 14.1 295.8 1.24 79

30 14.2 289.8 1.24 98

30 14.3 250.2 1.24 100

optimization based on this stochastic model leads to significant improvements against the tradi-

tional design based on deterministic models. A typical field test comparison, performed in some

China oilfields, has shown that this new model can speed up the average drilling rate to as high

as 12 percent. It is believed that the new stochastic model has a great application potential in

the oil industry.

REFERENCES

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