**4. Conceptual geomechanical model**

The conceptual geomechanical model already summarised (**Figure 5**), which might account for seismicity beneath Newdigate, caused by pressure decreases in the Portland reservoir resulting from production (or other activities) from the HH1 or BRX2Y wells, will now be developed quantitatively. The basis of this model (**Figure 5**) is as follows. The Upper Portland Sandstone reservoir adjoining these wells is assumed to make a subhorizontal hydraulic connection with the seismogenic Newdigate fault zone via a permeable fabric formed in calcite 'beef' in the stratigraphically adjacent Lower Portland Sandstone. The seismogenic fault is assumed highly permeable and to provide a downward hydraulic connection to the rocks beneath the Jurassic succession. These rocks are assumed to include the dolomitic conglomerate and Dinantian limestone, as in the HH1 well (**Table 1**),

which are themselves permeable. It is further assumed that the Newdigate seismicity has occurred at locations where these permeable lithologies are in contact across this fault. Pressure reduction in the Portland reservoir will thus be communicated via the 'beef' fabric, reducing the fluid pressure in this fault, which will cause flow from within the adjoining permeable lithologies into the fault. The associated reduction in fluid volume within these lithologies will cause them to compact. This will result in surfaces that were previously in contact across this fault to separate slightly, reducing the normal stress across the fault. This will 'unclamp' the fault (as in **Figure 5(b)**), moving it closer to the Mohr-Coulomb failure condition. The fault is itself assumed to be 'critically stressed', already near this failure condition, potentially enabling relatively small changes in the state of stress to cause coseismic slip (cf. [64, 65]).

(e.g., [77]), where k and η are the permeability of the medium and viscosity of

<sup>2</sup> <sup>þ</sup> *<sup>ϕ</sup>* <sup>1</sup> *BF* � 1 *BR*

> þ *ϕ BW*

where BR and ϕ are the bulk modulus and porosity of the rock, BF is the bulk modulus of the fluid, and BE is the 'effective' bulk modulus of the combination,

> � <sup>1</sup> � *<sup>ϕ</sup> BR*

*<sup>D</sup>* � *kBE*

which can be compared with Eq. (2); in practice, the difference in choice of elastic modulus (BE or M) makes little practical difference (see below); from Eqs.

> � *BE BR*

The pressure variations in the calcite 'beef' layer hydraulically connected to, and surrounding, the Brockham and Horse Hill Portland reservoirs can be assumed to have circular symmetry; analysis in terms of cylindrical polar co-ordinates is thus required. Thus, if the flow does not vary azimuthally or across the vertical extent h

> *P ∂r*<sup>2</sup> þ 1 *r ∂P ∂r*

As others (e.g., [79, 80]) have noted, if production at rate Q starts at time t = 0,

*E*1

where the subscripts B denote values of D, h and k appropriate for the layer of

exp ð Þ �*s s*

E1 is not supported directly in Microsoft Excel, but using its relationship to other functions (discussed, e.g., by [81]) it can be evaluated indirectly. This is possible

*r*2 4*DB t*

� �, (3)

*:* (4)

*<sup>η</sup>* , (5)

<sup>2</sup> *:* (6)

� �*:* (7)

� �, (8)

*ds:* (9)

*E*1ð Þ� *x* lim *<sup>ψ</sup>*!<sup>0</sup>Γð Þ *ψ*, *x* (10)

the fluid and M is the Biot modulus of the fluid-rock combination. M can be

*Seismicity at Newdigate, Surrey, during 2018–2019: A Candidate Mechanism…*

� *BE BR*

1 *BE*

> 1 *<sup>M</sup>* � <sup>1</sup> *BE*

*∂P <sup>∂</sup><sup>t</sup>* <sup>¼</sup> *<sup>D</sup> <sup>∂</sup>*<sup>2</sup>

<sup>Δ</sup>*<sup>P</sup>* <sup>¼</sup> *<sup>Q</sup> <sup>η</sup>*

*E*1ð Þ� *x*

'beef'. Here, E1 is the Exponential Integral Function, defined as

4*π kB hB*

∞ð

*x*

the variation in P, ΔP, after t = 0 takes the form

1 *<sup>M</sup>* � <sup>1</sup> *BR*

*DOI: http://dx.doi.org/10.5772/intechopen.94923*

expressed as

defined as

(3) and (4),

because

**81**

Costain [78] expressed D as

of the 'beef', Eq. (1) reduces to

Regarding the assumptions thus made, the presence of calcite 'beef' within the Portland Group sediments and its permeability have already been discussed. The permeability of faults is a major issue in Earth science (e.g., [66–70]). There is no information regarding the permeability of any strand of the Newdigate fault zone; however, although counterexamples exist (e.g., faults made impermeable by cemented fault gouge [71]), the view that faults are generally permeable, especially when critically stressed (e.g., [72]) is widely accepted, as is the precautionary principle that faults are assumed permeable, in the absence of contrary evidence, when assessing the possibility of subsurface fluid migration (e.g., [73, 74]). In the present study area, faults with offsets of tens of metres, where the permeable Portland Sandstone is juxtaposed against the impermeable Purbeck Anhydrite, act as seals for oil reservoirs (e.g., [26, 27]) and are obviously impermeable. However, low-offset faults with the permeable Portland Sandstone juxtaposed on both sides can be expected to be permeable. The question of the continuity of the Dinantian limestone from the HH1 area to the vicinity of the seismogenic strand of the Newdigate fault zone has already been discussed. The uncertainty regarding the state of stress in the Weald Basin is considered in the online supplement. As will become clear below, if the differential stress here is anything like as high as it is the Preese Hall area (after [11]), then any fault with the orientation of that which slipped will be very close to the Mohr-Coulomb failure condition.

The model thus includes three elements: pressure drawdown caused by production in wells, and its communication via the 'beef' fabric; compaction of the permeable rocks alongside the Newdigate fault caused by fluid withdrawal accompanying depressurization; and the associated Mohr-Coulomb failure analysis.

#### **4.1 Pressure drawdown accompanying production**

In general variations in fluid pressure, P, in a porous medium, are solutions to the diffusion equation

$$\frac{\partial \mathbf{P}}{\partial \mathbf{t}} = D\_{\cdot} \nabla^{2} P,\tag{1}$$

where t is time, ∇<sup>2</sup> is the Laplacian operator, and D is the hydraulic diffusivity of the medium (e.g., [75]). The value of D depends on properties of the medium and fluid and on solution details, such as boundary conditions for pressure and strain (e.g., [76]). For pressure diffusion,

$$D \equiv \frac{kM}{\eta} \tag{2}$$

*Seismicity at Newdigate, Surrey, during 2018–2019: A Candidate Mechanism… DOI: http://dx.doi.org/10.5772/intechopen.94923*

(e.g., [77]), where k and η are the permeability of the medium and viscosity of the fluid and M is the Biot modulus of the fluid-rock combination. M can be expressed as

$$\frac{1}{M} \equiv \frac{1}{B\_R} - \frac{B\_E}{B\_R} + \phi \left(\frac{1}{B\_F} - \frac{1}{B\_R}\right),\tag{3}$$

where BR and ϕ are the bulk modulus and porosity of the rock, BF is the bulk modulus of the fluid, and BE is the 'effective' bulk modulus of the combination, defined as

$$\frac{1}{B\_E} \equiv \frac{1-\phi}{B\_R} + \frac{\phi}{B\_W}.\tag{4}$$

Costain [78] expressed D as

$$D \equiv \frac{kB\_E}{\eta},\tag{5}$$

which can be compared with Eq. (2); in practice, the difference in choice of elastic modulus (BE or M) makes little practical difference (see below); from Eqs. (3) and (4),

$$\frac{1}{M} \equiv \frac{1}{B\_E} - \frac{B\_E}{B\_R^{-2}}.\tag{6}$$

The pressure variations in the calcite 'beef' layer hydraulically connected to, and surrounding, the Brockham and Horse Hill Portland reservoirs can be assumed to have circular symmetry; analysis in terms of cylindrical polar co-ordinates is thus required. Thus, if the flow does not vary azimuthally or across the vertical extent h of the 'beef', Eq. (1) reduces to

$$\frac{\partial P}{\partial t} = D \left( \frac{\partial^2 P}{\partial r^2} + \frac{1}{r} \frac{\partial P}{\partial r} \right). \tag{7}$$

