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Transport in Porous Media

, Volume 126, Issue 1, pp 115–137 | Cite as

Analysis of Travel Time Distributions for Uncertainty Propagation in Channelized Porous Systems

  • Olga FuksEmail author
  • Fayadhoi Ibrahima
  • Pavel Tomin
  • Hamdi A. Tchelepi
Article
  • 148 Downloads

Abstract

In the context of stochastic two-phase flow in porous media, one is often interested in estimating the statistics of fluid saturations in the reservoir. In this work, we show how we can efficiently compute the probability distribution of water saturation in channelized porous systems by pursuing the methodology proposed in Ibrahima et al. (Transp Porous Media 109:(1):81–107, 2015; Comput Geosci 22:(1):389–412, 2018). One of the key metrics of the developed distribution method, called the frozen streamline (FROST) method, is the logarithm of time-of-flight (log-TOF). Therefore, we dedicate a significant portion of the paper to study the cumulative distribution function and probability density function of the log-TOF. We also demonstrate that a Gaussian mixture model can be applied for parametrization of the probability distribution of the log-TOF random field, which is expected to lead to significant computational speedup. We compare the results of the saturation statistics obtained from both the FROST method and exhaustive Monte Carlo (MC) simulations of coupled two-phase flow and transport (i.e., streamlines may change with time), and we show that the agreement between the two approaches is good and improves with grid refinement. Finally, we compare FROST with MC simulations that correspond to the exact assumptions used in FROST (i.e., the streamlines are time independent), and we demonstrate excellent agreement.

Keywords

Stochastic porous media Two-phase flow Uncertainty propagation Saturation distribution Channelized systems 

1 Introduction

Simulation of multiphase flow in realistic porous systems often involves large-scale domains and complex media structures. For example, reservoir simulation aims at predicting the state of the reservoir by estimating the evolution in time of fluid saturation fields under active field development. The reservoirs can be modeled using millions of cells and can have complex geology. Due to scarce and expensive data acquisition processes, detailed knowledge of the geological information is rarely available. The uncertain properties of the media translate into significant uncertainties in how flow evolves with time, thus encouraging a stochastic treatment of the saturation fields (Dagan and Neuman 1997; Zhang 2002), especially through the study of saturation probability density functions (PDF) and cumulative distribution functions (CDF), or saturation statistical moments.

Stochastic treatment of subsurface flow has been an active area of research for a long time. Early efforts in analyzing stochastic subsurface flow and geostatistics can be found in Gelhar (1986) and Kitanidis (1997), leading to today’s extensive research on risk management and uncertainty quantification in subsurface flow (see, e.g., Tartakovsky 2013).

Four main approaches have been used in recent stochastic subsurface flow research. The most general one is the Monte Carlo (MC) method where flow statistics are obtained by first sampling the stochastic subsurface input, then running the forward model to obtain flow realizations, and finally aggregating those flow realizations into statistical metrics such as statistical moments or distributions. Due to its simplicity, MC is often used as a benchmark against more sophisticated methods (Chang et al. 2015). However, due to its poor statistical convergence properties, MC often requires a large number of samples to be meaningful, resulting in high computational cost. This issue has been addressed by exploiting structures in the stochastic problem, for instance, in Müller et al. (2013) with multilevel MC or in Dodwell et al. (2015) with multilevel Markov chain MC.

Another approach for estimating uncertainty is the statistical moment equation (SME) method, which is based on low-order approximations. In this method, deterministic partial differential equations (PDEs) for the statistical moments of the response variable are obtained by averaging of stochastic equations. The SME method has been used for two-phase flow for almost two decades, with early works by Zhang and Tchelepi (1999) and Zhang et al. (2000). While it was exposed in Jarman and Russell (2002) that the method is unreliable for large uncertainties, further refinements have been studied such as in Liu et al. (2007) or Likanapaisal (2010).

Stochastic spectral methods are also widely used for assessing the uncertainty in subsurface flow. This class of methods relies on solving PDEs for statistical moments by truncating a stochastic basis representation of the input data and solving for deterministic coefficients by projection. For instance, polynomial chaos was used in Sochala and Le Maître (2013) for one-dimensional Richard’s equations, but more recent developments tackle multiphase flow in higher dimensions using Galerkin methods (Pettersson and Tchelepi 2016) or astute transformations before applying a collocation method (Liao and Zhang 2013, 2014, 2016). These methods mainly suffer from the so-called curse of dimensionality, implying that the number of coefficients needed to be estimated grows exponentially with the dimension of the problem. The sparse grid approach has been among the recent techniques that attempt to alleviate this limitation (see, e.g., Liao et al. 2017).

Lastly, there are distribution methods focused on estimating the PDF or CDF of the fluid concentration or saturation. They are often achieved by, once again, deriving PDEs for the concentration/saturation PDF (see, e.g., Meyer et al. 2010; Tartakovsky and Broyda 2011; Venturi et al. 2013; Boso and Tartakovsky 2016) or CDF (see, e.g., Wang et al. 2013; Tartakovsky and Gremaud 2016). Other approaches directly estimate the concentration PDF using for instance particle-based MC simulations (see, e.g., Meyer and Tchelepi 2010; Meyer et al. 2013). While some of the PDE formulations can be solved exactly, often a closure approximation is needed, leading to limitations in the scope of uncertainties similar to those of SME methods.

