Elastic response of porous rock to accumulated slip on strike slip fault networks in geo-reservoirs

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Elastic response of porous rock to accumulated slip on strike slip fault networks in geo-reservoirs

Item Type Article

Authors Yalcin, Bora;Zielke, Olaf;Mai, Paul Martin

Citation Yalcin, B., Zielke, O., & Mai, P. M. (2023). Elastic response of porous rock to accumulated slip on strike slip fault networks in geo-reservoirs. International Journal of Rock Mechanics and Mining Sciences, 166, 105370. https://doi.org/10.1016/

j.ijrmms.2023.105370 Eprint version Publisher's Version/PDF

DOI 10.1016/j.ijrmms.2023.105370

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International Journal of Rock Mechanics & Mining Sciences 166 (2023) 105370

Available online 23 March 2023

Contents lists available atScienceDirect

International Journal of Rock Mechanics and Mining Sciences

journal homepage:www.elsevier.com/locate/ijrmms

Elastic response of porous rock to accumulated slip on strike slip fault networks in geo-reservoirs

Bora Yalcin

^∗

, Olaf Zielke, P. Martin Mai

Physical Science and Engineering Division, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia

A R T I C L E I N F O

Keywords:

Triangular dislocation Poro-elasticity Fault roughness Fault length vs. slip scaling Porous medium permeability

A B S T R A C T

Subsurface reservoirs are generally highly fractured, whereby fractures constitute a natural fluid flow path and define the preferential flow direction. Slip accumulated during the faulting process alters also the petrophysical properties of the host rock. Although the mechanical alteration of the host rock, and related porosity and permeability changes, due to fault slip has been previously described, a predictive physics-based model has not been scaled with fault length in reservoirs to provide an initial porosity permeability alteration model yet. In this study, we develop a predictive model to quantify how accumulated fault slip changes porosity and permeability in a porous medium, by combining deformation modeling based on triangular dislocations and linearized poro-elasticity equations. We applied our model the Ghawar field fault map and rock-types.

We conducted a Monte-Carlo simulation, varying fault roughness and accumulated slip, to quantify the corresponding variation in porosity and permeability using a 5 km long strike-slip fault and three different rock- types. Our Monte-Carlo simulation shows that long-term accumulated slip on rough strike-slip fault surfaces change porosity by±1%, leading to an absolute permeability change of up to 22.5%. We further used these results as a benchmark for the elastic response of porous rocks to accumulated slip scaled to certain fault length. Using these benchmark results for Ghawar field reservoir rocks, we determined the slip-related porous medium permeability changes for every fault on the Ghawar fault map, accounting for their length, location, and orientation. In doing so, we found that fault roughness, slip amount, and shear sense all affect the medium’s permeability, creating substantial permeability anisotropies. Locally, these anisotropies are further enhanced by superposition of permeability changes of individual faults that constitute the fault system. We suggest the resulting permeability distribution model should be used as the initial permeability model for porous media in fractured reservoirs.

1. Introduction

Hydrocarbon and geothermal reservoirs are of great socio-economic importance. Such subsurface reservoirs comprise porous and perme- able rocks that contain hydrocarbons and latent heat, respectively.

Optimizing sustainable resource extraction from these reservoirs is a central topic in the respective scientific fields and industries.¹ Pri- mary parameters controlling fluid flow within the reservoir are the rock’s porosity and permeability. To first order, these parameters (as well as others that affect fluid-flow properties) are often assumed to be homogeneous and isotropic in the volume under consideration.² While conceptually helpful, such conditions are rarely found in nature.

In fact, a range of geologic processes and properties may change a rock’s porosity and permeability in space and time. For example, most reservoirs are fractured (e.g., Ref. 3). These fractures form complex hierarchical networks that strongly influence fluid flow capacity and direction within a reservoir (e.g., Ref.4).

∗ Corresponding author.

E-mail address: [email protected](B. Yalcin).

A fracture network with high permeability (relative to the host rock) promotes flow along its surfaces. Therefore, knowledge about fracture and rock properties that affect fluid flow in the reservoir is of great economic importance. A fracture network also changes the petrophysical properties of the host rock itself. This situation arises, for example, when the fractures that form the fracture network accumulate slip over time (Fig. 1) and therefore deform the host rock. For example, Farrell and Healy⁵ found that pore space in the host rock is deformed (elon- gated) near a fault surface and that deformation is proportional to the amount of accumulated slip. Flodin et al.⁶ reported a decrease in host rock permeability near fault surfaces that were embedded in aeolian sandstones. Micarelli et al.⁷ reported an inverse relationship between a fault’s accumulated slip amount and the permeability of the highly porous carbonate host rock.

These qualitative observations describe off-fault deformations in great detail, however, they cannot be extrapolated to other locations

https://doi.org/10.1016/j.ijrmms.2023.105370

Received 18 November 2021; Received in revised form 6 February 2023; Accepted 28 February 2023

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International Journal of Rock Mechanics and Mining Sciences 166 (2023) 105370 B. Yalcin et al.

Fig. 1. (a) Map view of the conceptual architecture of a strike slip fault, comprising a main fault-slip plane, a fault core, small-scale fractures, and an intensely fractured damage zone. (b) Close-up view of a 3D block diagram depicting beyond the damage zone. Conceptually, the off-fault media is created due to repeated slip-induced inelastic and elastic deformation. With decreasing distance to the slip plane, grain-size and pore-dimensions are changing due to accumulated strain.

or different reservoir conditions, hence lacking predictive capabilities.