As others (e.g., [79, 80]) have noted, if production at rate Q starts at time t = 0, the variation in P, ΔP, after t = 0 takes the form

$$
\Delta P = \frac{Q \,\eta}{4 \,\pi k\_B \, h\_B} E\_1 \left(\frac{r^2}{4D\_B t}\right), \tag{8}
$$

where the subscripts B denote values of D, h and k appropriate for the layer of 'beef'. Here, E1 is the Exponential Integral Function, defined as

$$E\_1(\varkappa) \equiv \int\_{\varkappa}^{\infty} \frac{\exp \left( -s \right)}{s} ds. \tag{9}$$

E1 is not supported directly in Microsoft Excel, but using its relationship to other functions (discussed, e.g., by [81]) it can be evaluated indirectly. This is possible because

$$E\_1(\mathfrak{x}) \equiv \lim\_{\psi \to 0} \Gamma(\psi, \mathfrak{x}) \tag{10}$$

where Γ(ψ, x) is the Upper Incomplete Gamma Function. As Schurman [82] has noted, this means that an Excel formula providing a good approximation to E1(x) can be written as

$$\text{EXP}(\text{GAMMALN}(\text{B1})) \* (\mathbf{1} - \text{GAMMA}.\text{DIST}(\text{A1}, \text{B1}, \mathbf{1}, \text{TRUE}) \tag{11}$$

where cell A1 contains x and cell B1 contains a small positive number (say, 10�<sup>8</sup> ), representing ψ. Values of E1 thus calculated were checked against the tables by Harris [83] and Stegun and Zucker [84] and against the power series approximations for E1(x) for the limit of x ≪ 1,

$$E\_1(\varkappa) \approx -\chi - \ln\left(\varkappa\right) + \varkappa - \frac{\varkappa^2}{4} + \dotsb \tag{12}$$

*tD* <sup>¼</sup> *<sup>r</sup>*<sup>2</sup> 4*DB*

*Q ηDB* Δ*t*

.

The Newdigate fault is envisaged as extremely permeable, such that a pressure

variation ΔP applied to any point of it by via the layer of 'beef' is transmitted downward, with no significant time delay, to depths where it transects the permeable Dinantian limestone, the presumed seismogenic layer. This model fault is vertical, the permeable seismogenic layer being assigned thickness HD, hydraulic diffusivity DD, permeability kD, and porosity ϕD. Depressurization at the point where the 'beef' layer intersects this fault is thus inferred to cause a reduction in groundwater pressure ΔPO at each point below on the fault within the permeable seismogenic layer. The resulting groundwater pressure variation in the Dinantian limestone will be governed by Eq. (1). However, if this variation

is independent of vertical position and position parallel to the fault, the

*∂P <sup>∂</sup><sup>t</sup>* <sup>¼</sup> *<sup>D</sup> <sup>∂</sup>*<sup>2</sup>

will require solution, where t is time and x is distance from the fault.

*<sup>δ</sup><sup>P</sup>* <sup>¼</sup> <sup>Δ</sup>*PO erfc <sup>x</sup>*

*P*

2 ffiffiffiffiffiffiffiffiffiffi ð Þ *D t* <sup>p</sup> !

As a solution to Eq. (19), the drawdown δP in groundwater pressure within such

erfc() denoting the Complementary Gaussian Error Function. For z > 0, erfc(z) decreases as z increases, reaching �0.0047 when z = 2. As Detournay and Cheng [77] noted, this condition indicates an effective outer limit to significant pressure

*xM* <sup>¼</sup> <sup>4</sup> ffiffiffiffiffiffiffiffiffiffi

As time progresses, as a result of continuing production from a well that is hydraulically connected to the model fault, an ever-widening volume of rock, in the x direction perpendicular to the model fault, will thus become depressurized, water previously stored within this volume being released into the fault. **Figure 10**

In general, poroelastic strain responses to changes in fluid pressure can be highly complex (e.g., [77, 86–89]). Segall [87] noted that in the limit where Δσkk = 0, a

�1 .

The maximum pressure variation ΔPM at distance r and time tD is thus

Δ*PM*ð Þ¼ *r*

*Seismicity at Newdigate, Surrey, during 2018–2019: A Candidate Mechanism…*

indicating that ΔPM varies inversely with r<sup>2</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.94923*

**4.2 Compaction alongside the Newdigate fault**

one-dimensional variant

a permeable layer is given (e.g., [78]) by

illustrates this effect for D = 1 m<sup>2</sup> s

**83**

variations, at distance xM from the model fault, where

*:* (17)

*<sup>π</sup> kB her*<sup>2</sup> , (18)

*<sup>∂</sup>x*<sup>2</sup> , (19)

ð Þ *D t* <sup>p</sup> *:* (21)

(20)

(e.g., [79]), where γ = 0.5772156649 … is Euler's Constant, and for the limit x ≫ 1,

$$E\_1(\varkappa) \approx \frac{\exp\left(-\varkappa\right)}{\varkappa} \left(1 - \frac{1}{\varkappa} + \cdots\right) \tag{13}$$

(e.g., [85]), and were found to be accurate to four significant figures or better. This method for evaluating E1(x) for all values of x is, thus, more accurate in general than the overall approximation formula proposed by Barry et al. [85].

In the near-wellbore volume the pressure variation will be in the Portland sandstone, not in the 'beef'. Thus, from Eq. (8), at time t the variation at the well rim, of radius rW, is ΔPW where

$$
\Delta P\_W = \frac{Q \,\eta}{4 \,\pi k\_P h\_P} E\_1 \left(\frac{r\_W^{-2}}{4D\_P t}\right),
\tag{14}
$$

where the subscripts P denote values of D, h and k appropriate for the Portland sandstone. Given the high value of DP for the Portland sandstone, for most of the durations of the production pulses at BRX2Y and HH1, rW 2 /(4 DP t) ≪ 1. One may thus approximate E1(x) using Eq. (12). Indeed, x will be so small that only the -ln(x) term need be considered. The resulting logarithmic dependence of ΔPW on t means that ΔPW remains roughly constant, as observed during the HH1 well test (see supplement).

Using the same general approach, a brief episode of production at rate Q starting at time t = 0 and ending at time Δt causes a pressure variation given by

$$
\Delta P = \frac{Q \,\eta}{4 \,\pi k\_B h} \left( E\_1 \left( \frac{r^2}{4D\_B t} \right) - E\_1 \left( \frac{r^2}{4D\_B (t - \Delta t)} \right) \right). \tag{15}
$$

Using the definition of E1 in Eq. (9), for t ≫ Δt, Eq. (8) can be approximated as

$$
\Delta P \approx \frac{Q \,\eta \,\Delta t}{4 \,\pi k\_B h \, t} \, \exp\left(\frac{-r^2}{4D\_B t}\right). \tag{16}
$$

This equation can be differentiated with respect to t; by setting the resulting partial derivative to zero one may solve for the time delay tD for the maximum pressure variation, thus:

*Seismicity at Newdigate, Surrey, during 2018–2019: A Candidate Mechanism… DOI: http://dx.doi.org/10.5772/intechopen.94923*

$$t\_D = \frac{r^2}{4D\_B}.\tag{17}$$

The maximum pressure variation ΔPM at distance r and time tD is thus

$$
\Delta P\_M(r) = \frac{Q \,\eta \, D\_B \,\Delta t}{\pi \, k\_B \, h \, e \, r^2},
\tag{18}
$$

indicating that ΔPM varies inversely with r<sup>2</sup> .

### **4.2 Compaction alongside the Newdigate fault**

The Newdigate fault is envisaged as extremely permeable, such that a pressure variation ΔP applied to any point of it by via the layer of 'beef' is transmitted downward, with no significant time delay, to depths where it transects the permeable Dinantian limestone, the presumed seismogenic layer. This model fault is vertical, the permeable seismogenic layer being assigned thickness HD, hydraulic diffusivity DD, permeability kD, and porosity ϕD. Depressurization at the point where the 'beef' layer intersects this fault is thus inferred to cause a reduction in groundwater pressure ΔPO at each point below on the fault within the permeable seismogenic layer. The resulting groundwater pressure variation in the Dinantian limestone will be governed by Eq. (1). However, if this variation is independent of vertical position and position parallel to the fault, the one-dimensional variant

$$\frac{\partial P}{\partial t} = D \frac{\partial^2 P}{\partial \mathbf{x}^2},\tag{19}$$

will require solution, where t is time and x is distance from the fault.

