A Method to Determine Stress-Dependent Filtration Properties of Fractured Porous Rocks by Laboratory Test Data

. Within the dual porosity model the authors develop and theoretically, using synthetic data, substantiate the method for determining mass exchange coefficient and stress-dependent permeability of fractured porous reservoir rocks. The proposed filtration test circuit consists of three sequential measurements of flowrate in a specimen subjected to the varied external stress σ at the inlet fluid pressure P : by standard scheme ( Q 0 ) and with plugging of fissures at one ( Q 1 ) and at the other end ( Q 2 ) of the specimen. The model of the experiment is created, and the analytical solution is obtained for the direct problem on steady-state flow: dependences of Q 0 , Q 1 and Q 2 on σ and P . The input data are synthesized by superimposition of multiplicative noise on the exact solution of the direct problem. The synthesized data are used to derive the inversion relations for calculating the permeabilities k 1 and k 2 of fissures and matrix as well as the mass exchange coefficient by Q 0 , Q 1 and Q 2 . Using LS method, the dependences k 1 ( σ ) and k 2 ( σ ) are reconstructed. The numerical experiments reveal low stability of inversion by input data. Thus, it is necessary to perform a cycle of measurements at the increasing input pressure with subsequent averaging of the results.


Introduction
Justification of opening and mining schemes in hydrocarbon extraction, estimation of well stability, inversion of logging data -this is a far from being complete list of problems solvable using knowledge on poroperm properties of reservoir rocks [1][2][3]. The source of such information can be laboratory experiments, field tests and integrated geophysical logging [4,5]. The obtained data are interpreted in the framework of hydrodynamic models usually selected based on the petrographic analysis of core samples [6]. * Corresponding author: lanazarova@ngs.ru Productive strata in many hydrocarbon reservoirs have porous and fractured structure [7]. For description of fluid flow in such strata, in [8] a dual porosity model is proposed for a medium composed of elements of two types -matrix and fissures (Fig. 1). Each type is provided with a set of the governing parameters (pressure, permeability, porosity, etc.) and the law of mass exchange between the constituent elements. Later on, this approach was generalized to any types of continuum [9][10][11] by introduction of notion of a representative elementary volume and was successfully applied in studying heat and mass transfer processes in multi-phase systems [12,13], including oil and gas reservoirs [14][15][16][17] and coal rock masses [18][19][20][21][22]. One of the stages in verification of such models is estimation of permeability and porosity of constituent elements, as well as determination of an empirical function M describing mass exchange between matrix and fissures. The objective is reached using data of laboratory [23,24] and full-scale [25][26][27] tests, as well as calculated effective parameters of media with regular [28,29] and stochastic [30,31] structure. In [8,25], it is assumed that the function M can only be determined in the transient mode of well operation using the pressure build-up curve.
The laboratory [32][33][34][35][36] and in-situ [37][38][39] researches prove the essential dependence of rock permeability on stresses. With increasing depth, this effect exerts stronger influence on efficiency of geophysical log data interpretation aimed to identify producing intervals and estimate poroperm properties of reservoir rocks [40,41]. This paper, within the model of the fractured porous medium [8], develops and theoretically substantiates the method which makes it possible to determine the empirical function of mass exchange between matrix and fissures as well as the dependence of their permeabilities on stresses by the data of stationary filtration tests.

Model of Fractured Porous Medium
Evolution of hydrodynamic fields in a fractured porous medium is described with the dual porosity model [8] including: the continuity equations 0 ) ( ) ( (1)  [34,35,42,43] show that under low pressures, the permeability dependence on stresses is well approximated by the exponential function ) exp( K m and m α are empirical constants. In case of comparable pressure and stresses, σ in (4) should be interpreted as an effective stress In [8,43] it is proposed to describe intensity of crossflows between matrix and fissures by the linear function where β is the mass exchange coefficient. In [8,25] it is mentioned that β depends on the specific surface of matrix rocks. When fluid, matrix and fissures are the weakly compressible media, system (1)-(3) can be linearized and, with regard to (5), reduced to two equations

Formulation and Solution of Boundary Problem
Let a cylindrical specimen (base area S, length l) be subjected to a constant stress σ over lateral surface (Fig. 2). We consider a one-dimensional steady-state flow described by the system of equations obtained from (6) where p 1 and p 2 are the functions of the coordinate x directed along the specimen axis. The permeability-stress dependence is included in (7) through (4). As per the study objective, it is required to develop the filtration test pattern such that to find the mass exchange coefficient and constants in empirical dependences (4) of stress-  (7) needs not less than three 'quantums' of information obtained on the same specimen. The three sequential measurement designs are presented below with relevant boundary conditions for (7).
T1. On the left end, fissures are plugged with a penetrant, and using a high resolution image (obtained, e.g., with electron microscope), the area S 1 of these fissures is determined. On the right end x = l, the fluid pressure P i is set; on the left end, the atmospheric pressure is assigned T2. The right end fissures are plugged and then In all tests T0, T1 and T2, the fluid flow rates Q 0 , Q 1 and Q 2 are measured on the left end in the steady-state flow mode.
Omitting cumbersome intermediate calculations, Table 1 presents the final result, namely, the pressure distribution in matrix and fissures, as well as the fluid flowrate at x = 0.
. Figure 3a depicts the pressure in matrix in the test design T2 at P i = 0.

Synthesis of Input Data
We set some values for the permeabilities  Table 2 presents the input data (columns 2, 3 and 4) generated using (14) at ε = 0.01.

Determination of Poroperm Parameters of Fractured Porous Medium
, then, dividing (11) and (13) by (9), we obtain The Macluarin expansion of the tangent at small γ reduces (15) to a system of equations with respect to γ and ψ 1 2 which has an analytical solution (columns 5 and 6, Table 2)  Table 2). It turns out that even at the small measurement error of the flowrate, the determination error of k 1 and β can exceed 70%. The cause of such errors lies in the contrast of the permeabilities of matrix and fissures as they can differ by 1-3 orders of magnitude [25]. For better accuracy, it is necessary to measure flow rates at several values of the input pressure P i and to assume the required values as average quantities (last line in Table 2).

Determination of Stress-Dependent Permeabilities of Matrix and Fissures
For the model parameters accepted in 4.2 but at ε = 0.02, using (14) Table 3. Virtual experimental results on reconstruction of the dependence between the fissure permeability k 1 and stresses.  Table 4. Virtual experimental results on reconstruction of the dependence between the matrix permeability k 2 and stresses.  obtained at all values of the input pressure P i (last lines in Tables 4 and 5), the variation factor is never higher than 5% at any random values ) , ( n i s ω in (14) (line 2 in Fig. 4).

Conclusion
The authors have developed and theoretically, based on virtual experimentation, substantiated the method that makes it possible to quantitatively estimate the mass exchange factor β and to establish the empirical dependence of permeability on stresses in fractured porous reservoir rocks. The interpretation of the synthetic input data is implemented within the model of continuum with dual porosity, for which the exact solution of the problem on steady-state fluid flow is obtained. The analytical expressions are derived for determining permeabilities of fissures and matrix as well as the coefficient β by flow rates measured in three different measurement designs. As yet the value of β has been determined only in an operating well by the pressure build-up curve while the proposed procedure allows such estimating in laboratory using standard equipment.