It was shown in recent work (Ibrahima et al. 2018, 2016) that estimating the saturation PDF can be achieved efficiently by considering two key metrics: the PDF of the logarithm of time-of-flight (log-TOF), the time it takes for a particle to travel from a source to a specified position; and the statistics of the equivalent injection time (EIT), which accounts for the two-phase flow effects. This method, referred here as the FROST method, builds a fast and accurate numerical method for uncertainty propagation so as to provide a reliable risk assessment of fractional flow in the field. Hence, the study of log-TOF distributions in highly heterogeneous porous systems is the main topic of this work.

Most of the uncertainty quantification approaches in the literature assume underlying stationarity for the input random models. For example, it is common to assume that the permeability field is a log-Gaussian with exponential covariance (Liao et al. 2017; Pettersson and Tchelepi 2016; Zhang et al. 2000; Meyer et al. 2010). Yet, geological realism is an increasingly important aspect of the uncertainty assessment in multiphase flow simulation. Geological properties of some subsurface systems are known to have a multi-scale and heterogeneous character which are often difficult to describe analytically. For example, in order to reproduce complex geological structures, such as channels, one has to consider multiple point geostatistics (MPS) approaches. MPS is based on a concept of training images that itself represent the statistical model for the system under consideration (see, e.g., Deutsch 2002 or more recently Mariethoz and Caers 2014). However, uncertainty quantification is especially complicated when the underlying permeability field is captured using the MPS, in particular for channelized systems obtained from a training image. In this study, we are particularly interested in this type of model and the performance of the FROST method within this MPS setting.

The paper proceeds as follows. Section 2 describes the equations for the two-phase flow problem and the boundary conditions. Then, Sect. 3 summarizes the FROST distribution method employed for estimating the saturation statistics. Section 4 describes the numerical settings for studying the log-time-of-flight statistics in uncertain channelized porous systems. Consequently, Sect. 5 investigates the probability distribution of the log-TOF under these settings and explains how the equivalent injection time statistics obtained and used. Section 6 illustrates the performance of the FROST distribution method by benchmarking some of its results with MC simulations. Section 7 provides an estimation of the computational savings of the FROST approach. Lastly, the final conclusions are given in Sect. 8.

2 Two-Phase Flow Model

We are interested in the flow settings where a one phase, e.g., oil (o), is being displaced by another phase, e.g., water (w), in a porous system with spatially varying permeability, \(k({\mathbf {x}})\), and porosity, \(\phi ({\mathbf {x}})\). We assume that the phases are incompressible and neglect gravity, buoyancy and capillary effects. A common model used to represent these settings is the Buckley–Leverett model where the pressure, p, and saturations, \(S_\alpha \) (\(\alpha =\mathrm {o}, \mathrm {w}\)) are governed by a coupled system of equations: mass conservation and Darcy’s equation (see, e.g., Aziz and Settari 1979).

Darcy’s flux for each phase is proportional to the pressure gradient (capillary pressure and gravity are neglected):
$$\begin{aligned} \begin{aligned} {\mathbf {u}}_\mathrm{\alpha } = -\frac{k({\mathbf {x}})k_\mathrm{r\alpha }(S_\mathrm{\alpha })}{\mu _\mathrm{\alpha }}\quad \nabla p = -\lambda _\mathrm{\alpha }\nabla p, \quad \alpha = \mathrm {o,w}. \end{aligned} \end{aligned}$$
(1)
Here \(\mu _\alpha \) are the viscosities of the phases, \(k_\mathrm{{r}\alpha }\) the relative permeabilities, and \(\lambda _{\alpha }\) are the mobilities.
The total flux, \({\mathbf {u}}_\mathrm{t}\), the sum of the fluxes of each phase, honors the following incompressibility condition:
$$\begin{aligned} \nabla \cdot {\mathbf {u}}_\mathrm{t} = \nabla \cdot ({\mathbf {u}}_\mathrm{w} + {\mathbf {u}}_\mathrm{o}) = q_\mathrm{t}, \end{aligned}$$
(2)
where \(q_\mathrm{t}\) represents a total source (sink) term.
With the water fractional flow function, \(f_\mathrm{w}\), defined as
$$\begin{aligned} f_\mathrm{w} = \frac{\lambda _\mathrm{w}}{\lambda _\mathrm{w} + \lambda _\mathrm{o}}, \end{aligned}$$
(3)
and the condition \(S_\mathrm{w} + S_\mathrm{o} = 1\), the conservation equation for water reads:
$$\begin{aligned} \phi ({\mathbf {x}})\frac{\partial S_\mathrm{w}}{\partial t} + \nabla \cdot \left( f_\mathrm{w}(S_\mathrm{w}) \cdot {\mathbf {u}}_\mathrm{t}\right) = q_\mathrm{w}, \end{aligned}$$
(4)
where \(q_\mathrm{w}\) is a source (sink) term for water. The source or sink terms represent the effect of wells. Equation (4) is supplemented by the initial and boundary conditions:
$$\begin{aligned} \begin{aligned}&S_\mathrm{w}({\mathbf {x}},t) = s_{\mathrm{wi}}, \quad \forall {\mathbf {x}} \text { and } t=0,\\&S_\mathrm{w}({\mathbf {x}},t) = s_\mathrm{b}, \quad {\mathbf {x}}\in \varGamma _\mathrm{inj} \text { and } t>0, \end{aligned} \end{aligned}$$
(5)
where \(s_{\mathrm{wi}}\) is the initial water saturation in the reservoir, and \(s_\mathrm{b}\) is the saturation at injection well or boundary, \(\varGamma _\mathrm{inj}\).

