School of Industrial and Information
Engineering
Master Thesis in Energy Engineering
Department of Civil and Environmental Engineering
ANALYTICAL EXPRESSIONS FOR
MACRODISPERSION IN
THREE-DIMENSIONAL
RANDOMLY HETEROGENEOUS
NON-GAUSSIAN CONDUCTIVITY
FIELDS
Candidate: Ceresa Laura
Student Number: 841932
Supervisor: Giovanni Porta
Co-Supervisor: Alberto Guadagnini
External Supervisor: Philippe Ackerer
Academic Year: 2017-2018
School of Industrial and Information
Engineering
Master Thesis in Energy Engineering
Department of Civil and Environmental Engineering
ANALYTICAL EXPRESSIONS FOR
MACRODISPERSION IN
THREE-DIMENSIONAL
RANDOMLY HETEROGENEOUS
NON-GAUSSIAN CONDUCTIVITY
FIELDS
Candidate: Ceresa Laura
Student Number: 841932
Supervisor: Giovanni Porta
Co-Supervisor: Alberto Guadagnini
External Supervisor: Philippe Ackerer
Academic Year: 2017-2018
particular thanksgiving goes to my friend Giuseppe, who has been accompanying me for a very long time and directly contributed in helping to overcome life’s obstacles, and to my classmate and friend Lorenzo, who shared this experience with me. I am grateful to all of you for your continuous presence by my side. My academic realization is also yours.
Introduction
Nowadays many technological applications have fostered the interest of groundwater literature towards the stochastic modeling of flow and transport, especially across non-Gaussian log-conductivity fields which well describe many natural formations featuring heterogeneity and wide spatial variability of the associated hydro-geological properties. Indeed, the study of subsurface transport has recently assumed a key role in hydrology due to the rising importance of designing oil and gas facilities and to the purpose of environmental safety tasks. The necessity of modeling these phenomena stochastically has been further strengthened by the difficulties of handling heterogeneous formations deterministically,[1] which has addressed many research efforts
towards the development and validation of stochastic models for describing subsurface transport in porous media. In agreement with this tendency, this master thesis proposes first to validate numerically the recently developed Generalized sub-Gaussian (GSG) model[2] by Riva et all, which predicts the behaviour of macrodispersion coefficients in
two-dimensional uniform mean flow fields (with longitudinal mean head gradient J ) with inert solutes by means of analytical analogues.[3] Secondly, a personal effort goes to the
purpose of extending this model to the case of three-dimensional domains, in accordance with the main role played by the analytical solutions to provide a theoretically consistent framework for benchmarking numerical estimates. Numerical simulations have been performed according to the Monte Carlo method,[4] which is based upon the principle of
generating multiple synthetic realizations of conductivity fields (Y ) over which solving the governing flow and transport equations, in order to build a sufficiently wide collection of observation to manage the problem statistically.
This work initially focuses on two-dimensional rectangular domains at progressively increasing degree of heterogeneity, which is measured in terms of log-conductivity
for σY2 < 1, in agreement with the range of validity of the first-order approximation (σ2
Y << 1) introduced by Dagan[1] and recalled by Riva, Guadagnini and Neuman.[3]
This matching worsens with increasing fields heterogeneity, but always predicts the correct order of magnitude for longitudinal macrodispersion. On the other hand, transverse macrodispersion analogues better capture the trends exhibited by numerical results, always predicting their asymptotic drop to zero in the absence of pore-scale diffusion. The sole difference among distinct σY2 fields is the peak value attained by transverse dispersion at early times, which grows at increasing levels of heterogeneity. This investigation has been then extended to the case of three-dimensional domains with σ2
Y below unity, which shows just slight discrepancies between analytical and
numerical estimates, especially during the pre-asymptotic regime, probably associated with the worse capability of the analytical model to capture all the physics behind the problem. However, the adoption of this model to describe non reactive transport in three-dimensional heterogeneous fields reveals successful outcomes in the range of validity of the first-order approximation, well capturing both the pre-asymptotic and asymptotic trends exhibited by numerical macrodispersions. Furthermore, it guarantees the correct prediction of the associated asymptotes at least in terms of magnitude and enables investigating the effect of different α (parameter which governs the subordinator variance σ2U = (2 − α)2) on the time evolution of macrodispersion coefficients.
Theoretical framework and numerical algorithms
The core of the Generalized sub-Gaussian (GSG)[2] model lays in defining the Spatial Random Function (SRF) Y (x), representing hydraulic log-conductivity lnK(x), as the summation between its spatial mean KG (coinciding with the ensemble mean under the
function of Y0 and Y (functions of the scalar lag r for isotropic media) to those of G: σ2Y = e2(2−α)2σ2G ; IY = IG k = e −(2−α)2 IG ; CY(r) = CG(r) k = σY2 k e − r IG With k = e(2−α)2
denoting a mathematical constant arbitrarily introduced for convenience. The effects of non-Gaussian frequency distributions of Y (x) on two-dimensional subsurface transport have been explored by Riva, Guadagnini and Neuman[3] and extended in this
work to three-dimensional domains following the procedure explained by Dagan in his book "Flow and Transport in Porous Formations":[1] given the isotropic covariance
function of log-conductivities CY(r), one can derive the auxiliary functions P (r) and
Q(r) anddQ/dr to mathematically handle the problem. Then, head autocovariance C h(r)
and conductivity - head cross-covariance CY h(r) can be computed according to the
problem dimensionality. Once both are explicit, one moves to the covariance matrix of velocity vectors, limiting here the focus to the sole diagonal components (longitudinal and transverse Cvx,vx(r) and Cvy,vy(r) = Cvz,vz(r), respectively). The latter ones are
indeed necessary to derive the diagonal components of particles displacement vectors covariance CXxx(r) and CXyy = CXzz(r), which lastly lead to the desired expressions of
directional macrodispersion analogues Dxx(r) and Dyy(r) = Dzz(r). This conceptual
procedure can be mathematically translated into the following set of equations, which have to be sequentially solved in order to derive macrodispersion analogues. The cross-covariance between hydraulic conductivity and head is first computed as follows:
CY h(r) = ( − J r Z r 0 zCY(z)dz 2D −J r2 Z r 0 z2CY(z)dz 3D
Then, the general head variogram γh(r) can be derived (with r = [rx; ry; rz]):
Ch(r) = ChT(|r| = ry) + (J · r)2 J2r2 ChL(|r| = rx) − ChT(|r| = ry) = −γh(r)
Ch(r) = J2 P (r) −2 r dQ(r) dr 3D ChT(r) = J 2 r dQ(r) dr
The diagonal components of local velocities and particles displacements covariances can be subsequently derived as:
Cvx,vx = K2 G φ2 J2CY(r) + 2J ∂ ∂rCY h(r) + ∂2γ h(r) ∂r2 Cvy,vy = KG2 φ2 ∂γh(r) ∂r = Cvz,vz CXxx(r) = 2 v2 ADV Z r 0 r − zCvx,vx(z)dz CXyy(r) = 2 v2 ADV Z r 0 r − zCvy,vy(z)dz = CXzz(r)
And directional macrodispersion analogues can be computed as:
Dxx(r) = vADV 2 dCXxx(r) dr Dyy(r) = vADV 2 dCXyy(r) dr = Dzz(r)
The analytical derivation of the above equations beginning from the physical meaning of each covariance is fully described in the Appendices, whereas, the mathematical steps followed for their solution in the cases of two and three-dimensional transport are explained in Chapters 2 and 4, respectively, finally yielding the following expressions for normalized macrodispersion analytical analogues (D∗i = Di
vADVIY): Dxx∗ (r) = σ2Y 1 − 3kIY r 1 2 − k2IY2 r2 + kIY r kIY r + 1e − r kIY ! 2D σ2 24k 4I4 Y − 4k2IY2 + 1 − 24k4IY4 + 24k3IY3 + 8k2IY2e−kIYr 3D
r r r r r r
These expressions enable benchmarking the numerical estimates obtained with the Monte Carlo method following these steps:
• Generation of multiple synthetic realizations of log-conductivity fields with a Sequential Gaussian Simulator,[5] adapted to handle statistical subordination to
generate sub-Gaussian Y fields;[6]
• Solution of the flow equation at prescribed fixed-head and no-flow boundary conditions at the inlet/outlet and lateral (and vertical in 3D) boundaries, at fixed deterministic porosity. Once multiple flow fields are available, a stabilization study for the statistical moments associated with transverse velocity has to be performed, iteratively, until they stabilize (converge to approximately constant values). This enables determining the suitable number of simulations necessary to ensure the reliability of the spatial moments associated with transverse velocity and looking for the influence width of the domain due to the presence of the boundaries (corresponding to the distance up to which the sample variance of transverse velocity is hindered by the effect of the boundary conditions). Indeed, lateral boundaries constrain transverse velocities to vanish in their proximity, which renders the associated sample moments over the number of simulations deterministic rather than stochastic. A similar effect is felt in proximity of the fixed-head boundaries, as demonstrated by the numerical results obtained in this thesis (shown in Chapters 3 and 4);
• Once the influence region has been set, this area has to be excluded from that of the particles injection window to avoid biasing macrodispersion estimates. Indeed, the selected algorithm to simulate transport - Random-Walk Particle Tracking (RWPT) - requires sampling stochastic velocities to produce reliable estimates according to the Monte Carlo approach. Considering that transport is purely advective, RWPT actually becomes Advective Particle Tracking (APT). Several
with the purpose of stopping transport simulations before particles start leaving the domain.
