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Uncertainty quantification for reservoir geomechanics
Accepted Manuscript Uncertainty quantification for reservoir geomechanics Leonardo Cabral Pereira, Marcelo S´anchez, Leonardo Jos´e do Nascimento Guimar˜aes PII: DOI: Reference:
S2352-3808(16)30090-9 http://dx.doi.org/10.1016/j.gete.2016.11.001 GETE 51
To appear in:
Geomechanics for Energy and the Environment
Received date: 1 November 2015 Revised date: 4 November 2016 Accepted date: 6 November 2016 Please cite this article as: Pereira LC, S´anchez M, Guimar˜aes. Uncertainty quantification for reservoir geomechanics. Geomechanics for Energy and the Environment (2016), http://dx.doi.org/10.1016/j.gete.2016.11.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Highlights
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HIGHLIGHTS
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Reservoir geomechanics incorporating uncertainty quantification
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Non-probabilistic methods, evidence theory for handling epistemic uncertainties.
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Analyses via analytical solutions of injection pressure for fracture propagation and
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reservoir compaction
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Comparison of deterministic, probabilistic and non-probabilistic methods
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Effect of adding increasingly information on uncertainty reduction
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*Manuscript (pages & lines numbered) Click here to view linked References
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SUBMITTED TO:
Geomechanics for Energy and the Environment
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DATE:
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TITLE:
4th November 2016
Uncertainty Quantification for Reservoir Geomechanics
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AUTHORS: Leonardo Cabral Pereira1, Marcelo Sánchez*2, Leonardo José do Nascimento Guimarães3 AFFILIATIONS:
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Dr Marcelo Sánchez Associate Professor Zachry Department of Civil Engineering Texas A&M University College Station Texas 77843-3136, USA Telephone: (+1) 979 862 6604 Fax: (+1) 979 862 7696 E-mail: [email protected]
Petrobras - Petróleo Brasileiro S.A, Brazil Zachry Department of Civil Engineering, Texas A&M University, College Station, US 3 Federal University of Pernambuco, Recife, Brazil 2
*CORRESPONDING AUTHOR:
Keywords: reservoir geomechanics, epistemic uncertainty, analytical solution, non-probabilistic methods, uncertainty quantification.
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ABSTRACT
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Geomechanics plays a critical role in oil and gas reservoir problems. One of the challenges related
33
to reservoir geomechanics is that information about sediments and rock properties is generally
34
very limited. Furthermore, these geomaterials are typically highly heterogeneous. Therefore, a
35
formal framework for dealing with the uncertainties associated with this type of problem is much
36
required. Uncertainty quantification can be performed either via probabilistic or non-probabilistic
37
methods. The latter approach is generally better suited for handling epistemic (or reducible)
38
uncertainties, which are the more usual in geomechanical reservoir problems. A non-probabilistic
39
framework based on the evidence theory was adopted in this work for dealing with the
40
uncertainties related to the material properties. The contribution focuses on how additional
41
information can be incorporated in a consist fashion in the analysis to reduce the uncertainties.
42
Two analytic solutions associated with typical reservoir geomechanics problems, such as the
43
estimation of the fracture pressure and subsidence, were adopted as the case studies. These
44
application cases shown that the evidence theory was able to reduce the uncertainties associated
45
with these problems when additional information about material properties became available.
46
Another advantage of the adopted framework is that experts’ opinion related (for example) to the
47
reliability of the experimental data gathered from different sources could be explicitly incorporated
48
in the study. It is shown that this type of approach can be instrumental in assisting the decision-
49
making process.
50
2
51
1.
52
Reservoir geomechanics encompasses aspects related to rock mechanics, structural geology and
53
petroleum engineering. Some typical geomechanical problems associated with petroleum and gas
54
production are, amongst others: pore pressure prediction, well design, reservoir compaction, CO2
55
(or H20) injection to enhance oil production, reactivation of geological fault, and hydraulic
56
fracturing. A common characteristic to these problems is that information about sediments and
57
rocks properties is generally scarce. These are also highly heterogeneous geological materials, so
58
significant variations of their geomechanical properties are anticipated. The combination of these
59
factors implies that huge uncertainties are generally associated with problems involving reservoir
60
geomechanics. Therefore, it is essential to develop proper frameworks for handling the
61
uncertainties associated with this kind of analysis.
62
Uncertainty can be defined as the lack of exact knowledge, regardless the cause of this ignorance
63
[1]. Each decision (or set of decisions) is associated with several factors and thus it is highly
64
affected by the uncertainties involved in a particular problem [2]. Uncertainties are typically
65
present in all the stages of a reservoir geomechanics project, from the adoption of simplifying
66
assumptions, to the selection of the constitutive models to be used in the analysis, and the adoption
67
of the input parameters. The need to identify the different sources and types of uncertainties has
68
become a subject of increasing interest in recent years. Different classifications of uncertainties
69
can be found in the literature (e.g., [3]; [4]; [5]). The definitions presented below have been adopted
70
in this work.
71
The dual nature of uncertainties can be described as [6]:
72
INTRODUCTION
i) aleatory uncertainty, associated with random behavior of the system, and
3
73
ii) epistemic uncertainty, associated with the lack of knowledge about a system and/or its
74
properties.
75
Uncertainty type i) above is also known as: stochastic, or type A, or irreducible uncertainty; while
76
uncertainty type ii) is also called subjective, or type B, or reducible uncertainty. In reservoir
77
geomechanics the uncertainties are generally associated with the lack, or limited, information
78
available in terms of material properties and initial stresses. Therefore, it can be considered that
79
these uncertainties are manly epistemic in nature.
80
Both probabilistic and non-probabilistic methods can be used to quantify uncertainties. Non-
81
probabilistic approaches are more appropriate when dealing with epistemic uncertainties (e.g. [6]).
82
A methodology based on the non-probabilistic evidence theory (e.g. [6], [7]) was adopted in this
83
work to handle the uncertainties related to reservoir geomechanics problems. An aspect of interest
84
in this kind of problem is how to reduce the uncertainties associated to problem under study by
85
incorporating any additional information that may become available during the analysis. To study
86
this aspect two relatively simple analytical solutions associated with the problems of fracture
87
propagation pressure and reservoir compaction were selected as application cases. These equations
88
were solved deterministically as well as using classical probabilistic and non-probabilistic
89
methods.
90
The paper is organized as follows. First, the main aspects related to the analytical solutions adopted
91
in this work are discussed. Second, the basic components of the evidence theory adopted for the
92
uncertainty quantification analyses are introduced. Third, the main components and outputs of the
93
application cases are discussed. Finally, the main conclusions of this work are presented.
94 95
4
96
2
97
This section presents the main components and equations associated with the two application cases
98
analyzed in this paper: fracture propagation pressure, and reservoir compaction.
99
2.1.
ANALYTICAL SOLUTIONS
Injection above the fracture propagation pressure
100
Water injection is one of the most common methods to maintain and/or enhance the production of
101
petroleum reservoirs. This technique is very effective to assist oil extraction once the reservoir
102
production by primary mechanisms is exhausted. It is also a convenient production strategy in the
103
early stages of the field exploitation to enhance oil production. During the water injection process,
104
the injectivity can decline because of specific rock and fluids features, geometry of the injectors
105
and producers wells, precipitation of salts, or by the presence of solid particles in the injection
106
water. One of the best ways to avoid injectivity loss is to inject above the fracture propagation
107
pressure [8].
108
A semi-analytical model to predict fracture propagation during water flooding processes was first
109
proposed by Hagoort et al. [9]. However, the thermally induced stresses (one of the most relevant
110
factors) was not considered in this model. Consequently, the transfer of heat between the injected
111
fluid and the formation was not discussed. Afterwards, a three model regions considering the
112
behavior of water flow together with the fracture mechanics was developed by [10]. This model
113
has been widely cited and applied in problems related to reservoir geomechanics (e.g. [11]; [12];
114
[13]). The formulations proposed by de Souza et al. [8] and Perkins and Gonzalez [10] to calculate
115
the reservoir fracture pressure is presented as follows.
116
2.1.1
Fracture pressure calculation
5
117
The fracture pressure in vertical wells for the case of a predominant normal fault regime
118
considering the effect of the penetrating fluid, and thermally induced stress changes, can be
119
calculated through (e.g. [14]; [15]):
120
Pfrat
h TPfrat max (3 h H eTa Pe T ) TPfrat (1 eTa )
(1)
121
where h is the minimum horizontal stress; H is the maximum horizontal stress; eTa is an elastic
122
parameter calculated as (1 2 (1 )) , where α and υ are the Biot and Poisson coefficients,
123
respectively; Pe is the reservoir initial pressure; is the pore pressure factor in tensile failure
124
criterion; T is the tensile strength of the reservoir rock, and TPfrat is the thermally induced
125
stress increment because of the temperature difference between the injected and formation fluids,
126
which can be calculated as follows:
127
TPfrat
(Tr Ti ) E ff 1
(2)
128
where β is the thermal expansion coefficient; E is the Young modulus; Tr is the reservoir
129
temperature, Ti is he injected fluid temperature; and ff is the elastic term form factor, which
130
typically ranges between 0.5 and 1, depending on the temperature front [10].
131
The total vertical stress can be calculated as:
132
v w LA r Sot (3)
6
133
where w is the water unit weight; r is the lithostatic (total) unit weight; LA is the water depth;
134
and Sot is the overburden thickness. The minimum total horizontal stress can be estimated by the
135
following expression:
136
h K0 ( v Pe ) Pe
(4)
137
where K0 correlates the minimum horizontal stress with the vertical stress, which is also known as
138
trust coefficient (also known as coefficient of earth pressure at rest). Note that K0 is used (strictly)
139
to relate the horizontal effective stress with the vertical effective stress. In this work K0 was also
140
used with the stresses definitions adopted in equations (1) and (4), which are based on Biot’s
141
stresses.
142
2.1.2
143
To illustrate the use and significance of the previous equations a simple application example is
144
presented in this section. Typical parameters and properties for oil reservoirs were adopted in this
145
analysis. They are listed in Table 1. Note that a quite simple approach was adopted in this paper
146
to calculate the stress state and to estimate some model parameters. The technique suggested in
147
this paper could also be coupled with more advanced methods (e.g. [16]) to deal with uncertainties
148
in reservoir geomechanics.