3 FROST Distribution Method

We briefly review the steps from Ibrahima et al. (2015) that lead to the frozen streamlines (FROST) distribution method for saturation fields. The frozen streamlines approximation is reasonable for highly heterogeneous porous media where the heterogeneities dominate the flow assuming small changes in well flow rates. Streamlines are also significantly impacted by the change in boundary conditions, so here we assume they constant in time.

Indeed, we are interested in understanding the probability distribution of two-phase flow under a stochastic channelized permeability field. To address this problem, we first recast Eq. (4) into multiple 1-D problems using the streamline representation (Fig. 1). The characteristic trajectory of a streamline, frozen at \(t=0\), is as follows:
$$\begin{aligned} {\mathbf {x}}_{\mathrm{sl}}(r) = {\mathbf {x}}_\mathrm{{sl},0} + \int _{0}^{r}{\mathbf {u}}_\mathrm{t}\left( {\mathbf {x}}_{\mathrm{sl}}\left( r'\right) ,0\right) \text {d}r', \end{aligned}$$
(6)
where \({\mathbf {x}}_{\mathrm{sl},0}\) is the starting position (usually injection well or boundary), and r is the position along the streamline. Each streamline can be viewed as the centerline of the streamtube with cross section A(r), which has no communication with the neighboring streamtubes. In the following, we use the terms streamline and streamtube interchangeably.
Fig. 1

Example of the streamline representation for a quarter five-spot problem with one injector and one producer. A few streamlines are represented by the red lines. An example of a streamtube with the cross section of A(r) is shown in gray

