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9th European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004

Abstract

A new full-3D balanced restoration technique for geological structures is used to estimate the strain tensor and to define fractured zones applying failure criteria. This approach based on the elastic solid theory of continuous media gives the tensor of deformations. In simple cases such as thin plates, this new approach is in agreement with results predicted by the theory. Strain tensor invariants (dilatation coefficient, principal strains, etc.) coupled with mechanical properties of rock types are used to characterize the distribution of fracture orientations. The method is applied on the Split Mountain Anticline Case Study (Utah, USA), a natural outcropped clastic reservoir. The final goal of this research is to improve oil and gas recovery in fractured reservoirs by a better estimation of the permeability tensor.

Keywords: Strain tensor, Elasticity, Faulted Reservoirs, Split Mountain.

Introduction

3D unfolding techniques are common practices to understand history of fracturing in complex faulted reservoirs, especially for oil and gas exploration. Using the elastic solid theory of continuous media, we propose to derive several structural attributes (dilatation coefficient, principal strains, etc…) that may be used to define fractured zones using failure criteria. This approach quantifies deformations and gives good results both in theoretical models and real case-studies. It also provides an estimate of strains and stresses during deformation. This general theoretical framework provides solutions to complex problems such as determination of strains resulting from tectonic events and for predicting faulted zones on reservoirs. The next step is to use these structural attributes as secondary variables to estimate the fracture density and finally the permeability of a reservoir.

A021 CHARACTERIZING FRACTURED RESERVOIRS USING STRUCTURAL ATTRIBUTES

J.J. ROYER, P. MURON, R. MOYEN, L. MACE and J.L. MALLET

Gocad Research Group, Ecole Nationale Supérieure de Géologie, rue doyen Roubault ,54501 Vandoeuvre, France

CRPG-CNRS, 15 rue d des Pauvres, 54501 Vandoeuvre-Les-Nancy, France e-mail : royer@crpg.cnrs-nancy.fr

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Figure 1: Parametric 3D restoration: Deformed structure (A) embedded (B) into a parametric space (u, v, t) where t corresponds to geological time (depositional isochrones) u and v represent the paleo geographical coordinates at depositional time; Restored structure (C) after flattening the top horizon

Theoretical background

Deformation analysis

The strain tensor ε (x) characterizes the deformation of elastic solid in the neighborhood of location x between time t0 and t. It is related to the variation of the metric tensor from the present state G(x) to the restored state G0(x) by: ε (x) = ½ [G(x) - G0(x)] (see [1], [2], [3], [4]). The metric tensor measures distances between points as the solid is deformed through time.

Several structural parameters such as cubic dilatation, principal strains, principal strain directions and elongation coefficients can be derived from the metric tensors G(x) and G0(x) ([3], [4]). In this work, the metric tensor G0(x) is assumed to be diagonal, while G(x) is estimated either from displacement restoration vectors (see [5]) or, equivalently, from a curvilinear coordinate system attached to the deformed object (called Geochron Model see [4],[8]). The components of the strain tensor may differ depending on the coordinate systems. For instance, the mixed components of ε (x) in the initial (before deformation) and final coordinate system (also called “Eulerian strain tensor” [3]), noted respectively S0 (x) and S(x), are given by:

S(x) = G-1(x) ε (x) = ½ [I - G-1(x). G0(x)] S0 (x) = G0-1(x) ε (x) = ½ [G0

-1(x). G(x) - I] (1)

It is used to evaluate the cubic dilatation coefficient θ (x) that measures the variation of volume of an infinitesimal element of volume dV at location x at time t during deformation:

θ (x) = (dV(x) - dV0(x)) / dV0(x) =

[det(G(x))/det(G0(x))]1/2 - 1 = [det(S0(x))/det(S(x))]1/2 - 1 (2)

where det is the determinant. The cubic dilatation is an invariant geometric characteristic. In the case of small deformations, it can be shown [2] that: θ (x) = tr (S(x)) (where tr(S) is the trace of matrix S). The principal strains and their directions (called principal strain directions) are given, respectively, by the eigen values λi (i=1,2,3) and the eigen vectors vi of the Eulerian strain tensor S(x):

S(x) vi(x) = λi vi(x) ∀ i =1,2,3 (3)

The principal strain directions are orthogonal and independent of the coordinate system. The relative elongation coefficients li(x) along these principal directions are also invariant and are deduced from the eigen values λi(x) of S(x) as follows:

li(x) = [1 -2 λi(x)]-1/2 – 1 (4)

Deformed layer

Top

Interpreted fault

Horizon

Neutral Plan Restoration

Working Volume

A B C

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9th European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004

In the case of small deformations, it can be simplified into: li(x) = λi(x).