Numerical macrodispersion coefficients have been computed following the effective calculation scheme, which relies upon the time derivative of average particle displacement sample variance, according to:
DEF F i(t > 0) = 1 Ns Ns X k=1 1 2 d dt ˆ σ∆X2 i,k(t) t With ˆσ2 ∆Xi,k(t) t
representing the single simulation average directional displacement variance over NP particles, whose square root is typically associated with the spreading
effect felt by solute plumes. Time derivative is instead approximated with finite differences between two subsequent times (m + 1) − th and m − th:
d dt ˆ σ2∆X i,k(t) t ≈ ˆ σ2 ∆Xi,k(t) t=tm+1 − ˆσ2 ∆Xi,k(t) t=tm tm+1− tm
Synthetic results presentation
This work focuses first on two-dimensional rectangular domains at progressively increasing degree of heterogeneity, which is measured in terms of log-conductivity sills σ2
Y associated
with the isotropic exponential covariance of Y . Since the first-order approximation well applies for sills much below unity, it has been chosen to check first the degree of matching with numerical estimates for a low heterogeneity case (σ2Y ≈ 0.82) in order to validate the expressions of macrodispersion analogues. Then, two transport cases have been investigated, with σY2 ≈ 2 and σ2
pointed out by several studies, such as by de Dreuzy, Beaudoin and Erhel.[7] The flow equation has been solved setting fixed-head values hIN
0 = 6[m] and hOU T0 = 1[m] at the
inlet/outlet sections, respectively, 10−2[m/s] average conductivity and 10% porosity, which yield 10−4[m/s] advective velocity (Darcy’s law).
Figure 1: Rectangular, regularly meshed hydraulic conductivity domain and boundary
conditions to the flow equation Figure 2: Expected qualitative behaviour of transverse velocity sample variance along y
The influence width of the domain due to the presence of the boundaries ranges in [3−7]IY along y and [6−10]IY along x, depending on the conductivity sills (in agreement
with previous studies[7],[8] more heterogeneous domains are found to cause stronger
effects on transverse velocity variability). Then, after having placed the injection window adequately, the suitable number of particles and time step have been determined to simulate solute transport. It has been found that no significant evolution in dispersion trends is evidenced beyond NP ≈ 1000, which leads to assume 10000 particles are
enough to ensure broad sampling of the velocity fields, while uniform 10000[s] time steps have been set in order to ensure a sufficiently tight Di sampling against time.
Lastly, the number of simulations necessary to roughly stabilize average macrodispersion coefficients has been always set around 1000, even if it has been recognized that Ns
should ideally be increased in strongly heterogeneous cases. However, due to the huge computational effort required to reach higher orders of magnitude (Ns≈ 10000),
necessary to appreciate significant improvements in the statistical moments stabilization, the number of simulations has been always limited to nearly 1000. Indeed, despite observing that the stabilization of second order statistical moments of directional
(a) Stabilization of DL∗ against the number of Monte Carlo simulations, at parametric times
(b) Convergence study on DT∗ against dimensionless time, at parametric NP
Figure 3: Determination of transport simulations parameters for 2D strongly heterogeneous case
The same procedure has been followed to handle three-dimensional transport, which has been limited to the unique case with σ2
Y ≈ 0.82 due to the huge computational
effort required to run the flow solver across three-dimensional domains. This made necessary to slightly decrease the number of cells included per correlation length, reduced to IY ≈ 1.2[m] at fixed resolution s ≈ 0.24[m], while decreasing Nx to 300 and
Ny = Nz to 60. Keeping fixed the porosity and the flow boundary conditions as in
the two-dimensional counterpart yields an increase in the advective velocity by nearly one order of magnitude, which in turn decreases the average transit time of the solute plume across the domain to the same extent. This renders the time window narrower in 3D with respect to 2D cases, with time steps reduced to 100[s], consistent with the last observation time tN ≈ 25 at which particle tracking has been stopped. The number
of simulations necessary to visually stabilize average directional dispersions has been taken around 800, similarly to the corresponding 2D case.
The following results compare first several 2D cases at progressively increasing σ2 Y and
Figure 5: Normalized macrodispersion analogues against time at parametric α ∈ [1.5; 2] in low heterogeneity three-dimensional fields (circles represent the numerical check at α = 1.5)
Figure 6: Comparison between numerical directional dispersion coefficients in two and three-dimensional transport across low heterogeneity hydraulic conductivity fields
Conclusion
The above results clearly witness that, in agreement with previous studies on Gaussian permeability fields,[7] macrodispersion coefficients intrinsically depend on σY2 also in sub-Gaussian domains. Of course, more heterogeneous fields induce stronger spreading
mean trajectory in order to overcome extremely low conductivity regions. This effect vanishes in 2D fields at late time regime in the absence of pore-scale diffusion, whereas, an asymptotic transverse dispersion survives in 3D fields because flow lines transverse expansion results here unconfined (since the asymptote grows with σ2
Y, it results
negligible in the three-dimensional low heterogeneity case under study). The peaks of 3D transverse dispersions are instead shorter than in their 2D counterparts because particles can move here both vertically and laterally in order to overcome less conductive zones, which yields narrower transverse deflections from the average solute trajectory. Longitudinal macrodispersion shows instead relatively similar asymptotes whatever the problem dimensionality, provided that σ2
Y is small, and grows proportionally with the
latter. Any increase in the subordinator variance at fixed sill and correlation length (lower α) slows then down the rate of assessment to the asymptotic regime leaving the Gaussian-based Fickian asymptote unchanged, consistently with its dependence uniquely on the magnitude of σ2
Y. The most important result lays however in proving
that the analytical expressions derived on the basis of the first-order approximation well capture all these behaviours. An optimal agreement between numerical and analytical predictions has instead been observed at low conductivity sills, while higher σY2, despite bringing to non-negligible estimation errors on the attained longitudinal asymptote (nearly the 15% and 25% for σ2Y ≈ 2 and 3.3, respectively), lead to capture the correct orders of magnitude for both DL∗ and D∗T. The analytical model represents therefore a powerful tool to predict the time behaviour of numerical macrodispersion coefficients in mildly heterogeneous sub-Gaussian conductivity fields and constitutes the unique, strong theoretical basis for benchmarking the corresponding numerical estimates. It furthermore enables saving significant amounts of time that numerical Monte Carlo simulations would take to be run and allows capturing, even in more heterogeneous structures, the correct orders of magnitude of macrodispersions, encouraging in this way future studies eventually involving reacting transport in strongly heterogeneous
interest of groundwater literature towards the stochastic modeling of flow and transport. A growing attention has been recently devoted to the attempt of quantifying the effect of non-Gaussian frequency distributions exhibited by many hydro-geological properties, such as the hydraulic conductivity, on the subsurface transport of contaminants. The motivation behind this approach, against the classic deterministic one, is the wide spatial variability featured by those properties in natural formations, which prevents reducing transport to the simpler Gaussian case.