149
The calculated fracture pressure based on these properties and using the equations presented in the
150
previous section is equal to 60.42 MPa. This is a deterministic analysis and then unable to quantify
151
the uncertainties associated with this problem. In Section 4.1, this case is solved using the evidence
152
theory incorporating the quantification of uncertainties in the analysis.
153
Application case
<< Insert Table 1 here >>
7
154
2.2
155
The terms subsidence and compaction have different meanings. Compaction is related to the
156
thickness reduction in a given formation, while subsidence refers to a downward movement of the
157
surface. The phenomenon of subsidence occurs over a much larger area than the compaction one,
158
which is associated with the reservoir thickness. The pore pressure reduction during production
159
leads to important changes in the reservoir and adjacent rocks. Such changes in pore pressure affect
160
directly the compaction and subsidence phenomena, which in turns significantly impact on the
161
fluid flow in the reservoir.
162
Several authors have discussed the relevance of the understanding the phenomena associated with
163
reservoir compaction and subsidence during oil fields exploitation (e.g. [17]; [18]; [19]; [20]). In
164
particular, Ferreira [21] conducted an extensive review on cases of subsidence around the world
165
and compared some analytical models with numerical solutions using a finite difference
166
commercial code. In the following section the equations to estimate the compaction and subsidence
167
of an oil reservoir under depletion are presented.
168
2.2.1
169
The vertical displacement field proposed by Geertsma [17] considers the geometry of the disk-
170
shaped reservoir as illustrated in Figure 1.
Reservoir compaction
Vertical displacement calculation
171 172
<< Insert Figure 1 here >> The vertical displacement is calculated via:
cm hR C ( Z 1) (3 4 )C ( Z 1) 2 2 1/2 2 [1 C ( Z 1) ] [1 C 2 ( Z 1) 2 ]1/2 2CZ (3 4 ) p [1 C 2 ( Z 1) 2 ]3/2
uz (0, z ) 173
8
(5)
174
where (see Figure 1) Z = z/c; C = c/R and ε = -1 when z > c, and ε = +1 when z < c; being c is the
175
distance between the free surface and the point under analysis The elastic constant cm is defined
176
by:
cm
177
(1 2 ) 2G(1 )
(6)
178
where G is the shear modulus.
179
Note that this equation provides the vertical displacement at different positions, therefore it is possible to
180
estimate the subsidence (i.e. the displacement at the seafloor or groundfloor level, depending if the case
181
under study corresponds to an off-shore or an on-shore case, respectively), and the compaction (as the
182
difference in displacements between the top and the bottom of the reservoir). Also in this case a relatively
183
simple model was selected to show how the selected approach can deal with uncertainties in reservoir
184
geomechanics related problems. The proposed approach is independent of the adopted equation and
185
certainly more complex models can be used in future works.
186
2.2.2
187
An example of application using the properties listed in Table 2 is presented in this section to
188
illustrate the use of the Equation 5.
Application example
189
<< Insert Table 2 here >>
190
Figure 2 presents the variation of the vertical displacement with depth for a reservoir located at a
191
depth of 2900 meters. Note that the model predicts that the top of the reservoir will move down
192
while the reservoir base will move up. The reservoir thickness reduction is around 2.7 m (i.e.
193
displacement at the top ~ +1.40 m, minus the displacement at the bottom ~ -1.3 m), while the
194
subsidence is in the order of 20 centimeters.
9
195
This analysis is deterministic and again is not possible to quantify the uncertainties associated with
196
it. The inference of new sampled data for the reservoir compaction problem and their impact on
197
the uncertainty quantification is discussed in Section 4.2.
198
<< Insert Figure 2 here >>
199
3.
200
Uncertainty quantification (UQ) is a key component when analyzing problems with limited data
201
and/or high variability of the available information. In the last few years the development of new
202
and more efficient numerical techniques, the increase in computational power, and the massive
203
use of parallel computing, have facilitated the incorporation of UQ analyses in routinely studies.
204
As mentioned before, there are two main types of uncertainties: aleatory and epistemic
205
uncertainties. The probability theory is commonly adopted to evaluate both types of uncertainties.
206
However, recent studies have shown that probabilistic methods may not be the most convenient
207
ones for analyses involving epistemic uncertainties. Alternative approaches, based for example on
208
evidence theory seem more appropriate for handling reducible uncertainties [6].
209
The prior probability distribution (called ‘prior’ henceforth) of a variable is a basic information
210
required in analysis based on probabilistic methods. The “Principle of insufficient reason” [22] is
211
generally used when this information is not available. This implies that when the density functions
212
associated with the model parameters are not known, uniform distribution functions could be
213
assumed. This assumption is valid for random variables, but may be inaccurate when dealing with
214
epistemic ones.
215
An alternative approach to quantify epistemic uncertainties is based on the evidence theory. This
216
theory is particularly useful in those cases where there is not enough information to quantify the
217
uncertainty with a known density function. In the evidence theory the range of the variables is
UNCERTAINTY QUANTIFICATION
10
218
defined by intervals or sets of possible values. The definition of a range (or a set) of probable
219
values linked with a given variable has three important implications:
220
i) it is possible to incorporate in the analysis experimental information from different
221
sources that can be weighted according to the user confidence on particular test results;
222
ii) the principle of insufficient reason is not enforced. Furthermore there is not a pre-
223
established structure for the distribution of the input variables. Initial probabilities can be
224
arranged in sets, without having pre-established assumptions about the probabilities of
225
individual events;
226
iii) the additivity axiom is not mandatory. Therefore, the complementary sets of
227
probabilities measures do not have to add up to one. When the additivity axiom is
228
considered in analysis based on the evidence theory this framework converges to the
229
traditional probabilistic method [6].
230
By comparing the probability and the evidence theories it is possible to see that the benefit of the
231
latter is that it allows a less restrictive description of the uncertainties. However, a drawback of the
232
evidence theory is that uncertainty analyses are computationally more demanding than the ones
233
based on probabilistic method. The additional computational effort comes mainly from the
234
numerical optimization process needed to calculate the belief and plausibility functions. In the
235
following section some background information related to the evidence theory is discussed, more
236
details can be found elsewhere (e.g. [23], [24], [7]).
237
3.1
238
The two core concepts in evidence theory are: i) the basic probability assignment (BPA) or
239
weights, and ii) the focal elements. The BPA for a given set is the probability that can be assigned
240
to it. This probability cannot be decomposed into additional probabilities for subsets of the original
Belief and plausibility functions
11
241
set. The focal elements are those sets that have nonzero BPAs. This then leads naturally to a clear
242
distinction between an evidence space for a set of possible outcomes and a probability space for a
243
set of possible outcomes. The basic probability assignment is different from the classical definition
244
of probability. It is defined by mapping over an interval in which the basic assignment of the null
245
set is ‘0’ and the summation of basic assignments in a given set is ‘1’ [6].
246
The first work in evidence theory was done by Shafer [25], as an expansion of the one previously
247
performed by Dempster [26]. In a finite discrete space, the evidence theory or the Dempster-Shafer
248
theory (DST), can be interpreted as the generalization of the probability theory for the case in
249
which the probabilities are assigned to sets of values. When the evidence is sufficient to assign
250
probabilities to single events, the Dempster-Shafer model falls into the traditional probabilistic
251
formulation.
252
Let’s consider a model defined by:
253
y f (x)
(7)
254
where x [ x1 , x2 , x3 ,..., xnX ] is the vector of the input variables and y [ y1 , y2 , y3 ,..., ynY ] is the
255
vector associated with the model results. Thus, the uncertainty in y can be estimated in the
256
framework of the probabilistic theory by the Cumulative Distribution Function (CDF).
257
prob( y y) ( f (x) | y)d X dX i S1 ( f (x) | y) / nS n
P
(8)
258
where y is the expected value (i.e. the different outputs associated with each ones of the possible
259
realizations); nS is the number of samples; ( f (x) | y) = 1 if f (x) y and ( f (x) | y) = 0 if
260
f (x) y . Furthermore, the probability theory requires that the uncertainty associated with x
261
should be defined by a density function (e.g. normal, triangular or uniform). To characterize the
12
262
uncertainty in x is necessary to establish a probability space (XP; XP’;mPX) where XP is the sample
263
space of possible values of x; XP’ is the set of subsets of XP chosen appropriately (Sigma-algebra);
264
and mPX is a probability factor.
265
Conversely the evidence theory does not impose a rigid structure for the characterization of the
266
uncertainties. It is a flexible approach. The uncertainty in y is estimated by means of two
267
uncertainty functions, the belief and plausibility functions as follows [6]:
268
269
BelY ( y y)
PlY ( y y)
m
A Ay
A Ay 0
EY
( A)
(9)
mEY ( A)
(10)
270
At an intuitive level, belief is a measure of the amount of information that indicates that a statement
271
is true, and plausibility is a measure of the amount of information that indicates that a statement
272
could be true. Central to this concept is the idea that all the probability distributions that do not
273
violate the assumed properties of the evidence space are under consideration. The ‘ball-box’
274
analogy of the Dempster-Shafer theory can assist to a simple interpretation of this technique. In
275
this model is assumed that n unmarked steel balls are distributed among k boxes A1,. . . , Ak, with
276
mi representing the fraction of balls in Ai. The boxes are placed in a bigger box U and are allowed
277
to overlap. The position of each ball within its box is unspecified. In this model, the granular
278
distribution A describes the distribution of the balls among the boxes. Note that the number of balls
279
in Ai is unrelated to the ones in Aj. It is important to distinguish between the number of balls in Ai
280
and the ones in Aj, because these fractions might overlap (i.e. the boundary of the boxes are
281
penetrable). Now, given a region Q in U, it is possible to ask the following question: How many
282
balls are in Q? Because the information regarding the position of each ball is incomplete, the
13
283
answer to this question will be (in general) interval-valued. The upper bound can readily be found
284
by visualizing Q as an attractor, for example, a magnet. Under this assumption, the proportion of
285
balls drawn into Q is given by Eq. (10), which is the expression for plausibility in the Dempster-
286
Shafer theory. Similarly, the lower bound results from visualizing Q as a repeller. In this case, the
287
lower bound is given by Eq. (9); which is the expression for belief in the Dempster-Shafer theory.
