Time-Dependent Shape Factors for Secondary Recovery in Naturally Fractured Reservoirs

Abstract

An accurate interporosity flow equation incorporating a time-dependent shape factor is derived and verified for improved dual-porosity modeling of waterflooding in naturally fractured reservoirs. This equation expresses the interporosity exchange rate in terms of the oil phase pressure gradient in the matrix, fracture surface area, fluid effective permeability at the matrix/fracture interface, fluid viscosity and a variable matrix-block shape factor. This approach can accommodate the flow directed from matrix to fractures while representing the permeability of interconnected fractures as a tensor. The model equations are expressed in dimensionless form for convenient integration into present numerical simulators for accurate simulation of waterflooding in naturally fractured reservoirs. These features distinguish this model from the conventional sugar-cube modeling of naturally fractured reservoirs. Fine-grid numerical simulation of a matrix block is performed to verify the flow equation using the time-dependent shape factor. Numerical experiments with various size matrix blocks indicate the shape factor varies with time and converges to the steady-state shape factor value reported in previous studies for single-phase flow. The shape factor for single-phase flow converges to a steady-state value at a speed proportional to the reciprocal of total compressibility, while the shape factor for two-phase flow converges at a speed proportional to the slope obtained from the capillary pressure curve evaluated at the average water saturation present at the matrix/fracture interface. Therefore, the single-phase shape factor converges much more rapidly to its steady-state value than the two-phase shape factor. This study demonstrates that neglecting the time-dependency of the shape factor introduces significant errors in fractured reservoir simulation, and time-dependent shape factors are needed because of the variation of the water saturation front location in the matrix block, which moves from the fracture face to the block-center as the water flows from the fracture into the matrix. Introduction Most naturally fractured reservoirs are characterized by initial high production rates that later drop to much lower stable flow rates. The later period of production is controlled by the matrix/fracture interaction through the interporosity flow rate. During this period, the low-permeability matrix replenishes fractures with hydrocarbon fluids, which are produced from the high conductivity pathways at the wellbore.1 Fluid expansion is the primary recovery mechanism during primary recovery. Successful implementation of a secondary or tertiary recovery project in a naturally fractured reservoir requires that the matrix/fracture interaction be well understood. Some of the mechanisms investigated in order to estimate and predict the interporosity flow rate include gravity and capillary effects,2–4 reinfiltration and capillary continuity throughout the matrix blocks,5–8 and cocurrent and/or countercurrent imbibition phenomena.9–11 In waterflooding, capillary imbibition is generally the most important recovery mechanism.12

Role of Fracture Flow in Matrix-Fracture Transfer

“…As explained previously, it should be noted that the analytical solution of the nonlinear diffusion equation in the semi-infinite domain would be used in the finite domain with the constraint that such a solution is a good approximation for the time period when the saturation disturbance is still away from the sealed boundary. This is analogous to modeling the first flow period as called by Penuela et al 30 . In every simulation, a time period t final , corresponding to the time taken by one percent of the saturation disturbance at the open boundary to reach the other end, is calculated.…”

“…At a more fundamental level the question arises whether the relationship between capillary pressure and saturation that is valid for a volume element greater than representative elemental volume (REV), is appropriate in the region very close to the boundary. This question was partly addressed by Penuela et al 30 . They proposed that there is a critical mobile oil saturation at the matrix-fracture interface greater than the residual oil saturation that ensures the flow of oil from the matrix.…”

“…Kazemi et al 34 improved the description of two-phase flow by redefining the shape factor. Recently Penuela et al 30 have derived a time-dependent shape factor to calculate the matrix-fracture transfer function. Kazemi et al 9 have cautioned against interpreting the time constant of the empirical exponential recovery correlation as anything other than purely a curve-fitting parameter.…”

Effective Local-Global Upscaling of Fractured Reservoirs under Discrete Fractured Discretization

“…For example, the complex flow behavior between fracture and matrix medium is represented by the shape factor as a function of fracture spacing only [2][3][4]. Although a lot of research [5][6][7][8] aiming to obtain a more accurate shape factor under specific flow scenarios were carried out in those years, most of them were still based on idealized fracture distributions. Another drawback of DP models is they are not well suited for the modeling of a small number of large-scale fractures, which may dominate the flow.…”

Unified fractional differential approach for transient interporosity flow in naturally fractured media

“…Initially, only constant values of the shape factor were considered [6,[53][54][55][56]. Later, the time dependent character of the shape factor was discovered [57][58][59][60][61][62]. Different attempts to compare transient and pseudo steady-state models for solute transport were proposed by Haggerty and Gorelick [14] using the MRMT model.…”