As a solution to Eq. (19), the drawdown δP in groundwater pressure within such a permeable layer is given (e.g., [78]) by

$$
\delta P = \Delta P\_O \circ \text{rfc}\left(\frac{x}{2\sqrt{(Dt)}}\right) \tag{20}
$$

erfc() denoting the Complementary Gaussian Error Function. For z > 0, erfc(z) decreases as z increases, reaching �0.0047 when z = 2. As Detournay and Cheng [77] noted, this condition indicates an effective outer limit to significant pressure variations, at distance xM from the model fault, where

$$\mathfrak{x}\_{\mathsf{M}} = 4\sqrt{(Dt)}.\tag{21}$$

As time progresses, as a result of continuing production from a well that is hydraulically connected to the model fault, an ever-widening volume of rock, in the x direction perpendicular to the model fault, will thus become depressurized, water previously stored within this volume being released into the fault. **Figure 10** illustrates this effect for D = 1 m<sup>2</sup> s �1 .

In general, poroelastic strain responses to changes in fluid pressure can be highly complex (e.g., [77, 86–89]). Segall [87] noted that in the limit where Δσkk = 0, a

reduction in fluid pressure by δP will cause a contractional strain ε = α δP/BE where α is Biot's coefficient, defined as

$$a \equiv 1 - \frac{B\_E}{B\_R}.\tag{22}$$

rocks on either side of a vertical fault at the mid-point of a continuum model (such as that by [88]) were depressurised, these rocks would move towards the fault, the opposite sense of motion to what is required to provide the ability to make an assessment of the effect of asperities on the fault. It is for this essential reason that new theory is derived here for the pressure and stress response in the Dinantian

> ð*<sup>x</sup>*!<sup>∞</sup> *x*¼0

the factor of 2 taking into account that the rocks on both sides of the fault will move away from it. Evaluation of Δx requires the integral of erfc() (cf. Eq. (20)).

*erfc z*ð Þ*dz* <sup>¼</sup> *x erfc x*ð Þþ <sup>1</sup> � exp �*x*<sup>2</sup> ð Þ ð Þ

*erfc z*ð Þ*dz* <sup>¼</sup> <sup>1</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi *DT π*

� *erfc <sup>x</sup>*

! !

<sup>Δ</sup>*POD xδ<sup>t</sup>* exp �*x*<sup>2</sup> ð Þ *<sup>=</sup>*ð Þ <sup>4</sup>*D t*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>π</sup>*ð Þ *D t* <sup>3</sup>

At times t ≫ Δt, the pressure variation δP is given to a good approximation by

2

The maximum pressure variation at distance x occurs after a time delay tD

*tD* <sup>¼</sup> *<sup>x</sup>*<sup>2</sup>

It follows that the maximum pressure variation at distance x and time tD is given

2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ *D t*ð Þ � *<sup>δ</sup><sup>t</sup>* <sup>p</sup>

<sup>q</sup> *:* (29)

<sup>6</sup>*<sup>D</sup> :* (30)

Δ*εxxdx*, (24)

ffiffiffi

ffiffiffi

*<sup>π</sup>* <sup>p</sup> (25)

*<sup>π</sup>* <sup>p</sup> (26)

*:* (28)

s� �, (27)

limestone, rather than using existing published theory.

*DOI: http://dx.doi.org/10.5772/intechopen.94923*

Subject to the above model definition, Δx can be estimated as

*Seismicity at Newdigate, Surrey, during 2018–2019: A Candidate Mechanism…*

Δ*x* ¼ 2

From Abramowicz and Stegun [92], p. 299 and Weisstein [93],

*E*ð Þ� ∞

ð*<sup>z</sup>*!<sup>∞</sup> *z*¼0

<sup>Δ</sup>*<sup>x</sup>* <sup>¼</sup> <sup>4</sup>*γ α*Δ*PO BE*

this quantity being positive for a reduction in pressure by ΔPO.

2 ffiffiffiffiffiffiffiffiffiffi ð Þ *D t* <sup>p</sup> !

Costain [78] also showed that if, rather than persisting indefinitely, the pressure change ΔPO imposed at x = 0 persists for duration δt, the resulting pressure varia-

*E x*ð Þ� <sup>ð</sup>*<sup>z</sup>*¼*<sup>x</sup>*

so

tion δP is given by

given by

by δPM where

**85**

*z*¼0

Using Eqs. (20), (23) and (26), one obtains.

*<sup>δ</sup>P x*ð Þ¼ , *<sup>t</sup>* <sup>Δ</sup>*PO erfc <sup>x</sup>*

*δP x*ð Þ¼ , *t*

Contraction will occur in the vertical z direction as well as in the x direction. Partitioning the contractional strain as Δεxx = γ ε and Δεzz = (1 - γ) ε,

$$
\Delta \varepsilon\_{\infty} = \frac{\gamma \, a \, \delta P}{B\_E}. \tag{23}
$$

Many studies of poroelastic deformation have treated petroleum or geothermal reservoirs as inclusions embedded in surrounding rocks and have treated faults as idealised planes within the inclusion or its surroundings (e.g., [86, 88, 90, 91]). Because the present study aims to explore the effect of fault asperities, the fault cannot be treated in this idealised manner. The fault is instead envisaged as a vertical 'cut' in the poroelastic medium, which has (distant) boundaries in both directions perpendicular to the fault plane. In this configuration, with the outer ends of the blocks on either side of the fault fixed or 'pinned', depressurization of pore fluid will cause their inner ends, facing each other across the fault, to separate slightly, by distance Δx, as depicted schematically in **Figure 5(b)**. In contrast, if the

#### **Figure 10.**

*Graphs of the predicted variation in pressure δP / ΔPO with distance x from the model seismogenic fault, calculated using Eq. (20) for hydraulic diffusivity D = 1 m<sup>2</sup> s* �*1 , representing Dinantian limestone. (a) after times t of 1 hour, 12 hours, 3 days and 2 weeks. (b) after times t of 2 months, 1 year, 5 years, and 20 years.*

*Seismicity at Newdigate, Surrey, during 2018–2019: A Candidate Mechanism… DOI: http://dx.doi.org/10.5772/intechopen.94923*

rocks on either side of a vertical fault at the mid-point of a continuum model (such as that by [88]) were depressurised, these rocks would move towards the fault, the opposite sense of motion to what is required to provide the ability to make an assessment of the effect of asperities on the fault. It is for this essential reason that new theory is derived here for the pressure and stress response in the Dinantian limestone, rather than using existing published theory.

Subject to the above model definition, Δx can be estimated as

$$
\Delta \mathfrak{x} = 2 \int\_{\mathfrak{x} = 0}^{\mathfrak{x} \to \infty} \Delta \varepsilon\_{\mathfrak{x}\mathfrak{x}} d\mathfrak{x},\tag{24}
$$

the factor of 2 taking into account that the rocks on both sides of the fault will move away from it. Evaluation of Δx requires the integral of erfc() (cf. Eq. (20)). From Abramowicz and Stegun [92], p. 299 and Weisstein [93],

$$E(\mathbf{x}) \equiv \int\_{x=0}^{x=\infty} \text{erfc}(\mathbf{z}) d\mathbf{z} = \mathbf{x} \text{erfc}(\mathbf{x}) + \frac{(\mathbf{1} - \exp\left(-\mathbf{x}^2\right))}{\sqrt{\pi}} \tag{25}$$

so

$$E(\infty) \equiv \int\_{x=0}^{x \to \infty} \text{erfc}(x) dx = \frac{1}{\sqrt{\pi}} \tag{26}$$

Using Eqs. (20), (23) and (26), one obtains.

$$
\Delta \mathfrak{x} = \frac{4\gamma a \Delta P\_O}{B\_E} \sqrt{\left(\frac{DT}{\pi}\right)},
\tag{27}
$$

this quantity being positive for a reduction in pressure by ΔPO.