288
The combination of the intervals defined as uncertain values will outline the upper and lower
289
bounds of the response as described in the previous paragraph. The definition of the belief and
290
plausibility functions for the model outputs can be done through an optimization process, as
291
discussed in the following section. More details can be found elsewhere (e.g. [24], [6], [7]).
292
3.2
293
The belief and plausibility functions are defined for the input variables (x). For each input variable
294
it is necessary to specify intervals (and the associated bounds) and the corresponding weights (i.e.
295
BPAs). To calculate the belief and plausibility measures it is necessary to calculate the minimum
296
and maximum of the response function for each interval cell combination. For example, if a case
297
has 2 input variables and each of these is defined by 3 intervals, the number of combination cells
298
is equal to 9. This implies that epistemic analyses require the implementation of an optimization
299
algorithm to find the maximum and minimum for all the possible combinations considered. More
300
details about the implementation of typical optimization methods to perform Dempster-Shafer
301
analysis can be found elsewhere (e.g. [27], [28]; 23]; [7]). The intervals and their associated BPAs
302
are then propagated to obtain cumulative distribution belief and plausibility functions (i.e. CBF
303
and CPF, respectively). As mentioned above, belief is the lower bound on a probability estimate
304
that is consistent with the evidence, and plausibility corresponds to the upper bound on a
305
probability estimate that is consistent with the evidence.
Optimization algorithm
14
306
In this work the approach discussed in Cabral et al. [7] to compute the CBF and CPF is adopted,
307
which consists of the main following steps:
308
1) Combine intervals of each input variable into combination cells.
309
2) For each cell, two optimization problems have to be solved to find the lower (lb) and
310
upper (ub) bounds of the response.
311
3) Sort the lower bounds of the output response intervals obtained in Step 2 in ascending
312
order.
313
More details about this optimization process are presented in Cabral et al. [7].
314
3.3
315
To establish the type of analysis to be performed it is important to pay attention not only to the
316
availability of data but also to the origin and reliability of this information. There are two critical
317
aspects related to the combination of evidence obtained from different sources. The first one is
318
related to the type of evidence involved in the study and the second aspect is based on how to
319
handle conflicting evidence. Figure 3 presents four different types of evidence that impact on the
320
choice of how such evidences must be combined. They are called: inclusive evidence, consistent
321
evidence, arbitrary evidence and disjointed evidence [29].
Types of evidence
322
<< Insert Figure 3 here >>
323
The inclusive evidence can be represented by subsets that are contained in a larger subsets, which
324
in turns can be part of a larger one (Figure 3a). For example, this type of evidence can be related
325
to the case where the information for a given set is obtained from a larger period of time by
326
reducing the intervals of evidence. In the case of the consistent evidence, at least one element is
327
common to all subsets of the sample space (Figure 3b). The arbitrary evidence corresponds to the
328
situation where there is no element in common for all the subsets (Figure 3c), although there may 15
329
be some subsets of elements in common. Finally, the disjointed evidence is associated with the
330
case in which for any two subsets of the sample space there are no elements in common. (Figure
331
3d).
332
Each of these possible configurations of evidence from different sources has different implications
333
on the level of conflict associated with each situation. It is intuitive that for the case of disjointed
334
evidence, all sources of evidence are in conflict. For the case of arbitrary evidence, there are some
335
agreements between arbitrary sources, but there is no consensus on any of the elements. The
336
consistent evidence implies an agreement between at least one subset of evidence. Inclusive
337
evidence represents a situation in which every subset is included in a larger one, i.e., it is
338
guaranteed that there is a correlation between the evidences. However, there is inherent conflict in
339
the additional evidence supported by the larger subsets.
340
The traditional probability theory cannot deal with inclusive, consistent or arbitrary evidences,
341
without applying the definition of probability density functions for all the elements of the sets.
342
Furthermore, it is not possible to express the level of conflict between these sets of evidences. The
343
following examples show how the evidence theory is more flexible when dealing with the
344
scenarios discussed above. Furthermore, in the evidence theory, there are several ways to
345
incorporate imminent conflicts in the process of combining multiple sources of information. In the
346
next section, the application of this method is conducted based on the two problems discussed in
347
Section 2.
348 349
4.
350
In reservoir geomechanics there is a tendency to develop increasingly sophisticated constitutive
351
models and coupled formulations. This is a very valuable effort, because the behavior of sediments
DECISIONS MAKING PROCESS UNDER UNCERTAINTY
16
352
and rock involved in this type of problems is generally very complex. However, it seems also very
353
important to assess the impact of the variation of material parameters in the analysis and the
354
associated uncertainties. This section is aimed at studying decision making cases incorporating
355
uncertainty quantification via the evidence theory and adopting simple solutions, as the ones based
356
on analytic or semi-analytical solutions presented in Sections 2.1 and 2.2.
357
Decision-making is certainly the most important task of a manager and it often tends to be rather
358
difficult. The type of approaches used to make the decision ranges between two extreme cases.
359
One of them corresponds to the deterministic solutions (as the ones presented in the previous
360
Sections), where specific (determined) values are assigned to the different parameters and then it
361
is not possible to quantify the uncertainties. The other extreme corresponds to the pure uncertainty
362
analysis, where there is not previous information about the parameters associated with the case
363
under study. Those studies in which the risk can be assessed lie between these two extreme types
364
of analyses [30].
365
During reservoir operation it is possible that additional information related to the material
366
properties become available. It is of great interest to study how this new information may impact
367
on the uncertainty studies. In the following section an analysis under pure uncertainty (i.e. lack of
368
input data) is discussed first. Then, the incorporation of increasingly new input data in the analyses
369
and the corresponding effect in the uncertainty quantification is discussed.
370 371 372
4.1
373
The lack or limited of material data is a recurring problem associated with reservoir geomechanics.
374
One of the main reasons related to it has been the high cost associated with the gathering of the
Lack of input data
17
375
data from the field. Furthermore, the reduction of the geomechanical data gathered via seismic
376
logs or cores is quite complex and require high technical expertise.
377
In this section the calculation of the fracture pressure is used as an example of the decision making
378
process incorporating uncertainty quantification in a case in which the input data is quite scarce.
379
Afterwards, the discussion focuses on the comparisons between the results obtained via probability
380
and evidence theories. In this example, three experts were consulted to define the ranges of
381
parameters to be used in the UQ analysis. These experts defined the weights required for the use
382
of the evidence theory. The ranges of the parameters were defined by the material properties found
383
in analogous reservoirs. The uncertainty parameters are: Young’s modulus, Poisson’s coefficient,
384
Biot’s coefficient and thrust coefficient. The same weight value for each expert opinion was
385
assumed, therefore the BPAs are equal to 1/3. Table 3 presents the experts opinions related to these
386
parameters.
387
<< Insert Table 3 here >>
388
The variables quoted above are considered epistemic and the evidence theory is a proper
389
framework for the treatment of this kind of uncertainty. Figure 4 presents the CBF and CPF of the
390
fracture pressure calculation using the parameters shown in Table 3.
391
One of the main dilemmas associated with the use of the probability theory for uncertainty
392
quantification of epistemic variables is related to the choice of the probability density functions.
393
Figure 3 shows three CDF's associated with the choice of an uniform probability density function
394
(in dark blue), normal (in light blue) and a distribution that considers the BPAs as weights (in
395
green) in the respective intervals.
396
<< Insert Figure 4 here >>
18
397
This figure can be used to analyze the injection pressure that need to be applied for a given
398
reservoir to induce the fracture propagation. These plots can be used in different ways, for example
399
let’s assume that 0.50 is the largest possible probability that an injection project manager would
400
like to take to assure that injection pressure is equal or above the fracture pressure propagation and
401
in this way maintain a good injectivity. Considering first the CDFs (i.e. the ones obtained with the
402
probabilistic method), the needed increment in the injection pressure is around 69.63 MPa if
403
uniform or normal density functions are assumed (note that only for a probability of 0.5 these two
404
solutions predict the same pressure, for other probabilities the injection pressures estimated by
405
these two functions are different), while the other CDF predict a bit lower value of injection
406
pressure, around 68.65 MPa. As for the evidence theory, it is important to remember that Bel(U) is
407
the smallest possible probability that could occur for U over the set of indicated distributions, and
408
Pl(U) is the largest possible probability that could occur for U over the set of indicated
409
distributions. Therefore looking at the CBF and CPF curves, one may conclude that based on the
410
existing evidences it is believed that the fracture will propagate with a confidence of 0.5 if the
411
injection pressure is above 63.74 MPa. However the available evidence also informs that it is
412
plausible that an injection pressure as high as 75.02 MPa may be necessary to induce the
413
propagation of the fracture in this reservoir (also in this case with a confidence of 0.5).
414
It can be seen that the evidence theory provides more complete information to assist the decision
415
maker, with a range of variation of the possible values of injection pressures that will induce the
416
fracture propagation when a give probability is set. This interval of variation of the injection
417
pressure is a clear indicator of the uncertainties associated with the problem under study and with
418
the available data. The example discussed in the next section shows how the uncertainty can be
19
419
reduced when more information about the geomechanical properties of the reservoir is added into
420
the analysis.
421
4.2
422
Table 4 corresponds to a possible scenario for information availability over time for reservoir
423
compaction problem analyzed in Section 2.2. At the beginning, there was no information about
424
the mechanical properties of the reservoir. The parameters used in this first analysis were estimated
425
from the data observed in similar reservoirs. Then, information obtained from the first well drilled
426
became available. This data was obtained from log correlations. Results from laboratory tests were
427
available afterwards. Finally, another well was drilled and therefore more information about
428
reservoir properties was available.
New data inference
429
<< Insert Table 4 here >>
430
A crucial point at this stage of the analysis is to define how this new information is incorporated
431
in the process of uncertainty quantification. It is also very important to have the opportunity of
432
weighting the available information based on the reliability of this data. As discussed previously,
433
the evidence theory is flexible and allows defining probabilities by sets, which is very convenient
434
for handling the information associated with this type of problem. In the evidence theory the expert
435
can decide how to measure the significance of the existing information coming from different
436
sources (i.e. by the definition of the BPAs or weights). The weighting of the available data is based
437
on the reliability of the existing data, experts’ judgment, and previous experience.