The above parameters fully characterize the strain at each location x. They are independent of the coordinate system (also called invariant). In the following, we will refer to them as structural attributes. One of the important points is how to restore a structure in a consistent manner in order to estimate the deformation tensor from initial to final states. This will be done using two algorithms: the first one (equivalent to a minimum internal energy criterion) consists in minimizing the strain energy induced by deformations from initial to final states [5]; the second one is based on a local parameterization of volumes, in which the isochrones correspond to the depositional surfaces (called Geochron Model [8]). If the parameterization is consistent with the geology both approaches give very similar results.

Mechanical Model

The second important point is now to relate the strains to the internal forces - the stresses in the material - to predict fractures using a geotechnical model (failure criterion). For sake of simplicity, it is assumed that the material is linear, isotropic and elastic in which stresses σij are linearly proportional to strains εij (Hooke’s law). The form of the Hooke’s law in the isotropic case is given by:

σ ij = λ tr (ε) δ ij+ 2 µ ε ij (5)

where λ and µ are the Lame coefficients, δ ij is the Kronecker symbol (i.e. δ ii = 1; δ ij = 0, i ≠ j )

If the material is isotropic, the principal axes of stress and strain coincide.

Predict fractures using failure criteria

Several failure criteria (maximum stress, Mohr-Coulomb, Tresca, Von Mises, etc…) can be used to identify fractured zones in the restored formation. Mapping such criteria can help in characterizing fractured zones. In the case of isotropic homogeneous material, most of the failure criteria based on the stresses can be rewritten in terms of strains using Eq. (5) [4].

Mohr-Coulomb criterion predicts the failure of brittle materials. Failure occurs when the Mohr’s Circle at a point in the material exceeds the envelope created by the two circles for the uniaxial tensile and compression strength. In terms of strains, it can be rewritten:

(ε1 - ε3).µ / cos ϕ - (ε1 + ε3).µ. tg ϕ - tr(ε).λ.tg ϕ ≥ c (6)

where c and ϕ are respectively the internal cohesion and friction angle of the rock.

Tresca criterion predicts the yielding of ductile materials that occurs when the shearing along the maximum shear stress surface is greater than the yield shear stress τe (obtained from a uniaxial tensile test). In terms of strains, this criterion can be rewritten as:

(ε1 - ε3)2 ≤ τe2/µ2 (7)

The failure criteria written in terms of strain components are useful to predict possible fractured zones from deformations.

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(e) Observed fractures

Figure 2: Split Mountain Anticline Case Study (Utah, USA): (a) Location map (after [6]); (b) Open (red) and slip (green) observed faults mapped onto the topography; (c) topography map; (d) Geological Map; (e) Fractures observed on the field [Data courtesy from the Consortium Carfrac: FSH – CEP & M, ASGA - GOCAD, IFP, Total , GDF, Earth Decision Sciences – Ecole Polytechnique, CNRS – CRPG].

The Split Mountain anticline Case Study, Utah

The above methodology has been applied to the restoration of the “Split Mountain”, an anticline located near the eastern end of the Uinta Mountains, Utah. The main objectives are to derive structural attributes that can be related to fault occurrences observed in the field and to use these attributes to predict fractures.

