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Summary

Compositional variation in a rectangular two-dimensional (x,z) fractured porous medium containing a two-component single-phase fluid in the presence of a prescribed linear temperature field is considered. The work examines the effect of the fracture parameters: fracture aperture (or fracture permeability), fracture intensity, and fracture connectivity on the fluid compositional variation. Numerical results reveal that, for a high fracture aperture, a pronounced convective motion within the fractures takes place, whereas the composition is only affected beyond a certain fracture aperture. The effect of connected and discrete fractures on compositional variation is also studied; connected fractures influence the compositional variation much more than discrete fractures, as expected. The convection cells are mainly loops which develop within the connected fractures.

Introduction

Examination of the compositional variation of reservoir fluids in hydrocarbon reservoirs reveals various trends. In some reservoirs there is substantial variation of composition in the horizontal direction.¹ In many other reservoirs, there is a large variation of composition in the vertical direction.^2,3 There are also reservoirs that have no variation of composition in the entire reservoir; such reservoirs may have an oil column thickness of 1.5 km and a horizontal extension larger than 10 km.^4,5 The purpose of this work is understanding of compositional variation in fractured hydrocarbon reservoirs.

There are four distinct mechanisms that affect the variation of composition in the single phase in a hydrocarbon reservoir;⁶ (1) thermal diffusion, (2) pressure diffusion, (3) molecular diffusion, and (4) natural convection. Thermal diffusion is the tendency of a convection-free mixture to separate under a thermal gradient. Molecular diffusion is the tendency to mix due to concentration gradient. Pressure diffusion is separation by pressure gradient; it is negligible in the horizontal direction even when there exists natural convection, but may be pronounced in the vertical direction due to high vertical pressure gradient. Natural convection is the convective circulation due to density gradient. Density gradient is established due to temperature and concentration gradients. A pronounced effect of natural convection on compositional variation may occur in homogeneous media.⁶

There are very few studies that combine the effect of natural convection and diffusion on the compositional variation in hydrocarbon reservoirs. These studies include the works of Jacqmin,⁷ and Riley and Firoozabadi. ⁶ Both studies address compositional variation in homogeneous porous media. Jacqmin’s study is concerned with a wide variety of conditions, sloping reservoirs, and even two phases but does not include thermal diffusion. He uses a perturbation analysis where approximations are made to the governing equations. Based on his study, Jacqmin states that under certain conditions the fluid composition reorients itself in such a way as to inhibit convection. Riley and Firoozabadi⁶ studied the effect of thermal, pressure, and Fickian diffusion, and natural convection on compositional variation in a cross-sectional reservoir with a prescribed linear temperature field. The behavior is investigated using a method of successive approximations which iterates on Poisson’s equation. This behavior is then incorporated in a simplified perturbation solution which provides accurate results for horizontal composition variation. Riley and Firoozabadi⁶ show that a small amount of convection can cause the horizontal composition gradient to increase until a maximum is reached and then decays as 1/k.

The works of Jacqmin,⁷ and Riley and Firoozabadi⁶ cover homogeneous porous media. This work is concerned with a numerical study of natural convection and diffusion in fractured porous media. To the best of our knowledge, this problem has not been discussed previously. The conservation equations of mass and species and Darcy’s law, together with the boundary conditions and the matching conditions at the matrix block/fracture interface, are numerically solved. The numerical investigations are carried out for fracture permeabilities varying by five orders of magnitude which corresponds to a fracture aperture variation from 0.01 to 1.00 mm (fracture permeability and aperture are related via a cubic law). However, in view of the large number of parameters, no attempt was made to present a complete parametric study.

Mathematical Formulation

We consider two-dimensional fractured porous media with width b and height h (Fig. 1) saturated by a binary mixture of C₁ (methane)/nC₄(normal butane). The fractured porous media consist of matrix blocks and fractures of permeabilities k_m and k_f, respectively. The matrix and fracture porosities are assumed to be the same. We assume that the Oberbeck-Boussinesq approximation (see Chandrasekhar⁸) is valid in the range of temperature, pressure, and composition expected so that the density ? is constant (equal to ? to be defined) except in the buoyancy term ?₀ where it varies linearly with the temperature T and the weight fraction w : \rho (T,w)=\rho {0}[1 - \beta {T}(T - T {0}) - \beta {w}(w - w {0})].\eqno ({\rm 1}) In the above equation, ?₀, ?_T=(?1/? ₀ )(??/?T)_w and ?_w =(?1/?₀ )(??/?w)_T are the density at temperature T ₀ and weight fraction w ₀ the thermal expansion coefficient, and the compositional expansion coefficient, respectively. The coefficients ?_T and ? _w are calculated for the system C₁/nC₄, using the Peng-Robinson equation of state.⁹. Fig. 2 depicts the density ? vs. temperature and mole fraction ? of methane. Fig. 2 clearly shows the validity of the assumption of linear variation of ? vs. T and ? in the range of temperature, pressure, and composition expected; ? _T and ? _w are thus the slopes of those lines.