ISSN 1062-7391, Journal of Mining Science, 2016, Vol. 52, No. 3, pp. 424–431. © Pleiades Publishing, Ltd., 2016. Original Russian Text © L.A. Nazarova, L.A. Nazarov, 2016, published in Fiziko-Tekhnicheskie Problemy Razrabotki Poleznykh Iskopaemykh, 2016, No. 3, pp. 11–19.

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Evolution of Stresses and Permeability of Fractured-and-Porous Rock Mass around a Production Well L. A. Nazarovaa,b* and L. A. Nazarova,b a

Chinakal Institute of Mining, Siberian Branch, Russian Academy of Sciences, Krasnyi pr. 54, Novosibirsk, 630091 Russia *e-mail: [email protected] b Novosibirsk State University, ul. Pirogova 2, Novosibirsk, 630090 Russia Received February 14, 2016

Abstract—The authors model deformation and mass transfer in jointed and porous rock mass around a production well. The modeling based on the concept of a continuum with double porosity uses an original method with finite difference solution of mass transfer equations and analytical solution of pore elastoplasticity equations. From the numerical experiments, dimensions of irreversible deformation zones in the well bore zone grow with the parameter Bio. The estimate of the reservoir permeability decline in the course of operation, obtained from the pore elasticity and pore plasticity models, qualitatively agrees with the in situ observation data. Keywords: Fractured-and-porous rock mass, poroelasticity, double porosity, seepage, stress evolution, fracture zone, numerical modeling. DOI: 10.1134/S106273911603061X

INTRODUCTION

The concept of the hierarchical block structure of rock masses, put forward in [1] and developed in [2–4], is now a common notion. Geomechanical modeling of natural and man-made objects should account for various scale structural features of a geomedium. Geophysical methods can distinguish rock blocks with linear sizes of the order of centimeters in rock mass in the wellbore vicinity [5]. Smaller blocks are impossible to identify and difficult to include in geomechanical and hydrodynamic models of deformation and mass transfer. The first dual porosity model, suggesting different pressures in pores and fractures in unit volume, was presumably described in [6]. This modeling approach is fruitfully developed for oil and gas reservoirs [7–9] and coal beds [10–12]. Many researches into fluid flow in wellbore vicinity in fractured-and-porous media assume simplified hypothesis where an analytical solution is obtained using integral transformations and, then, is analyzed with the help of asymptotic methods at large and/or small times [13–15]. For porous media, there are the developed and implemented models to describe evolution of geomechanical, thermal and electro-hydro-dynamic fields in rock masses with multi-phase fluid flows [16–23]. This study addresses elastic and irreversible deformation of fractured-and-porous rock mass in the wellbore vicinity in a nearly depleted reservoir. 1. FRACTURED-AND-POROUS ROCK MASS MODEL. FORMULATION AND SOLUTION

Let at a time t = 0 a vertical well intersect a fluid-saturated fractured-and-porous reservoir with a thickness h at a depth H (Fig. 1). Such objects are assumed as solid media with dual porosity, m1 and m2 , and dual permeability, k1 and k 2 [6, 7]. Subscripts 1 and 2 mean blocks and fractures, respectively. 424

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Fig. 1. Schematic structure of fractured-and-porous medium and the computational domain.

Description of mass transfer involves: mass conservation equations  ∂ (m1 ρ ) + div( ρv1 ) − ρM = 0 ; ∂t  ∂ ( m2 ρ ) + div( ρv2 ) + ρM = 0 ; ∂t

(1) (2)

Darcy’s law  k vi = − i ∇pi ;

η

(3)

constitutive relations

ρ ( pi ) = ρ 0 (1 + βpi ) , (4)  where vi , ρ , β and η are the velocity, density, compressibility and viscosity of fluid, respectively, i = 1, 2 ; ρ 0 is the initial value of ρ ; M = C ( p1 − p2 ) is the flow from blocks to fractures, C = k1S /η , S is an empirical constant proportional to the block substance surface area. The increments of porosity and pressure relate linearly: dm1 = α 11 dp1 − α 12 dp 2 , (5) dm2 = −α21dp1 + α22dp2 , where α1i and α 2i are the comporessibilities of the matrix and fractures. Simple transformations reduce (1)–(5) to a system of two parabolic equations: q11

∂p1 ∂p 1 − q12 2 = div ( k1∇ p1 ) − M / β , ∂t ∂t βη

q22

∂p2 ∂p 1 − q21 1 = div(k2∇p2 ) + M / β , ∂t ∂t βη

(6)

where q11 = m1 + α11 / β ; q22 = m2 + α 22 / β ; q12 = α12 / β ; q21 = α 21 / β . It is noteworthy that the permeabilities in a general case depend on the stresses and pressure. We describe deformation of the fractured-and-porous rock mass using the equations of poroelasticity [12]: Σ = (λε − p)Ι + 2μΕ , (7) where Σ , Ε and Ι are the tensor of stresses and strains, and the unit tensor; λ , μ are the Lamé parameters; ε = trΕ ; p = (1 − B) p1 + Bp2 , B is the Bio parameter. In (7) the stresses and strains are JOURNAL OF MINING SCIENCE Vol. 52 No. 3 2016

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associated with the representative volume of the fractured-and-porous medium, and the equilibrium equations are satisfied for this volume: divΣ = 0 (8) alongside with the Cauchy relations:   Ε = 0.5(∇u + ∇u T ) , (9)  where u is the displacement vector; Т means transposition. The relations (6)–(9) make a closed system of equations to model one-phase fluid flow in a fractured-and-porous rock mass subjected to deformation. Let the upper and lower boundary of the reservoir be impermeable, h << H , and the horizontal in situ stress field components σ h be the same and described by the lateral earth pressure coefficient q.

