Compaction-Induced Porosity - CiteSeerX

Compaction-Induced Porosity/ Permeability Reduction in Sandstone Reservoirs: Data and Model for Elasticity-Dominated Deformation P.M.T.M. Schutjens, SPE, Shell/SINTEF Petroleum Research; T.H. Hanssen, SPE, M.H.H. Hettema, SPE, and J. Merour, Statoil; P. de Bree and J.W.A. Coremans, Shell; and G. Helliesen, Norwegian Petroleum Directorate

Summary Open literature and new experimental compaction data from five reservoir and 16 outcrop sandstones are used to delineate the nearelastic, inelastic, and failure domains in 3D-stress space for porosity classes of 5 to 15%, 15 to 25%, and 25 to 35%. Applications of this compaction-domain model include the analysis of the extent of the near-elastic domain (where elasticity theory can be used to describe and predict rock deformation), the pore-volume compressibility (Cpp), and the permeability reduction as a function of reservoir stress path. This is illustrated for a well-consolidated sandstone reservoir with an average porosity of approximately 18%. Two aspects of dynamic reservoir modeling in the nearelastic domain are addressed: calculation of Cpp from raw volumetric-compaction data as a function of isotropic total stress change, and the correction of Cpp for a nonhydrostatic reservoir stress path. Open-literature work combined with our experimental data indicates that the compaction-induced permeability reduction of 15 to 25% porosity sandstone in the near-elastic domain depends predominantly on the increase of the effective mean stress, not on the reservoir stress path. Introduction Reservoir depletion increases the stress carried by the load-bearing grain framework of the reservoir rock. It triggers micron-scale deformation mechanisms such as elastic (Hertzian) grain-contact spreading, microcrack growth and closure, cement breakage, grain rotation and sliding, and crystal plastic deformation in clay and mica grains.1–4 Rocks then often (but not always) show volume reduction (compaction), porosity loss, and acoustic velocity increase. Numerous problematic consequences of depletion-induced reservoir deformation/compaction have been reported (see Carbognin et al.5) for recent case studies. These include subsidence,6,7 changes in (and mostly reduction of) reservoir permeability,8,9 enhanced risk of casing damage,10,11 and earthquakes.12,13 In addition, the change in total stress associated with reservoir compaction may govern the integrity of the cap rock and (fault) seal, place constraints on the weight of the drilling fluid that optimizes wellbore stability, guide infill-drilling campaigns, and affect hydraulic fracture design.14–18 There are also beneficial effects of compaction, but these are more rare; the preferential compaction of porous layers and fault gouge may be an important additional driving force for hydrocarbon flow toward the wells (compaction drive19,20) and may delay or prevent water influx.21 From a reservoir management point of view, compaction-induced changes in porosity and permeability must be understood and modeled to optimize the drilling and completion strategy and the drawdown as a function of time and reservoir-fluid pressure. The aim of this paper is to present a model for compaction behavior and permeability change as a function of stress path and

Copyright © 2004 Society of Petroleum Engineers This paper (SPE 88441) was revised for publication from paper SPE 71337, first presented at the 2001 SPE Annual Technical Conference and Exhibition, New Orleans, 30 September–3 October. Original manuscript received for review 30 November 2001. Revised manuscript received 27 January 2004. Paper peer approved 3 March 2004.

202

porosity, based on published and new experimental data. The model can be used to investigate the effect of stress path on compaction and porosity and permeability change, even in the absence of core compaction data. Also, the model can help to determine if (new) compaction experiments are needed and how these should be conducted [e.g., at reservoir PT conditions, under reservoirrepresentative stress paths, or with simultaneous measurement of (horizontal) permeability]. Describing Stress State and Stress Path A model of compaction-induced permeability reduction requires knowledge of the compaction behavior as a function of stress state and stress path. Both are discused next, in a theoretical sense, for the 2D and 3D stress space. It serves as a basis to define three “compaction domains” characterized by specific micromechanisms and deformation behavior. Stress State. A common representation of stress state is based on Mohr-Coulomb failure in a 2D stress space, where the effective mean stress is