On the other hand, combining the mechanical and chemical alterations related to fault mechanics have been an interest for geomechanical studies (e.g., Ref.8). Although such studies provide flexibility to model the fault mechanics, they do not provide fracture scaling rules, hence lack suggesting suitable parameter initialization with respect to accumulated slip on faults. For example, slip-length scaling relations, along-fault slip distribution, and fault roughness are all well established in fault and fracture geology (e.g., Ref.9), yet they are not considered by default in currently available solvers. These rules are universal and useful to assess the amount of mechanical deformation on reservoir rock caused by accumulated fault slip.

In this study, we quantify the host rock’s porosity and permeability change as a function of accumulated slip. We parameterize the fault surface roughness, fault length vs accumulated slip scaling and accumulated slip distribution (as tapered) on fault surface. We apply dislocation theory to calculate stress and strain on the fault surface and in the rock volume for displacements occurring on complex fault geometries and with spatially variable slip on the fault. We coupled the dislocation calculations to the poro-elastic formulations by Zim- merman¹⁰ to quantify how fault slip affects the host rock’s porosity and permeability near the fault. We examine this effect for three rock types, with varying initial porosity, and assess the variability in the porosity and permeability change due to the epistemic uncertainty in fault properties.

The method section focuses on the mathematical derivation of porosity change due to fault slip. Then we describe the conversion from porosity to permeability using well accepted methods. Our model is then applied to a strike-slip fault 𝐷 = 5 km, considering three different reservoir rock types. In our numerical experiments, we compute the strain-induced permeability change expected in a faulted, porous medium. Finally, we scale the results to a real reservoir sector model case, where the fault map and rock-types are considered and the changes in permeability displayed. InFig. 2the flow chart summarizes the steps we are using to have the permeability distribution model using fault map and reservoir rock parameters.

Fig. 2. Flow chart outlining our modeling approach to compute the modified permeability of the porous medium due to fault slip. First we prescribe the fault parameters with its roughness and accumulated slip on it. Then we choose the porous media properties for the reservoir of interest. These are rock mechanical properties (Shear and Young’s modulus) and petrophysical properties (porosity and permeability) of reservoir rock-types. We enforce slip on the fault and receive strain influence matrix on solid media. We then convert the strain to change in porosity/permeability. To compare the permeability change with respect to the length of a particular fault, we perform a Monte-Carlo type simulation for each rock type by changing the fault parameters.

We consider the Monte-Carlo simulation results as a benchmark. Next, we fill in the permeability variations in the reservoir porous media using benchmark results and fault map.

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2. Model formulation and set-up

Our formulation relates volumetric strain due to slip on a fault surface, to porosity change in the surrounding porous medium. We parameterize the displacement that accumulated over time along the fault as a single, non-uniform distribution of slip. This slip distribution is then used to compute the corresponding deformation of the porous, elastic medium. By modifying the linearized poro-elasticity equations of Zimmerman,¹⁰ we develop an expression that links slip-induced strain to porosity changes. The derivation assumes that the difference in deformation (elastic behavior) between porous and non-porous materials is represented by pore space deformation (assuming an otherwise identical host rock composition). However, the derivation carries the limitation of not explicitly calculating the grain strain component of the bulk strain for the faulted porous material. Therefore, we constrain the grain strain with the Biot constant (e.g., Refs.11,12). Finally, we converted the change in porosity to a change in permeability, using established power-law relations.

2.1. Triangular dislocation, discretization and prescription of rough fault and porous medium

We employed elastic dislocation theory and used analytical ex- pressions for triangular dislocations by Nikkhoo and Walter¹³ to calculate the slip-induced changes in the medium’s stress and strain state (e.g., Ref. 14). We choose a triangular dislocation approach since this method allows simulating complex fault geometries and slip distributions (e.g., Ref.15). In our case, we use the approach to compute the medium’s response due to slip along a rough, penny- shaped fault surface with a radially symmetric slip distribution that is tapered to zero towards its edges. Laboratory experiments suggests that limited cracks in continuum space are penny-shaped. We can use the same analogy as limited dimensions of fault in an infinite brittle crust for reservoir scale faults (e.g., Ref. 16). Therefore, the penny-shaped crack is the simplest and most general representation to use in our physical model.

We design the benchmark model setup of fault geometry for 5 km diameter, penny shaped, strike-slip fault embedded in porous rock. We discretize this circular fault with approximately equilateral triangles (80 m side length). We then define the solid medium – the porous reservoir rock – by its rock mechanical parameters. We prescribe the porous rock by its shear modulus which defines the solid material’s resistance to shear and by its Young’s modulus which measures how much volumetric deformation the rock body will accommodate by shear related forces. We then define the homogeneous rock body into the observation points which are regularly distributed to every 40 meters to capture slip-induced strain on porous rock (Fig. 3). Note that the triangular solution we adopted provides correct result at distances beyond the shadow of one discrete triangle dimension (80 m in our benchmark case)¹³ that is, if the fault has triangle meshes with 80 meters side lengths, observation points at fault distances less than 80 meters may not be fully accurate and hence these points are ignored in the final analysis and interpretation.

