Modeling the Cooling of the Whin Sill, North England CHARLIE KENZIE

Department of Earth Science, University of Durham 2013  

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1. INTRODUCTION

1.1 Geological setting The Whin Sill complex is a suite of igneous basaltic intrusions emplaced during the late Carboniferous in northern England (Johnson & Dunham 2001), which stretches on land around 120km in a north-south direction and around 80km in an east-west direction. The Whin Sill, a horizontal body intruded between stratigraphical successions, is fed by four major vertical intrusions or feeder dykes (Anderson 1951 & Goulty et al 2000), (Fig.1). The intrusion of the basaltic magma, was at a sufficient temperature to alter the country rocks by contact metamorphism (Robinson 1972) generally resulting in spotting of mudstones, and at higher temperatures, re-crystallization of pure limestone to marble and of impure limestone to metamorphic rocks containing garnet, andausite, cordierite and wollastonite (Johson & Dunham 2001). 1.2 Setting up the model The intrusion of a hot layer into cooler country rock, and the temperature distribution through the layers over time, can be modeled numerically using the forward Euler time-stepping method, found by discretizing the one-dimensional heat diffusion equation: Heat diffusion equation

Discretizing the equation yields

The time step formula was implemented into a MatLab code and can be seen in the appendix. We start our model assuming that the magma was intruded with an initial temperature of approximately 1000°C and is preceded by a further latent heat release equivalent to 300°C due to thermal cooling [1]. Therefore we model the sill as having an “effective” intrusion temperature of 1300°C. The sill, which is 70m thick, was intruded approximately 1.5km below the surface (Robinson 1972), so we choose to set up the heat diffusion model in a “box” with a depth range between 1000 and 2000m (Fig.2). We estimate the temperatures of the country rock at the top and bottom of the “box” are 30°C and 60°C respectively [2] and add these to the initial boundary conditions of the model appropriately (see appendix code lines 1-19). 1.3 Discretization and time stepping Since we are modeling a sill with thickness of 70m, we chose a discretization step of 10m, which allows sensible precision without overloading the program. Additionally, since time integration methods are only conditionally stable, it is important to choose a time step that is no larger than the critical time step, otherwise the numerical solution may become unstable (Askes et al 2011). Therefore in this case, we used a von Neumann stability analysis [3] to calculate the stability criterion (equations 3 – 6) and set the time step to a fraction of this criterion (appendix code lines 22 & 23). This stops the error from increasing as computations in the model are continued, in other words, it ensures a stable system.

ABSTRACT Explicit numerical integration is an effective way of modeling ordinary differential equations. We use a forward Euler time-stepping method to effectively model heat diffusion through the top 2km of the crust post intrusion of the Whin Sill around 300 Ma. The model proves effective at estimating the 500°C isotherm, which coincides with estimates of the metamorphic aureole. Additionally, the model shows that all the heat from the sill is dissipated between 45,000 and 50,000 years after emplacement and that there is no temperature flow through the surface during this time. Our model proves accurate over short time periods, but it is hard to say whether it is as accurate over longer time scales, i.e. thousands of years. Geological evidence and continued benchmarking with other studies is needed to be sure of the accuracy of our model.

[1] Latent heat release is not explicitly dealt with, instead it is just added to the intrusion temperature. For further reading on latent heat release the reader is reffered to Turcotte and Schubert (2002). [2] Temperatures in the country rock were estimated assuming a constant, homogenous thermal gradient of 3°C per 100m. [3] Our analysis is actually a simplified version of the complete Fourier analysis, and such calculation is out of the limits of this paper, however the reader is referred to Isaacson and Keller (1966) for full numerical analysis.

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Fig.2 Sketch of Whin sill intruded into impure limestone country rock. Temperatures in the rock at depth are calculated using the geothermal gradient. Dashed line shows the boundary “box” for the model.

Forward euler time step:

 

Error propagates as:

 

,which yields

 ,  where  

and results in the stability criterion given by

 

2. MODEL RESULTS

2.1 Running and testing the model The completed model showing heat diffusion through the crust post intrusion is shown in Fig.3i pp 5, and the accompanying MatLab code is displayed in the appendix. The sill shows a rapid transfer of energy into the surrounding rock in the first few decades after intrusion, and then starts to slow when time post intrusion progresses into the order of hundreds of years. It is important to thoroughly quantitatively test the model, so my model was benchmarked against an independent code (Greggs 2013). Exactly the same parameters were chosen for both models and they can be compared in Fig.2 (i) and (ii).

