INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, VOL. I, 75-86 (1983)

FINITE ELEMENT METHODS FOR GEOTHERMAL RESERVOIR SIMULATION

GEORGE ZYVOLOSKI

Los Alamos National Laboratory, Los Alamos, New Mexico 87545, U.S.A.

SUMMARY

Two finite element algorithms suitable for long term simulation of geothermal reservoirs are presented. Both methods use a diagonal mass matrix and a Newton iteration scheme. The first scheme solves the 2N unsymmetric algebraic equations resulting from the finite element discretization of the equations governing the flow of heat and mass in porous media by using a banded equation solver. The second method, suitable for problems in which the transmissibility terms are small compared to the accumulation terms, reduces the set of N equations for the Newton corrections to a symmetric system. Comparison with finite difference schemes indicates that the proposed algorithms are competitive with existing methods.

INTRODUCTION

In recent years the interest in geothermal energy has grown enormously. Parallel with this growth has been the need accurately to model geothermal reservoirs. Garg et aLY1 Pritchett et af.,' Faust and M e r ~ e r , ~ Thomas and P i e r ~ o n , ~ Coats' and Zyvoloski et aLY6 have presented finite difference based simulations which are capable of modelling one- and two-phase geother- mal reservoirs. Pruess et af.' have developed an integrated finite difference based code capable of simulating one- or two-phase flow. Most finite element schemes have been limited to compressed liquid geothermal reservoirs because of difficulties involved in modelling phase changes and two-phase flow. Mercer et a1.' discuss some of these difficulties.

It is the purpose of this paper to present some finite element schemes which are competitive with finite difference schemes while preserving the inherent flexibility of the finite element method.

MATHEMATICAL FORMULATION

Detailed derivations of the governing equations for two-phase geothermal reservoirs have been presented by several investigators (Mercer et aLY8 Mercer and Faust' and Brownell et af." for example) and therefore only a brief development will be presented here.

Conservation of mass is expressed by the equation

a A m -+ at v * f m +qm = 0

where the mass per unit volume A, is given by

0363-9061/83/010075-12$01.20 @ U.S. Government

Received 6 February 1981 Revised 17 July 1981

76 G. ZYVOLOSKI

and the mass flux fm is given by

fm = pv Vv + PI VI (3) Here q5 is the porosity of the matrix, S, and S1 are saturations, pv and p1 are densities, V, and Vl are flow rates per unit volume with the subscripts v and 1 indicating quantities for the vapour phase or liquid phase, respectively. Source or sinks (that is, bores or re-injection wells) are represented by the term qm.

Conservation of energy is expressed by the equation

aAe -+v * f e + q e = O at (4)

where the energy per unit volume A, is given by

Ae = (1 - 4 > P r u r + 4 ( S v ~ v u v + SIP~U~) ( 5 )

fe = pvhvVv+plh~ VI-KVT (6)

and the energy flux fe is given by

Here the subscript r refers to the rock matrix, ur, u, and uI are specific internal energies, h, and hl are specific enthalpies, K is an effective thermal conductivity, T is the temperature and qe is the energy contributed from sources and sinks.

To complete the governing equations it is assumed that Darcys Law applies to the movement of each phase and the capillary pressure is r~egligible:~

kR I v1= -- (Vp -p1g)

PI

Here k is the permeability, R , and R1 are the relative permeabilities, m, and pI are viscosities, p is the pressure and g represents the acceleration due to gravity. (For simplicity, the equations are shown for an isotropic medium, though this restriction does not exist in the computer code.) The relative permeabilities used here are a version of a form suggested by Corey and adopted by Faust and M e r ~ e r . ~

R , = (1 - STz)(l -Sf) RI = Sf

where Sf = (Sl--Slr--Svr)/(l -Sir-Svr) and Slr, S,, are the residual saturations at which the liquid phases respectively become immobile. Values of S1, = 0.3 and S,, = 0.05 have been used by other authors and are used in this work.

Using Darcys Law the basic conservation equations (1) and (4) can be rewritten

and

GEOTHERMAL RESERVOIR SIMULATION 77

Here the transmissibilities D, and D, are given by

D m = Dm1 + D m v and

The functional dependence of the thermodynamic and transport quantities i.n equations (1 1) and (12) on p and h is not given here. The formulae given by Mercer and Faust' were used and the reader can refer to that paper for details.