Following the terminology of Hewett and Yamada (1997), the total volumetric flow rate, \(u_\mathrm{t}^{\mathrm{sl}}\), along a streamtube is defined as
$$\begin{aligned} u_\mathrm{t}^{\mathrm{sl}} = |{\mathbf {u}}_\mathrm{t}|\,A(r), \end{aligned}$$
(7)
where A(r) is the streamtube cross section. Since
$$\begin{aligned} \frac{\partial {u_\mathrm{t}^{\mathrm{sl}}}}{\partial r} = 0, \end{aligned}$$
(8)
\(u_\mathrm{t}^{\mathrm{sl}}\) does not depend on r,  and for a specific streamline it is a function of time only.
Based on the definitions above and using the parametrization of saturation along a streamtube, we have
$$\begin{aligned} S_\mathrm{w}({\mathbf {x}},t)=~\tilde{S}_\mathrm{w}(r,t). \end{aligned}$$
(9)
We can reduce Eq. (4) to a set of independent 1-D problems formulated along the streamlines:
$$\begin{aligned} \phi (r)A(r)\frac{\partial {\tilde{S}_\mathrm{w}}}{\partial {t}} + u_\mathrm{t}^{\mathrm{sl}}(t) \frac{\partial {f_\mathrm{w}(\tilde{S}_\mathrm{w})} }{\partial {r}} = 0. \end{aligned}$$
(10)
Next, according to Ibrahima et al. (2015), the cumulative pore volume (CPV), V, and the cumulative injection volume (CIV), Q, per streamline are introduced as follows:
$$\begin{aligned} \begin{aligned}&V(r) = \int _{0}^{r}\phi (r')A(r')\text {d}r', \\&Q(t) = \int _{0}^{t}u_\mathrm{t}^{\mathrm{sl}}(t')\text {d}t'. \end{aligned} \end{aligned}$$
(11)
After applying the change of variables: \(\tilde{S}_\mathrm{w}(r,t)=\check{S}_\mathrm{w}(V,Q)\), Eq. (10) becomes
$$\begin{aligned} \frac{\partial {\check{S}_\mathrm{w}}}{\partial {Q}} + \frac{\partial {f_\mathrm{w}(\check{S}_\mathrm{w})}}{\partial {V}} = 0, \end{aligned}$$
(12)
with the initial and boundary conditions as
$$\begin{aligned} \begin{aligned} \check{S}_\mathrm{w}(V,0)=s_\mathrm{wi}, \quad \forall V \, \check{S}_\mathrm{w}(0,Q)=s_\mathrm{b}, \quad Q>0. \end{aligned} \end{aligned}$$
(13)
In the following, we assume both \(s_{\mathrm{wi}}\) and \(s_\mathrm{b}\) to be uniform in space.
The solution of Eq. (12) maps the uncertain reservoir characteristics (porosity and permeability) to the stochastic water saturation field. Equation (12) has an analytical similarity solution, \(\hat{S}_\mathrm{w}(Z)\), in terms of the ratio \(Z={V}/{Q}\). It can be shown (see Ibrahima et al. 2018) that subject to the frozen streamlines assumption, Z admits the following more convenient representation:
$$\begin{aligned} Z({\mathbf {x}},t) = \dfrac{\tau _0({\mathbf {x}})}{T({\mathbf {x}}, t)}, \end{aligned}$$
(14)
where
$$\begin{aligned} \tau _0({\mathbf {x}})=\int _0^{r({\mathbf {x}})}\frac{\phi (r')}{|{\mathbf {u}}_\mathrm{t}(r',0)|}\,\text {d}r' \end{aligned}$$
(15)
is the initial time-of-flight (TOF) to position \({\mathbf {x}}\), and
$$\begin{aligned} T({\mathbf {x}}, t) = \int _0^t\frac{u_\mathrm{t}^{\mathrm{sl}}({\mathbf {x}},t')}{u_\mathrm{t}^{\mathrm{sl}}({\mathbf {x}},0)}\,\text {d}t' \end{aligned}$$
(16)
is the so-called equivalent injection time (EIT) at time t. Due to the fixed streamline assumption (thus, A(r) is constant in time), \(T({\mathbf {x}}, t)\) can be computed solely from the total velocity field:
$$\begin{aligned} T({\mathbf {x}}, t) = \int _0^t\frac{|{\mathbf {u}}_\mathrm{t}({\mathbf {x}},t')|}{|{\mathbf {u}}_\mathrm{t}({\mathbf {x}},0)|}\,\text {d}t'. \end{aligned}$$
(17)
Therefore, the stochastic saturation field, \(S_\mathrm{w}\), is driven by the combination of the smooth stochastic fields \(\tau _0({\mathbf {x}})\) and \(T({\mathbf {x}}, t)\) via the nonlinear mapping \(\hat{S}_\mathrm{w}\). Thereby, the stochastic surrogate model for saturation is \(S_\mathrm{w}({\mathbf {x}},t) = \hat{S}_\mathrm{w}\left( Z({\mathbf {x}},t)\right) \), leading to the semi-analytical expression for the saturation field CDF:
$$\begin{aligned} F_{S_\mathrm{w}\left( {\mathbf {x}},t\right) }(s) = 1 - F_{\log Z({\mathbf {x}},t)}\left[ \log \left( \hat{S}_\mathrm{w}^{-1}\left( s\right) \right) \right] , \end{aligned}$$
(18)
where s is the saturation level, and \(\hat{S}_\mathrm{w}^{-1}\left( s\right) \) is the inverse mapping. The logarithm is introduced for better representation of the probability distribution as discussed in Ibrahima et al. (2018) and illustrated later in Sect. 5.
Next, as further explained in Sect. 5.3, we approximate the EIT by its mean, \(\bar{T}\)(t), independently of space, leading to the following final approximation for the saturation CDF:
$$\begin{aligned} F_{S_\mathrm{w}\left( {\mathbf {x}},t\right) }(s) = 1 - F_{\log \tau _0({\mathbf {x}})}\left[ \log \left( \hat{S}_\mathrm{w}^{-1}\left( s\right) \cdot \bar{T}(t) \right) \right] . \end{aligned}$$
(19)
Equation (19) allows us to compute any saturation statistical moment by simple integration. For instance, the FROST-based mean and variance for the saturation are given by the following formulae:
$$\begin{aligned} \begin{aligned} \bar{S}_\mathrm{w} ({\mathbf {x}},t)&= 1 - \int _{0}^{1} F_{S_{w}\left( {\mathbf {x}},t\right) }(s)\text {d}s, \\ \sigma _{S_\mathrm{w}}^{2}({\mathbf {x}},t)&= 1-\int _{0}^{1} F_{S_{w}\left( {\mathbf {x}},t\right) }(\sqrt{s})\text {d}s - \left( \bar{S}_\mathrm{w}({\mathbf {x}},t)\right) ^{2}, \end{aligned} \end{aligned}$$
(20)
which are used in the numerical assessment in Sect. 6.