The fault surface itself is not a planar surface, but instead comprises small-scale geometric roughness (Fig. 3). After we discretize the penny shape fault surface into a large number of triangles, we deviate strike and dip of the triangles in the out-of-plane direction to generate the fault surface roughness. We model the fault-surface roughness in the Fourier domain in terms of its two-dimensional power-spectral density function with Hurst exponent for dip 𝐻_𝑑𝑖𝑝 and strike𝐻_{𝑠𝑡𝑟𝑖𝑘𝑒}, which constrain the spectral decay of an anisotropic fractal random-field as a function of wave number.^17,18The roughness amplitude (in relation to the characteristic fault dimension𝑅) is prescribed via the pre-factor, 𝐶.^17,19

After prescribing the fault roughness, we apply a non-uniform (ra- dial) slip-distribution, linearly tapered at the fault edges, to the rough

fault surface. We then link the amount of accumulated slip to fault length using an empirical relationship for strike slip faults (maximum slip to fault length ratio) given as10⁻² ≤ 𝑑_𝑚𝑎𝑥∕𝐿 ≤ 1.5 ∗ 10⁻².^20,21 Therefore every discontinuous triangle mesh on the fault surface has a slip information in Burger’s vector form, where in the center of fault has maximum slip, which then tapers to zero towards the tip of the fault surface. The 5 km diameter fault thus can be characterized by long-term accumulated maximum slip in the range 50–75 m. Here, we adopt 50 m as maximum slip to remain in the conservative side of calculations.

2.2. Petrophysical and rock mechanical parameters for the case study Input parameters for our numerical modeling method are initial porosity, permeability and elastic moduli of faulted porous rock. In this study, we select model parameters based on rock-physical data from the Ghawar field Arab-D formation, Saudi Arabia. This field is considered the world’s largest oil reservoir based on the prolific production amount and duration. In the Ghawar field, fracture lineations from seismic images vary between 1 km to 8 km in length²²(minimum fault length solely depends on the seismic resolution, it is certain that structures with<1 kmlength do exist). Correspondingly, we assume a fault of diameter𝐷 = 5 kmembedded in three typical rock types found in Arab-D formation: Type-IA, Type-IB and Type II-B rock-types of Arab-D formation. These rock types have varying initial porosity, permeability and elastic moduli summarized inTable 1. Their porosity and permeability values are taken from Ref.23and the shear and bulk modulus are from taken from Ref.24.

2.3. The change in porosity with respect to fault slip

To calculate slip-induced porosity change, we first modify the linearized poro-elasticity equations by Zimmerman¹⁰ and then examine the effects of pure strike-slip motion on the porous medium. Derivations and calculations assume a full space, such that no elevation change occurs and hence no hydrostatic pressure variation need to be considered. In addition, we assume a homogeneous connected porous medium to illustrate the model conceptually. Alternative parametrizations of partially connected and heterogeneous pore-space models are beyond the scope of this study.

Porosity and specific volume sub-spaces can be expressed as follows:

𝑉_𝑏=𝑉_𝑝+𝑉_𝑔, 𝑉_𝑝 𝑉_𝑏 =𝜙,

𝑉_𝑔

𝑉_𝑏 = 1 −𝜙 (1)

where𝑉_𝑏is the bulk volume,𝑉_𝑝 represents the pore volume,𝑉_𝑔 is the grain volume, and𝜙denotes porosity. Small changes in bulk volume can be expressed by the summation of small changes in pore volume and grain volume𝛿𝑉_𝑏=𝛿𝑉_𝑝+𝛿𝑉_𝑔. Dividing all terms by bulk volume 𝑉_𝑏leads to:

𝛿𝑉_𝑏 𝑉_𝑏 =𝜙𝛿𝑉_𝑝

𝑉_𝑝 + (1 −𝜙)𝛿𝑉_𝑔

𝑉_𝑔 (2)

In the deformed pore space, pore pressure may locally increase above hydrostatic equilibrium, however, in a connected pore system, pore pressure retains hydrostatic equilibrium.²⁵Hence, we assume constant pore pressure,𝑃_𝑝. Taking derivatives of Eq.(2)with respect to the slip- induced stress𝜎_𝑓 under the assumption of constant pore pressure, and recalling the ‘‘compression positive’’ sign convention,𝜀_{𝑏,𝑝,𝑔} =𝑣𝑜𝑙(𝜀) = 𝜀₁₁+𝜀₂₂+𝜀₃₃, the resulting change in strain is given by:

(𝜕𝜀_𝑏

𝜕𝜎_𝑓 )

𝑃𝑝

=𝜙 (𝜕𝜀_𝑝

𝜕𝜎_𝑓 )

𝑃𝑝

+ (1 −𝜙) (𝜕𝜀_𝑔

𝜕𝜎_𝑓 )

𝑃𝑝

(3) where𝜀_𝑏 is bulk strain, 𝜀_𝑝 is pore strain and𝜀_𝑔 is grain strain. Pore strain can be expressed as the difference between grain and bulk strain by Eq.(3)in following form:

𝜀_𝑝=𝜀_𝑏− (1 −𝜙)𝜀_𝑔

𝜙 . (4)

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Fig. 3. (a) Model setup showing the circular fault, discretized by a triangular mesh and embedded in an elastic full-space, on which the tapered slip distribution is resolved.

Observation points are defined on a fault-perpendicular plane at the depth level of the center of the circular fault. In the calculations, the fault-surface mesh and observation-point spacing are much denser than displayed here. (b) Each triangular dislocation mesh describes a discontinuous slip amount which lies on plane A. Plane A bounded by three vertices𝐴₀, 𝐴₁, 𝐴₂on which vertices the slip has a vertical component with𝛽angle and horizontal component𝛼angle. The amount of slip is described with Burger’s vector𝑏_𝑘 (c) Conceptual 3D view of the bulk volume composed of grains (blue spheres) and pores (white space between grains). The applied 2D dislocation function on the geometrically rough fault determines the components of the slip-induced strain for every observation point. We use the volumetric strain to calculate the change in porous volume.