␴=

Sv + Sh − pp, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1) 2

and the maximum shear stress is

␶=

Sv − Sh ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2) 2

Sv and Sh are principal stresses, with v ⳱ vertical ⳱ maximum stress. The states of stress plot as points, and directional information can be achieved in a Mohr circle plot on the normal and shear stresses on any plane through the rock. Depletion will be accompanied by growth of the stress circles and movement of their center to the right (higher effective mean 2D stress). Rock failure is predicted at the stress state at which the circle touches the trend of the failure stress data. The orientation of the fracture plane can be calculated with respect to the principal stresses using Mohr circle theory.22 It is sometimes desirable to include the intermediate total stress SH in the analysis. For instance, in the case of pervasive inelastic deformation, common in soils and weak rocks like porous chalks, the volumetric strain and associated porosity reduction, as well as the yield stress, are functions of SH.23 An additional advantage of the 3D stress representation is that it allows the calculation of the parameters in the Drucker-Prager plasticity criterion, the elastic and plastic strain energy functions, and the mean shear stress.24,25 One of the aims of this study is to better understand permeability reduction caused by compaction-induced changes in rock microstructure; these are brought about by the combined effect of changes in Sv, SH, Sh, and pp. We will therefore use the 3D stress space and further assume that ⌬SH⳱⌬Sh. In 3D stress space, the axes and yield conditions are functions of the invariants of the stress tensor. In this paper, the x-axis is the mean effective stress peff, defined as one-third of the first invariant of the stress tensor minus the pore-fluid pressure: June 2004 SPE Reservoir Evaluation & Engineering

peff =

␴v + ␴H + ␴h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3) 3

The y-axis is the differential stress Q, defined as the difference between the maximum and minimum total stress: Q = Sv − Sh = ␴v − ␴h. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4) Q is obtained from the second invariant of the deviator stress Q = Sv − Sh = 公3公J2, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5) 24

where

1 J2 = 关共Sv − Sh兲2 + 共Sv − SH兲2 + 共SH − Sh兲2兴. . . . . . . . . . . . . . . . (6) 6 Note that states of stress in 3D stress space plot as single points, rather than as circles. No directional information can be obtained on the magnitude of normal and shear stress on a plane at an angle to the principal stress axes (as in a 2D Mohr circle plot through the pole method).26 Stress Path. This can be described by the ratio of the change in effective minimum horizontal stress (⌬␴h) to the change in effective vertical stress (⌬␴v): ⌬␴h , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7) ⌬␴v

K=

or, through the horizontal stress-path coefficients,

␥h =

⌬Sh , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8) ⌬pp

␥H =

⌬SH , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9) ⌬pp

and vertical stress-path coefficient

␥v =

⌬Sv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10) ⌬pp

␥v and ␥h often can be determined directly from field measurements, whereas determination of K requires additional assumptions, treated below. K can be related to equivalent linear slopes, ␩, in the peff vs. Q stress plane by27 ␩=

⌬Q 3共1 − K兲 ⌬␴v − ⌬␴h . . . . . . . . . . . . . . . . . . . . (11) = = 2 ⌬peff 1 1 + 2K ⌬␴ + ⌬␴ 3 v 3 h

Reservoir depletion will, in most cases, be accompanied by a change of the total stress state.14,18,28,29 Consequently, for a stresspath analysis, both pore pressure and total stress change must be taken into account. The depletion-induced changes in effective vertical and horizontal stress are defined following the theory of poroelasticity: ⌬␴v = ⌬Sv − ␣⌬pp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12) ⌬␴H = ⌬SH − ␣⌬pp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (13) and ⌬␴h = ⌬Sh − ␣⌬pp. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (14) The parameter ␣ is the Biot-Willis coefficient, and it describes the relative contribution of total stress and pore pressure to the rock deformation.30 Using these definitions in Eq. 7 gives K=

