Simulation of multiphase flow in fractured reservoirs using a fracture-only model with transfer functions

Summary This paper discusses new techniques for the modeling and simulation of naturally fractured reservoirs with dual-porosity models. Most of the existing dual-porosity models idealize matrix-fracture interaction by assuming orthogonal fracture systems (parallelepiped matrix blocks) and pseudo-steady state flow. More importantly, a direct generalization of single-phase flow equations is used to model multiphase flow, which can lead to significant inaccuracies in multiphase flow-behavior predictions. In this work, many of these existing limitations are removed in order to arrive at a transfer function more representative of real reservoirs. Firstly, combining the differential form of the single-phase transfer function with analytical solutions of the pressure-diffusion equation, an analytical form for a shape factor for transient pressure diffusion is derived to corroborate its time dependence. Further, a pseudosteady shape factor for rhombic fracture systems is also derived and its effect on matrix-fracture mass transfer demonstrated. Finally, a general numerical technique to calculate the shape factor for any arbitrary shape of the matrix block (i.e., nonorthogonal fractures) is proposed. This technique also accounts for both transient and pseudosteady-state pressure behavior. The results were verified against fine-grid single-porosity models and were found to be in excellent agreement. Secondly, it is shown that the current form of the transfer function used in reservoir simulators does not fully account for the main mechanisms governing multiphase flow. A complete definition of the differential form of the transfer function for two-phase flow is derived and combined with the governing equations for pressure and saturation diffusion to arrive at a modified form of the transfer function for two-phase flow. The new transfer function accurately takes into account pressure diffusion (fluid expansion) and saturation diffusion (imbibition), which are the two main mechanisms driving multiphase matrix-fracture mass transfer. New shape factors for saturation diffusion are defined. It is shown that the prediction of wetting-phase imbibition using the current form of the transfer function can be quite inaccurate, which might have significant consequences from the perspective of reservoir management. Fine-grid single-porosity models are used to verify the validity of the new transfer function. The results from single-block dual-porosity models and the corresponding single-porosity fine-grid models were in good agreement. Introduction A naturally fractured reservoir (NFR) can be defined as a reservoir that contains a connected network of fractures (planar discontinuities) created by natural processes such as diastrophism and volume shrinkage (Ordonez et al. 2001). Fractured petroleum reservoirs represent over 20% of the world's oil and gas reserves (Saidi 1983), but are, however, among the most complicated class of reservoirs. A typical example is the Circle Ridge fractured reservoir located on the Wind River Reservation in Wyoming, U.S.. This reservoir has been in production for more than 50 years but the total oil recovery until now has been less than 15% (www.fracturedreservoirs.com 2000). It is undeniable that reservoir characterization, modeling, and simulation of naturally fractured reservoirs present unique challenges that differentiate them from conventional, single-porosity reservoirs. Not only do the intrinsic characteristics of the fractures, as well as the matrix, have to be characterized, but the interaction between matrix blocks and surrounding fractures must also be modeled accurately. Further, most of the major NFRs have active aquifers associated with them, or would eventually be subjected to some kind of secondary recovery process such as waterflooding (German 2002), implying that it is essential to have a good understanding of the physics of multiphase flow for such reservoirs. This complexity of naturally fractured reservoirs necessitates the need for their accurate representation from a modeling and simulation perspective, such that production and recovery from such reservoirs be predicted and optimized.

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Multirate-Transfer Dual-Porosity Modeling of Gravity Drainage and Imbibition

SPE Journal ◽

10.2118/93144-pa ◽

2007 ◽

Vol 12 (01) ◽

pp. 77-88 ◽

Cited By ~ 38

Author(s):

Ginevra Di Donato ◽

Huiyun Lu ◽

Zohreh Tavassoli ◽

Martin Julian Blunt

Keyword(s):