Table 1

Petrophysical and rock mechanical parameters of Ghawar Arab-D reservoir.

Rock Type Petrophysical Parameters Rock Mechanical Parameters

Grainstone (Type I-A) 𝜙= 24.1% B = 22 GPa

K = 1354 mD G = 11 GPa

Mud-lean Packstone (Type I-B) 𝜙= 27.3% B = 22 GPa

K = 319 mD G = 9 GPa

Packstone (Type II-B) 𝜙= 21.0% B = 19 GPa

K = 41.3 mD G = 13 GPa

Calcite Grain 𝜙= 0% B = 129.53 GPa

K = 0 mD G = 35 GPa

Using the porosity definition 𝜙 = 𝑉_𝑝∕𝑉_𝑏 and taking the logarithmic derivative leads to:

𝑑𝜙 𝜙 =𝑑𝑉_𝑝

𝑉_𝑝 −𝑑𝑉_𝑏

𝑉_𝑏 =𝜀_𝑝−𝜀_𝑏 (5)

Substituting Eq.(4)into Eq.(5)then provides the following relationship:

𝛥𝜙= (1 −𝜙_𝑜)(𝜀_𝑏−𝜀_𝑔). (6)

where𝜙_𝑜stands for the original/initial porosity and𝛥𝜙is the change in porosity as a function of initial porosity, bulk strain and grain strain.

The information on grain strain is still missing to solve Eq.(6). To set the amount of grain strain, we apply the strain–energy concept, considering the energy stored by a system undergoing elastic deformation.

For linear elastic materials, strain energy is given as (e.g., Ref.26):

𝐸= 1 2𝑉 𝜎𝜀= 1

2𝑉 𝐵𝜀² (7)

where𝑉 is volume,𝜀is strain and𝜎is stress,𝐵=𝜎∕𝜀is bulk modulus.

For a given system (here, the finite volume of the porous rock,Fig. 3.b), the work done by fault slip can be calculated via the strain and bulk modulus of the reservoir unit. We formulate the poro-elastic strain here based on equilibrium thermodynamic approach similar to Ref.27, such that work done to the porous rock by fault slip is responsible to strain the grains inside the volume of interest. This can be expressed by the following approximation:

𝐸_{𝑏𝑢𝑙𝑘}≅𝐸_{𝑔𝑟𝑎𝑖𝑛} (8)

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Fig. 4. (a) We enforce the fault length scaled slip to a 5 km strike-slip fault and display the associated grain strain to bulk strain ranges for three rock types of the Ghawar Arab-D formation. Type I-A and Type I-B have the same Young’s modulus and hence the slope of gran strain to bulk strain is same. However the maximum amount of strain is higher in Type I-A then the Type I-B because of the difference in shear modulus between those two types of rocks. Type I-A has higher shear modulus than the Type I-B which creates more resistance to shear/slip that leads more strain. On the other hand Type II-B shows with a higher slope and higher maximum strain because of relatively lower Young’s modulus and relatively higher shear modulus. (b) We converted bulk volumetric strain to change in porosity using initial porosity of the rock types and the grain to bulk strain ratios. The differences in the shear, Young’s modulus and initial porosity depicts the final change in the porosity with respect to fault slip.

For any discrete volume, this constraint allows to estimate the grain strain as a function of bulk strain as

𝜀_𝑔 𝜀_𝑏 =

√ 𝐵_𝑏

𝐵_𝑔(1 −𝜙_𝑜) (9)

For volumetric strain, the change in porosity is a factor of initial porosity and the difference between bulk and grain strain of porous rock. The triangular dislocation model allows to compute the bulk strain with respect to elastic moduli of the porous medium as in Eq.(6), but it does not provide a direct estimate of grain strain. However, because we are dealing with slip induced deformation in linear elastic system, we can redefine the grain strain in terms of bulk strain as follows:

𝛥𝜙= (1 −𝜙_𝑜) (

1 −

√ 𝐵_𝑏 𝐵_𝑔(1 −𝜙_𝑜)

)

(𝜀_𝑏) (10)

and hence the formula simplifies to

𝛥𝜙=𝐶(𝜀_𝑏) (11)

where C includes information about grain strain to bulk strain ratio and initial porosity and it is called Biot constant (e.g., Ref. 12) and can be estimated by the initial porosity, the Young’s modulus of grain and porous rock. InFig. 4we see the porous medium responses of three different rock types for the same slip amount as grain strain component for every bulk strain value.

Our derivation in Eq. (11) explains the change in porosity with respect to slip-induced strain, as a function of initial porosity, volumetric strain and rock mechanical properties. However, this equation is dimensionless and does not describe yet the complex fault information, such as, its variation in strike and dip along the slip surface or the slip distribution, which are the factors of the slip-induced volumetric strain.

