**Drainage Patterns of Subvertical Wells intersecting Naturally Fractured Carbonate Reservoir Layers**

**Shaho Bazr-Afkan¹, Stephan Matthai¹, and Caroline Milliotte²**

*¹Institute of Reservoir Engineering, Department of Mineral Resources and Petroleum Engineering, Montan University of Leoben, Leoben, Austria
²Fractured Reservoir Dynamics (FRD), Rosegger Str. 11, Leoben, Austria
*

In the Middle East and elsewhere, the identification of optimal well patterns for naturally fractured carbonate reservoirs (NFRs), still heavily relies on classic reservoir engineering concepts like
*drainage radius* and *area* (e.g., Muskat & Meres, 1936). This is astonishing because these concepts represent strong conceptual idealisations even when applied to much less heterogeneous reservoirs where fluids flow pervasively in layered porous rocks. These concepts were simplistic by necessity: they had to be amenable to mathematical analysis prior to the advent of digital computing.

Today, water breakthrough-, pressure diffusion- and tracer test data (e.g., Schechter et al., 2003) as well as flow patterns observed in computationally demanding discrete fracture and rock matrix simulations (e.g., Belayneh et al., 2006), show that drawdown in NFRs is anything but radial. Also, PLT- and temperature logs, and isotopic tracer data indicate that flow is highly localised and often remotely sourced (e.g., Aguilera, 1995). Ultimate recovery factors also are extremely low (Allan & Sun, 2003). These observations raise the question whether such conceptual models can be applied to NFRs and which techniques could substitute for them in field development planning.

Here we present two-dimensional numerical (slightly compressible 2-phase flow) drainage simulations carried out with a new hybrid simulator (Bazr Afkan and Matthai, 2011): The (transient) fluid-pressure equation is solved implicitly using the finite element approach, calculating element-wise constant velocity vectors by post-processing. Then the transport equation is solved explicitly. Conservation of fluid mass is obtained by solving it with a complementary node-centered finite volume discretisation using multi-dimensional flux limiting. Interim nodal saturations are aggregated into piecewise constant element-based values by solving sets of auxiliary equations. This variable placement has the advantage that saturation jump discontinuities can evolve across material interfaces. Also, there is no smearing of saturation values during domain boundary parallel flow as was observed in older node-centered finite-element finite-volume methods (Nick and Matthai, 2011). Capillary-driven cross-interface flow is handled via transfer functions that simultaneously take into account the mitigating effects of viscous flow-induced saturation gradients in the rock matrix. Special constitutive relationships are applied to the fractures: Rate-dependence and viscous coupling among phases is considered through application of the relative passibility (*not permeability!*) model of Fourar and Lenormand (2001). It is closely related to the Lockhard – Martinelli model for two-phase flow in pipes and therefore has the minimum number of adjustable parameters of the models screened for this purpose. Special provisions are also made to make our new IMPES scheme more computationally efficient: (1) finite volumes are tagged and only active ones are computed at any time step; (2) lower-dimensional finite elements are used to represent the fractures (cf., Juanes et al., 2000), (3) saturation shock-height and characteristic saturations are precomputed and guide the flux limiting.

The simulator is applied to unstructured spatially adaptive finite-element finite-volume meshes resolving two-dimensional km-scale outcrop-analog Discrete Fracture and Matrix (DFM) models of fractured subhorizontal reservoir layers in detail. Sensitivity analyses illustrate from where and over what horizontal distance fluids can be drawn into wells intersecting- and off-side fractures and how the balance of viscous to capillary forces changes along the fluid pathways. Our investigation considers a range of fracture geometries (multiple sets of clustered fractures juxtaposed on diffuse ones), and plausible states of
*in situ* stress used to compute orientation and length-dependent fracture apertures.

Analogous to Schechter et al.’s (2002) field observations, the simulation results indicate a strong anisotropy of drainage and rate dependence of flow and saturation patterns. Source to well times for fluid batches do not necessarily correlate with distance to well, but proximity to larger interconnected fractures. Water encroachement from the far periphery of the well is marked by distal flow patterns where fracture-saturation often reflects capillary pressure equilibrium with the rock matrix. In contrast, proximal flow patterns are marked by fracture flow that is so fast that any drained oil is displaced immediately. Regarding field applications, these results indicate, that the closer the water front encroaches on a production well, the lower its rate needs to be to delay water coning.

In addition to patterns of flow, we visualise the recovery behaviour by means of velocity – saturation histograms, displaying how average saturation varies with flow velocities in the different volume fractions of each model, distinguishing different matrix block sizes, and fracture classes (large-aperture clustered to diffuse small aperture fractures). Respective histograms serve as accurate snapshots of recovery efficiency and its variation through well life.

**References**

Aguilera, R., “Naturally Fractured Reservoirs.” 2^{nd} ed., PennWell Publishing Company, Tulsa, Oklahoma (1995).

Allan, J. and Sun, Q. S., “Control on recovery factor in fractured reservoirs: lessons learned from 100 fractured fields.” SPE 84590, Annual Tech. Conv., Denver, CO, USA, Octobre 5-8 (2003).

Bazr Afkan, S. and Matthai, S. K., “A New Hybrid Simulation Method for Multiphase Flow on Unstructured Grids with Discrete Representations of Material Interfaces.” IAMG Salzburg, doi:10.5242/iamg.2011.0120 (2011).

Belayneh, M., S. K. Matthäi, M. J. Blunt, and S. Rogers, “Comparison between deterministic with stochastic fracture models in water-flooding numerical simulations.” AAPG Bulletin, v. 93 (2009).

Fourar, M. and Lenormand, R., “Inertial effects in two-phase flow through fractures.” Oil & Gas Science and Technology – Rev. IFP, Editions Technip, 55:3, 359-268 (2000).

Juanes, R., Samper, J. and Molinero, J., “A general and efficient formulation of fractures and boundary conditions in the finite element method.” Int. J. Num. Meth. in Eng., 00, 1-25 (2000).

Matthai, S. K. and Nick, H. M., “Comparison of three FE-FV numerical schemes for single- and two-phase flow simulation of fractured porous media.” TIPM 90, 421-444 (2011).

Muscat, M. and Meres, M.W., “The flow of heterogeneous fluids through porous media.” Physics 7:9, 346-363 (1936).

Schechter, D., Putra, E., Baker, R. O., Knight, W. H, McDonald, W. P., Leonard, P. and Rounding, C., “Waterflood performance in the naturally fractured Spraberry trend area, west Texas., Naturally Fractured Reservoir Conference, Oklahoma City, June 3-4 (2003).

*AAPG Search and Discovery Article #120034©2012 AAPG Hedberg Conference Fundamental Controls on Flow in Carbonates, Saint-Cyr Sur Mer, Provence, France, July 8-13, 2012*