5 km

(d) Geological Map (c) Topography

N

E-W faulting located at E-W of Uinta mountains

Field observations

open faultsSlide faults

(a)(b)

NPlanes(Strike)

Axial N = 46

20

20

20

20

NPlanes(Strike)

Axial N = 33

20

20

20

20

NPlanes(Strike)

Axial N = 32

20

20

20

20

NPlanes(Strike)

Axial N = 42

NPlanes(Strike)

Axial N = 57

NPlanes(Strike)

Axial N = 58

20

20

20

20

NPlanes(Strike)

Axial N = 20

20

20

20

20

NPlanes(Strike)

Axial N = 15

NPlanes(Strike)

Axial N = 26

20

20

20

20

NPlanes(Strike)

Axial N = 31

NPlanes(Strike)

Axial N = 69

20

20

20

20

NPlanes(Strike)

Axial N = 41

NPlanes(Strike)

Axial N = 73

NPlanes(Strike)

Axial N = 70

20

20

20

20

NPlanes(Strike)

Axial N = 37

NPlanes(Strike)

Axial N = 54

NPlanes(Strike)

Axial N = 66

NPlanes(Strike)

Axial N = 73

NPlanes(Strike)

Axial N = 86

NPlanes(Strike)

Axial N = 81

NPlanes(Strike)

Axial N = 61

20

20

20

20

NPlanes(Strike)

Axial N = 68

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9th European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004

Geological Setting

The “Split Mountain” anticline, in Dinosaur National Monument, Utah, is a manifestation of Late Mesozoic to early Tertiary compression within the Cordilleran region (Fig. 2). It formed during the Eocene uplift of the Eastern Uinta Mountains. Interest in fractures at Split Mountain Anticline raises from its similarity to faulted anticline reservoirs in which fractures affect the efficiency of secondary recovery [6], [9]. It therefore constitutes a natural analogue of faulted reservoirs.

Structural field studies were conducted by a consortium involving IFP, TotalFinaElf, ASGA and Earth Decision Sciences. Geological units within the outcrop were identified and described according to their lithology, thickness and style of deformation. Two groups of fratures, strip slip and extension fractures (Fig. 2b), respectively, were recognized [10].

Available Data Set

A 3D model of the Split Mountain formation was built in gOcad using: digital elevation models, outcrop observations, geological maps, and dip measurements from field study, area photos and remote sensing data. The top and bottom surfaces of the main sandstone formations were built after integrating all these data into a comprehensive database (see Fig 3.a).

Then, a mechanical study was performed by IFP and Earth Decision Sciences on the Split Mountain sandstone formation in order to predict the faulted zones. It provided faulted zones classified according to their type (extension or compression), their orientation and their intensity. These results were used to compare the structural attributes derived from the GeoChron (orientation, dilatation coefficient, etc.) with the calculated fractures.

Computing Structural Attributes

The Top and Bottom surfaces of the Weber sandstone formation were used to build a 3D model of the Split Mountain anticline (Fig. 3a). A parametric representation was chosen after several trials in order to locate a realistic reference neutral surface. Various structural attributes were computed systematically at each location: (i) the three local Principal Strains (ε1, ε2, ε3); (ii) their orientation (v1, v2, v3); (iii) the dilatation coefficient θ (Fig. 3d). These parameters were compared with the available observed fracture data set (Fig 3e and 3f). The same attributes were also calculated minimizing the strain energy induced by deformations from initial (top of sandstone flattened) to final states (present position). The obtained displacements are reported in Fig. 3g together with the dilatation coefficient. This example shows that the minimizing strain energy and the Geochron methods give very similar results indicating that the parameterization is consistent.

Results

Two strain regimes can be observed at Split Mountain from the dilation coefficient (see Fig. 3d): (i) an extension type of deformation (in red) on the top of the anticline with an EW orientation in the central part of the structure, and a NW orientation on the east flank of the formation; and (ii) a compression type of deformation (in blue) on the longitudinal lateral N and S flanks of the anticline. The main extension strain directions (θ>0) are consistent with the orientation of the extension fractures reported by the mechanical study, while slip fractures θ<0) occurred mainly in compression zones (Fig 3d). The fault intensity seems to be correlated with the dilatation coefficient.