Finally the triangular dislocation and its associated strain field 𝜀_𝑖𝑗 at any point M is given as²⁸:

𝜀_𝑖𝑗(𝑀) =

∑3

𝑘=1

∑3

𝑙=1

𝐸̌_𝑖𝑗𝑘(𝑀 , 𝐴_𝑙, 𝐴_𝑙+1, 𝛼_𝑙, 𝛽_𝑙)𝑏_𝑘=𝐸_𝑖𝑗𝑘𝑏_𝑘. (12)

𝐸̌_𝑖𝑗𝑘represents the strain influence matrix due to triangular dislocation. We defined the discontinuous slip vector, angular dislocation, by the azimuthal angle𝛼and dip angle𝛽. This vector lies on an infinite plane A. The uniform displacement discontinuity is defined by the triangle whose corners are𝐴₀, 𝐴₁, 𝐴₂and the normal vector is given by the Burgers vector𝑏_𝑘. For a fault model made of𝑛triangular dislocation elements, the strain influence matrix at any point M is calculated by superposition²⁸:

𝜀_𝑖𝑗(𝑀) =

∑𝑛 𝑚=1

𝐸^𝑚_𝑖𝑗𝑘𝑏^𝑚_𝑘 (13)

Finally, we write the representation for porosity change with respect to accumulated fault slip at any point by combining Eqs.(11)and(13) as follows:

𝛥𝜙(𝑀) =𝐶(𝑀)◦

∑3

𝑖=1

∑3

𝑗=1

∑𝑛

𝑚=1

𝐸^𝑚

𝑖𝑗𝑘𝑏^𝑚

𝑘𝛿_𝑖𝑗 (14)

In Eq.(13), C(M) includes information about petrophysical and rock mechanical properties of rock type at point M. The Kronecker delta,𝛿_𝑖𝑗, and i,j summations denote for summing over components of volumetric strain (𝜀₁₁, 𝜀₂₂, 𝜀₃₃) at point M. The Hadamard product (elementwise matrix multiplication, ◦) of C(M) and slip-induced volumetric strain matrix gives the change in porosity with respect to accumulated slip on a arbitrarily oriented fault with certain length.

2.4. Porosity permeability relations for evolving pore space under loading The slip-induced modification in the pore volume will affect the permeability of the porous medium, hence we need to relate the change in porosity to change in permeability. The Kozeny–Carman (KC)²⁹ equation relates absolute permeability 𝐾 to pore volume (porosity 𝜙) and pore shape. The KC equation is used to estimate changes in permeability due to changes in porosity for chemical or mechanical processes.³⁰

A widely used but simplified version of the KC equation is given as a power-law relation, describing the permeability change for evolving pore space in experimental and modeling studies.³⁰Assuming that the specific surface area and turtosity are proportional, the KC power-law relation can be simplified to:

𝐾 𝐾₀ =

(𝜙 𝜙₀

)𝑝

(15) where𝑝is the power-law coefficient. An experimental study of mechanical pore deformation (compaction) reports a good fit using power-law with 2.5 ≤ 𝑝 ≤ 3.³¹ In similar studies, 𝑝 is estimated between 3 and5³²for mechanical pore compaction. We used in this paper the𝑝 power coefficient varying between 2.5 and5to capture the uncertainty introduced by the changing𝑝. However better estimation of this power coefficient will be with permeability measurements under deformation tests of particular rock-types of interest.

3. Numerical simulations and results

In this section, we provide three numerical examples to present the results. First, we display how the model formulation works in order to visualize the conceptual model and its formulation step by step. We then discuss how the strain and corresponding change in porosity may differ for a perfectly planar fault versus rough fault surfaces. In the process, we emphasize the importance of rough fault surfaces and their effects on the porous medium.

We incorporate the existing epistemic uncertainty of fault geometry and (elastic) medium properties into our study by adopting a Monte Carlo approach, randomly varying fault geometry and relevant properties of the three rock types mentioned before. We discuss the acceptable ranges of those randomized/varying variables based on published studies. The simulation results serve as the benchmark for the

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Fig. 5. (a) Graphical demonstration of the model formulation and work-flow. We subtract the volumetric grain strain from volumetric bulk rock strain to obtain the pore strain at observation points. We then use initial porosity to calculate modified porosity values on the reservoir rock. (b) The highly affected region, that is the highest change in porosity, from fault slip related deformation is closest to fault. A large number of observation points are only minimally affected by this deformation which means the sensible alteration in porous media is local. Note that we used Type-IA rock type petrophysical and rock mechanical properties with a 5 km long fault embedded on the medium for this example.

Other parameters for this realization is as follows; the grain strain to bulk strain ratio is𝜀_𝑔∕𝜀𝑏= 0.473and the fault roughness parameters are𝐶= 0.005,𝐻_𝑑𝑖𝑝= 0.8,𝐻_{𝑠𝑡𝑟𝑖𝑘𝑒}= 0.8.

reservoir application. Using these benchmark results and the fault map, we build a permeability distribution model. The flow chart in Fig. 2 summarizes the steps we are using to have the permeability distribution model using fault map and reservoir rock parameters.

3.1. Conceptual model

Based on the accumulated fault slip and the degree of fault surface roughness, volumetric strain accumulates and modifies the porous medium on the deformed space along a strike-slip fault. In Fig. 5.a we summarize and display our model formulation. As we enforce the accumulated slip on 5 km rough fault we obtain the strain influence matrix for porous medium. In this conceptual model the solid medium is prescribed by its shear and Young’s modulus defined for Type- IA rock-type. Then we sum deviatoric strains to calculate volumetric strain at each point. We estimate the grain strain based on Young’s Modulus of calcite that constitutes to porous limestone unit Type-IA, also Type-IA’s Young’s modulus and its initial porosity based on Eq.(9) (Biot poroelasticity relations). We display in Fig. 5.a the difference between bulk strain and grain strain that represents the amount of pore strain. With known initial porosity of Type-IA rock-type we convert the volumetric pore strain into the porosity change in the porous medium.