Transfer Functions ◽

Fractured Reservoirs ◽

Transport Equations ◽

Oil Field ◽

Dual Porosity ◽

Gravity Drainage ◽

Stagnant Region ◽

Dual Porosity Model ◽

The Matrix ◽

Porosity Model

Summary We develop a physically motivated approach to modeling displacement processes in fractured reservoirs. To find matrix/fracture transfer functions in a dual-porosity model, we use analytical expressions for the average recovery as a function of time for gas gravity drainage and countercurrent imbibition. For capillary-controlled displacement, the recovery tends to its ultimate value with an approximately exponential decay (Barenblatt et al. 1990). When gravity dominates, the approach to ultimate recovery is slower and varies as a power law with time (Hagoort 1980). We apply transfer functions based on these expressions for core-scale recovery in field-scale simulation. To account for heterogeneity in wettability, matrix permeability, and fracture geometry within a single gridblock, we propose a multirate model (Ponting 2004). We allow the matrix to be composed of a series of separate domains in communication with different fracture sets with different rate constants in the transfer function. We use this methodology to simulate recovery in a Chinese oil field to assess the efficiency of different injection processes. We use a streamline-based formulation that elegantly allows the transfer between fracture and matrix to be accommodated as source terms in the 1D transport equations along streamlines that capture the flow in the fractures (Di Donato et al. 2003; Di Donato and Blunt 2004; Huang et al. 2004). This approach contrasts with the current Darcy-like formulation for fracture/matrix transfer based on a shape factor (Gilman and Kazemi 1983) that may not give the correct average behavior (Di Donato et al. 2003; Di Donato and Blunt 2004; Huang et al. 2004). Furthermore, we show that recovery is exceptionally sensitive to parameters that describe the physics of the displacement process, highlighting the need to make careful core-scale measurements of recovery. Introduction Di Donato et al.(2003) and Di Donato and Blunt (2004) proposed a dual-porosity streamline-based model for simulating flow in fractured reservoirs. Conceptually, the reservoir is composed of two domains: a flowing region with high permeability that represents the fracture network and a stagnant region with low permeability that represents the matrix (Barenblatt et al. 1960; Warren and Root 1963). The streamlines capture flow in the flowing regions, while transfer from fracture to matrix is accommodated as source/sink terms in the transport equations along streamlines. Di Donato et al. (2003) applied this methodology to study capillary-controlled transfer between fracture and matrix and demonstrated that using streamlines allowed multimillion-cell models to be run using standard computing resources. They showed that the run time could be orders of magnitude smaller than equivalent conventional grid-based simulation (Huang et al. 2004). This streamline approach has been applied by other authors (Al-Huthali and Datta-Gupta 2004) who have extended the method to include gravitational effects, gas displacement, and dual-permeability simulation, where there is also flow in the matrix. Thiele et al. (2004) have described a commercial implementation of a streamline dual-porosity model based on the work of Di Donato et al. that efficiently solves the 1D transport equations along streamlines.

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Multiphase Flow in Fractured Reservoirs

Advances in Transport Phenomena in Porous Media ◽

10.1007/978-94-009-3625-6_12 ◽

1987 ◽

pp. 551-629 ◽

Cited By ~ 3

Author(s):

Ole Torsaeter ◽

Jon Kleppe ◽

Teodor Golf-Racht

Keyword(s):

Multiphase Flow ◽

Fractured Reservoirs

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The "Checker Model," An Improvement in Modeling Naturally Fractured Reservoirs With a Tridimensional, Triphasic, Black-Oil Numerical Model

Society of Petroleum Engineers Journal ◽

10.2118/10977-pa ◽

1985 ◽

Vol 25 (05) ◽

pp. 743-756 ◽

Cited By ~ 5

Author(s):

Dimitrie Bossie-Codreanu ◽

Paul R. Bia ◽

Jean-Claude Sabathier

Keyword(s):