⌬␴h ⌬Sh − ␣⌬pp = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (15) ⌬␴v ⌬Sv − ␣⌬pp

In the special case in which depletion is not accompanied by a change in total vertical stress (␥v⳱0, that is, no “stress arching”; see Adachi et al.31), Eq. 15 leads to

␥h = ␣共1 − K兲 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (16) and to ⌬Q = ⌬Sv − ⌬Sh = ⌬␴v − ⌬␴h = ⌬␴v共1 − K兲 = −␣⌬pp 共1 − K兲. . . . . . . . . . . . . . . . . . . . . . (17) June 2004 SPE Reservoir Evaluation & Engineering

The increase in mean effective stress ⌬peff is, by including ␣, ⌬peff =

⌬Sv + ⌬SH + ⌬Sh − ␣⌬pp. . . . . . . . . . . . . . . . . . . . . . . . (18) 3

In the special case of depletion under a constant total stress (⌬Sv⳱⌬SH⳱⌬Sh⳱0), depletion/compaction occurs under an isotropic increase in effective stress, and Eq. 18 reduces to ⌬peff = −␣⌬pp. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (19) In the general case of a depletion-induced decrease in the sum of the total stresses, Eq. 18 shows that ⌬peff ⬍ −␣⌬pp. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (20) For the special case of K0.5) undergo pore collapse with grain-size reduction (cataclasis) and are compactant.4,36,37 This forms the cap in stress space (see Fig. 1). Eventually, after a porosity loss of several porosity units, grain-contact stresses will start to decrease again, and a renewed stress-hardening phase can occur, though with a completely different microstructure from the original one, and thus with different compaction behavior from that during the first strain hardening.

* In experiments performed at a constant loading rate (⌬S/⌬t=constant), this yield point will also be visible as the point at which the compressibility starts to increase. However, in contrast to the constant-strain-rate test, no maximum stress will be achieved—and the compressibility will continue to increase (i.e., compaction weakening3) until the sample fails.

Compaction Data. Failure-stress data from 16 outcrop sandstones are shown in Fig. 2. The Mohr-Coulomb failure trend is nearlinear (dark symbols). Data from different sandstones are in good

Fig. 2—Failure-stress data for 16 outcrop sandstones, with data taken from recent publications by Wong, Boutéca, Crawford, and coworkers. Dark symbols: fracture-plane development. Open symbols: pore collapse. Note the effect of porosity on cap behavior. Sandstone type is listed in the legend. Capitals in legend point to data reference (e.g., W97 is Wong et al., 1997). 204

June 2004 SPE Reservoir Evaluation & Engineering

Fig. 3—Reservoir sandstones: effect of porosity on failure stress. No distinction is made between samples from the five reservoirs. No data for the cap behavior are available. In consolidated sandstones, cap stresses have large peff values of several hundred MPa (Wong et al., 1997), outside the range of interest of most petroleum rock-mechanics applications.

agreement. The area enclosed by the cap-stress envelope decreases with increasing porosity. Failure-stress data for sandstones from five hydrocarbon reservoirs are shown in Fig. 3 (information on these reservoirs is listed in Table 1). Data from different reservoirs overlap, so no distinction is made between them. The low-porosity data (squares) plot at higher Q stress levels than medium- and high-porosity data (closed and open circles). Fig. 4 illustrates yield stress* and dilation stress compared to failure stress in 15 to 25%-porosity sandstones. At a given peff stress state, dilatancy occurs at higher Q values than yield and at Q values similar to failure.

* In the data used here, yield stress is defined as the stress level at which the instantaneous axial compressibility exceeds the minimum axial compressibility (as measured during first loading) by 10%. This was viewed as the point at which nonlinearity was statistically significant (i.e., not caused by apparatus or data logging artifacts, producing apparent variations in compressibility with stress).