As part of the theoretical framework described, this master thesis focuses on the study of conservative transport in mildly heterogeneous hydraulic conductivity fields under mean uniform steady flow conditions. The crucial goals are to validate, by means of numerical simulations, the analytical expressions for Fickian macrodispersion analogues[3] developed in the context of the Generalized sub-Gaussian (GSG) model[2] by Riva, Guadagnini and Neuman for infinite two-dimensional domains and extend them to three-dimensional fields. Numerical results are post-processed with the Monte Carlo method, adopting a Sequential Gaussian Simulator (SGSIM)[5] to generate conductivity
fields conditioned to prescribed ensemble means and covariance functions, later rendering them sub-Gaussian by means of statistical subordination.[6] The flow equation is solved
numerically for large regular grids and transport is simulated via Random-Walk Particle Tracking (RWPT) applied to pure advection.
This work shows first synthetic results on two-dimensional domains at increasing degree of heterogeneity, extending later to slightly heterogeneous three-dimensional fields. These results yield good agreements between analytical and numerical estimates for mild heterogeneity, the matching worsening for markedly heterogeneous fields. Yet the analytical model reveals a successful outcome, at least for assessing the correct orders of magnitude, if heterogeneity keeps moderate. Analytical estimates can thus reasonably replace numerical ones in several cases, depending on the level of accuracy required,
Key words
• non-Gaussian hydraulic conductivity fields • heterogeneity
• stochastic modeling of conservative transport • macrodispersion
l’interesse della ricerca nel campo dell’ingegneria idraulica alla modellazione stocastica dei processi di flusso e trasporto nel sottosuolo. In particolare, si dedica sempre più attenzione allo studio del trasporto sotterraneo di contaminanti in campi di conducibilità idraulica non Gaussiani. L’adozione di modelli stocastici trova ampia giustificazione nell’estrema variabilità spaziale che i parametri idrogeologici mostrano in natura nei mezzi porosi eterogenei, rendendo impossibile ricondurre i fenomeni di trasporto al comune caso Gaussiano.
Questa tesi si propone, nel contesto sopra descritto, di studiare il trasporto conservativo di soluto in campi di conducibilità idraulica moderatamente eterogenei, subordinato a condizioni di flusso mediamente uniformi e stazionarie. L’obbiettivo primario è quello di convalidare numericamente le espressioni analitiche di macrodispersione[3] sviluppate mediante il modello Generalizzato sub-Gaussiano (GSG)[2] da Riva, Guadagnini e
Neuman per domini bidimensionali infiniti ed estenderle al caso tridimensionale. I risultati numerici sono elaborati attraverso il metodo Monte Carlo. La generazione dei campi avviene con un Simulatore Gaussiano Sequenziale (SGSIM)[5] che condiziona a valori medi e funzioni di covarianza prefissati. I campi di conducibilità idraulica sono quindi resi sub-Gaussiani attraverso una subordinazione statistica.[6] L’equazione di
flusso è risolta con metodi numerici e il trasporto simulato con algoritmi di Random-Walk Particle Tracking (RWPT) applicati a casi puramente avvettivi.
Questo testo si incentra inizialmente sull’analisi di casi bidimensionali di trasporto in campi progressivamente più eterogenei, focalizzandosi infine su un dominio tridimensionale leggermente eterogeneo. Tali simulazioni consentono di confrontare i risultati numerici e analitici circa l’evoluzione temporale dei coefficienti di macrodispersione, che rivelano un buon grado di concordanza, specialmente per campi poco eterogenei, e una discrepanza crescente all’aumentare dell’eterogeneità. É altresì osservabile che, a patto di limitare la trattazione al caso di domini moderatamente eterogenei, il modello analitico produce stime valide (almeno in termini di ordini di grandezza), trovando così applicazione
Parole chiave
• campi di conduttività idraulica non Gaussiani • eterogeneità
• modellazione stocastica del trasporto conservativo • macrodispersione
Extended Summary . . . i
Abstract . . . xi
Sommario . . . xiii
1 Introduction 1 2 Theoretical framework and numerical methodology 7 2.1 Analytical model . . . 7
2.1.1 Generalized sub-Gaussian model . . . 7
2.1.2 Flow and transport in two-dimensional sub-Gaussian hydraulic conductivity fields . . . 12
2.2 Numerical algorithms . . . 19
2.2.1 Random field generator . . . 21
2.2.2 Flow code . . . 22
2.2.3 Transport process . . . 27
3 Two-dimensional transport 33 3.1 Less heterogeneous field case of study . . . 36
3.1.1 Low heterogeneity transport case (σ2 Y < 1) . . . 36
3.2 Heterogeneous fields cases of study . . . 48
3.2.1 Mildly heterogeneous transport case (σ2 Y ≈ 2) . . . 48
3.2.2 Strongly heterogeneous transport case (σ2 Y ≈ 3.3) . . . 53
3.3 Macrodispersion comparison in increasingly heterogeneous fields . . . 59
(σ2Y < 1) . . . 67
5 Results discussion and conclusion 79
Appendix A: Auxiliary functions . . . I Appendix B: Small perturbation expansion applied to mean uniform flow cases . . . III
List of Figures . . . .XVIII List of acronyms . . . XIX
Bibliography and Websites . . . XIX Symbols and Alphabetical Index . . . .XXIII
Introduction
The study of subsurface flow and solute transport is nowadays a topic of growing interest which embeds knowledges from several disciplines and finds application in many engineering and environmental fields. Indeed, several facilities, such as nuclear waste repositories or carbon storage, as well as EOR or IOR techniques for oil and gas applications, require the adoption of subsurface transport models to predict systems future evolutions and enable designing the associated facilities.
Even considering the upstream level of the oil and gas production cycle, such models are necessary to manage the exploration and appraisal phases which include several reservoir and development feasibility studies. Also, the development phase makes use of those concepts to deal with the design of drilling wells facilities, as the production one does to rely with reservoir management and wells optimization techniques, among which the previously cited EOR and IOR play a main role.
Figure 1.2: The Svalin oil field linked to the Grane oil and gas platform: the reservoir is produced via pressure depletion and pressure support from a regional aquifer.[10]
Moving to environmental safety tasks, transport simulations in the subsurface finds applications in wellhead protection studies and analyses of contaminants migration. In all the mentioned cases, the key point is that simulating tracers or, in general, fluids transport along with a main flow direction, which is determined by existing pressure gradients, is fundamental to forecast flow rates and travel times across prescribed domains. These information are necessary to perform optimization studies while designing all the facilities typical of the oil and gas production industry and enable facing environmental protection issues associated with drilling activities.
the past achievements to increasing complexity contexts, such as the stochastic modeling of subsurface flow phenomena across more heterogeneous structures. Indeed, a strong spatial variability of hydro-geological properties characterizing porous media (which in nature rarely exhibit spatial homogeneity) impedes relying upon the common frequency distributions, like the Gaussian one, that have been often adopted in the past to study natural phenomena under the Monte Carlo approach. This master thesis has been conceived to give a contribution to the effort of extending the well validated stochastic approach of modeling hydrological phenomena to the realistic case of moderately heterogeneous formations, focusing in particular on the case of mildly heterogeneous hydraulic conductivity fields which are modeled as sub-Gaussian ones. More in detail, the attention will be devoted to the estimation procedures for large-scale (space averaged) parameters commonly employed to mathematically model flow and transport by means of differential governing equations as the steady-state flow and advective-dispersive ones. The main objective is the assessment of Fickian macrodispersion coefficients, by means of numerical simulations and following the Monte Carlo procedure, that will be compared with the recently built analytical estimates by Riva, Guadagnini and Neuman[3] for two-dimensional domains under mean-uniform, steady flow conditions. The goal is to understand the degree of consistency of the latter, which inevitably makes use of simplifying non-realistic assumptions (such as the occurrence of an infinite unbounded domain), with those obtained by means of numerical simulations that, despite being usually reliable, lack a strong theoretical basis and require a considerable computational effort. The key idea is therefore to check first the reliability of the analytical asymptotic estimates, moving later to observe the quality of the pre-asymptotic predictions and lastly to understand the threshold levels of K fields spatial heterogeneity at which the analytical model significantly fails.
