Up-scaling issues in Fractured media

Les Rencontres scientifiques d'IFP Energies nouvelles Flows and mechanics in natural porous media from pore to field scale. Pore2Field 16-18 November 2011, IFP Energies nouvelles (France)

Up-scaling issues in Fractured media *

Benoit Noetinger , Bernard Bourbiaux, Matthieu Delorme, Andre Fourno, Nolwenn Jarrige, Chakib Kada-Kloucha, Nina Khvoenkova, Arnaud Lange, Catherine Ponsot-Jacquin and Abdelaziz Snoussi IFPEN, 1-4 avenue de Bois Préau 92852 Rueil-Malmaison, France, e-mail : [email protected] * Corresponding author Abstract — Up-scaling issues in Fractured media — Fractured media (FM) are an extreme case of heterogeneous media. Fractures or faults of small volume control the large-scale flow pattern and so the hydrocarbons recovery. In most cases, fluids stored in the reservoir flows through the connected fracture network. This so-called localization phenomena implies using specific simulation and up-scaling methods, especially if history matching and sensitivity studies are to be performed for oil recovery optimization. In this paper, we will review some up-scaling issues connected to the emerging Discrete Fracture Network (DFN) description, connected with EOR questions. New issues related with non conventional reservoir exploitation will be addressed.

INTRODUCTION About 30 % of known hydrocarbon reserves are contained in naturally fractured formations. Recovery factors may vary considerably, due to difficulties to get a reliable characterization of the fracture positions, shapes and flow properties [1-3]. Connectivity (percolation) effects are emphasized, a flow localization phenomena (flow through the connected fracture network) can be observed. This implies strong sensitivity to coupled processes (like geomechanical damage of the conducting fractures etc..). Large uncertainties are expected, that increase the risks and associated exploitation costs. Simulating the dynamic behavior of such reservoirs, using as much as possible a workflow accounting for both fine scale data and geological considerations, permits to estimate these risks, and to improve the final recovery. Integrated softwares like FracaFlow provide workflows allowing to build 3D DFN models using geological, seismic and well data. Additional modules enable to simulate transient well tests and to facilitate the fracture properties calibration.

Figure 1: A fractured reservoir (left), zoom on a grid block to be upscaled (right)

As soon as the fracture model is built, gradual deformations of the DFN may be used to improve the history matching, [38-40]. This integrated fractured reservoir modeling workflow has been subject of continuous developments for two decades [4, 5, 41]. Dynamic simulations at the field scale are still based on continuous approaches, due to computing limitations. So, an up-scaling step still remains inside the workflow to capture the DFN information. The continuous model is generally a set of double porosity/double permeability equations [6] that captures the dual nature of flows in fractured reservoirs: the fluid flows to the wells via the fracture network. The "2φ/2k" equations read as:  ∂Pf ( x,t ) Kf  = ∇. ⋅ ∇Pf ( x,t )  + Q mf ( x, t) + f ( x,t ) Φ f V f ctf ∂t   µ   P ( x , t )  K  ∂ Φ V c m = ∇. m ⋅ ∇Pm ( x,t )  − Q mf ( x, t)  m m tm ∂t  µ  

Here, Kf and Km are second order full tensors (in principle symmetric). The flux Qmf(x,t) characterizes the flow between matrix and fractures and f(x,t) the external fluid exchanges via wells, here connected with the fracture medium only (unless connection with the matrix is possible). These equations may be derived from first principles [6-11]. Developing closure formulae relating Qmf(x,t) to Pm and Pf is still an active area of investigation. Generally, a steady state linear formulation is retained : -2 Qmf(x,t) = αL Km/µ ×(Pm - Pf). Here, dimensionnless α is the so-called shape factor, and L the average block size.

Les Rencontres scientifiques d'IFP Energies nouvelles Flows and mechanics in natural porous media from pore to field scale. Pore2Field 16-18 November 2011, IFP Energies nouvelles (France) The extrapolation to multiphase compositional flows with gravity is done in an ad'hoc phenomenological manner by splitting the exchange matrix-fracture flux Qmf(x,t) into different contributions referring to the involved physical mechanisms of transfer, as detailed later on. This allows to assess the role played by gravity, capillarity, viscous drag, and also molecular diffusion when dealing with gas-oil systems. In EOR applications, or in unconventional reservoir exploitation, matrix hydrocarbon fluids recovery is essentially difficult, generally characterized by a slow kinetics, which justifies to improve considerably the description of matrix to fracture transfers. Streamline methods, fast computation techniques based on graph theory were recently proposed [44, 45], but at present times, these techniques are not implemented in standard workflows based on dual porosity simulation.