Then the model (6)–(9) becomes axially symmetric and can be written as: ∂ ( q11 p1 − q12 p 2 ) D1 ∂ ⎛ ∂p ⎞ = ⎜ rG ( s ) 1 ⎟ − M / β ; ∂t ∂r ⎠ r ∂r ⎝

∂(q22 p2 − q21 p1) D2 ∂ ⎛ ∂p ⎞ = ⎜ rG(s) 2 ⎟ + M / β ; ∂t ∂r ⎠ r ∂r ⎝ ∂σ rr σ rr − σ θθ + =0; ∂r r σ rr = λε + 2με rr − p ; σ θθ = λε + 2μεθθ − p ; ε rr = ∂u / ∂r , ε θθ = u / r .

(10)

(11) (12)

The relation (10) is introduced with the dependence k i = k i0 G ( s ) of the permeability on the equivalent stress s = trΣ / 3 + p , where G ( s) = exp(γ s / μ ) ( γ is an empirical constant) [24, 25]; Di = ki0 / βη . The formulated boundary conditions for (10) and (11) are given by: σ rr (rw , t ) = − p w ; σ rr (rc , t ) = −σ h ; p1 (rw , t ) = p2 (rw , t ) = p w ; p1, r (rc , t ) = p2, r (rc , t ) = 0 ,

(13) (14) (15) (16)

where p w is the wellbore pressure; rc is the external boundary; σ h = qσ V ( σ V = ρ r gH is the lithostatic pressure, ρ r is the enclosing rock density; g is the gravitational acceleration). At the initial time: p1 (r ,0) = p2 (r ,0) = pc , (17) and pc = (1 + 2q)σ V / 3 as follows from [26]. The relations (16) mean that the reservoir is on depletion. The implementation of the system (10)–(17) uses the modified numerical and analytical technique from [27]: at each time Eqs. (1) given (15)–(17) are solved using an implicit finite difference scheme and matrix procedure [28]. Finally, at the known pressure p(r , t ) the system (11)–(13) has an analytical solution: σ rr (r , t ) = [δΨ (rc , t ) − σ h + pw ]L− (r ) − δΨ (r , t ) − pw , σ θθ (r , t ) = [δΨ (rc , t ) − σ h + pw ]L+ (r ) + δ [Ψ (r , t ) − p(r , t )] − pw , r

where δ = 2μ /(λ + 2μ ) ; Ψ (r , t ) = r −2 ∫ ξp(ξ , t )dξ ; L± (r ) = (1 ± rw2 / r 2 ) /(1 − rw2 / rc2 ) . rw

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2. IRREVERSIBLE DEFORMATION IN THE WELLBORE VICINITY

At great depths, even when heavy drill mud is used, irreversible deformation zones F can appear in the wellbore vicinity [20, 21]; porosity and permeability of the reservoir change jump-wise in these zones [29]. We assume the simplest failure criterion [30]: τ max = τ * , (18) where τ max = 0.5 | σ rr − σ θθ | is the maximum shearing stress; τ * is the ultimate shear strength. In the domain rw ≤ r ≤ r* ( r* is the radius of F), the equilibrium equation (11) is fulfilled; joint solution of (11) and (18) under the boundary condition (13) yields: r (19) σ rr (r , t ) = − p w − 2τ * ln , σ θθ (r , t ) = σ rr (r , t ) − 2τ * . rw Description of deformation of the domain r* ≤ r ≤ rc uses the equation of poroelasticity (11) and (12), the general solution of which takes on the form: σ rr ( r , t ) = A1 + A2 r −2 − δΨ ( r , t ) , (20) −2 σθθ (r, t) = A1 − A2r + δ [Ψ(r, t) − p(r, t)] . The unknown functions of time A`1 , A2 and r* are found so long as: —the radial stresses are continuous at the boundary of F and elastic zone; —the criterion (18) is fulfilled for (20) at τ = τ * and the boundary condition (14) is included. Then: r A1 (t ) = 0.5 p (r* , t ) − τ * − p2 − 2τ * ln * ; rw A2 (t ) = r*2 [τ * + Ψ (r* , t ) − 0.5 p (r* , t )] , determination of r* (t ) (assuming r* << rc ) uses a transcendental equation: r (21) 2τ * ln * + δ [Ψ (r* , t ) − 0.5 p (r* , t )] = σ h − τ * − p w . rw At t = 0 the pressure p = pc ; then (21) yields an irreversible deformation zone condition σ h > p w + τ * , which allows estimate of a related depth: p +τ z* = w * . qρ r g In case of drill mud with the density ρ w : z* =

τ*

q( ρ r − ρ w ) g

.