The mixture enthalpy h is related to the other variables by the formula

The source and sink terms in equations (1) and (4) arise from bores, and assuming that the total mass withdrawal q, for each bore is specified, then the energy withdrawal qe is determined as follows:

qe = qvhv +qIhl (17)

where qv = aq,, ql = (1 -a)qm and

(+ = 1 / ( 1 + L) PIRIP PvRvW1

The form of equation (18) shows the importance of the ratio of relative permeabilities RI/Rv in controlling the discharge composition. The limited field data available (see GrantI2) show that there may be significant differences between the relative permeability formulae used here in equations (4) and (10) (and by other authors) and field relative permeabilities.

FORMULATION OF FINITE ELEMENT EQUATIONS

The finite element equations are generated using the Galerkin formulations. For a detailed presentation of the finite element method the reader is referred to Zienkie~icz. '~ In this method the flow domain, s1, is assumed divided into finite elements and the variables P and h, along with the accumulation terms A, and A, are interpolated on each element

These approximations are introduced in equations (1) and (4) and the Galerkin formulation (described by Zienkiewicz and Parekh14) is applied. The following equations are derived

78

where

G. ZYVOLOSKI

Equations (21) and (22) need some comment. D: and D p indicate an upstream weighted transmissibility (after Dalen). This technique has worked well in the low order elements (3-node triangle, 4-node quadrilateral) where the schemes resemble difference techniques. Research is ongoing to determine the applicability of using upstream weighted transmissibilities when using 8-node quadratic elements. The upstream weighting is determined by comparing the velocities at the nodes i and j. The matrix [C] is diagonalized by choosing integration points at the element vertices. This also corresponds to a finite difference approach. As Dalen15 points out, more general finite difference techniques may be formulated, but only at the expense of considerable sophistication. The finite element technique generates high order and irregular meshes routinely. In the usual finite element practice the matrices [TI and [C] are built by an assembly of corresponding element matrices. In the computer code corresponding to the development outlined above, the equations are developed by node. This allows for easy determination of the upstream node. It further makes for easy implementation of solution algorithms that were developed for finite difference schemes.

SOLUTION ALGORITHMS

I. In this algorithm the full Newton-Raphson method is implemented. The system of finite element equations may be written as:

: + 1

= -

K

The method solves this unsymmetric band system of equations using the SGBFA subroutine package of the LINPACK library developed at the Argonne National Laboratory.

GEOTHERMAL RESERVOIR SIMULATION 79

11. This algorithm is an extension of the one presented by Zyvoloski et aL6 The equations solved at each iteration can be represented in block form as follows:

Here matrices such as [aF,/ap] contain the derivatives of Fmi, with respect to all the variables Pii. However, the contribution to the Jacobian matrix from the conduction terms is neglected and therefore the effect on the solution process of inclusion of the conduction terms merely corresponds to additional terms added to the energy source term 4e.

To obtain a computationally efficient scheme for solution the derivatives of the trans- missibilities with respect to pressure and enthalpy are neglected. With this important modification the matrices [aF,/ah] and [aFe/ah] become diagonal while [aFm/ap] and [aFe/ap] are tridiagonal. This simplification means that the equations for the correction {ah) and {ap} can be separated, giving the equation

and

As a consequence of the diagonal nature of [8F,/ah] and [aFJah], equation (27) represents a symmetric banded system of equations which is solved using an active column profile solver. Once {Ap} is known {Ah} is obtained by direct evaluation using equation (26). Although there is an approximation made in the solution process by neglecting the derivatives of the trans- missibilities with respect to p and h it is important to note that this involves no approximation in the final solution of equation (25). The coefficient matrix and the right-hand side of equation (25) are updated at each iteration using the latest information to calculate transmissibilities and other quantities.

There is some additional flexibility in the above algorithms. The programs are set up such that the algorithms presented above may be applied line-by-line or block-by-block. These variants are not discussed here. They are the subject of ongoing research and results based on these algorithms will be presented at a later date.