Costain [78] also showed that if, rather than persisting indefinitely, the pressure change ΔPO imposed at x = 0 persists for duration δt, the resulting pressure variation δP is given by

$$\delta P(\mathbf{x}, t) = \Delta P\_O \left( e \eta \mathbf{f} \overline{\mathbf{c} \left( \frac{\mathbf{x}}{2 \sqrt{(Dt)}} \right)} - e \eta \mathbf{f} \overline{\mathbf{c} \left( \frac{\mathbf{x}}{2 \sqrt{(D(t-\delta t))}} \right)} \right). \tag{28}$$

At times t ≫ Δt, the pressure variation δP is given to a good approximation by

$$\delta P(\mathbf{x}, t) = \frac{\Delta P\_O D \mathbf{x} \,\delta t \,\exp\left(-\mathbf{x}^2 / (4Dt)\right)}{2\sqrt{\pi (Dt)^3}}.\tag{29}$$

The maximum pressure variation at distance x occurs after a time delay tD given by

$$t\_D = \frac{\mathfrak{x}^2}{6D}.\tag{30}$$

It follows that the maximum pressure variation at distance x and time tD is given by δPM where

*Earthquakes - From Tectonics to Buildings*

$$
\delta P\_M(\mathbf{x}, t) = \frac{3\sqrt{6}\,\Delta P\_O D}{\sqrt{(\pi e^3)} \mathbf{x}^2}. \tag{31}
$$

will thus be evaluated where σN, τ and c are the resolved normal stress, shear stress and coefficient of friction on the fault plane, and P is the fluid pressure in the fault. Φ = 0 marks this condition, with Φ < 0 indicating frictional stability. This analysis can also be visualised using the standard Mohr circle construction, as a graph of τ against effective normal stress σN', defined as σ<sup>N</sup> P (see below). From Eq. (37), other factors remaining constant, a decrease in P will 'clamp' a fault, making it more stable, and an increase in P will 'unclamp' a fault, potentially resulting in seismicity. The latter change is the accepted mechanism for the widespread occurrence in recent years of seismicity caused by fluid injection (e.g., [95– 98]). On this basis, one might conclude that a decrease in the groundwater pressure

However, it is generally accepted that the mechanics of faults, notably whether they are stable or can slip seismically, are determined by the properties of strong patches – asperities – where the opposing fault surfaces are in frictional contact (e.g., [99]). A fault surface consisting, on a microstructural scale, of a fractal size distribution of asperities with a small proportion of the fault surface in contact, in proportion to the normal stress applied to the fault, can mimic the effect, on a macroscopic scale, of a constant coefficient of friction (e.g., [100, 101]). Brown and Scholz [102] showed that natural rock surfaces follow fractal scaling for surface features of height up to 0.1 m. Laboratory simulations of faulting often include asperities on a microstructural scale (e.g., [103, 104]). Most recently, the view has gained ground that the physics of co-seismic faulting is likewise governed by processes on a microstructural scale (e.g., [105, 106]). For example, McDermott et al. [106] deduced that asperities can be patches of fault with areas of no more than a few square metres, their properties being determined by mineral grains with dimensions of microns. Because these strong patches with fault surfaces in contact occupy only a small proportion of a fault surface, they act as stress concentrations. For example, in the laboratory experiments by Selvadurai and Glaser [104], millimetre-sized asperities with micron-sized heights occupy a very small proportion of the fault area; in one experimental run, a decrease in the mean normal stress across the fault area by 0.3 MPa caused decreases in the normal stress affecting

**Figure 5(b)** illustrates how a small increase in separation of fault surfaces, Δx, can destabilise a fault through its effect on contact between asperities. Moving from configuration (i) to configuration (ii), two of the three asperities depicted will no longer contribute to fault stability. The third one will experience a significantly reduced normal stress, as a result of the increased separation of the fault surfaces. This will reduce the maximum shear stress that this asperity can sustain, in accordance with Eq. (37), whereas the shear stress that it is required to sustain to keep the fault stable will increase because the other asperities no longer contribute. Overall, it can thus be seen how a small increase in separation of fault surfaces might bring a fault significantly closer to the condition for slip, and might indeed

As others (e.g., [107, 108]) have noted, a general calculation of this 'unclamping' effect for a fault with a general size-distribution of asperities would be very complex; this is thus not attempted here. A simplified calculation is instead presented, assuming that a patch of fault is prevented from slipping by a single asperity. For this patch, of area A, the normal stress and shear stress are σ<sup>N</sup> and τ; the single asperity has cross-sectional area a, uncompressed height b, and Young's modulus E. The effect of σ<sup>N</sup> compresses this model asperity to height d (**Figure 11**). The tip of this asperity will act as a stress concentration, with normal stress and shear stress (A / a) σ<sup>N</sup> and (A / a) τ and coefficient of friction c. The areas of fault where the wall rocks are not in contact initially contain fluid with pressure PO (**Figure 11(a)**),

within the Newdigate fault cannot be the cause of the local seismicity.

*Seismicity at Newdigate, Surrey, during 2018–2019: A Candidate Mechanism…*

*DOI: http://dx.doi.org/10.5772/intechopen.94923*

individual asperities by 10 MPa.

result in coseismic slip.

**87**

These results, in Eqs. (30) and (31), can be compared with those for the radially symmetric case in Eqs. (17) and (18). In both cases, tD is proportional to the square of distance and inversely proportional to D, but differs by a numerical factor. Alternative empirical analysis by Hettema et al. [94], based on 'rules of thumb' rather than derivation from first principles, predicts a value for tD that likewise differs by a numerical factor, but does not predict pressure variations. The pressure variation varies inversely with the square of distance for both the radially symmetric case and for the one-dimensional case (cf. Eqs. (18) and (31)).

Δx can thus be calculated using the exact formula for δP (Eq. (28)) as

$$
\Delta \mathfrak{x} = \frac{4\gamma a \Delta P\_O}{B\_E} \sqrt{\left( \left( \frac{Dt}{\pi} \right) - \left( \frac{D(t-\delta t)}{\pi} \right) \right)},\tag{32}
$$

and using the approximate formula (Eq. (29)) as

$$
\Delta \mathbf{x} = \frac{\gamma a \Delta P\_O \delta t}{B\_E} \sqrt{\left(\frac{D}{\pi t}\right)}.\tag{33}
$$

Segall [87] reported that, for the Δσkk = 0 boundary condition applicable for this analysis,

$$D \equiv \frac{k}{\eta} \frac{2\mu(1-\nu)}{(1-2\nu)} \frac{B(1+\nu)}{3a(1-\nu) - 2Ba^2(1-2\nu)},\tag{34}$$

where μ is the shear modulus of the rock and B is its Skempton coefficient.

One may likewise quantify the loss of volume ΔV as a result of the vertical compaction Δz of the Dinantian limestone. By analogy with Eq. (24), an upper bound to Δz can be estimated as

$$
\Delta z = \int\_{x=0}^{x=H} \frac{(1-\chi)a\,\Delta P\_O}{B\_E} dz,\tag{35}
$$

where H is the thickness of the Dinantian limestone. If the volume of limestone thus affected has dimensions Ly parallel to and Lx perpendicular to the model fault, so ΔV=Lx � Ly � Δz or

$$
\Delta V = \frac{(\mathbf{1} - \mathbf{y}) \, a \, \Delta P\_O H L\_\mathbf{x} L\_\mathbf{y}}{B\_E}.\tag{36}
$$

#### **4.3 Mohr-Coulomb failure analysis**

The tendency for coseismic slip on the seismogenic fault is analysed using the standard Mohr-Coulomb approach. The Mohr-Coulomb failure parameter Φ:

$$
\Phi = \pi - c(\sigma\_N - P),
\tag{37}
$$

*Seismicity at Newdigate, Surrey, during 2018–2019: A Candidate Mechanism… DOI: http://dx.doi.org/10.5772/intechopen.94923*

will thus be evaluated where σN, τ and c are the resolved normal stress, shear stress and coefficient of friction on the fault plane, and P is the fluid pressure in the fault. Φ = 0 marks this condition, with Φ < 0 indicating frictional stability. This analysis can also be visualised using the standard Mohr circle construction, as a graph of τ against effective normal stress σN', defined as σ<sup>N</sup> P (see below).

From Eq. (37), other factors remaining constant, a decrease in P will 'clamp' a fault, making it more stable, and an increase in P will 'unclamp' a fault, potentially resulting in seismicity. The latter change is the accepted mechanism for the widespread occurrence in recent years of seismicity caused by fluid injection (e.g., [95– 98]). On this basis, one might conclude that a decrease in the groundwater pressure within the Newdigate fault cannot be the cause of the local seismicity.