The next chapter of this thesis deeply describes the theoretical framework under investigation, beginning from the basic assumptions and the problem specifications under which the analytical solution for macrodispersion analogues has been derived. A detailed presentation of the most significant mathematical steps followed by Riva et all[2][3] has
been carried out for the case of two-dimensional domains, whereas, its extension to three-dimensional fields has been autonomously developed and reported in Chapter 4.
The second chapter also includes a detailed description of the numerical methodology and the computational codes employed to carry out numerical simulations with the Monte Carlo method,[4] whose presentation for selected cases of study constitutes the
main body of this work. In particular, Chapter 3 focuses on two-dimensional flows across increasingly heterogeneous hydraulic conductivity fields, at prescribed layering and head gradient conditions adopted to solve the flow equation and for almost purely advective trasport (consistently with the theoretical framework of the analytical model). Lastly, due to the huge computational effort required to run numerical simulations across three-dimensional domains, this condition has been investigated for a unique illustrative case of study with limited level of conductivity heterogeneity across the domain, the associated results being shown in Chapter 4. This last case has been studied on a stretched domain due to computational time issues associated with the huge times required for the flow solver to handle three-dimensional fields.
It will be shown through this work how the analytical model provides optimal predictions for macrodispersions in hydraulic conductivity fields characterized by low heterogeneity, the matching with numerical estimates progressively worsening at increasing σ2Y. It will indeed be proven that longitudinal asymptotic macrodispersion grows as a function of the log-conductivity sill, which constitutes an hardly describable effect by means of the first-order approximation upon which the analytical analogues are built (beyond σ2
Y ≈ 2, the linear increase with σY2 predicted by the analytical model for longitudinal
macrodispersion analogues does not apply any more). The behaviour of transverse macrodispersion is rather better captured by the analytical model, at least in terms of asymptotic behaviours, in two-dimensional domains lacking the effects of pore-scale diffusion (the analytical analogue correctly predicts its drop to zero at late time regime). As shown by Beaudoin and de Dreuzy,[8] the decay to zero is not implied any more by the
absence of molecular diffusion if flow and transport are modeled in the three-dimensional space, where the presence of very low conductivity regions acts deflecting the flow lines structure and allows an unconfined transverse expansion of the latter. However, it will be proven in Chapter 4 how this effect remains practically negligible for fluid transport taking place across low heterogeneity fields (conductivity sill below unity).
comparison between all the achieved results for two and three-dimensional transport is lastly shown in Chapter 5.
Theoretical framework and numerical
methodology
2.1
Analytical model
The Generalized sub-Gaussian model (GSG) developed by Riva et all[2] to describe the statistics of certain non-Gaussian hydro-geological variables has found application in the literature to study subsurface flow and transport[3]in natural formations characterized by heterogeneity and wide spatial variability of the associated hydro-geological properties. The details about the statistical model are explored in the first part of this section, the following part describing its application to elementary cases of transport.
2.1.1 Generalized sub-Gaussian model
Figure 2.1: Frequency distribution of the increments for a sub-Gaussian variable (continuous curves) and its Gaussian counterpart (dashed lines), with the same mean and variance, at distinct lags[2]
The Generalized sub-Gaussian model developed by Riva et all[2] enables
describing a type of frequency distribution, exhibited by many hydro-geological variables Y and their spatial (or temporal) increments ∆Y , which cannot be reconciled with the common Gaussian ones. Indeed, it is often observed in natural formations that the increments of some parameters, such as porosity and, to larger extents, hydraulic log-conductivity Y = ln(K) , show sample probability densities characterized
by sharper peaks and heavier tails than their Gaussian counterparts. This behaviour, which becomes more accentuate as the separation distance r between pairs decreases (the parameter ρG in figure 2.1, whose physical meaning is described in the following
paragraphs, increases), is well captured by the mentioned Gaussian sub-Gaussian model, which provides a formal description to the statistics of Y . In the context of this master thesis, the generic stochastic variable Y will always represent the sub-Gaussian log-conductivity (Y = ln(K)), in agreement with the objective of studying conservative solute transport in heterogeneous hydraulic conductivity fields. The latter is therefore modeled as a Spatial Random Function (SRF), defined in terms of its ensemble mean E[Y (x)] (corresponding to the spatial average under the ergodic hypothesis, i.e. E[Y (x)] = ln(KG), with KGrepresenting the geometric mean of hydraulic conductivities
over the domain) and zero-mean random fluctuation Y0(x) about the mean value, with x = [x, y, z] representing the vector of physical space coordinates. The crucial point lays in defining the first-order perturbation as:
Y0(x) = Y (x) − E[Y (x)] = U (x) · G(x) (2.1)
Where U (x) and G(x) represent the log-normal subordinator, independent on G, and the zero-mean single-scale Gaussian SRF, respectively. The associated probability densities are: f (u) = √ 1 2πu(2 − α)e − ln2 u 2(2−α)2 with U ∼ N LN (0, (2 − α)2) (2.2) f (g) = √ 1 2πσG e− g2 2σ2 G with G ∼ N (0, σ2 G) (2.3)
The constant α governs the subordinator variance σ2
U = (2 − α)2 and, ultimately, how
far the sub-Gaussian Y0 (as well as Y ) is from the Gaussian counterpart. This parameter is limited by the value 2, which represents the ideal condition of the sub-Gaussian Y0 becoming Gaussian, consistently with the subordinator variance approaching zero. The more |α| is far from this limit, the more markedly non-Gaussianity affects the frequency distributions of the variable and its increments.
The Gaussian SRF is characterized by an exponential isotropic covariance function :
CG(r) = σG2ρG(r) = σG2 · e − r
IG (2.4)
With integral correlation length IG =
R+∞
0 ρG(r)dr and sill σ 2
G = CG(0) , ρGrepresenting
the correlation function of G(x).
It has been shown by Riva et all[3] that the variable’s marginal PDF fY0(y0) originating
from Y0 bivariate probability density, denoted as fY0 1Y20(y 0 1, y 0 2), with Y 0 1 = Y 0(x 1) and Y20 = Y0(x2), is Normal-Log-Normal (NLN) typed: fY0(y0) = Z +∞ −∞ fY0 1Y20(y 0 1, y 0 2)dy 0 2 = 1 2π(2 − α) Z +∞ 0 1 u2e −1 2( ln2 u σG (2−α)2+ y02 u2)du (2.5)
If one calculates the q − th order statistical moments of Y0, defined as:
E[Y0q] = Z +∞
−∞
Y0q · fY0(y0)dy0 (2.6)
Immediately sees that odd order ones are null, as expected observing the symmetry of Y0 probability density curve (figure 2.2), while variance and kurtosis assess to positive values.
It can be observed that the standardized Kurtosis is always ≥ 3, meaning that the marginal frequency distribution of Y0 is always leptokurtic, except for the limiting Gaussian case α → 2 (normokurtic curve) :
κY =
E[Y04] E[Y02] = 3e
4(2−α)2 ≥ 3 (2.7)
The sub-Gaussian variance results instead:
Figure 2.2: Empirical frequency distribution of the zero mean random fluctuation of a sub-Gaussian variable (dotted red curve) and its Gaussian counterpart (dashed black line), with the same mean and variance
Thus, σ2
G is amplified by the action of the subordinator, giving rise to σY2 > σG2 ∀α 6= 2.