2.3 Changing the time step The time step (dt) is given by a fraction of the stability criterion (dtmax) see (appendix code line 23) and can be altered by changing the fraction as required. Fig.3i is the model computed using a time step with a value one tenth of that of the stability criterion. If we increase the value of the time step so that it is a significant fraction of the stability criterion, the model does not show any change, and remains stable. However, if the time step is increased to a value equal to that of the stability criterion, then the model starts to show signs of instability (Fig.3iii) and the thermal diffusion isotherm is not as well defined and becomes stepped. If the time step is increased to a value greater than that of the stability criterion, the model becomes unstable and is unable to complete numerical time stepping (Fig.3iv).

A lower time step increases the time for the model to run, and since a higher value does not seem to inhibit the model in a significant way, we chose a time step with a value of half the stability criterion for further investigation of the model. 2.3 Depth of 500°C isotherm The 500°C isotherm is of geological importance since it is the temperature at which impure limestone metamorphoses to wollastonite (Robinson 1972). To model the depth of the isotherm over time we calculated between which nodal points the 500°C isotherm is crossed and then linearly interpolated from the nodal point temperatures to find the depth of the isotherm

Fig.1 Overview map of the Whin Sill complex and it’s associated dykes. Outcrops are shown in black. Taken from Liss et al (2004).

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(see appendix). The isotherm depth through time is shown in Fig. 4 below. The point at which the isotherm is furthest away from the sill, or in other words the maximum thickness of a rim of country rock experiencing temperatures above 500°C, is shown by the model to be 28m and this is reached at a time

, or approximately 35 years, after intrusion. 2.4 Extending the model to the surface To investigate whether heat released from the intruded sill reached the surface we extend our model to the surface (Fig.5). If the model is run for 45,000 years (Fig.6), the heat energy released from the sill eventually reaches the geotherm, and its energy exhausted. Over this time, none of the thermal energy reaches the surface, and the model does not record any surface temperature change.

3. DISCUSSION

3.1 Numerical accuracy As discussed previously the time step is chosen to be a fraction of the critical time step in order to stop errors compiling as the model continues to carry out computations. This helps to improve the accuracy of the numerical analysis. Additionally, when benchmarked against another model (Greggs 2013), the two models showed to be almost identical (Fig.2i & Fig.2ii), giving a good indication that our model is numerically accurate. 3.2 Width of metamorphic aureole Investigations into the depth over time of the 500°C isotherm reveal that the model provides an accurate prediction of the metamorphic aureole described by Robinson (1972). Robinson states that a maximum temperature curve after 25 years would give a contact temperature in the region of 720°C, with a decrease away from the contact at approximately 17°Cm-1. Thus, the 500°C isotherm can be calculated to be 30m away from the sill, which coincides with the results from our model (Fig.4) that showed a maximum width of 28m. Additionally, our model predicts the maximum width of the

500°C isotherm to have occurred approximately 35 years after intrusion, which again corresponds with Robinson’s (1972) findings. This highlights that our model results are in agreement with other authors, and increases our confidence that the model is numerically accurate. 3.3 Heat flow at the surface We can run the model until all the heat released from the sill is exhausted, and to the point that heat distribution in the crust returns to the normal geothermal gradient. Although running the model for long periods of time is less accurate (this is discussed more thoroughly in section 3.4) it is still unlikely that any surface heat flow occurred because our model shows that the intrusion temperature of the sill would have to be significantly larger in order to cause heat flow at the surface. Additionally, other authors such as Sayer (2013) ran a similar model for over 30,000 years and saw no evidence of surface heat flow. 3.4 Assumptions of the model and error Hydrothermal convection In areas of the crust that experience high temperatures, it is not uncommon for convective heat transport to dominate over conduction (Loosveld and Etheridge 1990). In such cases steam production can cause a massive temperature depression in comparison to pure heat transfer by conduction (Berner & Georg, 2010). Hydrothermal processes are not accounted for in our model but it seems unlikely that this caused any errors when the model was run over a short time period since our model proves accurate over the first few decades due to cross-referencing with other authors (Robinson 1972) [4], (see section 3.2). Additionally, in cases where a sill is intruded into rocks with a low permeability, hydrothermal convection can be reasonably ignored (Wang et al) and since impure limestone is commonly argillaceous, it is less likely that hydrothermal activity occurred, and it seems reasonable to ignore the effects of hydrothermal convection in our model.

[4]Yes, we cross-referenced our results against a model from Robinson (1972) and found that our models agreed. However, it is possible that his model also lacked any consideration of hydrothermal activity. Unfortunately, a discussion of Robinson’s model is out of the limits of this paper.