EXAMPLE CALCULATIONS

The first example is a model of a two-phase highly permeable geothermal reservoir originally proposed by Toronyi and Farouq Ali16 and solved by a number of authors (Mercer and Faust, Thomas and Pierson4). The model reservoir is shown in Figure 1. The input for the problem is presented in Table 1. As with Thomas and Pierson, time steps of 10 days with an initial time step of 8.3 days were used. The final state corresponds to 19 percent of the original water mass removed. Figure 1 shows the results obtained in this study using quadrilateral and triangular elements and compared with the results obtained by Thomas and P i e r ~ o n . ~ The trangular grid was obtained by bisecting the quadrilateral elements in the top left direction. These results were obtained with algorithm 1 as described in the previous section. Good

80

,183.182 ,173.1 73 .15 .152 ,119 120 ,148.1 49 ,164.164

G. ZYVOLOSKI

,183

,183

( O N ( 1828,O) r I I I I I I I

.182 ,173 ,173 ,152.1 53 .119 ,119 ,148,149 ,164.1 63 ,183 ,173 ,152 .119 ,148 ,161 ,182 ,173 ,173 ,152 ,153 1 17 ,116 ,148 ,150 .164 ,163 .183 .173 .152 .118 .148 .161

,183

,183

,182 ,173 ,173 ,152 ,153 ,119 ,120 ,148 ,149 ,164 . I 6 3 .183 ,173 .152 .120 ,148 ,161 .I82 ,173 ,173 ,152 ,152 ,120 .122 ,149 .148 ,164 ,164 ,183 ,173 ,152 .121 ,148 ,161

,183.182 ,173.173 ,152 .152 ,121 .123 .149.148 ,164.164 ,183 ,173 ,152 ,122 ,148 ,161

1 I I 1 I I I I ( 1828,182.8)

ReF81 InJADRIUTERAL (0,182.8)

I TRIANGLE

Figure 1. Comparison of results of Toronyi's problem

agreement is evident. This is somewhat surprising considering the elements have an aspect ratio of 10. The same problem run with algorithm 2 and smaller time steps was necessary to complete the simulation. However, run times were comparable. A discussion of computer costs and run times is presented with the second example.

The second example problem considered is a model of a geothermal field which has recharge occurring across a constant pressure and enthalpy boundary. The model reservoir is depicted in Figure 2. The input data are presented in Table 11. This problem was proposed recently as a Department of Energy geothermal code test pr0b1em.l~ The simulation is for 10years. Because of the relatively long simulation time and the stability afforded by the recharge (constant pressure) conditions, a variable time stepping algorithm was employed in this example. The logic is as follows. After each time step the number of iterations was compared to some fixed value, in this case eight, and if the iterations were less than this amount then the next time step was made a multiple (1.2 in this example), of the last time step. This scheme had an advantage of limiting time step increases when the solution was changing quickly, such as phase changes. Further logic was also employed to decrease the time step increment if the iterations exceeded 25 cycles. In this example the FE (finite element) quadrilaterals took 41 time steps to model the 10-year production, while the FD (finite difference) scheme took 49 time steps. Computer times were 47.4 and 51.3 cpu sec respectively. To get an idea of the temporal truncation error, a more accurate 284-step simulation was also performed. The

Table I. Parameters for Toronyi example

Parameter Symbol Value

Permeability Thermal conductivity Porosity Rock density Rock specific heat Aquifer length Aquifer width Initial water saturation Initial pressure Discharge

9.869 x m2 1.730 w/(m "C)

0.05 2563 kg/m3 1010 J/kg"C

1828 m 1828 m

0.2 44.816 bar

0.082021 kg/(sec m)

GEOTHERMAL RESERVOIR SIMULATION 81

( 3 0 0 ) CONSTANT PRESSURE AND ENTHALPY p =36 bar

PRODUCTION h = ,676 MUKg

(62.5.62.5)

OBSERVATION WELL

(162.5,137.5) 0 E

Figure 2. Solution domain for DOE test problem

pressure and temperature of the production and observation wells, as well as the outlet enthalpy are presented in Figures 3-5. As can be seen from the figures, the FD and FE schemes plot very close to each other while the FE scheme is closer to the presumably more accurate solution. It is interesting to note here that the accurate solution was run with algorithm 2 using the approximate Jacobian. Even though many more time steps were used the cpu time was only 114 sec. It is also important to note that the truncation error was felt most strongly in the output temperatures and flowing enthalpies, the quantities which are of most interest in power plant sizing.

The third example is a model of a fractured geothermal reservoir similar to that being studied at Los Alamos National Laboratory. It further represents a problem which is very tractable with finite difference methods. The model reservoir is presented in Figure 6. A graded mesh was used with elements varying in width from 0.002 m near the fracture face to 10 m at the boundary opposite the fracture; the lengths of the elements varied from 25 m to

Table 11. Parameters for DOE test problem

Parameter Symbol Value

Permeability Thermal conductivity Porosity Rock density Rock specific heat Aquifer length Aquifer width Initial pressure Initial temperature distribution:

[240"C r s 1 0 0

k 2.5 x 10-14mZ K 1 w/(m "C) 4 0.35 Pr 2563 kg/m3 C, 1010 J/kg "C - 300 m - 200 m

P; 36 bar

1-100 r-100 T(x , y, 0) = 1 240- 160 ( - 2oo )'+go(=) "C 100

82

38

36

G. ZYVOLOSKI

- - - -

40r I 1 I I I I I I I I

OBSERVATION WELL - .............