1

UP-SCALING FOR FRACTURED MEDIA

1.1

Statement of the problem

Most reservoir studies begin by single phase flow calculations. This allows one to estimate recoverable reserves, to optimize well location and the associated drainage area, to compute productivity index, coning problems, tracer behavior etc... Basically, at the smallest scale, the Laplace equation has to be solved for pressure.

ϕ(x )c t

∂p(x, t )  k (x , t )  = ∇. ⋅ ∇p(x , t ) + f ( x , t ) ∂t  µ 

also proposed, as well as MINC techniques [17, 18]. Up to our knowledge, no team was able to treat large 6 problems involving about 10 fractures, even without any coupling with the matrix. Techniques allowing to restrict the problem to unknowns located at the intersections between fractures, and selected nodes in the matrix were proposed by [19-20] in 2D, and recently in 3D [21]. A theoretical justification of these models was recently given [22]. This transforms the initial DFN into a resistor/capacitor network that permits direct well-test simulation. The coupling with the matrix gives a set of discrete equations formally analogous to a 2φ 2k model, in which the fracture part corresponds to the resistor capacitor network squaring with the actual DFN. Finally, an approach in which the DFN is kept close to the wells, and an averaged 2φ 2k model used far from the well has been developed at IFPEN [23]. It is a first step toward multiscale techniques. In summary, the DFN is essentially used to parameterize the 2φ 2k model via an upscaling step: DFN + Matrix →Kf, Km and α. The α can be computed using fast random walk computing avoiding explicit meshing of the matrix [24, 25].

1.2

Fast methods

These Kf, Km and α. parameters can vary with position, if the fracture pattern is not stationary. Their computation can be considerably accelerated using fast sorting methods based on the concept of the connectivity index Ic= ( = average). Such indicators permit to select the up-scaling method: analytical or fully numerical. In this last case, the resistor/capacitor network corresponding to the DFN gives a direct up-scaling method [26,27].

Tracer or heat displacement may be modeled by solving the associated convection dispersion equation:

ϕ(x )

∂c(x , t ) + ∇.( u (x , t )c(x , t )) = ∇.(d (u (x , t )) ⋅ ∇c(x , t )) ∂t

In these expressions, d(u(x,t)) is the dispersion tensor and u x , t ) = − k (x , t ) ⋅ ∇p(x , t ) is the Darcy velocity. In FM µ the coefficients k(x,t) and d(u(x,t)) are highly oscillating functions, reflecting the disorder of the underlying DFN. This implies meshing difficulties, and very badly conditioned problems. Direct resolution of these equations using direct meshing of both the detailed DFN and the matrix using finite volume and/or finite elements schemes were proposed [12-15], including two phase flow. A difficulty is to build an acceptable 2D mesh of each fracture of the DFN, coupled to 3D elements for the matrix. Smeared fractures methods [16], keeping structured meshes were

Figure 2: Fast computation of Ic maps provides useful information about the choice of up-scaling method, and connectivity properties.

Les Rencontres scientifiques d'IFP Energies nouvelles Flows and mechanics in natural porous media from pore to field scale. Pore2Field 16-18 November 2011, IFP Energies nouvelles (France)

1.3

Representative Elementary Volume

Using the 2φ 2k model, one assumes that the Representative Elementary Volume (REV) size ζ of the FM is smaller than the grid size used to solve the 2φ 2k equations. It may be shown that ζ controls also the uncertainties: small ζ implies that fluctuations due to the DFN disorder are smoothed out. So, the resulting large scale model results depend mainly on the values of the input petrophysical parameters. In the opposite case, large fluctuations can be expected, and the detailed geometry of the DFN must be accounted for, implying large uncertainties and probably some difficulties to history-match the model. In practice, ζ is not evaluated, and using a 2φ 2k model at a scale smaller than ζ remains done virtually, even if it must be used with caution. Percolation theory [28] suggests that ζ is directly related to Ic. This concept could play a major role to determine "Quality Control" indicators of coarse grid solutions.