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Homogeneous country rock and sill Although we see it reasonable to ignore hydrothermal processes in our model when run over short time periods, we have no evidence that suggests that hydrothermal convection remained inactivate after longer periods of time. Sudden activation seems unlikely if we assume a homogenous and continuous layer of country rock however; such an ideal homogenous layer is unlikely to exist in reality where stratigraphy changes both laterally and vertically. Such an assumption is another limitation of our model and highlights that our model becomes increasingly inaccurate as it is run over longer time periods. Additionally, we assume that the sill is intruded as a concordant tabular horizontal body, although in reality, the sill is seldom concordant over wide areas and usually falls in the stratigraphical succession in gentle transgressions and abrupt jumps to new levels (Johnson and Dunham 2001). This assumption will have caused error in our model from the start and will have carried through the computation. Again, the longer the model is run for the more inaccurate it will become. 3.5 Conclusion and need for further study Uncertainties such as those highlighted above indicate the need for external constraints, such as geological data, in order to be sure of the accuracy of our model. The model provides a relatively crude estimate of heat diffusion through the crust post intrusion, because the model assumes a homogenous crust and a perfectly horizontal sill. Additionally, the model assumes that heat is transferred purely by conduction, which is not realistic especially if the crust contains more stratigraphy than just impure limestone. As there is no way of testing the models accuracy over longer time periods, it would be useful to try to benchmark this model against one that effectively models diffusion over hundreds of years. Additionally, it would have been good to test the numerical stability of the model against an analytical solution, but such a solution is out of the limits of this paper. More generally the model may have been

more successful if it didn’t try to model the whole of the Whin sill but rather a specific section, where more informed geological constraints could be implemented into the model.

4. REFERENCES ANDERSON, E. M. (1951). The Dynamics of Dyke

Formation. In E. M. Anderson, The dynamics of faulting and dyke formation with applications to Britain (pp. 40-44). Edinburgh: Oliver and Boyd.

ASKES, H., Nguyen, D. C., & Tyas, A. (2011, June). Increasing the critical time step: micro-inertia, inertia penalties and mass scaling. Computational Mechanics, 47(6), 657-667.

BERNER, U., & Georg, D. (2010). Influence of sill intrusions on the hydrology and thermal maturity of sediments - Modelling heat flow and organic geochemical alterations. EGU General Assembly (p. 13089). Vienna, Austria: Smithsonian Astrophysical Observatory, NASA.

GREGGS, E. (2013) Numerical modelling of the cooling of the Whin sill Department of Earth Sciences, University of Durham

GOULTY, N. R., Peirce, C., Flatman, T. D., Home, M., & Richardson, J. H. (2000). Magnetic survey of the Holy Island Dyke on Holy Island, Northumberland. Proceedings of the Yorkshire Geological Society, 53(2), 111-118.

ISAACSON, E., & Keller, H. B. (1966). The von Neumann Stability Test. In E. Isaacson, & H. B. Keller, Analysis of Numberical Methods (pp. 523-531). New York: John Wiley & Sons.

JOHNSON, G. A., & Dunham, K. C. (2001). Emplacement of the Great Whin Dolerite Complex and the Little Whin Sill in relation to the structure of northern England . Proceedings of the Yorkshire Geological Society, 53(3), 177-186.

LOOSVELD, R. J. H. & Etheridge, M. A. (1990) A model for low-pressure facies metamorphiscm during crustal thickening. Journal of Metamorphic Geology, 8(3) 257-267.

ROBINSON, D. (1972). Metamorphic Rocks. In G. A. Johnson, & G. Hickling, Geology of Durham County (pp. 119 - 123). Durham: Natural History Society of Northumberland.

SAYER, M. J. (2013). Numerical modelling of the cooling of the Whin sill. Department of Earth Sciences, University of Durham.

TURCOTTE, D. L., & Schubert, G. (2002). Geodynamics. Geodynamics (2nd Edition).

WANG, S., Zhao, M., & Qi, T. (2012). Heat-Transfer-Model Analysis of the Thermal Effect of Intrusive Sills on Organic-Rich Host Rocks in Sedimentary Basins. In I. Ahmad, Earth Sciences (pp. 91-98). InTech Sciences.

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Fig. 6 Model extended to the surface when run for 45,000 years, thermal energy released from sill reaches geotherm and is exhausted.

Fig. 4: Graph showing the depth of the 500°C isotherm through time. The sill has a depth of 1500m, and the dotted lines show the point in time at which the isotherm is furthest away from the sill, in other words. The model shows a maximum width of 28m at a time of around 35 years post intrusion.

Fig. 5 Same model of heat diffusion through the crust, after 319 years post intrusion of a sill at 1300°C, but this time the model is extended to the surface.