- p ................... - .........................................

PRODUCTION WELL 28 - 26 - 24 - 22 - 20 I I I I 1 I 1 I I

- - - -

0 365 730 1095 1460 1825 2190 2555 2920 3285 3650

Figure 3. Pressure histories for DOE test problem

100m. This leads to a very unfavourable aspect ratio (5000) for the 4-node element; the standard finite element solution failed to finish the problem. If the integration points were changed from the standard 2-by-2 Gauss type to integration points which were equally weighted at the 4 nodes, the standard 5-point difference formula resulted. With this change the 10-year simulation was run in 28 time steps and a CPU time of 71 sec. As before, a logic based on the iteration count was used to increase the time step size. In this case if the iterations were

160: 1 50 0 365 730 1095 1460 1825 2190 2555 2920 3285 3650

DAYS

FD (284 TIME STEPS) ............... FD (49 TIME STEPS) -. - . - . . FE (41 TIME STEPS)

Figure 4. Temperature histories for DOE test problem

GEOTHERMAL RESERVOIR SIMULATION 83

1 .201 I 1 I 1 I I 1 I I I 4 1.10

::::Il I I , I I , 0.70 0 365 730 1095 1460 1825 2190 2555 2920 3285 3650

DAYS

FD (284 TIME STEPS) .-.-..---..--. FD (49 TIME STEPS) FE (41 TIME STEPS)

Figure 5. Outlet enthalpy for DOE test problem

less than 8, the time step was multiplied by 1-414. The temperature results are presented in Figure 7. Also shown in Figure 7 are the results of a more accurate 87-time-step solution (157 sec cpu time). Considerable temporal truncation error is evident. It is interesting to note that a plot of outlet temperature vs. time (not given) would show identical drawdown curves. Algorithm 2 also performed poorly on this problem. The time steps were constrained with

(10

PRODUCTION WELL

INJECTION WELL

0)

GRAVITY

1

,2000)

Figure 6. Solution domain for fracture flow problem

84

2600-

G . ZYVOLOSKI

28 TIME STEPS 87 TIME STEPS

- . . . . . . . . .

, 180

190

? I I I 1 I

0 20 40 60 80 1 L x 34001

Figure 7. Temperature field for fracture flow problem

0

this algorithm to lop4 days for the first 50 time steps in which it used a comparable amount of time as algorithm 1 used to finish the problem.

DISCUSSION

The three examples presented above illustrate some of the strengths and weaknesses of the finite element representation of flow in porous media. The finite element method performed slightly better than the finite difference method for the DOE test problem (example 2). The third example demonstrated the well-known result that the aspect ratio for elements must be kept about the same order of magnitude for good results. The code performed well on this

Table 111. Parameters for fracture flow example

Parameter Symbol Value

Permeability

Thermal conductivity Porosity Rock density Rock specific heat Aquifer depth Aquifer width Fracture length Fracture width Initial pressure Initial temperature Discharge production

injection

k

K

4

m2 matrix m2 fracture

2.9 W/(m "C) 0.001 matrix fracture

2700 kg/m3 1000 J/kg "C

1000 (2400 m-3400 m) 100 m 300 m

0.002 m hydrostatic

40.0+ 0.055 (depth) 0.05 kg/(sec m)

0.05 kg/(sec m) at 118C

GEOTHERMAL RESERVOIR SIMULATION 85

problem using an integration scheme which resulted in the 5-point difference scheme. This implementation is not restricted to rectangular grids like the difference formulae, and thus is more general. How it performs in different grid orientations is worth investigating, as well as an experimental investigation of the spatial discretization error.