3. CALCULATION ANALYSIS

The source data for the model calculation were: H = 2000 m, rw = 0.1 m, rc = 200 m,

ρ r = 3000 kg/m3, pw = 0.1 MPa, η = 0.004 Pa·s, m1 = m2 = 0.1 , β = 2 ⋅ 10 −9 Pa–1, k10 = 5 mD , k 20 = 30 mD , μ = 30 GPa. Typical for terrigenous reservoirs, the compressibilities α11 = 10 −10 Pa–1, α11 = 2α 22 , α12 = α 21 = 0.1α11 of blocks and fractures were set in accordance with [13]. The pressure distribution in blocks and fractures, p1 and p 2 , respectively, at different times and at C = 10 −4 and 5·10–4 Pa–1 is demonstrated in Fig. 2. As the permeability of blocks is much smaller than the permeability of fractures, p 2 sufficiently long exceeds p1 at low C . With an increase in C , as is expected, the values of p1 and p 2 differ only at the early stage of the well operation. JOURNAL OF MINING SCIENCE Vol. 52 No. 3 2016

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Fig. 2. Pressure in blocks (solid lines) and fractures (dashed lines) at (a) C = 10 −4 Pa–1 and (b) C = 5 ⋅10 −4 Pa–1.

Fig. 3. Distribution of (a) σ rr and (b) σ θθ in the wellbore vicinity at different time in accord with the models of poroelasticity (dashed lines) and poro-elastoplasticity (solid lines).

Figure 3 shows the distribution of the stresses σ rr and σ θθ at different times at q = 1 and τ * = 12 MPa. A decrease in the pressure in the wellbore vicinity due to the reservoir depletion results in an increase in the compression (porous elasticity model), and σ θθ grows faster than σ rr in this case, which causes an excess of the maximum shearing stress τ max over the limit τ * . It is worthy of saying that with the higher Bio parameter, τ max faster reaches τ * .

Fig. 4. Time-dependent change in the irreversible deformation zone radius r* at the varied lateral earth pressure coefficient and Bio parameter. JOURNAL OF MINING SCIENCE Vol. 52 No. 3 2016

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Figure 4 shows the change in the size of the irreversible deformation zone, r* (t ) , at τ * = 12 MPa and varied values of the lateral earth pressure coefficients q (solid lines— B = 0.7 , dashed lines— B = 0.3 ); with the reservoir depletion, the rate of the increase in r* lowers. The calculations yield that the zone F grows in size with the Bio parameter: when B > 0.5 the portion of p2 in p dominates and the pressure in fractures reduces faster than in blocks as the latter have much lower permeability. The lab tests of rocks from the Bazhenov formation [31], which are the representative samples of fractured-and-porous media [32], showed that Poisson’s ratio v changed from 0.05 to 0.45 depending on clay content. Figure 5 gives the curves of r* = r* (t ) at q = 0.8 , τ * = 10 MPa for different v (solid lines— B = 0.7 , dashed lines— B = 0.3 ): with an increase in v, the size of F grows, whereas the effect of the value B on r* lowers. The full-scale observations tell on decrease in the reservoir permeability in the course of operation. The laboratory tests [29] revealed that permeability of a specimen sharply worsened at the post-limit deformation stage as compared with the stage of elasticity. Figure 6 illustrates the drop of the permeability G (s) for γ = 600 at varied times and q = 1 and 0.8 (dashed lines—model of poroelasticity, solid lines—model of porous plasticity). The resultant decrease in k with time quantitatively conforms with the in situ data. At the same time, in the model of poroplasticity, the irreversible deformation zone r ≤ r* possesses higher permeability than the elastic zone. It is required to use different values of γ in these zones, or to use more complex constitutive relations of porous plasticity.

Fig. 5. Size of the zone F in case of different Poisson’s ratios ν.

Fig. 6. Time-dependent change in the permeability in the wellbore vicinity at γ = 600 : (a) q = 1.0 ; (b) q = 0.8 .

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CONCLUSIONS

The authors have developed a geomechanical model to describe reversible and irreversible deformation induced in the fractured-and-porous rock mass by fluid flow in the producing wellbore vicinity in the reservoir under depletion. The proposed numerical–analytical method allows onedimensional implementation of the model and uses a finite difference scheme to solve equations of permeability and, later on, equations of poroelasticity and poroplasticity in quadratures. The condition of the irreversible deformation zone is defined, and the zone radius r* is assessed as function of the horizontal in situ stress and the strength of rocks. The numerical experiments have found that: — r* grows with an increase in the Bio parameter and Poisson’s ratio; —a decrease in the pressure in the wellbore vicinity causes an increase in the compressive stresses, and the shearing component grows in this case faster than the radial component, which results in an excess of the maximum compression stress over its limit value and initiates a failure zone. ACKNOWLEDGMENTS

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