Together with the analytical form of fY0(y0), also the marginal frequency distribution
of the increment ∆Y (x1, x2) = ∆Y0(x1, x2) = U (x1) · ∆G(x1, x2) has been derived
by Riva et all under the assumption of second order statistic stationarity:
f∆Y(∆y) = Z +∞ −∞ fY0 1Y20(y 0 2+ ∆y 0 , y20) = 1 2π2(2 − α)2 r π 2 Z +∞ 0 Z +∞ 0 e− 1 2( ln2u1 σG+ln2σGu2 (2−α)2 + ∆y02 u21+u22−2u1u2ρG) u1u2pu21+ u22− 2u1u2ρG du1du2 (2.9)
Once again, the sole even order statistical moments survive, the focus being particularly on E[(∆Y )2], which enables deriving closed-form mathematical expressions for the
variogram (specular of the covariance under validity of the intrinsic hypothesis) of the sub-Gaussian Y . Under the assumption of statistic isotropy, which makes the dependence move from x1, x2 to the lag r = |x1 − x2|, the following applies:
E[(∆Y (r))2] = 2e(2−α)2(e(2−α)2 − ρG(r))σ2G (2.10)
It is now recalled that the variogram γY(r) of Y is defined as :
γY(r) =
E[(∆Y (r))2]
2 (2.11)
It is linked by definition to the Y covariance, under the assumption of weak statistic stationarity, by:
CY(r) = σY2 − γY(r) (2.12)
As well as the covariance of G is linked to γG by:
CG(r) = σG2 − γG(r) (2.13)
Making the correlation function of G(x) explicit as:
ρG(r) = 1 −
γG(r)
And substituting 2.14 in 2.10, and the resulting expression in 2.11, one finds: γY(r) = σG2e (2−α)2 (e(2−α)2 − 1) | {z } nugget +e(2−α)2γG(r) = σY2 − e(2−α)2(σG2 − γG(r)) | {z } CG(r) (2.15)
Substituting equation 2.15 into 2.12, it is finally possible to correlate the covariance function of the sub-Gaussian Y with that of the normal SRF from which the former has been generated:
CY(r) = e(2−α)
2
CG(r) = E2[U ]CG(r) (2.16)
This expression evidences that the subordinator amplifies without destroying the covariance structure of G.
The last parameter that has been expressed as a function of its Gaussian counterpart is the integral correlation scale of Y . Recalling its definition:
IY =
Z +∞
0
ρY(r)dr (2.17)
and introducing, following Riva et all,[2] the constant k = E[UE2[U ]2] = e
(2−α)2 , it can be written: CY(r) = E[U2]σ2G | {z } σ2Y/k ρG(r) (2.18) ρY(r) def = CY(r) σ2 Y = 1 ke − r IG (2.19)
Substituting equation 2.19 into 2.17 finally yields the desired expression which relates IY with IG: IY = IG k = e −(2−α)2 IG < IG ∀α (2.20)
Which highlights that the integral correlation scale of G is rather dampened by the action of the subordinator, which reduces the distance up to which the center-cell values of K are still reciprocally correlated.
2.1.2 Flow and transport in two-dimensional sub-Gaussian hydraulic conductivity fields
The effects of non-Gaussian heterogeneity on subsurface transport have been investigated for steady mean-uniform flows by Riva, Guadagnini and Neuman,[3] who have extended
the analytical results developed by Dagan,[1] as part of his discussion on the direct problem (determination of head fields h(x) in fixed domains of prescribed hydro-geological properties) for Gaussian permeability fields, to the case of sub-Gaussian domains. The general procedure to derive the expressions of macrodispersion analogues is briefly recalled in this section, referring the reader to Appendices A and B for more details. The analyzed framework is a non-Gaussian case of conservative (non reactive) solute transport in an unbounded, infinite two-dimensional domain with the main flow direction parallel to the longitudinal axis. Transport, modeled at the macroscale, is advection-dominated, with molecular diffusion Dd kept absent (Peclet number P e = Lx· vADV/Dd approaching infinitive, with Lx representing the characteristic length of the
problem). Advection and mechanical dispersion are thus the sole mechanisms, physical and fictitious, respectively, which drive the solute transport along the domain. The latter is represented by a sub-Gaussian log-conductivity field with deterministic prescribed porosity φ . Such field is statistically described by its ensemble mean, coinciding with the spatial geometric average under the ergodic hypothesis (KG = eE(Y )= QNi (Ki)
1/N
), and the exponential isotropic covariance function with sill σ2
Y and correlation range
IY. The problem is formulated assuming the validity of the intrinsic hypothesis, which
enables considering K field statistics completely defined by unique values of ensemble mean and variance (at least the first two orders statistic moments being constant regardless the physical space coordinate). Furthermore, in order to apply the small perturbation approach (which is described in details in Appendix B) introduced by Dagan[1] on Y0 and ∇h, which appears in the steady-state subsurface flow equation,
Riva et all[2] approximate Y0 truncating its domain of definition from (−∞; +∞) to (−a; a], with |a| 0, to render the even-order statistic moments of Y0 convergent.
the concept of Darcy velocity (q(x)):
∇Tq(x) = 0 ; q(x) = v(x)
φ = − k(x)
φ · ∇h(x) (2.21)
The steady-state flow equation becomes:
∇T(K(x) · ∇h(x)) = ∇T(eY (x)· ∇h(x)) = 0 (2.22)
Recalling that the flow is uniform in the mean, one can define the mean hydraulic gradient J as E[∇xh(x)] = ∇xE[h(x)]. Expressing Y (x) = E[Y (x)] + Y0(x) and
h(x) = E[h(x)] + h0(x) and expanding all the variables in series of σ2
Y up to the
first-order in σ2
Y,[1] it is possible to rearrange the expressions of conductivity - head
cross-covariance CY h(r) , head autocovariance Ch(r) and velocity components covariance
matrix Cvivj(r), with main focus on its diagonal components Cvxvx(r) and Cvyvy(r).
One can then derive the analytical expressions for longitudinal and transverse particles displacement covariances CXxx(r) and CXyy(r) , which in turn allow determining the
corresponding macrodispersion analogues Dxx(r) and Dyy(r) . Considering that the
mean flow is longitudinal (along x), the dependence of dispersion coefficients on physical time can be successfully converted into longitudinal travel distance dependence if one takes the lags oriented along with the main flow direction. Indeed, the separation distance is directly linked with the physical time by means of the mathematical expression describing the mean flow trajectories: r = vADVt .
The set of equations used to derive the mentioned covariances, which come from rearrangements of Dagan’s results[1] applied to two-dimensional domains, are shown in
details in Appendix A and B. This procedure leads to the following results:
CY(r) =
σ2Y k e
− r
kIY (2.23)
Two auxiliary functions P (r) and Q(r) are introduced to mathematically handle the derivation of conductivity - head cross-covariance and head autocovariance beginning from their relationships with CY(r), which exhibit a dependence on certain powers of
functions of the conductivity isotropic covariance CY(r), as described in Appendix A: 1 rn d dr rndP (r) dr = −CY(r) (a.3) 1 rn d dr rndQ(r) dr = P (r) (a.4)
Referring in this section to two-dimensional domains (n = 2), the above equations lead to: dP (r) dr = − 1 r Z r 0 zCY(z)dz (2.24) dQ(r) dr = 1 r Z r 0 zP (z)dz (2.25)
The auxiliary functions become (upon solving the above integrals):
P (r) = kIY2σ2Y(1 − E − e−kIYr − ln ( r kIY ) + Ei(− r kIY )) dP (r) dr = σ 2 YIY (kIY r + 1)e −kIY r − kIY r (2.26) dQ(r) dr = r 2P (r) − k3σ2 YIY4 4r 6 − r 2 k2I2 Y − 2e−kIYr (3 + 3 r kIY + r 2 k2I2 Y ) (2.27)
Where E ≈ 0.5772 denotes the Euler-Mascheroni constant , ln(z) the natural logarithm of z and Ei(z) =R+∞
−z − e−t
t dt the exponential integral of z , which is well approximated
by the infinite series expansion, often truncated at the second term, limz→0Ei(z) ∼
E + ln z for z approaching zero.