However, it is generally accepted that the mechanics of faults, notably whether they are stable or can slip seismically, are determined by the properties of strong patches – asperities – where the opposing fault surfaces are in frictional contact (e.g., [99]). A fault surface consisting, on a microstructural scale, of a fractal size distribution of asperities with a small proportion of the fault surface in contact, in proportion to the normal stress applied to the fault, can mimic the effect, on a macroscopic scale, of a constant coefficient of friction (e.g., [100, 101]). Brown and Scholz [102] showed that natural rock surfaces follow fractal scaling for surface features of height up to 0.1 m. Laboratory simulations of faulting often include asperities on a microstructural scale (e.g., [103, 104]). Most recently, the view has gained ground that the physics of co-seismic faulting is likewise governed by processes on a microstructural scale (e.g., [105, 106]). For example, McDermott et al. [106] deduced that asperities can be patches of fault with areas of no more than a few square metres, their properties being determined by mineral grains with dimensions of microns. Because these strong patches with fault surfaces in contact occupy only a small proportion of a fault surface, they act as stress concentrations. For example, in the laboratory experiments by Selvadurai and Glaser [104], millimetre-sized asperities with micron-sized heights occupy a very small proportion of the fault area; in one experimental run, a decrease in the mean normal stress across the fault area by 0.3 MPa caused decreases in the normal stress affecting individual asperities by 10 MPa.

**Figure 5(b)** illustrates how a small increase in separation of fault surfaces, Δx, can destabilise a fault through its effect on contact between asperities. Moving from configuration (i) to configuration (ii), two of the three asperities depicted will no longer contribute to fault stability. The third one will experience a significantly reduced normal stress, as a result of the increased separation of the fault surfaces. This will reduce the maximum shear stress that this asperity can sustain, in accordance with Eq. (37), whereas the shear stress that it is required to sustain to keep the fault stable will increase because the other asperities no longer contribute. Overall, it can thus be seen how a small increase in separation of fault surfaces might bring a fault significantly closer to the condition for slip, and might indeed result in coseismic slip.

As others (e.g., [107, 108]) have noted, a general calculation of this 'unclamping' effect for a fault with a general size-distribution of asperities would be very complex; this is thus not attempted here. A simplified calculation is instead presented, assuming that a patch of fault is prevented from slipping by a single asperity. For this patch, of area A, the normal stress and shear stress are σ<sup>N</sup> and τ; the single asperity has cross-sectional area a, uncompressed height b, and Young's modulus E. The effect of σ<sup>N</sup> compresses this model asperity to height d (**Figure 11**). The tip of this asperity will act as a stress concentration, with normal stress and shear stress (A / a) σ<sup>N</sup> and (A / a) τ and coefficient of friction c. The areas of fault where the wall rocks are not in contact initially contain fluid with pressure PO (**Figure 11(a)**), the fault being stable with Φ = -ΔΦ. The reduction in fluid pressure to PO - ΔPO is followed by poroelastic separation of the fault wall rocks by distance Δx, which brings the fault to the condition for slip (Φ = 0). One may thus state versions of Eq. (37) at the tip of the model asperity for these 'before' and 'after' conditions:

$$
\sigma \frac{A}{a} - c \left( E \frac{(b-d)}{b} - P\_O \right) = -\Delta \Phi,\tag{38}
$$

indicating that this poroelastic unclamping effect requires Δx / b to be comparable to ΔΦ / E. Eq. (44) may also be rearranged, recalling from Eq. (40) that

The geomechanical consequences of this model can now be illustrated using the Mohr circle construction (**Figure 12**), for a model fault with c = 0.6. A model stress field is adopted, similar to that deduced by Westaway [11] at 2400 m depth for the

σ<sup>h</sup> = 37.880 MPa, with hydrostatic groundwater pressure P = 23.544 MPa. As **Figure 12(a)** indicates, the state of stress on a vertical model fault (characterised by effective normal stress of magnitude σN' = σN-P = 22.332 MPa and shear stress τ = 12.199 MPa), with normal vector oriented at 57° to the maximum principal stress, plots below the Coulomb failure envelope, indicating stability, the differential stress Δσ being 26.608 MPa. If P within this fault decreases by 10 kPa to 23.534 MPa and this causes, via the poroelastic mechanism described above, in a reduction in the magnitude of σN' by 2 MPa to 20.332 MPa, the resulting state of stress is depicted in **Figure 12(b)**. This condition, with τ still 12.199 MPa, now plots on the Coulomb failure envelope, indicating instability. The fault normal vector is now oriented at 60° to the maximum principal stress, reflecting the slight rotation of the stress field in the vicinity of the fault, caused by the poroelastic reduction in σN, which accompanies an increase in Δσ to 28.260 MPa. The state of stress on the model fault has thus effectively shifted left by 2 MPa on the Mohr-Coulomb plot, moving towards the failure envelope by 1.2 MPa, these adjustments being interrelated via Eq. (45) given c = 0.6. This poroelastic adjustment to the stress field thus involves reducing the magnitude of σN' keeping τ constant, rotating the near-fault stress field and increasing Δσ. It is different from what occurs during 'fracking' of impermeable rocks, where an increase in P causes a Mohr circle of constant diameter (indicating constant Δσ) to shift leftward until part of its circumference touches

Application of the above theory to the Newdigate case study requires determi-

*E* � 3*BR*ð Þ 1 � 2 *ν* (46)

2 1ð Þ � *<sup>ν</sup> :* (47)

(48)

nation of many model parameters, representing properties of key lithologies (Portland Sandstone, calcite 'beef', and Dinantian limestone) and of the Newdigate fault. To facilitate this, it is noted that the elastic moduli that appear in the foregoing

*μ* � 3*BR*

Hydraulic conductivity K and permeability k are also interrelated thus:

*<sup>K</sup>* � *<sup>k</sup>ρ<sup>W</sup> <sup>g</sup> η*

1 � 2 *ν*

Preese Hall case study, with σ<sup>H</sup> = 64.488 MPa, σ<sup>V</sup> = 54.300 MPa, and

*Seismicity at Newdigate, Surrey, during 2018–2019: A Candidate Mechanism…*

ΔΦ ¼ *c*Δ*σN:* (45)

Δσ<sup>N</sup> = E Δx / (2 b), to give

*DOI: http://dx.doi.org/10.5772/intechopen.94923*

the failure envelope.

and

**89**

**4.4 Estimation of model parameters**

analysis are interrelated via standard formulas, such as

and

$$
\pi \frac{A}{a} - c \left( E \frac{(b - d - \Delta \mathbf{x}/2)}{b} - P\_O + \Delta P\_O \right) = \mathbf{0}.\tag{39}
$$

Subtraction of Eq. (38) from Eq. (39) gives

$$
\Delta\Phi = -\varepsilon(\Delta P\_O - \Delta\sigma\_N) = -\varepsilon\left(\Delta P\_O - \frac{E\Delta x}{2b}\right),
\tag{40}
$$

or, substituting Δx from Eq. (27),

$$
\Delta\Phi = c \,\Delta P\_O \left( \frac{Ea}{b \, B\_E} \sqrt{\left(\frac{Dt}{\pi}\right)} - 1 \right). \tag{41}
$$

One may also combine Eqs. (27) and (40) by eliminating ΔPO, to obtain

$$
\Delta \mathfrak{x} = \frac{2\Delta\Phi}{c \left(\frac{E}{b} - \frac{B\_E}{2\gamma a} \sqrt{\left(\frac{\pi}{Dt}\right)}\right)} \cdot \tag{42}
$$

In the limit of high D t / b, at time t ≫ ts, where

$$t\_s = \frac{\pi b^2 B\_E^2}{4\chi^2 a^2 E^2 D},\tag{43}$$

this equation simplifies to

$$
\Delta \mathfrak{x} \approx \frac{2b\,\Delta\Phi}{cE},
\tag{44}
$$

#### **Figure 11.**

*Schematic model asperity used to calculate effects of poroelastic fault unclamping. (a) Initial state, with fluid pressure within the fault PO and the asperity (of uncompressed height b and cross-sectional area a) compressed to height d by the normal stress across the fault. (b) Modified state, with fluid pressure within the fault reduced to PO-ΔPO and the asperity compressed to height d-Δx/2 as a result of the separation Δx between the wall rocks on both sides of the fault caused by their poroelastic compaction.*