June 2004 SPE Reservoir Evaluation & Engineering

Based on experimentally observed deformation behavior and micromechanical considerations, the reservoir sandstone data are used to define the boundaries of the three compaction domains described previously: (1) a near-elastic domain in which inelastic deformation makes up less than, typically, 20% of the total strain at a given stress; (2) an inelastic domain in which yielding, dilatancy, acoustic emission, and/or permeability changes announce a strong rise in inelastic deformation at the expense of elastic deformation; and (3) a failure domain, with (a high chance of) failure by shear localization or pervasive pore collapse. Fig. 5 shows the composition domains for the 15 to 25%porosity data. The top of the failure domain is delineated by a polynomial function splined through the data shown in Fig. 3. Its lower bound is a polynomial spline function through the data describing the upper boundary of the yield-stress data (Fig. 4). The lower boundary of the inelastic domain is a straight line drawn through lower yield-stress data. The near-elastic domain falls under this line. The compaction domains of the low-porosity (5 to 15%) and high-porosity (25 to 35%) reservoir sandstone are shown in Figs. 6 and 7, respectively, plotted at the same scale as Fig. 5. In high-porosity rocks, it is often difficult to make the transition between near-elastic, inelastic, and failure, and a pore-collapse domain may be invoked (Fig. 7); see Refs. 3 and 4 for two case studies of this behavior. Pore Compressibility and Porosity Reduction One application of the compaction-domain model is to delineate the stress space in which the theory of linear poro-elasticity is expected to give good predictions of rock mechanical behavior,38 and where it should not be used. We will use it here to determine the pore compressibility and porosity reduction from experiments in which rocks are hydrostatically stressed at a constant pore pressure. Pore Compressibility. Adopting the notation scheme by Zimmerman,30 in which the first subscript indicates the relevant volume change (b⳱bulk volume; p⳱pore volume) and the second subscript indicates the pressure that is varied [c⳱pc⳱(Sv+SH+Sh)/3, p⳱pp], the bulk volume and pore compressibilities are Cbc = −

冉 冊

1 ⭸Vb Vb ⭸pc

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (22) ⌬pp=0

205

Fig. 4—Reservoir sandstones (15 to 25% initial porosity): yield, dilatancy, and failure-stress data.

and Cpp =

冉 冊

1 ⭸Vp Vp ⭸pp

; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (23) ⌬pc=0

Cbc is the volumetric (or bulk) compressibility of a rock stressed externally at constant pore pressure, and it involves a change in volume that can be measured macroscopically. The pore compressibility Cpp expresses the effect of pore-pressure variations on the pore volume at a constant total stress. Compaction is also governed by the compressibility of the individual grains (often termed the grain compressibility or rockmatrix compressibility), defined by Cr =

冉 冊

1 ⭸Vr Vr ⭸p

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (24) ⌬pc=⌬pp

where Cr relates to the Biot-Willis parameter ␣ by

␣=1−

Cr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (25) Cbc

Zimmerman30 obtains Cpp =

Fig. 5—Reservoir sandstones of 15 to 25% porosity: nearelastic, inelastic, and failure domains. The continuous line bounds the failure domain; the dashed line bounds the inelastic domain. The near-elastic domain is located below the bottom (near-horizontal) boundary of the inelastic domain.

Cbc − Cr共1 + ␾0兲 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (26) ␾0

where ␾0 is the porosity at a reference stress state (e.g., before the reservoir depletion or sample loading). Rewriting Eq. 25,

Cpp =

Cbc 关␣ − ␾0共1 − ␣兲兴. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (28) ␾0

Pore volume and porosity are scalar properties. However, during depletion, their magnitudes change as a result of 3D changes in the effective stress, which is a tensor. Cpp, as defined in Eq. 23, describes the change in pore volume as a function of depletion under constant total stress. However, as mentioned earlier, depleting reservoirs generally will experience a simultaneous change of the total stress state. We will assume here that the pore-volume reduction is a linear function of the increase in effective mean stress, independent of whether this occurs through an isotropic increase in effective stress (K⳱1) or through a nonisotropic increase in effective stress (K