1

Uncertain Predictions of Flow and Transport in Random Porous Media: The

Implications for Process Planning and Control

Shlomo Orr, MRDS, Inc.

Traditional predictions of flow and transport in

porous media are based on mass balance

equations in the form of partial differential

equations (PDEs), where the flux at every point

is defined by Darcy’s law, q = -K∇h, i.e., the

flux is proportional to hydraulic head gradient,

where K is the hydraulic conductivity of the

medium (a tensor or a scalar; essentially, a

material property); it is further assumed that

Darcy’s law applies to transient multiphase flow

in three dimensions [14,26,36,55]. The solutions

of these PDEs constitute groundwater models,

oil reservoir simulators, geothermal models, and

models of flow and transport in soils/vadosezone. Due to the similarity between the linear

Darcy’s law and Ohm’s law in electricity,

Fourier law in heat conduction, and Hooke’s law

in elasticity, such models (or PDE solutions) are

similar and commonly interchangeable between

these fields.

Natural porous formations are heterogeneous,

and display spatial variability of their geometric

and hydraulic properties. This variability is of

irregular and complex nature. It generally defies a

precise quantitative description because of

insufficient information on all relevant scales

[9,18,26, 29,30,32,33,86,91]. In practice, only

sparse measurements are available (limited by cost

of drilling and monitoring). Under lack of

exhaustive information, the higher the variability,

the higher is the uncertainty. Geostatistics is

commonly used to analyze and interpolate

between measurements in mining and oil

explorations, as well as hydrology and soil

sciences, using methods such as “kriging”, where

the uncertainties in “krigged” values are also

quantified [33,35,36,42,56-58,76,77,81,83,84,87,

104,105]. Frequently, these data are collected on

different scales that may differ from the required

scale of predictions. The task of quantitatively

relating measurements and properties on different

scales is difficult and intriguing [4,5,7,13,27,29,

30,38-40,46,59,78,86,91,101,108,109]. Lack of

information in both observed results (output) and

measured material properties (parameters) causes

uncertain predictions. Spatial variability and

uncertainty have lead engineers and geologists to

use probabilistic theories that translate the

uncertainty to a random space function (RSF) or a

random field, consisting of an ensemble of

(infinite number of) equally probable

“realizations” of parameter values, all having the

same spatial statistics, particularly correlation

structure [107,76,77,23,33,35,36,42,56,58,83,84,

85,87,91]. Imbedded in this approach is a

geostatistical model of an assumed joint pdf. In

practice, only the first two moments are

considered, with an underlying assumption of

multivariate normal distribution; in particular, the

theoretical semi-variogram (or simply,

“variogram”) - the reciprocal of the covariance

function, and the mean and variance of the pdf.

Since these joint moments are inferred from

spatial data, the assumption of ergodicity (i.e.,

assuming that the ensemble and spatial statistics

are identical – a theorem that cannot be proven on

real data) must be invoked, which, in turn, implies

some kind of stationarity (or statistical

homogeneity) [33,35,36,49]. Further, in order to

determine the variogram model from available

spatial data, an inverse method has to be used to

estimate the parameters of this variogram;

sensitivity to data errors on one hand, and

identifiability problems (of model parameters) on

the other hand [81,87,104,105] lead to uncertainty

in the geostatistical model itself, which is usually

ignored (in fact, the common practice is to fit the

variogram model to the experimental variogram

by eye and by subjective judgement of model

type, degree of stationarity (drift), and statistical

anisotropy). Another ignored uncertainty is in the

“measured” hydraulic conductivity value that are

actually inferred from hydraulic tests interpreted

by simplistic models that assume local

homogeneity, which is somewhat inconsistent

with the RSF approach.

The use of RSF to predict behavior of

uncertain systems is not limited to flow in porous

media; great efforts have been devoted in all

science and engineering fields to (a) estimate or

predict mean behavior of the system under

different stresses, and (b) compute the uncertainty

associated with these predictions (expressed by the

variance-covariance of the solution), i.e., the first

two statistical moments of system output

[4,5,7,10-12,15,16-22,25,28,31-33-35,44,47-51,

57,62,63,68-71,73-78,82-93,97,99-101,103,106,

107,110,111,112,114]. The resulted approximate

solutions are usually limited to simple geometry
2

and boundary conditions, and to moderate to low

variability, as they mostly rely on variations of

small-perturbation methods. Recently developed

approximations for higher variability in material

properties using integro-differential

representations have been limited to simple 2D

geometry and simple boundary conditions

[91,114], with little use, yet.