*Seismicity at Newdigate, Surrey, during 2018–2019: A Candidate Mechanism… DOI: http://dx.doi.org/10.5772/intechopen.94923*

indicating that this poroelastic unclamping effect requires Δx / b to be comparable to ΔΦ / E. Eq. (44) may also be rearranged, recalling from Eq. (40) that Δσ<sup>N</sup> = E Δx / (2 b), to give

$$
\Delta\Phi = c \,\Delta\sigma\_N.\tag{45}
$$

The geomechanical consequences of this model can now be illustrated using the Mohr circle construction (**Figure 12**), for a model fault with c = 0.6. A model stress field is adopted, similar to that deduced by Westaway [11] at 2400 m depth for the Preese Hall case study, with σ<sup>H</sup> = 64.488 MPa, σ<sup>V</sup> = 54.300 MPa, and σ<sup>h</sup> = 37.880 MPa, with hydrostatic groundwater pressure P = 23.544 MPa. As **Figure 12(a)** indicates, the state of stress on a vertical model fault (characterised by effective normal stress of magnitude σN' = σN-P = 22.332 MPa and shear stress τ = 12.199 MPa), with normal vector oriented at 57° to the maximum principal stress, plots below the Coulomb failure envelope, indicating stability, the differential stress Δσ being 26.608 MPa. If P within this fault decreases by 10 kPa to 23.534 MPa and this causes, via the poroelastic mechanism described above, in a reduction in the magnitude of σN' by 2 MPa to 20.332 MPa, the resulting state of stress is depicted in **Figure 12(b)**. This condition, with τ still 12.199 MPa, now plots on the Coulomb failure envelope, indicating instability. The fault normal vector is now oriented at 60° to the maximum principal stress, reflecting the slight rotation of the stress field in the vicinity of the fault, caused by the poroelastic reduction in σN, which accompanies an increase in Δσ to 28.260 MPa. The state of stress on the model fault has thus effectively shifted left by 2 MPa on the Mohr-Coulomb plot, moving towards the failure envelope by 1.2 MPa, these adjustments being interrelated via Eq. (45) given c = 0.6. This poroelastic adjustment to the stress field thus involves reducing the magnitude of σN' keeping τ constant, rotating the near-fault stress field and increasing Δσ. It is different from what occurs during 'fracking' of impermeable rocks, where an increase in P causes a Mohr circle of constant diameter (indicating constant Δσ) to shift leftward until part of its circumference touches the failure envelope.

#### **4.4 Estimation of model parameters**

Application of the above theory to the Newdigate case study requires determination of many model parameters, representing properties of key lithologies (Portland Sandstone, calcite 'beef', and Dinantian limestone) and of the Newdigate fault. To facilitate this, it is noted that the elastic moduli that appear in the foregoing analysis are interrelated via standard formulas, such as

$$E \equiv \Im B\_{\mathbb{R}} (\mathbb{1} - \mathcal{D}\nu) \tag{46}$$

and

$$
\mu \equiv \mathfrak{Z} B\_R \frac{\mathbf{1} - \mathbf{2}\nu}{\mathbf{2}(\mathbf{1} - \nu)}.\tag{47}
$$

Hydraulic conductivity K and permeability k are also interrelated thus:

$$K \equiv \frac{k \, \rho\_W \text{g}}{\eta} \tag{48}$$

present study region makes such distinctions moot; D will, therefore, be estimated

layer with ϕ 0.25 and k 200 mD. Lee [109] reported BR 13 GPa as typical for dry sandstone with ϕ 0.25. Taking BW 2.15 GPa for water (from [110]), using Eq. (4), BE for Portland Sandstone with pore space occupied by water is 6 GPa. At Horse Hill, Xodus [26] reported a 35 foot or 11 m section in Portland Sandstone with permeability up to 20 mD. The water in contact with the Portland reservoirs at both sites, at 600 m depth, is at 25 °C (cf. [111]), for which η = 0.9 mPa s [112], with ρ<sup>w</sup>

*Seismicity at Newdigate, Surrey, during 2018–2019: A Candidate Mechanism…*

For the Upper Portland Sandstone at Brockham, Angus [27] reported a 3 m thick

η = 11 cP or 11 mPa s [27]. Using Eq. (5), with k 200 mD, BE 6 GPa, and η 0.9 mPa s,

Many workers (e.g., [42, 113, 114]) have investigated the aperture, or width, of bedding-parallel fractures (typically filled with 'beef') in shale, as a guide to its hydraulic properties. In a study spanning several shale provinces, Wang [113] found fractures with width between 15 μm and 87 mm. Many of the wider ones could be seen to have opened by multiple increments, each adding a few tens of microns of width, prior to cementation due to growth of calcite. Permeability and fracture aperture can be interrelated by comparing the Darcy equation for laminar fluid flow, Q = (k A / η) dP/dx, and the Poiseulle equation for laminar flow between parallel boundaries, Q = (D W2 / (12 η)) dP/dx, which is a solution to the more general Navier–Stokes equation for fluid flow (e.g., [115]). Here Q is the volume flow rate, η the viscosity of the fluid, dP/dx the pressure gradient in the direction of flow, k the permeability of the medium, A the cross-sectional area of the flow, and W and D the width of the channel and its length in the direction perpendicular to the flow. Combining these two formulae, equating A to D W, gives k <sup>W</sup><sup>2</sup>

1

This formula gives the permeability equivalent to W = 15 μm as 20 D

[50], might be applicable to calcite 'beef' in the present study area. Identifying a suitable representative value for BR, the bulk modulus for calcite 'beef', is problematic, because of its strongly anisotropic character. 'Beef' is abundant within mudstones of the Neuquen Basin of Argentina (e.g., [116]). Sosa Massaro et al. [117] estimated a representative vertical Young's modulus and Poisson's ratio for this lithology as 15 GPa and 0.25; using Eq. (46) these parameters yield a bulk modulus of 10 GPa. Using Eq. (5), with k 900 mD, KB 10 GPa, and η 0.9 mPa s, D for calcite 'beef' can be estimated - subject to considerable uncertainty - as

). Overall, it is inferred that the 900 mD value, from Carey et al.

. The calcite 'beef', reported in the BGS borehole viewer online docu-

mentation as 'veins of secondary calcite', in the CF1 borehole occurred at depths of 2327–2329 feet or 709.3–709.9 m TVD, thus 626 m TVDSS. Although this interval was cored, the core was not analysed for permeability; however, core between 2300 and 2301 feet TVD yielded k 1650 mD. The top Portland in this borehole is at 1753 feet or 534 m TVDSS, the estimated base of the oil reservoir at 566 m TVDSS (**Figure 7**); this 'beef' layer is thus 100 m below the top Portland. Based on this information this layer is assigned a nominal thickness hB of 1 m for the purpose

Carbonate rocks such as the Dinantian limestone are likely to be complex, being fractured, so water storage within them will be in part by opening of fractures and in part by opening of pore space. Dinantian limestone typically has low matrix porosity (e.g., [118]), its ability to store and transmit groundwater being largely via fractures. Parameter values for Dinantian limestone include BR = 50 GPa and ϕ = 0.04, along with Young's modulus ER = 75 GPa and Poisson's ratio ν = 0.25, from Bell [119]. With this set of values, BE is 27 GPa, and α = 1–27/50 = 0.46 (Eq. (22)). Poisson's ratio ν ranges between 0.19 and 0.31 [119] so 0.25 is adopted, for which

. Under these conditions the oil at Brockham has

, which can be rounded to 1 m2 s

1 .

/12.

using Eq. (5).

(<sup>2</sup> <sup>10</sup><sup>11</sup> <sup>m</sup><sup>2</sup>

10 m<sup>2</sup> <sup>s</sup>

of modelling.