4 Numerical Example Settings

4.1 Geostatistical Input

To investigate log-TOF distributions in channelized systems, we generate 5000 unconditioned realizations of high-contrast, binary, channelized permeability system using the SNESIM algorithm within the geostatistical software SGeMS (Remy et al. 2009). A total of 5000 realizations was found to be sufficient for convergence of log-TOF PDF and CDF. A 2-D training image (Strebelle 2002) of size \(250\times 250\), shown on the left of Fig. 2, was used, while the generated realizations were of size \(100\times 100\). The constant permeability of 100 mD was assigned to the channels, while the permeability of the shale zones was 1 mD. Without loss of generality, porosity was assumed to be uniform, \(\phi =0.3\). Examples of generated realizations are shown on the right of Fig. 2.
Fig. 2

Training image (Strebelle 2002) that was used to generate realizations of a channelized permeability field (left). A few generated stochastic realizations (right)

4.2 Simulation Setting

We consider line injection of water in a 2-D horizontal reservoir of 5000 m \(\times ~5000\) m \(\times ~1\) m from the left boundary and a production from the opposite right boundary (Fig. 3). The injection is set in such a way that the mean flow coincides with the principal direction of the channels. Injection is performed with the constant total rate of 1000 \(\hbox {m}^{3}/\hbox {day}\), while production proceeds at a constant pressure. We use an unfavorable viscosity ratio, \(M=\mu _\mathrm{o}/\mu _\mathrm{w} =2.0\), and quadratic functions of saturation for the relative permeabilities, \(k_\mathrm{{r}\alpha } = s_\alpha ^2\) (\(\alpha =\mathrm {o,w}\)).
Fig. 3

Simulation setting: line injection at constant rate on the left boundary and line production at constant pressure on the right boundary

5 TOF and EIT Statistics

5.1 Probability Distributions of Log-TOF with Finite-Volume and Streamline Solvers

We investigate the properties of the log-TOF distribution obtained from the underlying uncertain channelized permeability field and injection settings presented in Sect. 4. We evaluate two different approaches to numerically sample the TOF.

The first approach (SL) uses a high-resolution streamline tracing and is implemented within the 3DSL simulator (Thiele et al. 1996; Thiele 2003). To trace streamlines, this method employs the analytical Pollock’s algorithm (Pollock 1988) that assumes piecewise linear approximation of the velocity field in each coordinate direction. Streamlines are launched from each gridblock face containing an injector in proportion to the flux out of a face, such that more streamlines are traced through high flow velocity regions. However, with this strategy, not every gridblock in the domain will contain a streamline for a fixed total number of streamlines launched. A missed gridblock is assigned a streamline which is then traced backward in the velocity field toward an injector (Batycky et al. 1996; Batycky 1997; Aarnes et al. 2005). When multiple streamlines pass through the gridblock, weighted averaging is applied in order to map different properties, like time-of-flight or saturation, from the streamlines to the background grid. The weighting factors are computed based on the volume of each streamline segment in the cell (Batycky 1997). In our numerical examples, we exhaustively traced the streamlines through the entire computational domain, and the average number of streamlines for simulation grid of \(100\times 100\) cells was around 2000 streamlines.

The second approach (FV) employs a single-point upwind finite-volume scheme (SPU) (Shahvali et al. 2012) and is implemented within the MATLAB Reservoir Simulation Toolbox (MRST) (Lie 2014). This scheme corresponds to the lowest order discontinuous Galerkin scheme developed by Natvig et al. (2007) and Natvig and Lie (2008) for the time-of-flight equation. However, it was shown in Shahvali et al. (2012) that the SPU scheme is less accurate than truly multi-D upstream weighting schemes (MDU). Also, (Kozdon et al. 2009) demonstrated that MDU schemes reduce the transverse numerical diffusion and grid orientation effects relative to SPU. Higher-order discontinuous Galerkin methods applied to the time-of-flight equation work generally well and were also found to be more accurate than SPU (Rasmussen and Lie 2014).

The log-TOF distribution is estimated by MC sampling followed by kernel density estimation (KDE) (Botev et al. 2010). Given that we are interested only in estimating the initial log-TOF distribution, MC simulation appears a reasonable approach.

Figure 4 shows comparison of log-TOF fields obtained using FV and SL approaches for two random realizations. We can see that even though the log-TOF fields from two methods visually look quite similar, significant differences can be seen further from the injection boundary. Even though these differences might seem, at least visually, small, their impact is critical in terms of the probability distribution. Indeed, Fig. 5 shows the log-TOF CDF at different positions in the simulation domain, while Fig. 6 shows the log-TOF PDF at these positions. We can see that close to the line injection, the two distributions coincide but the difference quickly grows when moving closer to the production boundary. We found that grid refinement does not significantly change the log-TOF distribution with the FV solver.
Fig. 4

Comparison of log-TOF profiles using FV and SL approaches for two realizations. TOF is in PVI (pore volume injected). Log-TOF profiles using FV and SL approaches are shown together with the absolute difference between them and the underlying channelized permeability field. a Realization 1. b Realization 2

Fig. 5

Evolution of log-TOF CDF along the simulation domain. TOF is in PVI. The dashed lines are SL results, while the solid lines are FV results

Fig. 6

Evolution of log-TOF PDF along the simulation domain. TOF is in PVI. The dashed lines are SL results, while the solid lines are FV results

Based on these results, we can conclude that a high-resolution solver is crucial for log-TOF distribution estimation within the context of the FROST framework. In the following, we choose the SL approach, because of its speed, accuracy, and fitness in estimating the log-time-of-flight. The approach is used both for the sampling of the log-TOF and for the direct MC simulations.