Two other important points arise in the discussion of the examples. This is the temporal truncation error and the performance of algorithm 2, the approximate Jacobian algorithm. These two subjects are somewhat related. It is evident from the temperature history plot of example 2 and the temperature contour plot of example 3 that considerable truncation error occurs in large time step runs. Algorithm 2 usually takes about 4-8 times the time steps using automatic time stepping, at 2-3 times the cost of algorithm 1. Thus in many problems it may prove to be very useful in obtaining an accurate solution. It did prove very efficient in well tests in two-phase conditions in conjunction with finite difference methods.18

CONCLUSIONS

A finite element method for two-phase flow in porous media has been presented which is competitive with finite difference methods in terms of computational efficiency while still possessing geometric flexibilities. Two solution algorithms, one using the full Newton-Raphson iteration and the other employing an approximate Jacobian, have been presented and are useful in solving long-term reservoir problems. Algorithm 1, using the full Jacobian, allowed very large time steps with associated temporal truncation error. Algorithm 2 was found to be limited in time step size. Despite this drawback, this algorithm proved to be an efficient alternative to algorithm 1 in some problems.

ACKNOWLEDGEMENT

The author is grateful to Dr. M. J. OSullivan and Gloria Bennett for reading the manuscript and providing many helpful comments. Financial support for this work was provided by the U.S. Department of Energy.

REFERENCES

1. S . K. Garg, J. W. Pritchett and D. H. Brownell, Jr., Transport of mass and energy in porous media, Roc. Seeond United Nations Symp. Development and Use of Geothermal Resources, San Francisco (1975).

2. J. W. Pritchett, S. K. Garg, D. H. Brownell, Jr. and H. B. Levine, Geohydrological environmental effects of geothermal power production-phase 1, Report No. SSS-R-75-2733, Systems, Science and Software, La Jolla, California (1975).

3. C. R. Faust and J. W. Mercer, Mathematical modeling of geothermal systems, Proc. Second United Nations Symp. Development and Use of Geothermal Resources, San Francisco (1975).

4. L. K. Thomas and R. G. Pierson, Three dimensional reservoir simulation, SOC. Pet. Eng. J., 18,151-161 (1978). 5. K. H. Coats, Geothermal reservoir modelling, paper SPE 6892, 52nd Annual Fall Meeting of the SOC. Pet. Eng.

of AZME, Denver, Colorado (1977). 6. G. A. Zyvoloski, M. J. OSullivan and D. E. Krol, Finite difference techniques for modelling geothermal

reservoirs, Znt. J. Numer. Anal. Methods Geomech., 3, 355-366 (1979). 7. K. Pruess, R. C. Schroeaar, P. A. Witherspoon and J. M. Zerzan, SHAFT78, two-phase multidimensional

computer program for geothermal reservoir simulation, Lawrence Berkeley Laboratory Report No. 8264, Novem- ber (1979).

8. J. W. Mercer, Jr., C. R. Faust and G. F. Pinder, Geothermal reservoir simulation, Proc. Conf. Research for the Development of Geothermal Energy Resources, Pasadena, California (1974).

9. J. W. Mercer and C. R. Faust, Simulation of water- and vapor-dominated hydrothermal reservoirs, Paper SPE 5520, 50th Annual Fall Meeting of the SOC. Pet. Eng. of AZME, Dallas, Texas (1975).

10. D. H. Brownell, Jr., S . K. Garg and J. W. Pritchett, Computer simulation of geothermal reservoirs, Paper SPE 5381,45th California Regional Meeting of the SOC. Pet. Eng. of AZME, Ventura (1975).

86 G . ZYVOLOSKI

11. A. T. Corey, The interrelation between gas and oil relative permeabilities, Producers Monthly, 19,38-41 (1954) 12. M. A. Grant, Permeability reduction factors at Wairakei, Paper 77-HT-52, AIChE-AIME Heat Transfer Conf.,

Salt Lake City, Utah (1977). 13. 0. C. Zienkiewicz. T h e Finite Element Method. McGraw-Hill, London, 1977. 14. 0. C. Zienkiewicz and C. J. Parekh, Transient field problems-two and three dimensional analysis by

15. V. Dalen, Simplified finite-element models for reservoir flow problems, SOC. Pet. Eng. J., 19, 333-343 (1979). 16. R. M. Toronyi and S. M. Farouq Ali, Two-phase, two-dimensional simulation of a geothermal reservoir and

17. M. W. Molloy, Geothermal reservoir engineering code comparison project, Sixth Workshop on Geothermal

18. G. Zyvoloski and M. J. OSullivan, Simulation of a gas-dominated geothermal reservoir, Soc. Per. Eng. I., 20,

isoparametric finite elements, Int. J. Num. Meth. Eng., 2, 61-70 (1973).

wellbore system, SOC. Pet. Eng. J., 17, 171-183 (1977).

Reseruoir Engineering, Stanford University (1980).

52-58 (1980).