Our numerical experiments show that only a limited volume of the host rock lose significant porosity due to slip induced strain. InFig. 5.b we display distance to fault plane versus the absolute porosity change.

There is an exponential increase in the porosity change due to elastic response of porous rock to fault slip. The highest porosity changes occur at close distances to the fault surface. However, most of the rock medium (perpendicular distance to fault surface) has little strain (<

0.05%) revealed by a bivariate histogram inFig. 5.b. These observations imply that the effect of the porosity change is locally prominent close to fault surface.

3.2. Planar versus rough fault

In reality no fault is perfectly planar, therefore, we present numerical examples for rough faults in comparison to a planar fault surface.

To express the significance of the rough fault assumptions and how different the deformation may be of the surrounding relative to planar faults, we model the slip-induced porosity change model for planar and rough cases. As we enforce the accumulated slip on planar fault surface, the change in porosity is smoothly distributed on the compressed and expanded blocks of the strike-slip fault. As we look at the deformation distribution along the fault surface direction on the porous medium, the change in porosity gradually increases from the center of the planar fault surface to the tips. However, we do not see the same regular deformation – hence the change in porosity – distribution along the rough fault surface direction on the porous medium. On the rough fault surface the change in porosity can abruptly undulate and does not necessarily localize on the tips of the fault. Depending on the location of

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Fig. 6. We investigate planar (a) and rough fault (b) scenarios to understand how much fault-surface roughness affects the deformation of the surrounding porous medium. The rough fault surface redistributes the strain along the fault surface and amplifies the strain. As a result, for the rough fault case the change in porosity in the deformed medium is about 30% higher than in the planar case. The same slip distribution is enforced for both rough and planar faults, tapered to the tip with a 50 m maximum slip. For the rough fault, the roughness parameters are as follows;𝐶= 0.005,𝐻_𝑑𝑖𝑝= 0.8,𝐻_{𝑠𝑡𝑟𝑖𝑘𝑒}= 0.8. Also note that for both planar and rough fault cases, we used Type-IA rock type petrophysical and rock mechanical properties with 5 km long fault embedded on the medium for this exercise.

severe roughness the restraining and releasing bends show changes in porosity. For the same amount of slip, a rough fault may locally deform the surrounding more than a planar fault (Fig. 6). Moreover, in rough faults the deformed space is spatially more widely dispersed based on how local restraining or releasing bends are distributed across the fault surface. In this specific example, we show rough faults generate more strain compared to planar faults. Based on the degree of roughness the absolute porosity change is here 20% more than the planar fault case.

3.3. Benchmark and uncertainty quantification

The limited knowledge about the Earth subsurface lead us to use probabilistic estimation methods to assess how variations in rock and fault properties affect our modeling results. For this purpose, we conduct Monte Carlo simulation using a set of physically plausible values, rather than just inserting the uncertain variable with a single potential value (i.e. an average value). Correspondingly, we generate multiple results with changing fault parameters and evaluate the results as probability distribution.

In this case study, we consider the fault slip amount and the fault roughness parameters as random variables whose particular values is randomly selected within physically plausible ranges. Using the scaling relation of Kim et al.,³³ the tapered slip distribution for a 5 km long strike-slip fault would have a maximum slip between 50 m ≤ 𝑑_𝑚𝑎𝑥≤75 m. In geo-reservoirs, faults are generally imaged with limited resolution, hence lineations on seismic images and a roughness cannot be measured in most cases. That is why a statistical estimate is needed for fault roughness. A 5 km long fault would have a roughness with following constrained parameters according to formerly published data;

0.005 ≤𝐶 ≤ 0.015, 0.7≤𝐻_𝑑𝑖𝑝, 𝐻_{𝑠𝑡𝑟𝑖𝑘𝑒} ≤ 0.9(e.g., Ref. 34). Assuming all mentioned independent variables distributed uniformly (Fig. 7) we generate 1000 random realizations for each of three rock types.

Fig. 7. We draw random samples from a uniform distribution for each fault parameter to conduct Monte-Carlo type simulations of porosity change. We considered the varying 𝐶 index, 𝐻_𝑑𝑖𝑝 and 𝐻_{𝑠𝑡𝑟𝑖𝑘𝑒} indexes for fault roughness and the fault length scaled maximum slip in intervals as documented in literature. In addition to varying fault related parameters we also consider varying Kozeny Carman power coefficient p changing between 2.5 to 5.

Our numerical experiments provide several important observations and results. When enforcing the same fault slip embedded in the porous medium, higher bulk modulus of the material creates less strain and hence the least porosity/permeability change of the porous medium.

These experiments also show that a 5 km long strike-slip fault changes porosity by up to±1%(Fig. 8) which changes the initial permeability by

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Fig. 8. We generate 1000 realizations for each of the three rock types and investigate the effects of 5 km long strike-slip fault to its surrounding. We evaluated rock-types from the Arab-D formation in the Ghawar field, using their initial petrophysical and rock mechanical properties. (a) 5 km strike-slip fault changes by up to 1% of original porosity of Type-IA which can results over 400 mD permeability change proximity to the fault. (b) When enforcing the same slip-boundary condition, a lower shear modulus, Type-IB, results lower porosity change than the Type-IA. (c) For Type-IIB, higher shear modulus and lower initial porosity/permeability changes the permeability by up to 36.6% compared to the initial value.

up to±14%. However, for Type-IIB, a similar 1% porosity change causes at most±20%permeability change. It is evident that the initial porosity and permeability values play a significant role in resultant slip-induced permeability change in porous medium.