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(a) Model (b) Principal strains (ε1) (c) Principal strains (ε2)

(d) Dilatation (θ) (e) Open faults (f) Slip faults

(g) Dilatation (θ) (h) Open faults (i) Slip faults Figure 3: Computing structural attributes using 3D unfolding: (a) Initial 3D split Mountain model (vertical scale x 3) Top (colored with elevation z) and Bottom triangulated surfaces constrained by field observations used to build the solid model. Principal strains (ε1) (b) and (ε2) (c) computed from a Geochron parameterization of the solid model; (d) Dilatation coefficient (θ) calculated from a Geochron parametrization and mapped onto the sandstone layer of the “Split Mountain” anticline; Observed open (e) and slip (f) fractures mapped onto the dilation map; (g) Dilatation coefficient (θ) and displacements calculated from minimizing the strain energy induced by deformations from initial (top of sandstone flattened) to final states (present position); Observed open (h) and slip (i) fractures projected onto the dilation map. As expected, open fractures occur mostly on the top of the anticline where the rock is in extension (in red), while slip fractures are observed on the flank in zones in compression (in blue)

These results suggest that structural attributes might be very helpful in predicting fractures and occurrence mode. Coupled with a failure criterion, one can predict occurrence map of fractures together with their orientation. This procedure then gives new tools to improve the estimation of the secondary permeability. This is a challenge for the future development of this work.

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9th European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004

Conclusions

In this work, it is shown that the Geochron Model and the elastic solid theory can be used together to derive Structural Attributes from a parametric representation of a deformed geological structure. Results are in agreement with theoretical cases such as the thin plate [4]. It provides useful structural attributes (dilatation coefficient, principal direction of strains, etc.) that can be helpful: (i) to predict fracturation and sub-seismic faults; (ii) to improve estimation of secondary permeability. Future development of such approach should focus on the following topics: (i) improve failure criterion to take into account rock types; (ii) to derive fault occurrence map when coupled with failure criterion; (iii) couple mechanical and geostatistical approaches (structural attributes as secondary predictive variables in co-kriging) for a better estimate of permeability; (iv) continue to calibrate this approach onto real case tests such as reservoirs and natural analogues. The next step is to use these structural attributes as secondary variables to estimate the fracture density and finally the permeability of a reservoir.

Acknowledgements

This research work was performed in the frame of the gOcad research project. The companies and universities members of the gOcad consortium are hereby acknowledged. Part of the work was supported by FSH - CEP & M (Fond de Soutien des Hydrocarbures - Comité d'Études Pétrolières & Marines). The data set from Split Mountain was provided by IFP (IFP - Institut Francais du Pétrole). The authors express sincere thanks to W. Sassi and M. Guitton (IFP) for their help and constructive remarks and also to the other members of the CARFRAC project Total, GDF and Earth Decision Sciences.

References

[1] Gray, A. (1993). Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL, CRC Press.

[2] Sédov L. (1973). Mécanique des milieux continus. Edition MIR, Moscou.

[3] Mallet, J.L. (2002). Geomodeling. Applied Geostatistics Series. University Press, Oxford.

[4] Royer J.J., Cognot R., Moyen R. and Mallet J.L. (2003). Deformation and fracturation in Gocad. 23rd gOcad meeting, LIAD, Nancy, France, 30p.

[5] Muron, P. and Mallet, J. (2003). 3D Balanced unfolding: the tetrahedral approach. 23rd gOcad meeting, LIAD, Nancy, France.

[6] Silliphant, L. J., Engelder, T., and Gross, M. (2002). The state of stress in the limb of the Split Mountain anticline, Utah: constraints placed by transected joints. Journal of Structural Geology, 24, 155-172.

[7] Turcotte, D. and Schubert, G. (1982). Geodynamics: Application of Continuum Physics to Geological Problems. John Wiley and Sons, New York.

[8] Mallet, J.L., (2004). Space-Time Mathematical Framework for Sedimentary Geology. Journal of Mathematical Geology, 36, 1, 1-32.

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[9] Wilkins, S. and Gross, M.R., (2002). Normal fault growth in layered rocks at Split Mountain Utah: influence of mechanical stratigraphy on dip linkage, fault restriction and fault scaling. Journal of Structural Geology, 24, 1413-1429.

[10] Sassi, W., Guiton, M.L.E., Daniel, J.M., Faure, J.L., Mengus, J.M., Schmitz, J., Delisle, S., Leroy, Y.L.M., and, Masson, J. (2003). Mechanical reconstruction of fracture development in Weber sndstone formation, Split Mountain (Utah). Poster presented at the 2003 annual meeting of the AAPG in Salt Lake City.