5.2 Parameterization of Log-TOF PDF Using a Mixture of Gaussian Distributions

Figure 7 shows that at each position in the simulation domain, the log-TOF distribution can be approximated by a mixture of Gaussians, unlike the results in Ibrahima et al. (2018), where for more simple permeability fields (log-Gaussian and variogram-based fields) the log-TOF distribution was found to be always essentially Gaussian. For most positions, a bimodal Gaussian distribution is sufficient to capture the MC-based log-TOF PDF. However, for some locations, a bimodal fit does not give satisfactory results, and a trimodal Gaussian distribution appears to fit better. Therefore, in the following, we use the trimodal fit. Employing the fitting strategy may allow running only a fraction of the realizations to reliably estimate the log-TOF distribution (for instance, one-tenth of realizations were used in Ibrahima et al. (2018) for the Gaussian fit case). We evaluate the effect of using the fitted mixture of Gaussian distributions instead of the actual one in Sect. 6.
Fig. 7

Examples of mixture of Gaussian fits for log-TOF PDF. TOF is in PVI. Top: bimodal fits for positions (10, 50) and (90, 50). Bottom: bimodal versus trimodal fits for position (40, 50)

5.3 Statistics of Equivalent Injection Time

The equivalent injection time (EIT) is defined by Eq. (16). As explained in Ibrahima et al. (2018), as a first approximation, EIT can be seen as an almost deterministic time-dependent function and can be approximated by its mean value from few MC samples (usually one-tenth of the realizations is enough). Moreover, based on the results in Fig. 8, in this work we use a simplified space-independent linear approximation:
$$\begin{aligned} \bar{T}(t) \approx \alpha t + \beta , \end{aligned}$$
(21)
where \(\alpha \) and \(\beta \) are constant parameters (however, they are implicit functions of viscosity ratio and relative permeability curves) and can be determined from fitting the \(\bar{T}(t)\) curve with a linear function. The EIT is estimated using the FV solver only because the data required for its computation is not provided by the SL solver. We did not observe that the parameters \(\alpha \) and \(\beta \) are significantly impacted by the numerical effects discussed earlier for the log-TOF distribution.
Fig. 8

Behavior of the mean and standard deviation of EIT at 9 different positions in the simulation domain for 5 successive time steps. EIT is in PVI. Black dashed line in each plot denotes the linear fit for the mean EIT

6 Saturation Statistics

We compare the results of the saturation statistics obtained from the FROST distribution method and direct MC simulations. In the context of FROST, we also compare the results when using the direct estimation of log-TOF PDF (we then call this approach simply ‘FROST’) and when employing the fitting of the empirical log-TOF PDF with the trimodal Gaussian distribution (this approach is further referred to as ‘fit-FROST’). To apply the FROST distribution method, we numerically sample the log-TOF and compute the log-TOF PDF and CDF using KDE (in direct FROST method), or using Gaussian fit (in fit-FROST method).

Figures 9, 10, 11 show a comparison of the MC-based, FROST-based and fit-FROST-based water saturation statistics (specifically, mean and standard deviation) at three different times: \(t=0.3\), \(t=0.9\) and \(t=1.5\) PVI (pore volume injected). We observe a high variance of the saturation field, even at small times \(t=0.3\) PVI (see Fig. 9, right plots), indicating high geological uncertainty inherited from the training image reflected in the unconditional simulation of geostatistical realizations.

Overall, the agreement between FROST and MC in mean water saturation is satisfactory, with the main details of saturation variations captured by FROST. However, as it can be seen from Fig. 11, the FROST-based mean saturation has slightly slower propagation speed compared to the MC-based results, leading to the later mean breakthrough time at the line producer. The standard deviation results, however, have more obvious disagreements. Indeed, while FROST-based saturation standard deviations accurately capture the contrasts and shapes of its MC-based counterparts, they also seem to consistently overestimate the MC-based saturation standard deviations in regions of high variance. The results of FROST and fit-FROST agree quite well for all times of the simulation, but there are just a few small artifacts that can be observed in the fit-FROST results - they occur probably when the trimodal fit fails to capture the shape of the log-TOF PDF. These results motivate the future application of Gaussian mixtures for parametrization of the log-TOF distributions.

Next, in order to assess the limitations of grid resolution in this particular numerical example, we perform a grid refinement study. As was mentioned before, the base case employs a grid of size \(100\,{\times }\,100\). We complement the base case results by conducting similar computations (both for the FROST distribution method and direct MC simulations) on the two refined grids: \(200\,{\times }\,200\) and \(400\,{\times }\,400\). To preserve the same geostatistical input, channelized permeability realizations were downscaled to these finer grids using a piecewise constant interpolation.