On account of roughness and slip distribution ranges, the changes in the porosity and permeability show a wide span as absolute values in Fig. 7. For example, the P50 curve (50th percentile of porosity change) for Type-IIB displays 50% of the observation points in close distance to the fault surface have more than 0.3% absolute porosity change.

The P90 curve for Type-IIB shows that 90% of the observation point proximal to fault surface have less then±0.6%change in porosity while the remaining 10%experience larger porosity changes above 1%.

This exercise provides a benchmark for the change in porosity and permeability in the mentioned Ghawar field rock-types for 5 km length of strike-slip fault. Since the amount of deformation is scalable with the fault length we can scale up the benchmark results for any length, strike and location of the strike-slip faults in the reservoir by regarding the shear sense.

3.4. Geomodel case study using fault map and reservoir rock properties We build an algorithm to populate the benchmark results of porous media permeability population to an actual fault map with varying fault

lengths, orientation and locations. For the benchmark model setup the observation points had 40 meters spacing surrounding a N–S oriented, 5 km strike-slip fault with right lateral accumulated slip embedded in three of the Ghawar field Arab-D formation rock-types. The benchmark results of permeability changes for one fault can be re-sampled for different length, orientation and location of the faults in Ghawar field since the deformation is linearly proportional with the displacement on the fault.

First, we modify both the deformation value on the observation points and the distances between observation points with respect to the ratio of the benchmark fault length vs. the target fault length. Then we rotate these observation points based on the strike information of the fault in the map. Subsequently, we modify the locations of these new deformation population based on the fault location in the map.

By recurrently applying this method for every fault in the sector model we obtain observation points with coordinates at which we calculate modified permeability around every fault plane in the map.

As the length of faults change, the distance between the observation points will also differ due to the scaling method. To re-sample the new permeability property into the sector model, we first discretize the sector area into 2-D grid coordinates such that the minimum distance between the data can be sampled without any information loss.

Then we used a discontinuous ‘‘nearest neighbor’’ triangulation method

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Table 2

Fault map in the Ghawar Sector. All faults have left lateral shear. Accumulated slips are scaled with the fault length and maximum in the center of the faults. The slip distribution taper to zero on the tip of the fault surfaces.

Faults Length (km) Maximum slip (m) Strike (Azimuth) Fault Center Easting (km) Fault Center Northing (km)

1 3.23 32.3–48.4 90 3 −4.5

2 2.26 22.6–33.9 135 −1.5 −1.5

3 2.80 28–41.9 135 −4 0

4 2.90 29–43.5 135 3.5 0

5 1.94 19.4–29 135 −2 1

6 1.51 15.1–22.6 135 −4.5 1.2

7 3.23 32.3–48.4 135 −4.2 2.7

8 1.72 17.2–25.8 75 −4 2

9 1.83 18.3–27.4 75 −1.7 1.9

10 2.69 26.9–40.3 135 −1.3 1.8

11 3.33 33.3–50 60 −0.5 3.3

12 3.76 37.6–56.4 70 −2.8 3.7

13 2.04 20.4–30.6 130 −3.7 3.5

14 2.04 20.4–30.6 75 3.8 3.7

15 1.51 15.1–22.6 120 −2.4 4.2

Fig. 9. (a) Approximate geographic location of the Ghawar field. (b) Map view of a 10 by 10 km sector showing mapped faults in the Haradh section of the Ghawar reservoir.

Source:Modified after Ref.22.

for interpolation. We advocate for such discontinuous methods since the fault related deformation is only locally prominent. Applying this interpolation method on each fault and superposing the permeability property changes on the map, we constructed the sector model for permeability change model due to accumulated slip on complex fault networks.

To demonstrate the method in a complex fault network, we applied this permeability alteration approach to a section of the Ghawar field.

We adopted a part of the subsurface structure map that belongs to the top of Arab-D formation with interpreted fault lineations. The area is a 10 km by 10 km sector of central Haradh, in the Ghawar field and it is documented by Stenger et al.²²and summarized inFig. 9. The location, length, estimated slip amount and strike information of the faults are given in theTable 2. In Ghawar field, the shear sense for NE and NW trending faults is documented as left lateral²²and particularly in this sector we see these two main sets of fault trends. The fault network is more connected and denser on the North-West corner of the sector.

Although we could demonstrate any particular roughness or accumulated slip on faults, due to uncertainty subsurface data holds, we

focus on showing P10, P50 and P90 cases for three rock-types (Type- IA, Type-IB and Type-IIB) obtained from our Monte-Carlo simulation (Fig. 10). Instead of absolute permeability values of each rock-type we show the change in the permeability as a percentage deviation from the original rock-type permeabilities.

Combination of the shear sense being left lateral and high fracture intensity at the northwestern corner on the sector model creates an overall anisotropic pattern in the porous medium permeability. Such composition of fault network increase or decrease permeability by up to 25% of the original permeability at those locations. This example reveals that based on fault network arrangement the combined effect of this local, fault related, porous medium deformation can create strong spatial variability in permeability. Also the sudden changes from low to high permeability between tectonically compressed areas to extended parts may create bypass zones on injection production scenarios. There- fore, the accumulated-slip related permeability change may provide additional insight for accurate infill or selection of well locations when developing a new field. Further investigation with reservoir simulations may quantify the effects of such permeability anisotropy in the recovery profile.