438
In Table 4 it can be observed that for the analysis identified as ‘T0’, the BPA assigned to the
439
different properties is equal to 1. This is because there is only one source of information (i.e. the
440
experience from previous studies similar to this one). In the analysis coded as ‘T1’ (i.e. the case in
441
which information from the first well became available), a BPA of 0.8 was assigned to the data 20
442
coming from the new well, and 0.2 was assigned to the T0 data. This implies that according to the
443
expert opinion, the information from this reservoir is more reliable (i.e. higher weight value) than
444
the one coming from previous similar studies. It can also be observed that the reservoir parameters
445
coming from the well vary in a smaller range. Also, according to the experts’ opinion, the
446
information gathered in the laboratory and incorporated in the analysis ‘T2’is more reliable than
447
the one related to the previous studies, but significantly less dependable than the data associated
448
with the borewell. The weights assigned to the previous studies, well, and lab data are: 0.1, 0.7 and
449
0.2; respectively (note that the sum of the BPAs has to be always equal to 1). A similar reasoning
450
is followed in analyses T3, when data coming from another well is incorporated in the analyses.
451
Figure 5 shows the changes in the curves associated with the uncertainty quantification analyses
452
when new information is incorporated at different stages of the analysis. The flexibility of the
453
evidence theory for new data inference is clear in the evaluation of the results. It is not surprising
454
that new information about the material parameters reduces the uncertainty range associated with
455
the estimated subsidence. Initially, when few information was available (i.e. based mainly on
456
previous experiences in similar reservoirs), the range of possible subsidence values was quite wide,
457
i.e. between 0.05 and 0.9 m. After incorporating information from the first well, the uncertainty
458
decreased and the calculated subsidence was between 0.05 to 0.7 meters. Once the information
459
from the laboratory test became available, a similar trend was observed with a further reduction of
460
the uncertainty range. The minimum and maximum possible subsidence in this new study was in
461
the range between 0.05 and 0.52 m, respectively. Finally, after the information from the drilling of
462
the second well was incorporated, the uncertainty reduced even more, and the possible subsidence
463
values were between 0.05 to 0.18 m.
464
<< Insert Figure 5 here >>
21
465
It is worth mentioning that the use of the evidence theory in this case allowed the inference of new
466
data without the requirement of defining priors, as it is necessary in classical probability analyses.
467
It was observed that the curve related to the probability theory considering uniform distribution
468
kept constant during the subsequent analyses, because the range that defines the minimum and
469
maximum values of the parameters were not modified as new information became available, and
470
consequently the uniform distribution did not change (Figure 5).
471
5.
472
This work focused on the uncertainty quantification in problems related to reservoir geomechanics.
473
These types of problems are characterized by the limited (and in some occasions nil) information
474
associated with material properties. To quantify the uncertainties a formal framework based on the
475
evidence theory was adapted to oil reservoir problems. This non-probabilistic method is especially
476
well-suited for dealing with uncertainty that are epistemic in nature. This type of reducible
477
uncertainty is very common in reservoir geomechanics because it is generally associated with the
478
lack of information. The main components of the adopted approach was presented in this work
479
and then it was applied to solve typical problems related to oil reservoirs, such as the estimation
480
of the fracture propagation pressure and reservoir compaction. Relatively simple analytical
481
solutions were adopted for the analyses, because the main effort was on exploring the capability
482
of the framework for dealing with uncertainties.
483
In both analyses (i.e. estimation of fracture pressure and reservoir compaction) it was observed
484
that the evidence theory facilitates the decision making process because it provides more complete
485
information than the probabilistic theory. For example, when defining the fracture propagation
486
pressure, the evidence theory gives based on the available information and for a given probability,
487
a range of variation of possible injection pressures that may induce fracture propagation. This
SUMMARY AND CONCLUSIONS
22
488
interval of variation for the injection pressures is a clear indicator of the uncertainties associated
489
with this problem. It was also shown that the evidence theory is a suitable framework for
490
incorporating consistently additional evidences and reducing consequently the uncertainty when
491
more information about the reservoir becomes available. This feature was evident when analyzing
492
the reservoir compaction problem, where the range of variation of the estimated settlements was
493
reduced in subsequent analyses that incorporated increasingly additional data about the
494
geomechanical properties of the reservoir.
495
The use of the evidence theory based on CBFs and CPFs, guarantee that the information is
496
consistent with all the known data associated with the input parameters of the model. In the limit,
497
these two curves (i.e CBF and CPF) converge to a single cumulative distribution function curve,
498
when the data that completely describes the model is available. It is worth mentioning that the use
499
of evidence theory in this case allowed the inference of new data without the requirement of the
500
definition of priors (as generally happens in probabilistic theories). It was observed that the curve
501
related to the probability theory considering uniform distribution kept constant during the analysis
502
because the range that defines the minimum and maximum values of the parameters were not
503
modified as new information became available (and thus the uniform distribution did not be
504
change). Furthermore, the evidence theory was able to properly incorporate in the analyses the
505
experts’ opinion and to weight the available information based on the reliability of the data source.
506
ACKNOWLEDGEMENTS
507
The authors acknowledge the financial support from PETROBRAS, CNPq (Brazilian National
508
Research Consul) and Foundation CMG (Chair on Reservoir Simulation at UFPE).
23
509
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510
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Rahman K, Khaksar A. Fracture Growth and Injectivity Issues for Produced Water
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Casing Damage. SPE Annual Technical Conference and Exhibition. New Orleans, USA.
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Reservoir Compection And Recovery. 6th North America Rock Mechanics Symposium
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Compaction-Induced Porosity/Permeability Reduction in Sandstone Reservoirs: Data and
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Model for Elasticity-Dominated Deformation. SPE Annual Technical Conference and
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Ferreira F. Subsidence due to Reservoir Compaction: From Theory to Well Integrity Problems. DSc, Federal University of Rio de Janeiro. 2014.
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Savage L. The Foundations of Statistics. Dover Publications. 2012
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Helton J, Johnson J, Oberkampf W. An exploration of alternative approaches to the
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representation of uncertainty in model predictions. Reliability Engineering & System
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Helton J, Oberkampf W, Johnson J. Competing Failure Risk Analysis Using Evidence Theory. Risk Analysis: An International Journal. 2005;25:973-995.
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Shafer G. A Mathematical Theory of Evidence. Princeton University Press. 1976
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Dempster A. Upper and Lower Probabilities Induced by a Multivalued Mapping. The
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Annals of Mathematical Statistics. 1967;38(2):325-339. [27]
Waltz R, Morales J, Nocedal J, Orban D. An interior algorithm for nonlinear optimization
585
that combines line search and trust region steps. Mathematical Programming.
586
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nonlinear programming. Mathematical Programming. 2000;89:149-185. [29]
590 591 592
Byrd R, Gilbert J, Nocedal J. A trust region method based on interior point techniques for
Sentz K, Ferson S. Combination of Evidence in Dempster-Shafer Theory. Sandia National Laboratories. SAND 2002-0835. 2002
[30]
Ben-Haim Y. Info-Gap Decision Theory: Decisions Under Severe Uncertainty. Elsevier Science. 2006.
593
27
Figure
Sea or ground level
Figure 1: Geometry for determining the displacement field in the surroundings of a disc-shaped reservoir (adapted from Geertsma [17]).
Figure 2: Vertical displacement field along the depth. The reservoir is located at 2900 m depth.
.
A E
A
B
B
C
C
E
D
b)
a)
D
A
B
C
A C
E
E
B
D
D c)
d)
Figure 3: Types of evidence: a) inclusive; b) consistent; c) arbitrary and d) disjointed. The letters A, B, C, D and E represent diferent sources of evidence.
Figure 4: Fracture pressure calculation. Uniform probability density function (in dark blue), normal (in light blue) and a distribution that considers the BPA's as weights in the respective ranges (in green). Results obtained with evidence theory: CBF (in red) and CPF (in black).
Figure 5: Incorporation of new data over time. Uniform probability density function (green curves), and a distribution that considers the BPA's as weights (blue curves) in their respective ranges. Via theory of evidence it is estimated the CBF (in red) and the CPF (in black).
Table
Table 1: Input data used for the calculation of the reservoir fracture pressure
Geomechanical data Young modulus (E)
25 GPa
Biot’s coefficient (α)
0.8
Tensile strength (σT)
4.02 MPa
Poisson’s coefficient (υ)
0.25
Thrust coefficient (K0)
0.8
Pore pressure factor in tensile failure criterion (φ)
0.07 Length data
Water depth (LA)
2100 m
Overburden tickness (Sot)
2450 m
Reservoir thickness (hR)
100 m Temperature and pressure data
Reservoir temperature (Tr) Injected water temperature (Ti) Reservoir initial pressure (Pe)
60 oC 35 oC 48.05 MPa
Table 2: Required input data for the estimation of the vertical displacement field in a depleted reservoir
Geomechanical data Shear Modulus (G)
147.1 MPa
Biot’s Coefficient (α) Poisson’s Coefficient (υ)
1.0 0.286 Reservoir properties
Reservoir thickness (hR)
100 m
Top (c)
2900 m
Radius (R) Total depth (z)
1000 m 4000 m
Porepressure variation (depletion) (Δp)
13.1 MPa
Table 3: Ranges of possible values characterizing the uncertainty of the input variables. Each interval was defined by different experts’ opinions.
Parameter
Expert 1
Expert 2
Expert 3
E (GPa) υ
35 - 40 0.2 - 0.3
35-45 0.2 - 0.3
30 - 40 0.2 - 0.3
α
0.6 - 0.8
0.6 - 0,8
0.5 - 0.7
K0
0.7 - 0.9
0.75 - 0.8
0.7 - 0.8
BPAs
0.33
0.33
0.33
Table 4: Information (evidence) over time. The uncertainty quantification process was modified with each new piece of evidence.