With respect to mean behavior, it is especially

desired to define effective properties of

heterogeneous media [1-3,7,8,25,33,37-41,43,45,

52,54,60,61,64-66,71-73,75,77,78,91,93,97,98,

102,108,109,114]. With the two statistical

moments of system output, one hopes to

optimize and control systems such as oil

production, groundwater remediation, irrigation,

leaching, etc. However, approximating the

statistical behavior of a complex system of flow

in random porous media based on the statistics of

the hydraulic parameters is a formidable task, at

best [46,114, 91-93], because this implies

solving the stochastic flow and transport

equations (analogous to the Heat/Diffusion

Equation) or other stochastic PDEs in other

fields (ibid). Since a direct, explicit (closedformed) solution to the problem of random

parameters (or coefficients) is practically

impossible1 [74,93,112,114], only approximate

solutions have been reported in the literature for

relatively simple cases [ibid,28,31-33,49-50,83-

93,100-101]. Interest in this class of stochastic

differential equations has its origins in quantum

mechanics, wave propagation, turbulence theory,

random eigenvalues, and functional integration

[8,20,51,48,62,63,68-71,74,78,99,103,111]. Due

to the limited types of problems that can be

tackled by stochastic theories (closure

approximations), in practice, numerical

approximations in the form of high-resolution

 

1

 A solution to a stochastic PDE consists of

specifying the (joint) probability density function

(pdf) of the response, h(x), given those of K(x)

(and forcing functions and boundary conditions).

Unfortunately, one cannot obtain the joint

cumulative distribution function (CDF) of the

random response at all (infinite number of)

points. Even for a finite set of points, one cannot

obtain closed-form equations for a finite number

of moments. This problem can be circumvented

by either approximations (e.g., perturbation

methods, Neumann series) or by numerical

approximations, i.e., Monte Carlo simulations

(MCS).

Monte Carlo simulations (MCS) are used;

however, MCS require ample computer power

and CPU time [46,114,88-89] Orr [114]

describes and analyzes other difficulties and nonquantifiable uncertainty associated with MCS,

particularly the generation of correlated random

fields that are faithful to the geostatistical model,

and simulations of flow in highly

heterogeneous/erratic media.

The geostatistical model provides the statistics

of the parameters, particularly the permeability; in

MCS, it provides the spatial distribution of

parameter values for each realization. Based on

these values and assumed model structure (i.e., the

conceptual model of flow and transport, including

large geologic features, boundary- and initialconditions, sources and sinks), stochastic solutions

are approximated (analytically or numerically).

Since the model structure itself is frequently

uncertain due to (a) unknown boundary and initial

conditions, (b) extent of large-scale geologic

features, and (c) information pertaining to

geochemical reactions and phase transition, such

sets of 500-1000 MCS need to be repeated for

several, if not many alternative conceptual models

or equally probable model structures [116].

Subsequent optimization requires many repetitions

of each Monte Carlo simulation in order to build

the search space (i.e., generate sufficient number

of scenarios or trajectories); hence, rigorous

optimization under uncertainty is prohibitive in

terms of computer power and time for most

practical applications. In an attempt to optimize

best new well placement in an oil reservoir,

Guyaguler and Horne [53] were forced to perform

optimization on only 23 randomly selected,

“history matched” realizations in order to

overcome the obstacle of prohibitive computer

power and time, while continuously verifying their

results against a “truth” model (apparently based

on extensive calibration and some “effective”

properties). Indeed, the authors concluded that “a

decision based on a single realization (though with

perfect history match) may differ substantially

from the true optimum”[53, p.4]. As was shown

theoretically by Neuman and Orr [91], unique

effective properties (of random media) that are

data independent do not generally exist except for

a few special cases. This explains why parameter

estimates obtained by traditional inverse methods

tend to vary as one modifies the database and/or

the imposed stress [113,115]; consequently,

calibration of deterministic models may be

meaningless in term of predictive power.

Thus, on the way to optimal solutions using

stochastic predictions we already encounter
3

significant theoretical and practical barriers,

particularly, uncertain interpretations, model

limitations, prohibitive computer power, and nonexisting effective parameters – all of which render

these predictions highly uncertain, while only part

of this uncertainty is being quantified. Our

presentation will discuss these barriers, and will

bring two examples where traditional stochastic

approximations (i.e., approximate solutions of the

governing stochastic PDEs) are unable to provide

reliable practical solutions. One example deals

with the difficult problem of finding the best

location of the next well in an oil reservoir [53].).