The second derivative of Q(r) can be computed either directly deriving the first derivative, or making use of equation a.6 for two-dimensional domains (d2drQ(r)2 = P (r) −
1 r dQ(r) dr ): d2Q(r) dr2 = kσY2IY2 2 1 2 − E − ln ( r kIY ) + Ei(− r kIY ) + 3k 2I2 Y r2 − e − r kIY(2 + 3kIY r (1 + kIY r )) (2.28)
Making use of expressions 2.26, 2.27 and 2.28, conductivity - head cross-covariance and head autocovariance (variogram) can be expressed in closed-form as :
CYh(r) = J σY2IY kIY r (1 + r kIY )e− r kIY − 1 (2.29) γh(r) = γhT(r) + (J · r)2 J2r2 γhL(r) − γhT(r) = −Ch(r) (2.30) With: γhL(r) =J 2kI2 YσY2 2 −1 2+ E + ln( r kIY ) − Ei(− r kIY ) − 3k 2I2 Y r2 + (2 + 3 kIY r (1 + kIY r ))e − r kIY (2.31) γhT(r) =J 2kI2 YσY2 2 −3 2+ E + ln( r kIY ) − Ei(− r kIY ) + 3k 2I2 Y r2 − 3 kIY r (1 + kIY r )e − r kIY (2.32)
The variogram of hydraulic head, specular of its autocovariance under the assumptions of second order statistic stationarity and sill σ2
h = 0, can be decomposed into longitudinal
and transverse components,[1] identified by the corresponding orientation of the lag. Equations 2.31 and 2.32 can be combined to obtain the general expression of the head variogram: γh(r) = J2kIY2σ2Y 2 2(J · r) 2 J2r2 − 1f ( r kIY ) + g( r kIY ) (2.33) With: f ( r kIY ) = 1 2 − 3 k2I2 Y r2 + (1 + 3 kIY r + 3 k2I2 Y r2 )e − r kIY (2.34) g( r kIY ) = E − 1 + e−kIYr − Ei(− r kIY ) + ln( r kIY ) (2.35) And: γhL(r) =J 2kI2 YσY2 2 f ( r kIY ) + g( r kIY ) (2.36) γhT(r) =J 2kI2 YσY2 2 −f ( r kIY ) + g( r kIY ) (2.37)
The diagonal terms of velocity components covariance matrix are computed applying equations b.53 and b.59, which hold for lags aligned with the main flow direction:
Cvx,vx(r) = (v2ADVIY2σY2 3k r2 1 2 − 3 k2I2 Y r2 + 1 + 3 kIY r + 3 k2I2 Y r2 e − r kIY r > 0 vADV2 σ2Y 1 − 5 8k r = 0 (2.38) Cvy,vy(r) = v 2 ADVI 2 Yσ 2 Y k r2 9k 2I2 Y r2 − 1 2− 9 k2IY2 r2 + 9 kIY r + 4 + r kIY e−kIYr (2.39)
Observing that in limr→0Cvx,vx(r > 0) =
3 8kv
2
ADVσY2, which does not coincide with the
value of Cvx,vx(r = 0) except for the limiting Gaussian case (k = 1), leads to conclude that
the flow equation, solved for sub-Gaussian log-conductivity fields, yields a longitudinal velocity covariance which features a clear nugget(1−8k5 )−8k3v2
ADVσ2Y = (1− 1 k)v
2
ADVσY2.
Instead, transverse velocity covariance does not feel the effect of any nugget because the value attained in r = 0 coincides with the right limit limr→0Cvy,vy(r) =
1 8kv
2 ADVσY2.
In agreement with the procedure described by Dagan,[1] the following step is the
derivation of particle displacement covariance CX(t) under the assumptions of purely
advective transport and validity of the first-order approximation in σ2
Y. The rationale
behind this is that for σ2
Y = 0, i.e. for a homogeneous formation, particle displacement
covariance vanishes and spreading - whether present - occurs due to pore-scale dispersion solely. Of course, for small log-conductivity sills (σ2
Y << 1) it makes sense to consider
displacement covariances well represented by first-order asymptotic expansions in σ2 Y,
which means that both longitudinal and transverse displacement covariances can be approximated by O(σ2Y). Recalling the mathematical expression of particle displacement vectors:
dX(t)
dt = v(t) ; X(t) = X0(t = 0) + v(t) · t (2.40) And assuming the correlation between velocities at distinct times t1 and t2just dependent
on the time lag t1− t2 = t, the following equation has been derived by Dagan:[1]
CX(t) = Z t 0 Z t 0 Cv(t)(t1− t2)dt1dt2 = 2 Z t 0 (t − τ )Cv(t)(τ )dτ (2.41)
As long as the lags are kept along the mean trajectory r = vADVt, one can proceed via
variable substitution to switch from time to lag dependence (τ = v z
ADV, dτ = dz vADV): CX(r) = 2 Z r 0 r vADV − z vADV Cv(z)(z) dz vADV (2.42) = 2 v2 ADV Z r 0 r − zCv(z)(z)dz (2.43)
Being interested in longitudinal and transverse displacements covariances, the sole diagonal components of CX(r) are derived in this work:
CX(r) = CXxx(r) CXxy(r) CXyx(r) CXyy(r) (2.44) CXxx(r) = 2 v2 ADV Z r 0 r − zCvx,vx(z)dz (2.45) CXyy(r) = 2 v2 ADV Z r 0 r − zCvy,vy(z)dz (2.46)
The above expressions yield:
CXxx(r > 0) = 3kσ 2 YI 2 Y 1 2− E + 2 3 r kIY − ln( r kIY ) + Ei(− r kIY ) −k 2I2 Y r2 + kIY r kIY r + 1e − r kIY ! (2.47) CXyy(r) = kσ 2 YI 2 Y E − 3 2+ ln( r kIY ) − Ei(− r kIY ) + 3k 2I2 Y r2 − 3 kIY r kIY r + 1e − r kIY ! (2.48)
It is now possible to compute macrodispersion analogues applying the following equation, which relates the first-order approximation of displacement covariances to the structure of flow variables (Dagan[1]):
D(t) = 1 2 dCX(t) dt = vADV 2 dCX(r) dr (2.49)
With D(r) representing the macrodispersion tensor. Once again the focus is on the sole diagonal components:
D(r) = Dxx(r) Dxy(r) Dyx(r) Dyy(r) (2.50) Dxx(r) = vADV 2 dCXxx(r) dr (2.51) Dyy(r) = vADV 2 dCXyy(r) dr (2.52)
These equations yield the following expressions for longitudinal and transverse dimensional dispersion coefficients: Dxx(r) = vADVIYσY2 1 − 3kIY r 1 2 − k2IY2 r2 + kIY r kIY r + 1e − r kIY ! (2.53) Dyy(r) = vADVIYσY2 kIY r 1 2 − 3 k2I2 Y r2 + 1 + 3 kIY r + 3 k2I2 Y r2 e − r kIY ! (2.54)
Whose normalized formats are expressed as:
D∗xx(r) = Dxx vADVIY = σ2Y 1 − 3kIY r 1 2− k2IY2 r2 + kIY r kIY r + 1e − r kIY ! (2.55) Dyy∗ (r) = Dyy vADVIY = σ2Y kIY r 1 2− 3 k2I2 Y r2 + 1 + 3 kIY r + 3 k2I2 Y r2 e − r kIY ! (2.56)
With Dxx∗ (r) and Dyy∗ (r) representing longitudinal and transverse macrodispersion analogues, respectively . If one needs to switch from dimensional lag dependence to dimensionless times t∗, it is sufficient to replace the former with the latter, according to:
t∗ = t · vADV IY
= r IY
(2.57)
Which physically describes the number of correlation lengths traveled by the particles plume at each time instant by advection (referring to the local position of the plume
center of mass). Normalized macrodispersion analogues finally become: Dxx∗ (t∗) = σY2 " 1 −3k t∗ 1 2 − k2 t∗2 + k t∗ 1 + k t∗e −t∗ k # (2.58) Dyy∗ (t∗) = kσ 2 Y t∗ " 1 2− 3k2 t∗2 + 1 + 3k t∗ 1 + k t∗ e−t∗k # (2.59)
2.2
Numerical algorithms
Numerical simulations are carried out according to the Monte Carlo[4] approach, whose rationale is to produce a sufficiently large number N s of synthetic realizations of a stochastic process (here flow and transport across a conductivity field), such that the system’s behaviour is simulated by means of a wide collection of observations that enable approximating the real frequency distribution of the phenomenon. When such sample is sufficiently large, the statistical moments of the parameters of interest stabilize to asymptotic values and moments-based governing equations eventually relying upon those parameters become reliable with certain levels of confidence. In particular, since this thesis focuses on the solution of flow and transport equations across heterogeneous permeability domains, the Monte Carlo method requires generating multiple hydraulic conductivity K fields and solving the associated governing equations. Each permeability domain represents an equally probable realization of a Spatial Random Function (SRF) Y (x) = lnK(x) = E[Y (x)] + U (x)G(x), conditioned to certain probability densities, here Gaussian sub-Gaussian, by means of a Sequential Gaussian Simulator (SGSIM)[6][5]