T = 0 - Similar reservoirs Parameter Α
T0 0.6 - 1.0
Υ
0.2 - 0.4
G (MPa)
49 - 294
BPA0
1.0 T = 1 - Well drilled
Parameter Α
T0 0.6 - 1.0
T1 0.7 - 0.8
Υ
0.2 - 0.4
0.32 - 0.36
G (MPa)
49 - 294
245.2 - 313.8
BPA1
0.2
0.8 T = 2 - Lab Test
Parameter Α
T0 0.6 - 1.0
T1 0.7 - 0.8
T2 0.95 – 1.0
Υ
0.2 – 0.4
0.32 - 0.36
0.35 – 0.4
G (MPa)
49 - 294
245.2 - 313.8
196.1 - 214.6
BPA2
0.1
0.7
0.2
T = 3 - New Well Drilled Parameter
T0
T1
T2
T3
Α
0.6 - 1.0
0.7 - 0.8
0.95 – 1.0
0.75 – 0.9
Υ
0.2 – 0.4
0.32 - 0.36
0.35 – 0.4
0.3 – 0.35-
G (MPA)
49 - 294
245.2 - 313.8
196.1 - 214.6
294 – 295.2
BPA3
0.02
0.4
0.18
0.4
S2352-3808(16)30090-9 http://dx.doi.org/10.1016/j.gete.2016.11.001 GETE 51
To appear in:
Geomechanics for Energy and the Environment
Received date: 1 November 2015 Revised date: 4 November 2016 Accepted date: 6 November 2016 Please cite this article as: Pereira LC, S´anchez M, Guimar˜aes. Uncertainty quantification for reservoir geomechanics. Geomechanics for Energy and the Environment (2016), http://dx.doi.org/10.1016/j.gete.2016.11.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Highlights
1
HIGHLIGHTS
2 3
Reservoir geomechanics incorporating uncertainty quantification
4
Non-probabilistic methods, evidence theory for handling epistemic uncertainties.
5
Analyses via analytical solutions of injection pressure for fracture propagation and
6
reservoir compaction
7
Comparison of deterministic, probabilistic and non-probabilistic methods
8
Effect of adding increasingly information on uncertainty reduction
1
*Manuscript (pages & lines numbered) Click here to view linked References
1
SUBMITTED TO:
Geomechanics for Energy and the Environment
2 3 4
DATE:
5
TITLE:
4th November 2016
Uncertainty Quantification for Reservoir Geomechanics
6 7 8 9 10 11 12
AUTHORS: Leonardo Cabral Pereira1, Marcelo Sánchez*2, Leonardo José do Nascimento Guimarães3 AFFILIATIONS:
13 14 15 16 17
1
18 19 20 21 22 23 24 25 26 27 28 29 30
Dr Marcelo Sánchez Associate Professor Zachry Department of Civil Engineering Texas A&M University College Station Texas 77843-3136, USA Telephone: (+1) 979 862 6604 Fax: (+1) 979 862 7696 E-mail: [email protected]
Petrobras - Petróleo Brasileiro S.A, Brazil Zachry Department of Civil Engineering, Texas A&M University, College Station, US 3 Federal University of Pernambuco, Recife, Brazil 2
*CORRESPONDING AUTHOR:
Keywords: reservoir geomechanics, epistemic uncertainty, analytical solution, non-probabilistic methods, uncertainty quantification.
1
31
ABSTRACT
32
Geomechanics plays a critical role in oil and gas reservoir problems. One of the challenges related
33
to reservoir geomechanics is that information about sediments and rock properties is generally
34
very limited. Furthermore, these geomaterials are typically highly heterogeneous. Therefore, a
35
formal framework for dealing with the uncertainties associated with this type of problem is much
36
required. Uncertainty quantification can be performed either via probabilistic or non-probabilistic
37
methods. The latter approach is generally better suited for handling epistemic (or reducible)
38
uncertainties, which are the more usual in geomechanical reservoir problems. A non-probabilistic
39
framework based on the evidence theory was adopted in this work for dealing with the
40
uncertainties related to the material properties. The contribution focuses on how additional
41
information can be incorporated in a consist fashion in the analysis to reduce the uncertainties.
42
Two analytic solutions associated with typical reservoir geomechanics problems, such as the
43
estimation of the fracture pressure and subsidence, were adopted as the case studies. These
44
application cases shown that the evidence theory was able to reduce the uncertainties associated
45
with these problems when additional information about material properties became available.
46
Another advantage of the adopted framework is that experts’ opinion related (for example) to the
47
reliability of the experimental data gathered from different sources could be explicitly incorporated
48
in the study. It is shown that this type of approach can be instrumental in assisting the decision-
49
making process.
50
2
51
1.
52
Reservoir geomechanics encompasses aspects related to rock mechanics, structural geology and
53
petroleum engineering. Some typical geomechanical problems associated with petroleum and gas
54
production are, amongst others: pore pressure prediction, well design, reservoir compaction, CO2
55
(or H20) injection to enhance oil production, reactivation of geological fault, and hydraulic
56
fracturing. A common characteristic to these problems is that information about sediments and
57
rocks properties is generally scarce. These are also highly heterogeneous geological materials, so
58
significant variations of their geomechanical properties are anticipated. The combination of these
59
factors implies that huge uncertainties are generally associated with problems involving reservoir
60
geomechanics. Therefore, it is essential to develop proper frameworks for handling the
61
uncertainties associated with this kind of analysis.
62
Uncertainty can be defined as the lack of exact knowledge, regardless the cause of this ignorance
63
[1]. Each decision (or set of decisions) is associated with several factors and thus it is highly
64
affected by the uncertainties involved in a particular problem [2]. Uncertainties are typically
65
present in all the stages of a reservoir geomechanics project, from the adoption of simplifying
66
assumptions, to the selection of the constitutive models to be used in the analysis, and the adoption
67
of the input parameters. The need to identify the different sources and types of uncertainties has
68
become a subject of increasing interest in recent years. Different classifications of uncertainties
69
can be found in the literature (e.g., [3]; [4]; [5]). The definitions presented below have been adopted
70
in this work.
71
The dual nature of uncertainties can be described as [6]:
72
INTRODUCTION
i) aleatory uncertainty, associated with random behavior of the system, and
3
73
ii) epistemic uncertainty, associated with the lack of knowledge about a system and/or its
74
properties.
75
Uncertainty type i) above is also known as: stochastic, or type A, or irreducible uncertainty; while
76
uncertainty type ii) is also called subjective, or type B, or reducible uncertainty. In reservoir
77
geomechanics the uncertainties are generally associated with the lack, or limited, information
78
available in terms of material properties and initial stresses. Therefore, it can be considered that
79
these uncertainties are manly epistemic in nature.
80
Both probabilistic and non-probabilistic methods can be used to quantify uncertainties. Non-
81
probabilistic approaches are more appropriate when dealing with epistemic uncertainties (e.g. [6]).
82
A methodology based on the non-probabilistic evidence theory (e.g. [6], [7]) was adopted in this
83
work to handle the uncertainties related to reservoir geomechanics problems. An aspect of interest
84
in this kind of problem is how to reduce the uncertainties associated to problem under study by
85
incorporating any additional information that may become available during the analysis. To study
86
this aspect two relatively simple analytical solutions associated with the problems of fracture
87
propagation pressure and reservoir compaction were selected as application cases. These equations
88
were solved deterministically as well as using classical probabilistic and non-probabilistic
89
methods.
90
The paper is organized as follows. First, the main aspects related to the analytical solutions adopted
91
in this work are discussed. Second, the basic components of the evidence theory adopted for the
92
uncertainty quantification analyses are introduced. Third, the main components and outputs of the
93
application cases are discussed. Finally, the main conclusions of this work are presented.
94 95
4
96
2
97
This section presents the main components and equations associated with the two application cases
98
analyzed in this paper: fracture propagation pressure, and reservoir compaction.
99
2.1.
ANALYTICAL SOLUTIONS
Injection above the fracture propagation pressure
100
Water injection is one of the most common methods to maintain and/or enhance the production of
101
petroleum reservoirs. This technique is very effective to assist oil extraction once the reservoir
102
production by primary mechanisms is exhausted. It is also a convenient production strategy in the
103
early stages of the field exploitation to enhance oil production. During the water injection process,
104
the injectivity can decline because of specific rock and fluids features, geometry of the injectors
105
and producers wells, precipitation of salts, or by the presence of solid particles in the injection
106
water. One of the best ways to avoid injectivity loss is to inject above the fracture propagation
107
pressure [8].
108
A semi-analytical model to predict fracture propagation during water flooding processes was first
109
proposed by Hagoort et al. [9]. However, the thermally induced stresses (one of the most relevant
110
factors) was not considered in this model. Consequently, the transfer of heat between the injected
111
fluid and the formation was not discussed. Afterwards, a three model regions considering the
112
behavior of water flow together with the fracture mechanics was developed by [10]. This model
113
has been widely cited and applied in problems related to reservoir geomechanics (e.g. [11]; [12];
114
[13]). The formulations proposed by de Souza et al. [8] and Perkins and Gonzalez [10] to calculate
115
the reservoir fracture pressure is presented as follows.
116
2.1.1
Fracture pressure calculation
5
117
The fracture pressure in vertical wells for the case of a predominant normal fault regime
118
considering the effect of the penetrating fluid, and thermally induced stress changes, can be
119
calculated through (e.g. [14]; [15]):
120
Pfrat
h TPfrat max (3 h H eTa Pe T ) TPfrat (1 eTa )
(1)
121
where h is the minimum horizontal stress; H is the maximum horizontal stress; eTa is an elastic
122
parameter calculated as (1 2 (1 )) , where α and υ are the Biot and Poisson coefficients,
123
respectively; Pe is the reservoir initial pressure; is the pore pressure factor in tensile failure
124
criterion; T is the tensile strength of the reservoir rock, and TPfrat is the thermally induced
125
stress increment because of the temperature difference between the injected and formation fluids,
126
which can be calculated as follows:
127
TPfrat
(Tr Ti ) E ff 1
(2)
128
where β is the thermal expansion coefficient; E is the Young modulus; Tr is the reservoir
129
temperature, Ti is he injected fluid temperature; and ff is the elastic term form factor, which
130
typically ranges between 0.5 and 1, depending on the temperature front [10].