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Fig. 10. For three different rock types (Type-IA, Type-IB and Type-IIB) we show P10, P50 and P90 cases of permeability change due to accumulated slip on reservoir faults. The change in permeability in the porous media is normalized and displayed as percentage change. The highly fractured northwest corner and shows stronger permeability change based on left lateral shear sense (marked on upper left corner map) on the strike-slip faults . This realization shows combination of the faults orientation and spacing can create substantial anisotropy in the permeability pattern on the reservoir.

4. Discussion

We parameterized the fault roughness and the amount of accumulated slip affecting mechanical deformation, then correlated them with fault length. Since deformation is linearly proportional to dislocation (accumulated slip in this case), it also scales with fault length. This relationship provides a general approach for reservoirs characterized by faults of varying length. Our method comes at lower computational cost than applying the physical model and uncertainty analysis for all faults in the reservoir. Fault lines on the fault map are traces of faults at the depth of the reservoir. Therefore, it is appropriate to position the center of the penny shaped faults embedded on the thin reservoir horizon and then enforce accumulated slip on the fault surface. We have not considered horizontal or vertical transitions in the reservoir from one rock type to another. Moreover, our model has not considered either the burial history and related changes in the reservoir rock properties or the other alterations such as chemical alterations. This method assumes the strike-slip fault growth – hence the slip accumulation – happens after lithification is completed without chemical alterations.

To couple the chemical alterations with mechanical alterations one needs burial and diagenesis history. The diagenesis or fluid migration history through fractures and porous media is unique for each reservoir and needs targeted studies to for calibration, which is beyond the scope of this work. For example there are existing solvers with multi-physics coupling capability for fault related chemical precipitation/dissolution and mechanics of porous material,⁸but these require as input the paleo fluid chemistry, paleo stress and paleo temperature.

The numerical experiments carried out in this paper offer a clear understanding of how slip-related poro-elastic deformation affects the petrophysical properties of reservoir rocks. The slip-induced poro- elastic deformation and resultant porosity/permeability change can be quantified with known initial porosity, permeability, bulk modulus, shear modulus and length of the faults in the reservoir. It is known that faults have rough surfaces and including this information in the model shows that fault roughness redistributes and amplifies the strain, and hence the change in porosity/permeability. Experiments in this study suggest that slip-induced poro-elastic deformation and the resulting porosity/permeability change are most prominent close to the fault

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surface. This deformation is spatially distributed all along the fault on deformed spaces. The numerical experiments for different rock types indicate that based on different initial porosity and permeability, enforcing the same fault slip may result in significantly different permeability changes around it. These numerical experiments for different stiffness values indicate that a lower bulk modulus results in a higher porosity change when enforcing the same slip-boundary condition.

Similarly, a lower shear modulus means less strain – and hence less stress for elastic behavior – for any given slip discontinuity; this means for the same slip on fault, the rock deforms less and hence the absolute change in porosity would be less.

For complex fault networks we see that combination of shear sense and fracture intensity creates permeability anisotropy in the porous medium. The combination of deformation caused by interme- diate length faults in the reservoir accumulates to large permeability anisotrophy. To capture such deformation and related permeability variation in the reservoir propose this new method, which in fact is straightforward to apply since only fault map and rock properties are needed. We conjecture that our proposed approach is promising to build better static reservoir models.

5. Conclusions

This paper presented a generalized solution for slip-induced poro- elastic deformation and its effects on the fault-surrounding – porous media – permeability. We integrated this new formulation with triangular dislocation solution and showed the resultant porosity and permeability profiles for several known rock types with possible uncertainty ranges. Our numerical experiments document that, for any fault length, slip-induced poroelastic deformation can be modeled with known petrophysical and rock mechanical parameters of target reservoir. To demonstrate that, we scaled up the single fault porous medium interaction to a complex fault network for a known reservoir rock- types and its faults. Results show that complex fault arrangements, with their shear sense and intensity, generate sensible permeability anisotropy in the reservoir. In fact, this physical model application is a necessity to increase the reliability of the porosity and permeability property distribution models for all fractured reservoirs. We developed a complete workflow that can be implemented on any reservoir with its rock properties and fault map.

It is important to emphasize that this paper solely focused on volumetric strain response of porous medium to accumulated fault slip and its effects on permeability. We are aware that damage zone fractures, cataclastic deformation or off-fault chemical alteration are also responsible for additional changes in the off-fault permeability.

Future work is needed to combine these physical processes affecting off-fault permeability with this study on fault slip-induces permeability changes in porous medium.

Code availability section

Contact:[email protected] Program language: MATLAB

The source codes are available for downloading at the link:

https://www.researchgate.net/project/Slip-induced-porous-medium -deformation.

CRediT authorship contribution statement

Bora Yalcin:Conceived the study, Wrote model code for translating strain into porosity/permeability change, Carried out the analysis and produced the figures, Wrote the manuscript. Olaf Zielke:Wrote the model code for slip-induced strain, Wrote the manuscript.P. Martin Mai:Provided overall supervision.

Declaration of competing interest

The authors declare that they have no known competing finan- cial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

Data will be made available on request.

Acknowledgments

The authors would like to acknowledge the fruitful discussion with Dmitry Garagash and his suggestions. Also the authors would like to thank Jagdish Vyas for his proof read on index notation derivation.

Finally, the authors would like to thank the reviewers for their helpful comments, which improved the manuscript and the overall presenta- tion. The research reported in this publication was supported by fund- ing from King Abdullah University of Science and Technology (KAUST, grants BAS/1/1339-01-01, and REI/1/4502-01-01). Figures and tables in the main manuscript provide all data used in this investigation.

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