made available by the Polytechnic University of Milan. The random generator builds single-scale Gaussian fields G(x) via sequential conditioning, embedding the possibility of rendering them sub-Gaussian by means of statistical subordination,[11] i. e. multiplying each discrete value of G(x) assigned to grid cells by a random draw of the SRF U (x), taken log-normal with zero mean and variance σU2 = (2 − α)2, the resulting Y (x) field becoming sub-Gaussian. The flow equation is numerically solved for each conductivity field under prescribed boundary conditions and porosity φ (deterministic), the result being an equal number Ns of hydraulic head and seepage velocity fields. The last step
simulated by means of Random-Walk Particle Tracking (RWPT) applied to the case of pure advection, the output being N s collections of statistical moments associated with directional particles displacements at several times. The algorithms adopted for solving flow and transport equations have been provided from the Polytechnic University of Milan, in cooperation with the University of Strasbourg, as open source format codes. It is relevant to highlight that generating multiple log-conductivity domains and solving the associated governing equations N s times is necessary to ensure the stabilization of the first two orders statistical moments associated with velocity components and dispersion coefficients (expressed as functions of directional particle displacement variances σ2
∆Xi). The stabilization criteria adopted to determine whether,
or not, convergence to asymptotic values is reached are both qualitative (monitoring the graphical behaviour of statistic moments plotted against N s) and quantitative. The latter implies providing the confidence levels (1 − αc) at which the first order statistical
moment stabilizes within a certain interval of values, as described by Guadagnini and Ballio,[4] relying upon the classical methods of statistical inference. In particular, once
the variable of interest X, monitored in a fixed point, is attributed a sample cumulative density, one checks if this CDF can be approximated by a normal one (e. g. employing the Shapiro-Wilk Test of the Null Hypothesis for composite normality[12]). In that case, the confidence interval CI1−αc around the mean value for the generic random variable
X can be expressed as:
CI1−αc(X) = Xn− N0,1(1 − αc 2 ) (σX)n √ n ;Xn+ N0,1(1 − αc 2 ) (σX)n √ n n ≥ 30 Xn− tn−1(1 − αc 2 ) (σX)n √ n ;Xn+ tn−1(1 − αc 2 ) (σX)n √ n n < 30 (2.60)
Where Xnand (σX)nrepresent the sample mean and standard deviation over n collected
data and N0,1(1 − α2c) = z determines the value z such that:
P Z ≤ z = 1 −αc
2 ; with Z ∼ N (0, 1) (2.61) While tn−1 refers to the T-Student distribution with n − 1 degrees of freedom.
inequality: CI≥1−1 a2 (X) = Xn− a σX √ n;Xn+ a σX √ n (2.62)
Where σX represents the standard deviation of X.
2.2.1 Random field generator
The random generator builds two or three-dimensional log-conductivity domains (rectangles or parallelepipeds, respectively) according to a Sequential Gaussian Simulation
algorithm[5] . The code generates a random sequence (discrete values) gi = g(xi), with
i = 1, 2, ..., NG, associated to a finite number NG of points on a real domain, randomly
extracted taking into account the probability density function of G(x). Indeed, once physical data have been transformed into normal-score data, those values are allocated to the simulation domain. Then, a random path is generated across several grid nodes. An algorithm visits and assigns a new discrete gi value to each node of this path,
conditioning any new assignment to a sample, local conditional PDF, which is ideally rebuilt step by step considering all the previous data. This is done conditioning new values to the sample variogram of G(x) at target lags, this operation being repeated at several lag values until the first realization of log-conductivity field is complete. Since this thesis requires simulating transport across large computational domains, with number of cells showing an order of magnitude close to one million, it is necessary to dampen the computational time associated with fields generation. This objective is pursued limiting the number of already allocated values gi taken into account to
condition the following allocations, once a sufficient number of gi has already been
assigned. Once the first realization of G(x) is complete, NG random values ui, with
i = 1, 2, ..., NG, are drawn from the subordinator PDF U (x), which is independent on
G(x) and identically - in this thesis log-normally - distributed. Multiplying each gi
by ui, a sequence of discrete values yi = uigi is obtained, which constitutes a sample
realization of the sub-Gaussian log-conductivity SRF Y (x). Repeating these steps Ns
times, it is possible to obtain the desired number of Y fields necessary to produce reliable estimates according to the Monte Carlo approach. This number has to be
selected with an iterative procedure, starting with a limited number of simulations and progressively increasing it until the desired accuracy is achieved. A detailed description of this procedure is shown in the next paragraph, but it is crucial to anticipate from now that numerical estimates produced in this work will not rely upon perfectly achieved stabilizations due to the huge computational times necessary to solve the flow equation over huge domains. Indeed, referring to equations 2.60 and 2.62, it is evident that the reliability exhibited by sample means of target random variables X depends on the term
(σ√X)n
n . This means that increasing the accuracy of the sample mean assessed on the
basis of n Monte Carlo simulations significantly would require adding at least an equal number of simulations (for example, halving the width of the confidence interval about the mean value assessed on the basis of n Monte Carlo simulations requires switching to 4n Monte Carlo realizations at the following iteration). Since the computational domains considered in this thesis are huge, statistical moments stabilization is visually determined without acting on progressive orders of magnitude of N s whenever the latter exceeds 1000. This means that the number of simulations has been practically selected in the range [0; O(1000)] due to the enormous computational effort required for the purpose of this work.