**91**

1

1000 kg m<sup>3</sup> and g 9.81 m s<sup>2</sup>

D for Portland Sandstone is 1.3 m2 <sup>s</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.94923*

#### **Figure 12.**

*Mohr circle diagrams representing a model stress field at 2400 m depth, to indicate the sense of changes to the state of stress envisaged as causing the Newdigate seismicity. (a) for a model stress field with σHO = 64.488 MPa, σVO = 54.300 MPa, and σhO = 37.880 MPa, with σ<sup>H</sup> oriented at 57° to the normal direction to the fault. Hydrostatic groundwater pressure P = 23.544 MPa causes σ<sup>H</sup> ' <sup>O</sup> = 40.944 MPa, σ<sup>V</sup> ' <sup>O</sup> = 30.756 MPa, and σh ' <sup>O</sup> = 14.336 MPa, resulting in σ<sup>L</sup> ' <sup>O</sup> = 28.723 MPa and σM' <sup>O</sup> = 27.706 MPa. The resolved shear stress and normal stress on the model fault, τ = 12.199 and σN'<sup>O</sup> = 22.332 MPa, plot below the coulomb failure line for c = 0.6, with (from Eq. (37)) Φ = -1.2 MPa, indicating that the fault is stable under these conditions. (b) for revised conditions consistent with the set of processes in Fig. 5. Groundwater pressure adjusts by ΔP = -10 kPa, to Pf = 23.534 MPa, and the principal stresses adjust to σ<sup>H</sup> = 65.130 MPa and σ<sup>h</sup> = 36.870 MPa keeping σ<sup>V</sup> = σVO = 54.300 MPa, with σ<sup>H</sup> now oriented at 60° to the fault normal. As a result, σ<sup>H</sup> ' = 41.596 MPa, σV ' = 30.766 MPa, and σ<sup>h</sup> ' = 13.336 MPa, resulting in σ<sup>L</sup> ' = 28.566 MPa and σM' = 27.466 MPa. The resolved shear stress and normal stress on the fault, τ = 12.199 and σN' = 20.332 MPa, now plot on the coulomb failure line for c = 0.6, with Φ = 0, indicating that the fault is now frictionally unstable. The calculated increase in Φ by 1.2 MPa, for Δσ<sup>N</sup> = 2 MPa with c = 0.6, is consistent with Eq. (45).*

where g is the acceleration due to gravity. Different formulas for hydraulic diffusivity D, subject to different boundary conditions, have already been noted. However, the considerable uncertainty in model parameters for lithologies in the *Seismicity at Newdigate, Surrey, during 2018–2019: A Candidate Mechanism… DOI: http://dx.doi.org/10.5772/intechopen.94923*

present study region makes such distinctions moot; D will, therefore, be estimated using Eq. (5).

For the Upper Portland Sandstone at Brockham, Angus [27] reported a 3 m thick layer with ϕ 0.25 and k 200 mD. Lee [109] reported BR 13 GPa as typical for dry sandstone with ϕ 0.25. Taking BW 2.15 GPa for water (from [110]), using Eq. (4), BE for Portland Sandstone with pore space occupied by water is 6 GPa. At Horse Hill, Xodus [26] reported a 35 foot or 11 m section in Portland Sandstone with permeability up to 20 mD. The water in contact with the Portland reservoirs at both sites, at 600 m depth, is at 25 °C (cf. [111]), for which η = 0.9 mPa s [112], with ρ<sup>w</sup> 1000 kg m<sup>3</sup> and g 9.81 m s<sup>2</sup> . Under these conditions the oil at Brockham has η = 11 cP or 11 mPa s [27]. Using Eq. (5), with k 200 mD, BE 6 GPa, and η 0.9 mPa s, D for Portland Sandstone is 1.3 m2 <sup>s</sup> 1 , which can be rounded to 1 m2 s 1 .

Many workers (e.g., [42, 113, 114]) have investigated the aperture, or width, of bedding-parallel fractures (typically filled with 'beef') in shale, as a guide to its hydraulic properties. In a study spanning several shale provinces, Wang [113] found fractures with width between 15 μm and 87 mm. Many of the wider ones could be seen to have opened by multiple increments, each adding a few tens of microns of width, prior to cementation due to growth of calcite. Permeability and fracture aperture can be interrelated by comparing the Darcy equation for laminar fluid flow, Q = (k A / η) dP/dx, and the Poiseulle equation for laminar flow between parallel boundaries, Q = (D W2 / (12 η)) dP/dx, which is a solution to the more general Navier–Stokes equation for fluid flow (e.g., [115]). Here Q is the volume flow rate, η the viscosity of the fluid, dP/dx the pressure gradient in the direction of flow, k the permeability of the medium, A the cross-sectional area of the flow, and W and D the width of the channel and its length in the direction perpendicular to the flow. Combining these two formulae, equating A to D W, gives k <sup>W</sup><sup>2</sup> /12. This formula gives the permeability equivalent to W = 15 μm as 20 D (<sup>2</sup> <sup>10</sup><sup>11</sup> <sup>m</sup><sup>2</sup> ). Overall, it is inferred that the 900 mD value, from Carey et al. [50], might be applicable to calcite 'beef' in the present study area. Identifying a suitable representative value for BR, the bulk modulus for calcite 'beef', is problematic, because of its strongly anisotropic character. 'Beef' is abundant within mudstones of the Neuquen Basin of Argentina (e.g., [116]). Sosa Massaro et al. [117] estimated a representative vertical Young's modulus and Poisson's ratio for this lithology as 15 GPa and 0.25; using Eq. (46) these parameters yield a bulk modulus of 10 GPa. Using Eq. (5), with k 900 mD, KB 10 GPa, and η 0.9 mPa s, D for calcite 'beef' can be estimated - subject to considerable uncertainty - as 10 m<sup>2</sup> <sup>s</sup> 1 . The calcite 'beef', reported in the BGS borehole viewer online documentation as 'veins of secondary calcite', in the CF1 borehole occurred at depths of 2327–2329 feet or 709.3–709.9 m TVD, thus 626 m TVDSS. Although this interval was cored, the core was not analysed for permeability; however, core between 2300 and 2301 feet TVD yielded k 1650 mD. The top Portland in this borehole is at 1753 feet or 534 m TVDSS, the estimated base of the oil reservoir at 566 m TVDSS (**Figure 7**); this 'beef' layer is thus 100 m below the top Portland. Based on this information this layer is assigned a nominal thickness hB of 1 m for the purpose of modelling.

Carbonate rocks such as the Dinantian limestone are likely to be complex, being fractured, so water storage within them will be in part by opening of fractures and in part by opening of pore space. Dinantian limestone typically has low matrix porosity (e.g., [118]), its ability to store and transmit groundwater being largely via fractures. Parameter values for Dinantian limestone include BR = 50 GPa and ϕ = 0.04, along with Young's modulus ER = 75 GPa and Poisson's ratio ν = 0.25, from Bell [119]. With this set of values, BE is 27 GPa, and α = 1–27/50 = 0.46 (Eq. (22)). Poisson's ratio ν ranges between 0.19 and 0.31 [119] so 0.25 is adopted, for which

μ � B (Eq. (47)). No attempt is made here to determine the value of γ; a value of 0.5 will be assumed, consistent with depressurization causing closure of both vertical and horizontal fractures to an equivalent extent. Skempton's coefficient B has been reported as �0.4 for many limestone samples (e.g., [120–122]). Bell [119] noted a range of values of K for laboratory samples of Dinantian limestone, ranging from

*Seismicity at Newdigate, Surrey, during 2018–2019: A Candidate Mechanism…*

values ranging from �10�<sup>6</sup> m s�<sup>1</sup> to �10�<sup>2</sup> m s�<sup>1</sup> in karstified regions, which correspond (using Eq. (48)) to k ≥ 100 mD. Using the latter set of values, Eq. (2)

determined an upper bound to D for Dinantian limestone in the Peak District of central England by modelling the hydrology of Meerbrook Sough, a disused mine drainage adit that drains a � 40 km<sup>2</sup> area. His analysis reported an upper bound of

observed magnitude of seasonal fluctuations in flow, favouring a higher value of D.