The following simplified block diagram describes

the work flow as described by Guyaguler and

Horne [53]. Note that “data” are inferred values

from field tests. Note also that at each step, there

are inherent errors, inconsistencies, and

unaccounted (as well as counted) uncertainty.

Another example involves optimal control of

heap leaching in the mining industry [94,95].

In both cases:

1. Rigorous MCS is already prohibitive in

terms of computing resources

2. Models cannot capture the full complexity;

hence, predictions are unreliable

3. Rigorous optimization is impossible due to

time and computer limitations

In each of these cases, decisions have been made

at every step of the solution. Many of these

decisions are made subjectively, based on

experience, knowledge, thoroughness,

understanding (or conceptual models of the

process), computer resources, and time limitation,

possessed by the modeler. These factors affect and

are being affected by the degree of belief (by the

modelers) in each decision made.

Initially, sampling (network) design

decisions have to be made re sampling locations

(an optimization procedure on its own that

depends on the end results as well, i.e., a feedback

mechanism with the goal of minimizing prediction

uncertainty [119]). Then, the following decisions

(and sub-decisions) have to be made: (a) type of

model and/or curve fitting to use to determine the

hydraulic conductivity or permeability (a miniinverse model that could be ill-posed); in the case

of two-phase flow (e.g., oil-water in a reservoir,

water-air in a heap), several other decisions (or

assumptions) have to be made, and an ill-posed

inverse procedure must take place [117-121, and

author’s personal experience]; (b) determining and

eliminating outliers; (c) determining optimal lag

distance for the experimental variogram; (d)

determining the pdf, variogram model, and model

parameters; particularly, choosing between

Gaussian and Indicator models, judging between

drift and/or anisotropy, and determining the drift

(requires a complex inverse procedure, with

typical ill-posed cases; see [36,81,104,105]); (e)

determining dimensionality, domain size, and

mesh resolution with respect to correlation scales,

measurement scales, and property/parameter

representation scales; (f) determining conceptual

model (or model structure) of flow and transport,

including the parameters to be treated as random,

Collect “data”

and construct

a geostatistical

model

Generate a random

field with many

realizations, and

run MCS

Select several calibrated

simulations/realizations

and perform optimization

on each (for max NPV)

Compare

w/”truth”

model

Collect historical

data on heap

construction, and

construct a

geostatistical model

Generate a random

field with many

realizations, run

MCS, and compute

“ensemble” statistics

I. Perform optimization on

the ensemble mean output

(predicted production) for

max NPV

II. Perform optimization

on selected realizations

that resemble production

behavior (for max NPV)
4

the governing equations (PDEs, including reactive

transport and multiphase flow), and uncertain

boundary- and initial-conditions, boundary

locations, zones of specific character/features

(based on geologic and hydraulic information) –

i.e., the simulator (these decisions should be made

first, and re-evaluated as more information is

being analyzed); (g) for MCS: random number

generator (RNG; portable or not; RNG type; seed,

etc.); (h) random field generator (RFG), including

type and RFG parameters (in addition to the

variogram model and pdf); (i) number of

simulations (should be based on the behavior of

point output variance as a function of the number

of simulations, i.e., a trial & error procedure which

usually is not being done, including in the above

two cases), given the limited computer power); (j)

number of realizations required for optimization;

ideally, all (hundreds of) realizations are used; one

compromise may be reducing the number of

realizations [53]; another compromise would

consider mean system behavior (using the

resulting ensemble mean results, and optimize that

mean behavior (a major uncertain decision); (k)

type of optimization/search algorithms (l) number

of simulations (per realization) for constructing a

sufficiently dense search state-space; (m)

objective function and cost variables. The last

three decisions have to be made by all

optimization procedures.

In the well placement problem [53], due to

limited computer power, a decision was made to use

only 23 realizations (while 500-1000 simulations

are typically needed to provide meaningful

ensemble statistics [114,46]), and a decision was

made to calibrate randomly selected random

realizations, individually, which contradicts the

concept of random fields (or RSF), but may have

served a practical purpose (i.e., to find the

maximum NPV). Similar prohibitive computer

power problem prevails in the heap leaching

simulations. While unstable oil-water fronts in the

well placement case cannot be captured by the

simulator, unstable wetting fronts during heap

leaching cannot be captured even by high-resolution

two-phase models (like the one used by Orr and

Vesselinov [95]). In the latter case, lack of

information on essential reactive transport

properties, and unmonitored dynamic changes in

heap structure (particularly sealing of pore space by

clays and erosion products) cannot be determined

and modeled. Consequently, the simulators are

weak, missing on mean system behavior (or

predictions) with unaccounted uncertainty, resulting

in weak optimization and control.