Fig. 3(i): Graph of the model when run for 317 years showing the temperature distribution through time after the intrusion of the Whin sill. The time step for the model is one tenth of that of the stability criterion.(ii) Model provided by Greggs (2013) for the purpose of benchmarking the models. (iii) The model is run again until 317 years, but the time step is the same value as the stability criterion. The model shows signs of instability. (iv) Attempted run of the model when time step is just over the value of the stability criterion. Instability reached – the model fails to continue to numerically compute time steps.

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APPENDIX function Whin_Sill 1   2  %Writing initial parameters, constants and defining boundary conditions 3  h=2000; % Height of boundary box is 2km 4  z1=0000; % Depth to top of boundary box 5  z2=2000; % Depth to bottom of boundary box 6   7  kappa = 1e-6; % Thermal Diffusivity 8  Tsill=1300; % Sill temperature in ^oC 9  Ttop = 0; % Temperature at surface in ^oC 10  Tbottom=60; % Temperature at base of box 11  Tisotherm=500 %Temperature of 500^o isotherm 12  dz = 10; % Discretization step in meters 13  nz=h/dz+1; 14   15  z=z1:dz:z2; % Array for the finite difference mesh 16  t=0; % Set time to zero 17   18  %---------------------------------------------------------------------------- 19   20  % Time equations 21  dtmax=(dz^2)./(2.*kappa);% Stability criterion 22  dt=0.5*dtmax; % Time step as a fraction of the stability criterion 23  tmax=1e10; 24   25  % Number of time steps to reach maximum temperature (tmax) in years 26  nt=ceil(tmax/dt)+1; 27  time=zeros(1,nt); 28   29  % Array of NaN for 500 degree isotherm 30  Itherm=NaN(1,nt); 31   32  % Initial and boundary conditions: 33  % Initial temperature 34  Told=z.*3./100; 35  Told(1)=Ttop; 36  % Except at sill, where temperature=1300 37  Told(150:157)=Tsill; 38  zMP=(z(1,1:end-1)+z(1,2:end))/2; 39   40  % Start time stepping 41  for it=1:nt 42  %Update time in seconds 43  t=t+dt; 44  time(it)=t; 45  tyears=floor(t./(60*60*24*365)); 46   47  % Initial boundary conditions: 48  % Initial temperature at depth z 49  Tinitial=z.*3./100; 50  Tinitial(1)=Ttop; 51  % Except at sill, where the temperature = 1300 52  Tinitial(150:157)=Tsill; 53   54  % Numerical solution 55  Tnew = oneDdiff(Told, z, zMP, kappa, dt); 56   57  % 500 degree isotherm 58  Ttherm(it)= MYinterp (Tnew,z,Tisotherm); 59   60   61  

  7  % Plot solution: 62  figure(1), clf 63  plot (Tnew,-z,'b',Tinitial,-z,'k') 64  xlabel('T [^oC]') 65  ylabel('z [m]') 66  title([' T after ',num2str(tyears),'Years']) 67  axis([0,1400,-2000,-0000]) 68   69  % Prepare for next time step: 70  Told = Tnew; 71   72  end 73   74  tyrs = floor(t./(60*60*24*365)); % In years 75   76  % Plot 500 degree isotherm 77   figure(2), clf 78   plot(time,Ttherm) 79   xlabel('Time (Seconds)') 80   ylabel('z [m]') 81   title(['Depth of 500 degree isotherm']) 82   axis([0,4e9,-1450,-1550]) 83   84  %---------------------------------------------------------------------------- 85   86  %Subfunction to solve 1 dimensional diffusion 87   88  function fout = oneDdiff (fin,z,zMP,kappa,dt) 89  % Performs 1 diffusion time step on finite difference array (fin) of size nz, 90  discretization dz, using timestep dt, and diffusion coefficent kappa (line 9). 91  Additionally, fixed essential b.c. used. 92  dfdz=diff(fin,1,2)./diff(z,1,2); 93  d2fdz2=diff(dfdz,1,2)./diff(zMP,1,2); 94  dfdt=kappa.*d2fdz2; 95  %Applying fixed f boundaries 96  dfdt=[0 dfdt 0]; 97  fout=fin+dfdt*dt; 98   99  %-------------------------------------------------------------------------- 100   101  % Sub function to find depth of 500^oC isotherm 102  function finterp = MYinterp(f,x,f0) 103  % Find interval where f-f0 changes sign: 104  larger=f>f0; 105  if sum(larger) == 0; 106  finterp = NaN; 107  else ix=find(diff(larger)==1)+1; 108  % Interpolate to find where exactly on this interval f=f0: 109  finterp=-(x(ix)-(f(ix)-f0)/(f(ix)-f(ix-1))*(x(ix)-x(ix-1))); 110  end 111