Figure 2.3: Two-dimensional hydraulic log-conductivity field Y (x) = ln(K(x))
2.2.2 Flow code
The solution of the subsurface flow equation takes place for sufficiently large domains that should simulate, as close as possible, the infinite size condition which represents the basic
Indeed, as often done in the literature, the domain is here subjected to fixed-head and no-flow boundary conditions on the orthogonal and parallel faces to the main flow direction, respectively. Considering for explanatory purposes a two-dimensional domain, the previous statement can be mathematically translated into a partial differential equation subjected to an equal number of Dirichlet and Neumann boundary conditions:
∇T(K(x) · ∇h(x)) = 0 h(0, y) = hIN0 x = 0, ∀y h(Lx, y) = hOU T0 x = Lx, ∀y qy y=0 = −K dh dy y=0= 0 y = 0, ∀x qy y=Ly = −K dh dy y=Ly = 0 y = Ly, ∀x
This problem is solved via discretization of the domain and adoption of a sparse matrix solver (MA57) which applies a multifrontal method, i. e. a direct method based on a variant of Gauss-Seidel elimination,[13] to the linear system of NCELLS equations
resulting from rearranging the governing equation with finite element schemes. Indeed, the flow domain is composed of a total number of cells NCELLS = NxNy, with Nx =Lx/s
and Ny = Ly/s representing the number of nodes along x and y, respectively, s denoting
the uniform grid resolution .
Figure 2.4: Two-dimensional rectangular domain over which the flow equation is solved
Figure 2.5: Example of head field
Velocity components are then estimated at cells interfaces by means of a semi-analytical algorithm, such as the Pollock one,[14] until the fluid exits the domain. In order to
enhance the algorithm stability, the equivalent permeability at the interface is computed as the geometric average of center-cell values between neighbouring elements.
necessary to avoid extremely large conductivity contrasts between adjacent cells during simulations, in order to prevent the generation of inconsistent results with the physics of the problem (such as head fields not mirroring the prescribed fixed-head values), that should inevitably be disregarded. This requirement leads to fix minimum and maximum thresholds values of α and σ2
G, respectively, that can be simultaneously applied to
numerical simulations at αM IN = 1.5 and σ2G,M AX = 2, whose combination gives rise to
σ2
Y,M AX ≈ 3.3. Even if this limitation could seem in principle to affect the completeness
of this work, this is not the case. Indeed, considering the purpose of this text, one can observe that the macrodispersion analogues under validation are expected to apply rigorously only in the validity range of the first-order approximation, in agreement with Dagan[1] and Riva, Guadagnini and Neuman[3] (σ2
Y << 1). Relevant discrepancies
from numerical results are thus predicted even at σ2
Y << 3, which guarantees that the
mentioned limitations do not hinder the completeness of this work.
The approach followed to post-process numerical results needs to account for one last stuff. As discussed in many scientific papers, such as by Ramasomanana, Younes and Ackerer[15] or de Dreuzy, Beaudoin and Erhel,[7] the presence of the boundary conditions applied to the flow equation affects the stochastic nature of the problem, at least up to a certain distance from the boundaries . This aspect needs to be deeply investigated before running transport simulations in order to avoid biasing numerical macrodispersion estimates.
Figure 2.6: Influence regions of the domain due to the presence of the boundaries
domains randomly generated, velocity fields resulting from solving the flow equation across those domains should share the same stochastic nature in the absence of any external bias. This means that the second order sample statistical moment (variance) associated with v(x) over the number of Monte Carlo simulations should ideally assess to positive values across all the domain. While this condition is supposed to apply in the hypothetical unbounded case, it has been widely witnessed in the literature that it fails under the influence of the boundary conditions. Indeed, the presence of the boundaries hinders the spatial variability of local velocities in their proximity. The key point becomes thus to infer the width of the affected region to exclude it from particles injection window. Considering that the mean flow is horizontal and the lateral faces are subjected to no-flow boundary conditions, transverse velocity components are constrained to vanish at the lateral boundaries, which renders the velocities here purely longitudinal. However, despite the deterministically fixed orientation of velocity vectors, their longitudinal component still shows a degree of freedom associated with its magnitude. On the contrary, transverse velocity becomes deterministic close to the boundaries. Furthermore, being the longitudinal extension Lx much larger than the
transverse Ly, determining the transverse length of influence dy becomes predominant
over setting dx appropriately (because a larger margin exists in terms on longitudinal
correlation lengths included in the simulation domain). The correct procedure to appreciate the real extent of the affected region is therefore to check first the behaviour of transverse velocity sample variance along y and secondly along x (at fixed sections of the domain, arbitrarily taken to be identified by the middle axes x =Lx/2 and y = Ly/2,
respectively). This process requires first performing a stabilization study on vy sample
statistical moments in a monitoring point (selected at the center of the domain), up to the second order, to determine the suitable number of simulations which guarantee a correct convergence of such moments to approximately constant values.
Figure 2.7: Monitoring axes for the study of transverse velocity sample variance
The definition of sample variance and mean associated with transverse velocities is recalled below: ˆ σv2y(x) = 1 Ns− 1 Ns X i=1 vy(x) − µvy(x)Ns 2 (2.63) ˆ µvy(x) = 1 Ns Ns X i=1 vy(x) (2.64)
The former is expected to show a qualitative trend like in figure 2.8, i. e. to oscillate around approximately constant values in the central region of the domain and drop to zero in proximity of the boundaries.
Figure 2.8: Expected qualitative behaviour of transverse velocity sample variance along y
direction, allows determining the horizontal length, downstream of the inlet boundary, at which the injection window can be suitably located (dx in figure 2.6).
2.2.3 Transport process
Figure 2.9: Particles injection window in 3D space
Simulating subsurface transport is the last step necessary to come up with numerical macrodispersion estimates. Recalling the inert nature of the solute and referring to purely advective transport (P e → ∞), the classical Advective-Dispersive Equation (ADE), which is the governing equation for non reactive transport phenomena in porous
media modeled at the continuum scale, becomes purely advective:
AA φ∂c(x, t) ∂t = −φ∇AA x c(x, t)v(x) +A A φ XX XX XX XX XXX X ∇x D(x)∇xc(x, t) = −∇x c(x, t)v(x) (2.65) With c(x, t) denoting the solute concentration field, v(x) the local fluid velocity and D(x) the diffusive-dispersive tensor (absent by hypothesis). Referring to this equation, it is possible to highlight that transport takes place by pure advection, which is the sole mechanism that physically drives particles migration processes (here occurring along the fluid streamlines). Molecular diffusion, which physically represents the upscaled effect of Brownian random motions occurring at the pore-scale, is instead absent. The occurrence of certain mechanical dispersion (the purpose of investigation for this text)
simply arises from the medium heterogeneity, which prevents local fluid velocities from being uniform over the domain, but is not associated to any physical process actually taking place at the pore-scale (it is often referred to as a fictitious transport mechanism arising from the mathematical rearrangements applied during the upscaling procedure). Transport is numerically simulated according to the Lagrangian perspective, which requires following each particle inside the domain and tracking its motion. The algorithm selected is Random-Walk Particle Tracking , which is simplified to Advective Particle Tracking due to the absence of molecular diffusion. Particle displacement is simulated at each time instant according to the following rule:
∆X(dt) = X(t = t0 + dt) − X(t0) = v(t0)dt + XX XX XX XX p 2Dddt · · r (2.66)
With v(t0) defining the local velocity of the single particle at time t0, representing a
random draw from the standard Gaussian distribution, r a unit vector with random orientation, X(t) the particle position at time t and ∆X(dt) the particle displacement occurring in the interval dt. Obviously, the assumption of high Peclet numbers cancels out the diffusive term, which is simply kept null, and each particle position is tracked in time by means of a semi-analytical algorithm, like the Pollock one.[14] Each particle’s
pathway is computed as the summation among a series of analytically - determined path lines: known the starting location at t0, the particle is assigned to a grid cell, velocity
components at a cell face are computed and the potential exit faces selected and then, regardless the precise trajectory followed by the particle across the domain, the shorter time interval required to exit the cell is calculated and assumed to be the travel time for crossing the cell. This allows determining both the exit point from the cell and the associated velocity components. Repeating this procedure until the exit point lies outside the domain enables building an approximate pathway for the particle and, upon NP repetitions of the algorithm, tracking the motion of target numbers of particles
across the domain, so that the average plume behaviour can be successfully extrapolated. To this purpose and following other studies, such as the already mentioned scientific papers of Ramasomanana, Young and Ackerer[15] and de Dreuzy, Beaudoin and Erhel,[7]