For an ensemble of faults in different lithologies, Brodsky et al. [107] deduced

*<sup>b</sup>* <sup>¼</sup> *bOL<sup>ζ</sup>*

where <sup>ζ</sup> <sup>=</sup> �0.6 and bO <sup>=</sup> �10�<sup>3</sup> <sup>m</sup>0.4. For example, with L = 100 m, Eq. (49)

[127], where MW is moment magnitude. Next, the radius a of the equivalent circular seismic source was determined from Mo, assuming a nominal coseismic

> *Mo* <sup>¼</sup> <sup>16</sup>*δσ* ð Þ <sup>1</sup> � *<sup>ν</sup> <sup>a</sup>*<sup>3</sup> 3 2ð Þ � *ν*

(e.g., [128]). The corresponding diameter 2a is equated to fault length L to

*<sup>L</sup>* <sup>¼</sup> <sup>2</sup>*<sup>a</sup>* <sup>¼</sup> 3 2ð Þ � *<sup>ν</sup> Mo*

*for the pressure variations depicted in (b). Calculations use Eq. (27) with ΔPO 8.7 kPa (from (a)), and BE*

*which arises after 0.25 hours or 15 minutes. (d) Graph of ΔΦ for the patch of the Newdigate fault that slipped in the 14 February 2019 earthquake versus time for the variations in Δx depicted in (c). Calculations use Eq. (44), with the same parameters as for (c) plus b 18.9 mm, c 0.6, and E 75 GPa for the model asperity on the seismogenic patch of fault. Dashed line indicates ΔΦ = 6 MPa, which arises after 0.25 hours or 15 minutes.*

2 *δσ* ð Þ 1 � *ν* <sup>1</sup>*=*<sup>3</sup>

To investigate the area of the patch of fault that slipped in each earthquake, and to thus quantify the associated asperity height, seismic moment MO was first deter-

limestone, subject to considerable uncertainty. For comparison, Hornbach et al. [125] deduced that a poroelastic pressure pulse resulting from large-scale injection of waste water propagated for up to �40 km through the Ellenburger Formation, a karstified limestone of Ordovician age, in �6 years, resulting in earthquakes in the

vicinity of Dallas, Texas. Using Eq. (30) gives an upper bound to D for the

�1

�1

that the typical asperity height b in length L of a fault scales as

. Lewis et al. [123] reported much higher

. However, this analysis did not reproduce the

�<sup>1</sup> is appropriate for karstified Dinantian

, in reasonable agreement. Zhang et al. [126]

, (49)

(51)

*:* (52)

�<sup>1</sup> for this karstified Ordovician limestone.

log <sup>10</sup>ð Þ¼ *Mo=N m* 9*:*05 þ 1*:*5*MW* (50)

*, for Dinantian limestone. Dashed line indicates Δx = 0.00504 mm,*

. For comparison, Shepley [124]

�1

0.07 � <sup>10</sup>�<sup>9</sup> m s�<sup>1</sup> to 0.3 � <sup>10</sup>�<sup>9</sup> m s�<sup>1</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.94923*

50,000 m<sup>2</sup> day�<sup>1</sup> or � 0.6 m<sup>2</sup> <sup>s</sup>

Overall, it is inferred that that D � 1 m<sup>2</sup> <sup>s</sup>

Ellenburger Formation of �1.4 m2 <sup>s</sup>

reported a nominal value of 1 m<sup>2</sup> s

mined using the standard formula

characterise asperity height. One thus obtains

�*1*

*27 GPa, α = 0.46, and D again 1 m<sup>2</sup> s*

gives b � 16 mm.

stress drop δσ:

**93**

gives values for D ranging upward from �3 m<sup>2</sup> <sup>s</sup>

#### **Figure 13.**

*Modelling of hydraulic consequences of the phase of production from well HH1 starting in February 2019. (a) Graphs of -ΔP versus radial distance r within the layer of calcite 'beef' at 600 m depth, which is inferred to connect the Horse Hill oil reservoir with the Newdigate fault, at different times t after the start of production. Calculations use Eq. (8) with Q = 0.4 l s<sup>1</sup> , η 0.9 mPa s, hB 1 m, and DB 10 m<sup>2</sup> s 1 . Dashed lines indicate, for r = 3 km, ΔP = -6.5 kPa after t = 2.5 days and ΔP = -8.7 kPa after t = 3 days. (b) Graphs of -δP/ΔPO versus distance x perpendicular to the Newdigate fault, at different times after the pressure variation in (a) reached this fault. Calculations use Eq. (20) with DD 1 m<sup>2</sup> s 1 , for Dinantian limestone. (c) Graph of Δx versus time*

*Seismicity at Newdigate, Surrey, during 2018–2019: A Candidate Mechanism… DOI: http://dx.doi.org/10.5772/intechopen.94923*

μ � B (Eq. (47)). No attempt is made here to determine the value of γ; a value of 0.5 will be assumed, consistent with depressurization causing closure of both vertical and horizontal fractures to an equivalent extent. Skempton's coefficient B has been reported as �0.4 for many limestone samples (e.g., [120–122]). Bell [119] noted a range of values of K for laboratory samples of Dinantian limestone, ranging from 0.07 � <sup>10</sup>�<sup>9</sup> m s�<sup>1</sup> to 0.3 � <sup>10</sup>�<sup>9</sup> m s�<sup>1</sup> . Lewis et al. [123] reported much higher values ranging from �10�<sup>6</sup> m s�<sup>1</sup> to �10�<sup>2</sup> m s�<sup>1</sup> in karstified regions, which correspond (using Eq. (48)) to k ≥ 100 mD. Using the latter set of values, Eq. (2) gives values for D ranging upward from �3 m<sup>2</sup> <sup>s</sup> �1 . For comparison, Shepley [124] determined an upper bound to D for Dinantian limestone in the Peak District of central England by modelling the hydrology of Meerbrook Sough, a disused mine drainage adit that drains a � 40 km<sup>2</sup> area. His analysis reported an upper bound of 50,000 m<sup>2</sup> day�<sup>1</sup> or � 0.6 m<sup>2</sup> <sup>s</sup> �1 . However, this analysis did not reproduce the observed magnitude of seasonal fluctuations in flow, favouring a higher value of D. Overall, it is inferred that that D � 1 m<sup>2</sup> <sup>s</sup> �<sup>1</sup> is appropriate for karstified Dinantian limestone, subject to considerable uncertainty. For comparison, Hornbach et al. [125] deduced that a poroelastic pressure pulse resulting from large-scale injection of waste water propagated for up to �40 km through the Ellenburger Formation, a karstified limestone of Ordovician age, in �6 years, resulting in earthquakes in the vicinity of Dallas, Texas. Using Eq. (30) gives an upper bound to D for the Ellenburger Formation of �1.4 m2 <sup>s</sup> �1 , in reasonable agreement. Zhang et al. [126] reported a nominal value of 1 m<sup>2</sup> s �<sup>1</sup> for this karstified Ordovician limestone.

For an ensemble of faults in different lithologies, Brodsky et al. [107] deduced that the typical asperity height b in length L of a fault scales as

$$b = b\_{\mathcal{O}} L^{\zeta},\tag{49}$$

where <sup>ζ</sup> <sup>=</sup> �0.6 and bO <sup>=</sup> �10�<sup>3</sup> <sup>m</sup>0.4. For example, with L = 100 m, Eq. (49) gives b � 16 mm.

To investigate the area of the patch of fault that slipped in each earthquake, and to thus quantify the associated asperity height, seismic moment MO was first determined using the standard formula

$$
\log\_{10}(M\_o/Nm) = 9.05 + 1.5M\_W \tag{50}
$$

[127], where MW is moment magnitude. Next, the radius a of the equivalent circular seismic source was determined from Mo, assuming a nominal coseismic stress drop δσ:

$$M\_{\vartheta} = \frac{16\delta\sigma(1-\nu)a^3}{3(2-\nu)}\tag{51}$$

(e.g., [128]). The corresponding diameter 2a is equated to fault length L to characterise asperity height. One thus obtains

$$L = 2\,\text{a} = \left(\frac{3(2-\nu)M\_o}{2\,\delta\sigma(1-\nu)}\right)^{1/3}.\tag{52}$$

*for the pressure variations depicted in (b). Calculations use Eq. (27) with ΔPO 8.7 kPa (from (a)), and BE 27 GPa, α = 0.46, and D again 1 m<sup>2</sup> s* �*1 , for Dinantian limestone. Dashed line indicates Δx = 0.00504 mm, which arises after 0.25 hours or 15 minutes. (d) Graph of ΔΦ for the patch of the Newdigate fault that slipped in the 14 February 2019 earthquake versus time for the variations in Δx depicted in (c). Calculations use Eq. (44), with the same parameters as for (c) plus b 18.9 mm, c 0.6, and E 75 GPa for the model asperity on the seismogenic patch of fault. Dashed line indicates ΔΦ = 6 MPa, which arises after 0.25 hours or 15 minutes.*