We compare the water saturation CDFs obtained both with FROST method and direct MC simulations on these refined grids. Figure 12 shows water saturation CDFs for three positions in the domain at \(t=1.5\) PVI for the considered grids. Even visually it can be seen that for the three different grids, the CDFs obtained from FROST are identical. The MC results, on the contrary, show that convergence has not been reached yet. With grid refinement, the MC results get closer to the FROST results, especially at positions away from the domain boundaries. We also study the \(L^{2}\)-norm of the difference between the CDFs obtained with the FROST and MC methods for some specific positions in the domain, and we observe the decrease in this error as the grid was refined (see Fig. 13). The same is also true for the difference in mean water saturation fields for the two methods. For example, Fig. 14 shows the normalized \(L^{2}\)-norm of the mean saturation difference at three times (\(t=0.3\), 0.9 and 1.5 PVI) as a function of the gridblock size. Figures 15 and 16 show the water saturation statistics at \(t=0.9\) PVI and \(t=1.5\) PVI for the most refined grid of \(400\,{\times }\,400\). The slower front propagation is much less pronounced now with the FROST method. The MC and FROST results for the second statistical moment are closer, but still FROST overestimates it. With these results, we conclude that indeed grid refinement allows for better agreement of the FROST and MC results.

We also performed MC simulations using an analytical streamline solver (Müller et al. 2013). In this case, no global pressure updates are made during simulation, and the streamlines are fixed. The employed solver accepts only grids where the number of cells in each direction is some power of 2. So, to perform these simulations the input permeability realizations were cut to size of \(256\times 256\) from realizations of size \(400\times 400\). Figure 17 shows water saturation CDFs obtained with the FROST and MC methods at different positions in the domain at three different times – 0.2, 0.6 and 1 PVI. From this figure, we can see that under the fixed streamline assumption results obtained with the FROST method are in excellent agreement with the ones from the MC method.
Fig. 9

Comparison of MC-based (top row), FROST-based (middle row) and fit-FROST-based (bottom row) saturation statistics at \(t = 0.3\) PVI on \(100 \times 100\) grid. Left plots are average saturation estimates. Right plots are saturation standard deviation estimates

Fig. 10

Comparison of MC-based (top row), FROST-based (middle row) and fit-FROST-based (bottom row) saturation statistics at \(t = 0.9\) PVI on \(100 \times 100\) grid. Left plots are average saturation estimates. Right plots are saturation standard deviation estimates

Fig. 11

Comparison of MC-based (top row), FROST-based (middle row) and fit-FROST-based (bottom row) saturation statistics at \(t = 1.5\) PVI on \(100 \times 100\) grid. Left plots are average saturation estimates. Right plots are saturation standard deviation estimates

Fig. 12

Water saturation CDFs obtained with the FROST and MC methods at three different positions in the domain at \(t=1.5\) PVI for three subsequently refined grids. The dashed lines are the MC results, while the solid lines are the FROST results. a Grid \(100\times 100\). b Grid \(200\times 200\). c Grid \(400\times 400\)

Fig. 13

\(L^{2}\)-norm of the difference in CDFs of water saturation from the FROST and MC methods at different positions in the domain at \(t=1.5\) PVI as a function of the gridblock size

Fig. 14

\(L^{2}\)-norm (normalized by the total number of cells) of the difference in mean water saturation field from the FROST and MC methods at \(t=0.3\), 0.9 and 1.5 PVI as a function of the gridblock size. Notice the logarithmic scale on vertical axis

Fig. 15

Comparison of MC-based (top row) and FROST-based (bottom row) saturation statistics at \(t = 0.9\) PVI on grid \(400\times 400\). Left plots are the average saturation estimates, right plots are the saturation standard deviation estimates

Fig. 16

Comparison of MC-based (top row) and FROST-based (bottom row) saturation statistics at \(t = 1.5\) PVI on grid \(400\times 400\). Left plots are the average saturation estimates, right plots are the saturation standard deviation estimates

Fig. 17

Water saturation CDFs obtained with FROST and MC methods at different positions in the domain at three different times–0.2, 0.6 and 1 PVI–for grid size \(256\times 256\). The dashed lines are the MC results, while the solid lines are the FROST results. MC results are obtained with analytical solver along streamlines (the streamlines were fixed during simulation)