unknown parameters using the known history (pressure, rates data). Once a suitable parameterization accounting for the main seismic and geological constraints is built, inversion loops (e.g. CondorFlow) allow to find the best set of input parameters. The generic method is the minimization of an objective function F(θ), in which θ is the set of invertible parameters. Subsequent uncertainty analysis can be performed. In the case of FM, it can be necessary to come back to the DFN: one can match the fracture's conductivities, or even the topology of the DFN in case of large ζ. This means that the up-scaling step is part of the HM process [38, 39, 40]. DFN + Matrix →(Kf, Km, α)(θ) →2φ 2k model. Here, θ represents collectively the set of parameters used in the local scale description. So, F(θ) =F((Kf, Km, α)(θ)) and using the chain rule, one gets:

1.4

Toward multiscale methods in FM

Implementing multiscale methods gain increasing popularity among the reservoir simulation community [29]. The main idea is to be able to manage in the same model the multiple scales of the problem without replaying the whole workflow. The basic idea is to consider that the up-scaling step can be viewed as a part of the resolution of the discrete equations. A posteriori estimation of the accuracy of the solution manages automatic refinement of the computational grid close to wells or any other singularity of the problem. In the fractured case, the situation is quite different than in the single porosity case, because the 2φ 2k equations have not the same algebraic form than the local equations. It is possible to find up scaling methods 2φ 2k equations ==> to 2φ 2k equations [31]. Developing such methods imposes to be able to up-scale strongly anisotropic permeability tensors, and exchange factors αL-2. Renormalisation techniques allowing to build fast downscaling techniques [30], algebraic methods [31] for -2 the exchange factors αL provide good starting points. A key point remains the treatment of strong permeability anisotropies arising in FM. This would lead to methods in which fractures of lengths greater that a specified cut off will be kept explicitely, and the others averaged out.

2

UP-SCALING AND HISTORY MATCHING FM

In practice, in order to get a more predictive model, reservoir engineers proceed to history match (HM) the

∂F ∂F ∂K f ∂F ∂K m ∂F ∂α , that = ⋅ + ⋅ + ⋅ ∂θ ∂K f ∂θ ∂K m ∂θ ∂α ∂θ

allows to set-up a HM scheme at the DFN level. History match of (Kf, Km, α) can be first calculated by using an arbitrary method working with the 2φ 2k model. Then θ can be matched knowing (Kf, Km, α), by inverting the upscaling formula (Kf, Km, α)(θ) . The derivatives

∂K f ∂K m ∂α can be computed easily, e.g. , , ∂θ ∂θ ∂θ ∂K f 2 ∂C f , where ji corresponds to the flux = ∑ ji ∂θ DFN ∂θ th

flowing in the i fracture, computed during the up-scaling process at no extra CPU cost. Note that the localization phenomena can render ji very contrasted : in the dominant flow-paths, the influence of individual fractures can be over-amplified. Analogous consequences can be anticipated for the resulting uncertainties [41].

3

3.1

UP-SCALING AND OIL FRACTURED MEDIA.

RECOVERY

IN

Motivation

Accurate and predictive modelling of oil recovery from FM requires a realistic representation of fracture flows and an accurate prediction of matrix-to-fracture transfers as the result of changing matrix-boundary conditions imposed by fractures. Such improvements are essential

Les Rencontres scientifiques d'IFP Energies nouvelles Flows and mechanics in natural porous media from pore to field scale. Pore2Field 16-18 November 2011, IFP Energies nouvelles (France) especially if one wants to discriminate expensive EOR schemes yielding fairly-close recovery factors. Upscaling issues are inherent to most fractured reservoir engineering studies because sooner or later one has to resort to advanced recovery processes that are almost always multiphase and often compositional processes, sometime even thermal. Multiphase flows emphasize fracture network heterogeneity and matrix-fracture contrast, by comparison with single-phase flows or heat transfers. The dual-porosity modelling approach offers advantages over the single-porosity approach to that respect, because it avoids to lump matrix and fracture media contributions to oil production, via specificallydefined relative permeability functions ([32]).