Transfer Functions ◽

Fractured Reservoirs ◽

Fracture Network ◽

Fractured Reservoir ◽

Local Flow ◽

Fractured Medium ◽

Black Oil ◽

Matrix Blocks ◽

The Matrix ◽

Conventional Models

Abstract This paper describes an approach to simulating the flow of water, oil, and gas in fully or partially fractured reservoirs with conventional black-oil models. This approach is based on the dual porosity concept and uses a conventional tridimensional, triphasic, black-oil model with minor modifications. The basic feature is an elementary volume of the fractured reservoir that is simulated by several model cells; the matrix is concentrated into one matrix cell and tee fractures into the adjacent fracture cells. Fracture cells offer a continuous path for fluid flows, while matrix cello are discontinuous ("checker board" display). The matrix-fracture flows are calculated directly by the model. Limitations and applications of this approximate approach are discussed and examples given. Introduction Fractured reservoir models were developed to simulate fluid flows in a system of continuous fractures of high permeability and low porosity that surround discontinuous, porous, oil-saturated matrix blocks of much lower permeability but higher porosity. The use of conventional models that permeability but higher porosity. The use of conventional models that actually simulate the fractures and matrix blocks is restricted to small systems composed of a limited number of matrix blocks. The common approach to simulating a full-field fractured reservoir is to consider a general flow within the fracture network and a local flow (exchange of fluids) between matrix blocks and fractures. This local flow is accounted for by the introduction of source or sink terms (transfer functions). In this formulation, the model is not directly predictive because the source term (transfer function) is, in fact, entered data and is derived from outside the model by one of the following approaches:analytical computation,empirical determination (laboratory experiments), ornumerical simulation of one or several matrix blocks on a conventional model. To derive these transfer functions, imposing some boundary conditions is necessary. Unfortunately, it is generally impossible to foresee all the conditions that will arise in a, matrix block and its surrounding fractures during its field life. It would be helpful, therefore, to have a model that is able to compute directly the local flows according to changing conditions. However, to have low computing times, it is necessary to use an approximate formulation and, thus, to adjust some parameters to match results that are externally (and more accurately) derived in a few basis, well-defined conditions. By current investigative techniques, only a very general description of the matrix blocks and fissures can be obtained, so our knowledge of local flows is very approximate. This paper presents a modeling procedure that is an approximate but helpful approach to the simulation of fractured reservoirs and requires a few, simple modifications of conventional black-oil mathematical models. Review of the Literature Numerous papers related to single- and multiphase flow in fractured porous media have been published over the last three decades. On the basis of data from fractured limestone and sand-stone reservoirs, fractured reservoirs are pictured as stacks of matrix blocks separated by fractures (Figs. 1 and 2). The fractured reservoirs with oil-saturated matrices usually are referred to as "double porosity" systems. Primary porosity is associated with matrix blocks, while secondary porosity is associated with fractures. The porosity of the matrices is generally much greater than that of the fractures, but permeability within fractures may be 100 and even over 10,000 times higher permeability within fractures may be 100 and even over 10,000 times higher than within the matrices. The main difference between flow in a fractured medium and flow in a conventional porous system is that, in a fractured medium, the interconnected fracture network provides the main path for fluid flow through the reservoir, while local flows (exchanges of fluids) occur between the discontinuous matrix blocks and the surrounding fractures. Matrix oil flows into the fractures, and the fractures carry the oil to the wellbore. For single-phase flow, Barenblatt et al constructed a formula based on the dual porosity approach. They consider the reservoir as two overlying continua, the matrices and the fractures. SPEJ p. 743

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Streamline simulation of water-oil displacement in a heterogeneous fractured reservoir using different transfer functions

Oil & Gas Science and Technology – Revue d’IFP Energies nouvelles ◽

10.2516/ogst/2018004 ◽

2018 ◽

Vol 73 ◽

pp. 14 ◽

Cited By ~ 4

Author(s):

Mohammad Mesbah ◽

Ali Vatani ◽

Majid Siavashi

Keyword(s):

Transfer Function ◽

Transfer Functions ◽

Water Saturation ◽

Fractured Reservoirs ◽

Rock Properties ◽

Fractured Reservoir ◽

Commercial Software ◽

The Matrix ◽

Eulerian Grid ◽

Streamline Method

Main parts of oil and gas reserves are stored in fractured reservoirs. Simulation of multiphase flow in fractured reservoirs requires a large amount of calculations due to the complexity, reservoir scale and heterogeneity of the rock properties. The accuracy and speed of the streamline method for simulating hydrocarbon reservoirs at field scale make it more applicable than conventional Eulerian simulators using finite difference and finite element techniques. Conventional simulators for fractured reservoirs consume a great deal of time and expense and require powerful CPUs like supercomputers. This makes the development of a fast, powerful and precise simulation method of great importance. The present study was undertaken to develop a computational code as a streamline simulator to study waterflooding in a two-dimensional fractured reservoir with heterogeneous permeability using the Dual Porosity-Single Permeability (DPSP) model. In this simulator, the pressure equation is solved implicitly over an Eulerian grid and then the streamlines are generated using Pollock's semi-analytical method and are traced. At this point, the Time-Of-Flight (TOF) is developed and the saturation equations are mapped and solved explicitly along the streamlines. Next, the results are transferred back onto the Eulerian grid and the calculations are repeated until the simulation end time. In fractured reservoirs, the interaction between the matrix and fracture is included in the transfer functions. Transfer functions model fluid flow and production mechanisms between the matrix and fracture. They introduce source/sink equations between the matrix and fracture and they are distributed throughout the media. In the current study, a problem is simulated using streamline method and several important transfer functions. A new linear transfer function with a constant coefficient is introduced that is based on differences in water saturation between the matrix and fracture. The simulation results were then compared and a commercial software is applied to solve the same problem. The results of the streamline simulator were compared with those of the commercial software and showed appropriate accuracy for the newly introduced transfer function. The accuracy and efficiency of the streamline simulator for simulation of two-phase flow in fractured reservoirs using different transfer functions are confirmed and the results are verified.

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