7 Complexity Analysis of the FROST Method

Here, we apply a simple complexity analysis to estimate the computational savings of the FROST approach against the direct MC simulations for saturation statistics estimation. Let us have \(N_\mathrm{r}\) realizations, each of which needs to be run for \(N_\mathrm{t}\) time steps. Assuming that the cost of a single time step is \(\zeta _{\varDelta t}\), then the cost of running the MC simulations can be estimated as
$$\begin{aligned} \zeta _{\mathrm{MC}} = N_\mathrm{r} \cdot N_\mathrm{t} \cdot \zeta _{\varDelta t}. \end{aligned}$$
(22)
Similarly, the cost of the FROST is
$$\begin{aligned} \zeta _{{\mathrm{FROST}}} = N_\mathrm{r}^{\mathrm{TOF}} \cdot \gamma \cdot \zeta _{\varDelta t} + N_\mathrm{r}^{\mathrm{EIT}} \cdot N_\mathrm{t}^{\mathrm{EIT}} \cdot \zeta _{\varDelta t}, \end{aligned}$$
(23)
where the first term corresponds to the sampling of the log-TOF distributions, and the second term is the cost of the mean EIT function estimation. Notice that the first term in Eq. (23) no longer contains \(N_\mathrm{t}\), and that we introduced the factor \(\gamma \) to adjust the cost due to the fact that we do not need to solve for saturation, but need to compute the TOF instead. \(N_\mathrm{r}^{\mathrm{TOF}}\) can be substantially smaller than \(N_\mathrm{r}\) when fit-FROST is employed, so the reduced number of realizations can be used for the log-TOF distribution estimation (Ibrahima et al. 2018). However, here for simplicity we use \(N_\mathrm{r}^{\mathrm{TOF}}=N_\mathrm{r}\). \(N_\mathrm{r}^{\mathrm{EIT}}=\delta \cdot N_\mathrm{r}\) accounts for a reduced number of realizations used for the EIT estimation (\(\delta \le 1\)). \(N_\mathrm{t}^\mathrm{EIT}\) accounts for a potentially smaller number of time steps needed to obtain a good enough fit for \(\bar{T}(t)\); again for simplicity we take \(N_\mathrm{t}^{\mathrm{EIT}}=N_\mathrm{t}\).
Finally, the estimated speedup is
$$\begin{aligned} \frac{\zeta _{\mathrm{MC}}}{\zeta _{\mathrm{FROST}}} = \frac{N_\mathrm{t}}{\gamma + \delta \cdot N_\mathrm{t}}, \end{aligned}$$
(24)
and if \(N_\mathrm{t} \gg 1\), the speedup is \(\delta ^{-1}\). Based on our experience, running simulations for only 10% of the realizations provide a satisfactory estimation for the mean EIT, the corresponding estimated speedup is a factor of 10. For example, based on the performance profile reported in Batycky et al. (2010) for SPE 10 problem (Christie and Blunt 2001), with \(\gamma \approx 0.5\) and \(N_\mathrm{t}=25\), the speedup is 8.3. This is a very modest estimate, further computational savings can be achieved with fit-FROST or using \(N_\mathrm{t}^{\mathrm{EIT}} < N_\mathrm{t}\).

8 Discussion and Conclusions

In this work, we studied the probability distributions of log-time-of-flight in binary channelized systems. We showed that the log-time-of-flight PDF is multi-modal over the entire spatial domain, and that a trimodal fit obtained with Gaussian mixtures can often capture its key features. This suggests a surrogate model for fast estimation of the log-TOF PDF.

We then showed how one can use the log-TOF CDF to efficiently estimate the statistics of fluid saturations during line-injection waterflooding in these highly heterogeneous porous media. The uncertainty propagation method employed in this paper follows the one formulated in Ibrahima et al. (2018) and is based on the physical understanding of the stochastic problem. We found that a high-resolution solver is crucial for log-TOF distribution estimation in the FROST framework. We also performed a computational complexity assessment showing that the FROST method can be more attractive compared to the direct MC simulations. Results obtained from the presented numerical examples in Sect. 6 allow us to conclude that even in complex channelized systems, the FROST method is able to provide estimates of mean water saturation fields. These estimates were found to be in excellent agreement with the results from the reference MC simulations, which was further improved with a grid refinement.

However, the numerical example also illustrates that estimating high-order statistics of the saturation field in channelized systems, where the characteristic size of the channels is comparable with the size of the computational domain, remains a particularly challenging task for the FROST method. Indeed, the results for the standard deviation of water saturation show some discrepancy between the FROST and MC results. These discrepancies fundamentally originate from the notable differences between the FROST and MC saturation CDFs that remain persistent with grid refinement, even though a convergence study shows linear improvement with the gridblock size. Accordingly, further analyses are needed to capture the finer statistical details (second-order statistical moments and higher) of the saturation field and understand the slow convergence of reference MC under the grid refinement.

Nevertheless, under the fixed streamline assumption the FROST method is able to provide excellent estimates of the saturation probability distribution. In future work, we will consider modification of FROST to reduce the uncertainty by conditioning on time-dependent saturation measurements in some regions. For this problem, a conditional log-TOF distribution should be estimated and used in the FROST framework. This could potentially increase the prediction capability of the FROST method for regions outside the domain with available saturation measurements.

Moreover, in our future work we shall consider utilizing within the FROST framework the total travel time (residence time) from the inlet to the outlet. For example, (Krogstad et al. 2015) contains an application of total travel time for fast flow diagnostics. So, with this metric, it should be possible for FROST to estimate the statistics of oil production, water production, and water cut under uncertainty. This will greatly extend the utility of the FROST method to the industrial applications for managing and developing subsurface resources.

Notes

Acknowledgements

Authors would like to thank Marco Thiele and Matthias Cremon for providing help with the simulations in 3DSL. Authors also thank Florian Müller for the provided analytical streamline solver and Brad Mallison for the valuable comments on this manuscript. Authors are grateful to Stanford University Petroleum Research Institute for Reservoir Simulation (SUPRI-B) for the financial support of this work.

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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Energy Resources EngineeringStanford UniversityStanfordUSA

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