131
The total vertical stress can be calculated as:
132
v w LA r Sot (3)
6
133
where w is the water unit weight; r is the lithostatic (total) unit weight; LA is the water depth;
134
and Sot is the overburden thickness. The minimum total horizontal stress can be estimated by the
135
following expression:
136
h K0 ( v Pe ) Pe
(4)
137
where K0 correlates the minimum horizontal stress with the vertical stress, which is also known as
138
trust coefficient (also known as coefficient of earth pressure at rest). Note that K0 is used (strictly)
139
to relate the horizontal effective stress with the vertical effective stress. In this work K0 was also
140
used with the stresses definitions adopted in equations (1) and (4), which are based on Biot’s
141
stresses.
142
2.1.2
143
To illustrate the use and significance of the previous equations a simple application example is
144
presented in this section. Typical parameters and properties for oil reservoirs were adopted in this
145
analysis. They are listed in Table 1. Note that a quite simple approach was adopted in this paper
146
to calculate the stress state and to estimate some model parameters. The technique suggested in
147
this paper could also be coupled with more advanced methods (e.g. [16]) to deal with uncertainties
148
in reservoir geomechanics.
149
The calculated fracture pressure based on these properties and using the equations presented in the
150
previous section is equal to 60.42 MPa. This is a deterministic analysis and then unable to quantify
151
the uncertainties associated with this problem. In Section 4.1, this case is solved using the evidence
152
theory incorporating the quantification of uncertainties in the analysis.
153
Application case
<< Insert Table 1 here >>
7
154
2.2
155
The terms subsidence and compaction have different meanings. Compaction is related to the
156
thickness reduction in a given formation, while subsidence refers to a downward movement of the
157
surface. The phenomenon of subsidence occurs over a much larger area than the compaction one,
158
which is associated with the reservoir thickness. The pore pressure reduction during production
159
leads to important changes in the reservoir and adjacent rocks. Such changes in pore pressure affect
160
directly the compaction and subsidence phenomena, which in turns significantly impact on the
161
fluid flow in the reservoir.
162
Several authors have discussed the relevance of the understanding the phenomena associated with
163
reservoir compaction and subsidence during oil fields exploitation (e.g. [17]; [18]; [19]; [20]). In
164
particular, Ferreira [21] conducted an extensive review on cases of subsidence around the world
165
and compared some analytical models with numerical solutions using a finite difference
166
commercial code. In the following section the equations to estimate the compaction and subsidence
167
of an oil reservoir under depletion are presented.
168
2.2.1
169
The vertical displacement field proposed by Geertsma [17] considers the geometry of the disk-
170
shaped reservoir as illustrated in Figure 1.
Reservoir compaction
Vertical displacement calculation
171 172
<< Insert Figure 1 here >> The vertical displacement is calculated via:
cm hR C ( Z 1) (3 4 )C ( Z 1) 2 2 1/2 2 [1 C ( Z 1) ] [1 C 2 ( Z 1) 2 ]1/2 2CZ (3 4 ) p [1 C 2 ( Z 1) 2 ]3/2
uz (0, z ) 173
8
(5)
174
where (see Figure 1) Z = z/c; C = c/R and ε = -1 when z > c, and ε = +1 when z < c; being c is the
175
distance between the free surface and the point under analysis The elastic constant cm is defined
176
by:
cm
177
(1 2 ) 2G(1 )
(6)
178
where G is the shear modulus.
179
Note that this equation provides the vertical displacement at different positions, therefore it is possible to
180
estimate the subsidence (i.e. the displacement at the seafloor or groundfloor level, depending if the case
181
under study corresponds to an off-shore or an on-shore case, respectively), and the compaction (as the
182
difference in displacements between the top and the bottom of the reservoir). Also in this case a relatively
183
simple model was selected to show how the selected approach can deal with uncertainties in reservoir
184
geomechanics related problems. The proposed approach is independent of the adopted equation and
185
certainly more complex models can be used in future works.
186
2.2.2
187
An example of application using the properties listed in Table 2 is presented in this section to
188
illustrate the use of the Equation 5.
Application example
189
<< Insert Table 2 here >>
190
Figure 2 presents the variation of the vertical displacement with depth for a reservoir located at a
191
depth of 2900 meters. Note that the model predicts that the top of the reservoir will move down
192
while the reservoir base will move up. The reservoir thickness reduction is around 2.7 m (i.e.
193
displacement at the top ~ +1.40 m, minus the displacement at the bottom ~ -1.3 m), while the
194
subsidence is in the order of 20 centimeters.
9
195
This analysis is deterministic and again is not possible to quantify the uncertainties associated with
196
it. The inference of new sampled data for the reservoir compaction problem and their impact on
197
the uncertainty quantification is discussed in Section 4.2.
198
<< Insert Figure 2 here >>
199
3.
200
Uncertainty quantification (UQ) is a key component when analyzing problems with limited data
201
and/or high variability of the available information. In the last few years the development of new
202
and more efficient numerical techniques, the increase in computational power, and the massive
203
use of parallel computing, have facilitated the incorporation of UQ analyses in routinely studies.
204
As mentioned before, there are two main types of uncertainties: aleatory and epistemic
205
uncertainties. The probability theory is commonly adopted to evaluate both types of uncertainties.
206
However, recent studies have shown that probabilistic methods may not be the most convenient
207
ones for analyses involving epistemic uncertainties. Alternative approaches, based for example on
208
evidence theory seem more appropriate for handling reducible uncertainties [6].
209
The prior probability distribution (called ‘prior’ henceforth) of a variable is a basic information
210
required in analysis based on probabilistic methods. The “Principle of insufficient reason” [22] is
211
generally used when this information is not available. This implies that when the density functions
212
associated with the model parameters are not known, uniform distribution functions could be
213
assumed. This assumption is valid for random variables, but may be inaccurate when dealing with
214
epistemic ones.
215
An alternative approach to quantify epistemic uncertainties is based on the evidence theory. This
216
theory is particularly useful in those cases where there is not enough information to quantify the
217
uncertainty with a known density function. In the evidence theory the range of the variables is
UNCERTAINTY QUANTIFICATION
10
218
defined by intervals or sets of possible values. The definition of a range (or a set) of probable
219
values linked with a given variable has three important implications:
220
i) it is possible to incorporate in the analysis experimental information from different
221
sources that can be weighted according to the user confidence on particular test results;
222
ii) the principle of insufficient reason is not enforced. Furthermore there is not a pre-
223
established structure for the distribution of the input variables. Initial probabilities can be
224
arranged in sets, without having pre-established assumptions about the probabilities of
225
individual events;
226
iii) the additivity axiom is not mandatory. Therefore, the complementary sets of
227
probabilities measures do not have to add up to one. When the additivity axiom is
228
considered in analysis based on the evidence theory this framework converges to the
229
traditional probabilistic method [6].
230
By comparing the probability and the evidence theories it is possible to see that the benefit of the
231
latter is that it allows a less restrictive description of the uncertainties. However, a drawback of the
232
evidence theory is that uncertainty analyses are computationally more demanding than the ones
233
based on probabilistic method. The additional computational effort comes mainly from the
234
numerical optimization process needed to calculate the belief and plausibility functions. In the
235
following section some background information related to the evidence theory is discussed, more
236
details can be found elsewhere (e.g. [23], [24], [7]).
237
3.1
238
The two core concepts in evidence theory are: i) the basic probability assignment (BPA) or
239
weights, and ii) the focal elements. The BPA for a given set is the probability that can be assigned
240
to it. This probability cannot be decomposed into additional probabilities for subsets of the original
Belief and plausibility functions
11
241
set. The focal elements are those sets that have nonzero BPAs. This then leads naturally to a clear
242
distinction between an evidence space for a set of possible outcomes and a probability space for a
243
set of possible outcomes. The basic probability assignment is different from the classical definition
244
of probability. It is defined by mapping over an interval in which the basic assignment of the null
245
set is ‘0’ and the summation of basic assignments in a given set is ‘1’ [6].
246
The first work in evidence theory was done by Shafer [25], as an expansion of the one previously
247
performed by Dempster [26]. In a finite discrete space, the evidence theory or the Dempster-Shafer
248
theory (DST), can be interpreted as the generalization of the probability theory for the case in
249
which the probabilities are assigned to sets of values. When the evidence is sufficient to assign
250
probabilities to single events, the Dempster-Shafer model falls into the traditional probabilistic
251
formulation.
252
Let’s consider a model defined by:
253
y f (x)
(7)
254
where x [ x1 , x2 , x3 ,..., xnX ] is the vector of the input variables and y [ y1 , y2 , y3 ,..., ynY ] is the
255
vector associated with the model results. Thus, the uncertainty in y can be estimated in the
256
framework of the probabilistic theory by the Cumulative Distribution Function (CDF).
257
prob( y y) ( f (x) | y)d X dX i S1 ( f (x) | y) / nS n
P
(8)
258
where y is the expected value (i.e. the different outputs associated with each ones of the possible
259
realizations); nS is the number of samples; ( f (x) | y) = 1 if f (x) y and ( f (x) | y) = 0 if
260
f (x) y . Furthermore, the probability theory requires that the uncertainty associated with x
261
should be defined by a density function (e.g. normal, triangular or uniform). To characterize the
12
262
uncertainty in x is necessary to establish a probability space (XP; XP’;mPX) where XP is the sample
263
space of possible values of x; XP’ is the set of subsets of XP chosen appropriately (Sigma-algebra);
264
and mPX is a probability factor.