We see that along the track of approximate

solutions and partial optimization of oil reservoirs

and heap leaching operations based on

predetermined stochastic PDEs, the confidence of

modelers and decision-makers is being eroded

with each decision being made, depending on their

knowledge and degree of belief at each decision

point. By the end of this process, decision-makers

find themselves with very little confidence and

very little decision power. Commonly, in this

stochastic approach, the degree of belief is not

quantified, though it contributes to the total

uncertainty. Alternative fuzzy logic techniques do

quantify the uncertainty associated with the degree

of belief.

We therefore propose to replace these

formidable stochastic approaches by a simpler

yet intelligent stochastic control, particularly, the

multiresolution decision support system (MRDS,

which includes fuzzy logic as one of its

components) in order to reach more reliable and

efficient optimal solutions, with reduced,

accountable uncertainty, in real time, with

minimal computer resources. Moreover, unlike

the rigid stochastic PDEs, MRDS can be

naturally extended to optimal control of linked

processes. In the oil field case, this includes

exploration, all surface installations and

operations, delivery system, and distribution. In

the heap leaching case, this includes subsequent

solvent extraction and electrowinning, as well as

all antecedent processes – from exploration and

blasting to transportation, crushing,

agglomeration, conveying, placement, and

design of the irrigation systems.

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83. Neuman, S.P., Statistical characterization of aquifer

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84. Neuman, S.P., Role of geostatistics in subsurface

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85. Neuman, S.P., Stochastic continuum representation of

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86. Neuman, S. P., Universal scaling of hydraulic

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87. Neuman, S.P. and E.A. Jacobson, Analysis of nonintrinsic

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88. Neuman, S.P., S. Orr, O. Levin, and E. Paleologos, Theory

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89. Neuman, S.P., O. Levin, S. Orr, E. Paleologos, D. Zhang,

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90. Neuman, S.P., Eulerian-Lgrangian theory of transport in

space-time nonstationary velocity fields: 1. Exact

nonlocal formalism by conditional moments and weak

approximation, Water Resour. Res., 1993.

91. Neuman, S. P., & Orr S., Prediction of steady state flow in

non uniform geologic media by conditional moments:

Exact non local formalism, effective conductivities,

and weak approximations, Water. Resour. Res., 29 (2),

341-364, 1993.

92. Neuman, S. P., Tartakovsky, D. M., Wallstrom, T. C. &

Winter, C. L., Correction to “Prediction of steady state

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moments: Exact nonlocal formalism, effective

conductivities, and weak approximation”, Water

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93. Orr, S. and S.P. Neuman, "Operator and IntegroDifferential Representations of Conditional and

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paper, Journal of Stochastic Hydrology and Hydraulics,

Vol 8(2), pp. 157-172, 1994.

94. Orr, S. "Enhanced Heap Leaching - I. Insights”, ME/SME

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104. Samper, F.J. and S.P. Neuman, Estimation of spatial

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105. Samper, F.J. and S.P. Neuman, Estimation of spatial

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likelihood cross validation, 3. Application to

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Fracture flow modelling: Application of automatic

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Neretnieks, Hydrogeology, Selected Papers, Intern.

Assoc. Hydrogeol., Verlag Heinz Heise, Hannover,

Germany, 2, 115-167, 1990.

114. Orr, S., Stochastic Approach to Steady State Flow in

Nonuniform Geologic Media, Ph.D. dissertation,

University of Arizona, 1993.

115. Neuman, S.P., and A. Guadagnini, A new look at

traditional deterministic flow models and their

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116. James, A.L., and C.M. Oldenburg, Linear and Monte

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117. Carrera, J., Neuman, S.P, Estimation of aquifer

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199-210, 1986a.

118. Carrera, J., Neuman, S.P., Estimation of aquifer

parameter under transient and steady state conditions :

2. Uniqueness, stability, and solution algorithms.

Water Resources Research, v. 22, 2, p. 211-227,

1986b.

119. Carrera, J., Neuman, S.P., Estimation of aquifer

parameter under transient and steady state conditions :

3. Application to synthetic and field data. Water

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120. Carrera, J., State of the art of inverse problem applied to

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121. Gh. de Marsily, J.P. Delhomme, A. Coudrain-Ribstein,

A.M. Lavenue, Four Decades of Inverse Problems in

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Birthday, 30 pp., 1998.