3.2

Up-scaling fracture flows

Difficulties in upscaling fracture flows arise from large differences between the static geological fracture network and the "dynamic" network involved in flows and transfers. Moreover, that "dynamic" network is depending on the type of flow under consideration. Whereas singlephase well testing explores the connected and conductive network, multiphase flows explore a yet more limited sub-network, the so called "backbone" where the distribution of phases is controlled by gravity segregation, capillary effects among other factors. A major consequence is a high sensitivity of the fracture network upscaled flow properties, and hard-to-predict reservoir responses when fluids flow at the percolation threshold. Such effects are certainly important for chemical EOR. Nonetheless, in most dual-porosity models, relative permeability (kr) functions are considered as linear in the fracture (the so called X rel perms). Moreover, in order to improve simulation accuracy, in analogy with matrix reservoirs, pseudoïsation approaches can be followed, mainly to minimize numerical dispersion. In the presence of conductive faults, pseudo-kr and pseudocapillary functions can be defined to simulate the flow of segregated phases [32].

3.3

Up-scaling matrix-fracture exchanges

The Pseudo-Steady-State (PSS) formula : Qmf(x,t) = αL Km/µ ×(Pm - Pf) gives a coarse representation of matrix-fracture exchanges. It neglects transient effects, and does not dissociate the different mechanisms responsible for the multiphase, compositional and/or thermal transfers involved in advanced recovery processes. -2

Actually, reliable solutions for matrix-fracture transfers could be obtained via matrix sub-gridding approaches ([18], [34]). However, their use in industrial simulators for full-field applications does not seem yet as common practice because of CPU costs. Many modified forms of the above PSS equation are found in the literature, involving various shape factor expressions and/or tuned multiphase pseudo-functions. Their range of validity remains most often restricted to the reservoir parameters, the flow mechanisms and the production history under consideration. Being aware of these difficulties and limits, our preferred approach remains to date in splitting the matrix-fracture transfer into the contributions of each involved physical mechanism [35], and assigning them scaling factors. That is, the matrix-fracture transfer flux of a given phase p is expressed at cell scale as:

∆X∆Y∆Z 6 i 2 Ai (∆p + Ccp.∆pcp + Cgp.Gi + Cvp.∆p if ) Km λ p ∑ lxlylz i=1 li with ∆X, ∆Y, ∆Z the cell dimensions, li (lx, ly, lz) the lateral fp=

dimensions of the equivalent parallelepipedic matrix block, Ai the cross-section area of each of the 6 matrix block lateral faces i (i= x-, x+, y- y+, z-, z+) and λ ip a specific phase mobility function (ratio of relative permeability to viscosity). ∆p is the matrix-fracture pressure difference in a given reference phase and ∆pcp the capillary pressure difference between both media. Gi (i= z-, z+) represents an average value of the gravity i head applied on the matrix block and ∆p f the viscous pressure drop applied on the block in the direction considered. Ccp, Cgp and Cvp are scaling factors referring respectively to capillarity, gravity and viscous drive mechanisms. By setting those scaling factors to 0 or 1, one can assess the role of capillarity, gravity and viscous drive in matrixfracture exchanges. In addition, for typical transfers governed by capillary and gravity forces alone, the scaling factors can be expressed analytically in order to satisfy the vertical equilibrium of fluids at the end of transfer [36]. Recent progress was also made to simulate matrix-fracture molecular diffusion coupled with phase equilibria, in the context of an air injection recovery process [37]. To conclude, the prediction of matrix-fracture transfers in field flow simulators remains an open problem because it involves unsolved up-scaling problems at the matrix block scale. Research is ongoing to better simulate matrix-fracture transfer transients, while keeping the geological information on fracture network. Among incentives are the growing contribution expected in the near future from tight or oil-wet fractured reservoirs.

Les Rencontres scientifiques d'IFP Energies nouvelles Flows and mechanics in natural porous media from pore to field scale. Pore2Field 16-18 November 2011, IFP Energies nouvelles (France)

4

CONCLUSIONS

In this paper, we have emphasized the role of up-scaling techniques for flow simulations in FM. Applications range from HM to uncertainty analysis. Fully multi-scale approaches allowing a real-time flexible treatment of the multi-scale nature of FM will become essential. In the EOR context, the accurate modeling of matrix to fracture exchanges is critical: being able to flush the matrix avoiding the fracture shortcuts remains a challenging issue. The key is understanding the physics of fluids distribution and recovery at reservoir scale, but a preliminary is to dispose of flow-based up-scaling methodologies and tools to identify the relevant fracture model and set up the relevant FM exchange model.

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