265
Conversely the evidence theory does not impose a rigid structure for the characterization of the
266
uncertainties. It is a flexible approach. The uncertainty in y is estimated by means of two
267
uncertainty functions, the belief and plausibility functions as follows [6]:
268
269
BelY ( y y)
PlY ( y y)
m
A Ay
A Ay 0
EY
( A)
(9)
mEY ( A)
(10)
270
At an intuitive level, belief is a measure of the amount of information that indicates that a statement
271
is true, and plausibility is a measure of the amount of information that indicates that a statement
272
could be true. Central to this concept is the idea that all the probability distributions that do not
273
violate the assumed properties of the evidence space are under consideration. The ‘ball-box’
274
analogy of the Dempster-Shafer theory can assist to a simple interpretation of this technique. In
275
this model is assumed that n unmarked steel balls are distributed among k boxes A1,. . . , Ak, with
276
mi representing the fraction of balls in Ai. The boxes are placed in a bigger box U and are allowed
277
to overlap. The position of each ball within its box is unspecified. In this model, the granular
278
distribution A describes the distribution of the balls among the boxes. Note that the number of balls
279
in Ai is unrelated to the ones in Aj. It is important to distinguish between the number of balls in Ai
280
and the ones in Aj, because these fractions might overlap (i.e. the boundary of the boxes are
281
penetrable). Now, given a region Q in U, it is possible to ask the following question: How many
282
balls are in Q? Because the information regarding the position of each ball is incomplete, the
13
283
answer to this question will be (in general) interval-valued. The upper bound can readily be found
284
by visualizing Q as an attractor, for example, a magnet. Under this assumption, the proportion of
285
balls drawn into Q is given by Eq. (10), which is the expression for plausibility in the Dempster-
286
Shafer theory. Similarly, the lower bound results from visualizing Q as a repeller. In this case, the
287
lower bound is given by Eq. (9); which is the expression for belief in the Dempster-Shafer theory.
288
The combination of the intervals defined as uncertain values will outline the upper and lower
289
bounds of the response as described in the previous paragraph. The definition of the belief and
290
plausibility functions for the model outputs can be done through an optimization process, as
291
discussed in the following section. More details can be found elsewhere (e.g. [24], [6], [7]).
292
3.2
293
The belief and plausibility functions are defined for the input variables (x). For each input variable
294
it is necessary to specify intervals (and the associated bounds) and the corresponding weights (i.e.
295
BPAs). To calculate the belief and plausibility measures it is necessary to calculate the minimum
296
and maximum of the response function for each interval cell combination. For example, if a case
297
has 2 input variables and each of these is defined by 3 intervals, the number of combination cells
298
is equal to 9. This implies that epistemic analyses require the implementation of an optimization
299
algorithm to find the maximum and minimum for all the possible combinations considered. More
300
details about the implementation of typical optimization methods to perform Dempster-Shafer
301
analysis can be found elsewhere (e.g. [27], [28]; 23]; [7]). The intervals and their associated BPAs
302
are then propagated to obtain cumulative distribution belief and plausibility functions (i.e. CBF
303
and CPF, respectively). As mentioned above, belief is the lower bound on a probability estimate
304
that is consistent with the evidence, and plausibility corresponds to the upper bound on a
305
probability estimate that is consistent with the evidence.
Optimization algorithm
14
306
In this work the approach discussed in Cabral et al. [7] to compute the CBF and CPF is adopted,
307
which consists of the main following steps:
308
1) Combine intervals of each input variable into combination cells.
309
2) For each cell, two optimization problems have to be solved to find the lower (lb) and
310
upper (ub) bounds of the response.
311
3) Sort the lower bounds of the output response intervals obtained in Step 2 in ascending
312
order.
313
More details about this optimization process are presented in Cabral et al. [7].
314
3.3
315
To establish the type of analysis to be performed it is important to pay attention not only to the
316
availability of data but also to the origin and reliability of this information. There are two critical
317
aspects related to the combination of evidence obtained from different sources. The first one is
318
related to the type of evidence involved in the study and the second aspect is based on how to
319
handle conflicting evidence. Figure 3 presents four different types of evidence that impact on the
320
choice of how such evidences must be combined. They are called: inclusive evidence, consistent
321
evidence, arbitrary evidence and disjointed evidence [29].
Types of evidence
322
<< Insert Figure 3 here >>
323
The inclusive evidence can be represented by subsets that are contained in a larger subsets, which
324
in turns can be part of a larger one (Figure 3a). For example, this type of evidence can be related
325
to the case where the information for a given set is obtained from a larger period of time by
326
reducing the intervals of evidence. In the case of the consistent evidence, at least one element is
327
common to all subsets of the sample space (Figure 3b). The arbitrary evidence corresponds to the
328
situation where there is no element in common for all the subsets (Figure 3c), although there may 15
329
be some subsets of elements in common. Finally, the disjointed evidence is associated with the
330
case in which for any two subsets of the sample space there are no elements in common. (Figure
331
3d).
332
Each of these possible configurations of evidence from different sources has different implications
333
on the level of conflict associated with each situation. It is intuitive that for the case of disjointed
334
evidence, all sources of evidence are in conflict. For the case of arbitrary evidence, there are some
335
agreements between arbitrary sources, but there is no consensus on any of the elements. The
336
consistent evidence implies an agreement between at least one subset of evidence. Inclusive
337
evidence represents a situation in which every subset is included in a larger one, i.e., it is
338
guaranteed that there is a correlation between the evidences. However, there is inherent conflict in
339
the additional evidence supported by the larger subsets.
340
The traditional probability theory cannot deal with inclusive, consistent or arbitrary evidences,
341
without applying the definition of probability density functions for all the elements of the sets.
342
Furthermore, it is not possible to express the level of conflict between these sets of evidences. The
343
following examples show how the evidence theory is more flexible when dealing with the
344
scenarios discussed above. Furthermore, in the evidence theory, there are several ways to
345
incorporate imminent conflicts in the process of combining multiple sources of information. In the
346
next section, the application of this method is conducted based on the two problems discussed in
347
Section 2.
348 349
4.
350
In reservoir geomechanics there is a tendency to develop increasingly sophisticated constitutive
351
models and coupled formulations. This is a very valuable effort, because the behavior of sediments
DECISIONS MAKING PROCESS UNDER UNCERTAINTY
16
352
and rock involved in this type of problems is generally very complex. However, it seems also very
353
important to assess the impact of the variation of material parameters in the analysis and the
354
associated uncertainties. This section is aimed at studying decision making cases incorporating
355
uncertainty quantification via the evidence theory and adopting simple solutions, as the ones based
356
on analytic or semi-analytical solutions presented in Sections 2.1 and 2.2.
357
Decision-making is certainly the most important task of a manager and it often tends to be rather
358
difficult. The type of approaches used to make the decision ranges between two extreme cases.
359
One of them corresponds to the deterministic solutions (as the ones presented in the previous
360
Sections), where specific (determined) values are assigned to the different parameters and then it
361
is not possible to quantify the uncertainties. The other extreme corresponds to the pure uncertainty
362
analysis, where there is not previous information about the parameters associated with the case
363
under study. Those studies in which the risk can be assessed lie between these two extreme types
364
of analyses [30].
365
During reservoir operation it is possible that additional information related to the material
366
properties become available. It is of great interest to study how this new information may impact
367
on the uncertainty studies. In the following section an analysis under pure uncertainty (i.e. lack of
368
input data) is discussed first. Then, the incorporation of increasingly new input data in the analyses
369
and the corresponding effect in the uncertainty quantification is discussed.
370 371 372
4.1
373
The lack or limited of material data is a recurring problem associated with reservoir geomechanics.
374
One of the main reasons related to it has been the high cost associated with the gathering of the
Lack of input data
17
375
data from the field. Furthermore, the reduction of the geomechanical data gathered via seismic
376
logs or cores is quite complex and require high technical expertise.
377
In this section the calculation of the fracture pressure is used as an example of the decision making
378
process incorporating uncertainty quantification in a case in which the input data is quite scarce.
379
Afterwards, the discussion focuses on the comparisons between the results obtained via probability
380
and evidence theories. In this example, three experts were consulted to define the ranges of
381
parameters to be used in the UQ analysis. These experts defined the weights required for the use
382
of the evidence theory. The ranges of the parameters were defined by the material properties found
383
in analogous reservoirs. The uncertainty parameters are: Young’s modulus, Poisson’s coefficient,
384
Biot’s coefficient and thrust coefficient. The same weight value for each expert opinion was
385
assumed, therefore the BPAs are equal to 1/3. Table 3 presents the experts opinions related to these
386
parameters.
387
<< Insert Table 3 here >>
388
The variables quoted above are considered epistemic and the evidence theory is a proper
389
framework for the treatment of this kind of uncertainty. Figure 4 presents the CBF and CPF of the
390
fracture pressure calculation using the parameters shown in Table 3.
391
One of the main dilemmas associated with the use of the probability theory for uncertainty
392
quantification of epistemic variables is related to the choice of the probability density functions.
393
Figure 3 shows three CDF's associated with the choice of an uniform probability density function
394
(in dark blue), normal (in light blue) and a distribution that considers the BPAs as weights (in
395
green) in the respective intervals.
396
<< Insert Figure 4 here >>
18
397
This figure can be used to analyze the injection pressure that need to be applied for a given
398
reservoir to induce the fracture propagation. These plots can be used in different ways, for example
399
let’s assume that 0.50 is the largest possible probability that an injection project manager would
400
like to take to assure that injection pressure is equal or above the fracture pressure propagation and
401
in this way maintain a good injectivity. Considering first the CDFs (i.e. the ones obtained with the
402
probabilistic method), the needed increment in the injection pressure is around 69.63 MPa if
403
uniform or normal density functions are assumed (note that only for a probability of 0.5 these two
404
solutions predict the same pressure, for other probabilities the injection pressures estimated by
405
these two functions are different), while the other CDF predict a bit lower value of injection
406
pressure, around 68.65 MPa. As for the evidence theory, it is important to remember that Bel(U) is
407
the smallest possible probability that could occur for U over the set of indicated distributions, and
408
Pl(U) is the largest possible probability that could occur for U over the set of indicated
409
distributions. Therefore looking at the CBF and CPF curves, one may conclude that based on the
410
existing evidences it is believed that the fracture will propagate with a confidence of 0.5 if the
411
injection pressure is above 63.74 MPa. However the available evidence also informs that it is
412
plausible that an injection pressure as high as 75.02 MPa may be necessary to induce the
413
propagation of the fracture in this reservoir (also in this case with a confidence of 0.5).
414
It can be seen that the evidence theory provides more complete information to assist the decision
415
maker, with a range of variation of the possible values of injection pressures that will induce the
416
fracture propagation when a give probability is set. This interval of variation of the injection
417
pressure is a clear indicator of the uncertainties associated with the problem under study and with
418
the available data. The example discussed in the next section shows how the uncertainty can be
19
419
reduced when more information about the geomechanical properties of the reservoir is added into
420
the analysis.
421
4.2
422
Table 4 corresponds to a possible scenario for information availability over time for reservoir
423
compaction problem analyzed in Section 2.2. At the beginning, there was no information about
424
the mechanical properties of the reservoir. The parameters used in this first analysis were estimated
425
from the data observed in similar reservoirs. Then, information obtained from the first well drilled
426
became available. This data was obtained from log correlations. Results from laboratory tests were
427
available afterwards. Finally, another well was drilled and therefore more information about
428
reservoir properties was available.
New data inference
429
<< Insert Table 4 here >>
430
A crucial point at this stage of the analysis is to define how this new information is incorporated
431
in the process of uncertainty quantification. It is also very important to have the opportunity of
432
weighting the available information based on the reliability of this data. As discussed previously,
433
the evidence theory is flexible and allows defining probabilities by sets, which is very convenient
434
for handling the information associated with this type of problem. In the evidence theory the expert
435
can decide how to measure the significance of the existing information coming from different
436
sources (i.e. by the definition of the BPAs or weights). The weighting of the available data is based
437
on the reliability of the existing data, experts’ judgment, and previous experience.
438
In Table 4 it can be observed that for the analysis identified as ‘T0’, the BPA assigned to the
439
different properties is equal to 1. This is because there is only one source of information (i.e. the
440
experience from previous studies similar to this one). In the analysis coded as ‘T1’ (i.e. the case in
441
which information from the first well became available), a BPA of 0.8 was assigned to the data 20
442
coming from the new well, and 0.2 was assigned to the T0 data. This implies that according to the
443
expert opinion, the information from this reservoir is more reliable (i.e. higher weight value) than
444
the one coming from previous similar studies. It can also be observed that the reservoir parameters
445
coming from the well vary in a smaller range. Also, according to the experts’ opinion, the
446
information gathered in the laboratory and incorporated in the analysis ‘T2’is more reliable than
447
the one related to the previous studies, but significantly less dependable than the data associated
448
with the borewell. The weights assigned to the previous studies, well, and lab data are: 0.1, 0.7 and
449
0.2; respectively (note that the sum of the BPAs has to be always equal to 1). A similar reasoning
450
is followed in analyses T3, when data coming from another well is incorporated in the analyses.
451
Figure 5 shows the changes in the curves associated with the uncertainty quantification analyses
452
when new information is incorporated at different stages of the analysis. The flexibility of the
453
evidence theory for new data inference is clear in the evaluation of the results. It is not surprising
454
that new information about the material parameters reduces the uncertainty range associated with
455
the estimated subsidence. Initially, when few information was available (i.e. based mainly on
456
previous experiences in similar reservoirs), the range of possible subsidence values was quite wide,
457
i.e. between 0.05 and 0.9 m. After incorporating information from the first well, the uncertainty
458
decreased and the calculated subsidence was between 0.05 to 0.7 meters. Once the information
459
from the laboratory test became available, a similar trend was observed with a further reduction of
460
the uncertainty range. The minimum and maximum possible subsidence in this new study was in
461
the range between 0.05 and 0.52 m, respectively. Finally, after the information from the drilling of
462
the second well was incorporated, the uncertainty reduced even more, and the possible subsidence
463
values were between 0.05 to 0.18 m.
464
<< Insert Figure 5 here >>
21
465
It is worth mentioning that the use of the evidence theory in this case allowed the inference of new
466
data without the requirement of defining priors, as it is necessary in classical probability analyses.
467
It was observed that the curve related to the probability theory considering uniform distribution
468
kept constant during the subsequent analyses, because the range that defines the minimum and
469
maximum values of the parameters were not modified as new information became available, and
470
consequently the uniform distribution did not change (Figure 5).
471
5.
472
This work focused on the uncertainty quantification in problems related to reservoir geomechanics.
473
These types of problems are characterized by the limited (and in some occasions nil) information
474
associated with material properties. To quantify the uncertainties a formal framework based on the
475
evidence theory was adapted to oil reservoir problems. This non-probabilistic method is especially
476
well-suited for dealing with uncertainty that are epistemic in nature. This type of reducible
477
uncertainty is very common in reservoir geomechanics because it is generally associated with the
478
lack of information. The main components of the adopted approach was presented in this work
479
and then it was applied to solve typical problems related to oil reservoirs, such as the estimation
480
of the fracture propagation pressure and reservoir compaction. Relatively simple analytical
481
solutions were adopted for the analyses, because the main effort was on exploring the capability
482
of the framework for dealing with uncertainties.
483
In both analyses (i.e. estimation of fracture pressure and reservoir compaction) it was observed
484
that the evidence theory facilitates the decision making process because it provides more complete
485
information than the probabilistic theory. For example, when defining the fracture propagation
486
pressure, the evidence theory gives based on the available information and for a given probability,
487
a range of variation of possible injection pressures that may induce fracture propagation. This
SUMMARY AND CONCLUSIONS
22
488
interval of variation for the injection pressures is a clear indicator of the uncertainties associated
489
with this problem. It was also shown that the evidence theory is a suitable framework for
490
incorporating consistently additional evidences and reducing consequently the uncertainty when
491
more information about the reservoir becomes available. This feature was evident when analyzing
492
the reservoir compaction problem, where the range of variation of the estimated settlements was
493
reduced in subsequent analyses that incorporated increasingly additional data about the
494
geomechanical properties of the reservoir.
495
The use of the evidence theory based on CBFs and CPFs, guarantee that the information is
496
consistent with all the known data associated with the input parameters of the model. In the limit,
497
these two curves (i.e CBF and CPF) converge to a single cumulative distribution function curve,
498
when the data that completely describes the model is available. It is worth mentioning that the use
499
of evidence theory in this case allowed the inference of new data without the requirement of the
500
definition of priors (as generally happens in probabilistic theories). It was observed that the curve
501
related to the probability theory considering uniform distribution kept constant during the analysis
502
because the range that defines the minimum and maximum values of the parameters were not
503
modified as new information became available (and thus the uniform distribution did not be
504
change). Furthermore, the evidence theory was able to properly incorporate in the analyses the
505
experts’ opinion and to weight the available information based on the reliability of the data source.
506
ACKNOWLEDGEMENTS
507
The authors acknowledge the financial support from PETROBRAS, CNPq (Brazilian National
508
Research Consul) and Foundation CMG (Chair on Reservoir Simulation at UFPE).
23
509
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27
Figure
Sea or ground level
Figure 1: Geometry for determining the displacement field in the surroundings of a disc-shaped reservoir (adapted from Geertsma [17]).
Figure 2: Vertical displacement field along the depth. The reservoir is located at 2900 m depth.
.
A E
A
B
B
C
C
E
D
b)
a)
D
A
B
C
A C
E
E
B
D
D c)
d)
Figure 3: Types of evidence: a) inclusive; b) consistent; c) arbitrary and d) disjointed. The letters A, B, C, D and E represent diferent sources of evidence.
Figure 4: Fracture pressure calculation. Uniform probability density function (in dark blue), normal (in light blue) and a distribution that considers the BPA's as weights in the respective ranges (in green). Results obtained with evidence theory: CBF (in red) and CPF (in black).
Figure 5: Incorporation of new data over time. Uniform probability density function (green curves), and a distribution that considers the BPA's as weights (blue curves) in their respective ranges. Via theory of evidence it is estimated the CBF (in red) and the CPF (in black).
Table
Table 1: Input data used for the calculation of the reservoir fracture pressure
Geomechanical data Young modulus (E)
25 GPa
Biot’s coefficient (α)
0.8
Tensile strength (σT)
4.02 MPa
Poisson’s coefficient (υ)
0.25
Thrust coefficient (K0)
0.8
Pore pressure factor in tensile failure criterion (φ)
0.07 Length data
Water depth (LA)
2100 m
Overburden tickness (Sot)
2450 m
Reservoir thickness (hR)
100 m Temperature and pressure data
Reservoir temperature (Tr) Injected water temperature (Ti) Reservoir initial pressure (Pe)
60 oC 35 oC 48.05 MPa
Table 2: Required input data for the estimation of the vertical displacement field in a depleted reservoir
Geomechanical data Shear Modulus (G)
147.1 MPa
Biot’s Coefficient (α) Poisson’s Coefficient (υ)
1.0 0.286 Reservoir properties
Reservoir thickness (hR)
100 m
Top (c)
2900 m
Radius (R) Total depth (z)
1000 m 4000 m
Porepressure variation (depletion) (Δp)
13.1 MPa
Table 3: Ranges of possible values characterizing the uncertainty of the input variables. Each interval was defined by different experts’ opinions.
Parameter
Expert 1
Expert 2
Expert 3
E (GPa) υ
35 - 40 0.2 - 0.3
35-45 0.2 - 0.3
30 - 40 0.2 - 0.3
α
0.6 - 0.8
0.6 - 0,8
0.5 - 0.7
K0
0.7 - 0.9
0.75 - 0.8
0.7 - 0.8
BPAs
0.33
0.33
0.33
Table 4: Information (evidence) over time. The uncertainty quantification process was modified with each new piece of evidence.
T = 0 - Similar reservoirs Parameter Α
T0 0.6 - 1.0
Υ
0.2 - 0.4
G (MPa)
49 - 294
BPA0
1.0 T = 1 - Well drilled
Parameter Α
T0 0.6 - 1.0
T1 0.7 - 0.8
Υ
0.2 - 0.4
0.32 - 0.36
G (MPa)
49 - 294
245.2 - 313.8
BPA1
0.2
0.8 T = 2 - Lab Test
Parameter Α
T0 0.6 - 1.0
T1 0.7 - 0.8
T2 0.95 – 1.0
Υ
0.2 – 0.4
0.32 - 0.36
0.35 – 0.4
G (MPa)
49 - 294
245.2 - 313.8
196.1 - 214.6
BPA2
0.1
0.7
0.2
T = 3 - New Well Drilled Parameter
T0
T1
T2
T3
Α
0.6 - 1.0
0.7 - 0.8
0.95 – 1.0
0.75 – 0.9
Υ
0.2 – 0.4
0.32 - 0.36
0.35 – 0.4
0.3 – 0.35-
G (MPA)
49 - 294
245.2 - 313.8
196.1 - 214.6
294 – 295.2
BPA3
0.02
0.4
0.18
0.4