The prediction of long-term geothermal reservoir performance and the environmental impact of exploiting this resource are two important problems associated with the utilization of geothermal energy for power production. Our research effort addresses these problems through numerical simulation. Computer codes based on the solution of partial-differential equations using finite-element techniques are being prepared to simulate multiphase energy transport, energy transport in fractured porous reservoirs, well bore phenomena, and subsidence

Faust, C.

Mercer, J. W., Jr.

Pinder, G. F.

English

NASA Technical Reports Server

I. 
GEOTHERMAL RESERVOIR SIMULATION 
James W. Mercer, Jr., and Charles Faust 
U. S. Geological Survey 
Reston, Virginia 
George F. Pinder 
Princeton University 
Princeton, New Jersey 
The prediction of long-term geothermal reservoir perfor - 
mance and the environmental impact of exploiting this resource a r e  
two important problems associated with the utilization of geothermal 
energy for power production. Our research effort addresses these 
problems through numerical simulation. Computer codes based on 
the solution of partial-differential equations using finite -element 
techniques a r e  being prepared to simulate multiphase energy 
transport, energy transport in fractured porous reservoirs, well 
bore phenomena, and subsidence. 
INTRODUCTION 
Geothermal reservoir simulation, in general, refers  to reproducing and 
predicting the behavior of a geothermal reservoir by use of a model. 
discussion we consider only mathematical models which we define as: (1) a set  
of equations that describe the physical processes active in a geothermal r e se r -  
voir, and (2) the solution of these equations subject to boundary and initial 
conditions. 
ditions, the solution generally involves numerical techniques in conjunction 
with the digital computer. 
In this 
Because of the complex nature of the equations and boundary con- 
One simulates -a geothermal reservoir to estimate the quantity of recov- 
erable energy and the rate at which mass and energy may be extracted. 
achieve these objectives, known geological information obtained from both 
surface techniques and drilling is utilized to determine equation parameters, 
boundary, and initial conditions. During the ear l ier  stages of field develop- 
ment, this information is limited, and the simulation model will undoubtedly be 
crude. Through the simulation of a variety of producing schemes, however, 
the energy available and its rate of extraction may be roughly estimated. 
decision is made to develop the field further, the reservoir model may be used 
to help answer such engineering questions as optimal well location and spacing, 
or whether or not to reinject condensate. As more geological information 
becomes available through continued drilling, the simulation model may be 
updated to give a more accurate analysis of the geothermal reservoir. 
To 
If the' 
256 
https://ntrs.nasa.gov/search.jsp?R=19750012780 2020-03-22T23:15:30+00:00Z
,-. 
A. Pi.e+o.as Investigatiiin . ’- 
The li terature on the simulation of both heat and fluid flow in porous 
media is extensive. 
petroleum by in situ combustion o r  hot-fluid injection (e. g . ,  Refs. 1 and 2) .  
Another a rea  of interest is that associated with free  convection models (see, 
for example, Refs. 3 and 4).  If, however. only geothermal reservoir simu- 
lation i s  considered, the literature i s  relatively limited. 
Much of the literature is concerned with the extraction of 
Whiting and Ramey (Ref. 5 )  were the f i rs t  to apply a reservoir model to a 
geothermal system. 
but did not take into consideration the spatial dependence of the solution (it was 
a lumped parameter model). 
phase (hot-water) model coupled with a rock-fracture model. They attempted 
to demonstrate that geothermal energy could be extracted from hot dry rocks. 
Mercer (Ref. 7) applied a hot-water model to the Wairakei hydrothermal 
system. 
was unable to reproduce historical data past 1962 due to the formation of steam 
in the reservoir. Brigham and Morrow (Ref. 8), in an attempt to allow some 
spatial variation in their solution, developed three lumped-parameter models 
based on vapor liquid distribution. Toronyi (Ref. 9 )  developed a two-dimensional 
(areal or cross-section) reservoir model coupled with a wellbore model. His 
reservoir model allowed only two-phase flow, being restricted to the saturated- 
vapor pressure curve. 
Their model allowed for two-phase (steam-water) flow, 
Harlow and Pracht (Ref. 6 )  developed a single 
The model was limited spatially to the two horizontal dimensions and 
B. A General Simulator 
The above models could be thought of a s  contributions toward a general 
The development of such a model constitutes the principal objective of 
model capable of completely describing the response of a reservoir to exploita- 
tion. 
our current research effort. 
The physical properties of the fluid in a geothermal reservoir vary spa- 
tially and temporally to the extent that vertical velocities a r e  often of the same 
order of magnitude as horizontal velocities. Consequently, a general model 
must be a distributed parameter model with the capability of three-dimensional 
simulation. 
To answer important engineering questions, a general model must not 
only accommodate the transient flow of compressed hot water, steam-water 
mixtures, and super-heated steam, but also must allow for phase changes 
(see Fig. 1). Presently we a r e  neglecting species transport and chemical 
reactions that may occur in the geothermal system. 
Subsidence of the land surface is often encountered when large quantities 
of fluid a r e  removed from subsurface reservoirs. Consequently, a general 
model should have the capability of forecasting the spatial and temporal dis- 
tribution of subsidence resulting from geothermal development. Such a simu- 
lator must solve not only the fluid flow and energy transport equations but also 
the equations describing the elastic behavior of the reservoir skeleton. 
257 
base 
hold 
The governing 
,d on the physics 
in  selected case 
equations employed in reservoir simulation are generally 
of flow through porous media. While this approach may 
s, there a r e  many reservoirs that a r e  productive only 
because large fractures are encountered during drilling. To simulate such a 
fractured reservoir,  the classical  continuum approach to porous media flow 
must be modified and extended to accommodate the influence of fracture flow. 
While we are currently developing models which will include each of the 
important capabilities described above, we cannot herein describe all aspects 
of our research program. Consequently, we have elected to focus our attention 
on multi-phase energy transport, a topic which should be of interest  to anyone 
contemplating the development of a geothermal reservoir simulation capability. 
11. THEORETICAL DEVELOPMENT 
The general governing equations consist of mass,  momentum, and energy 
balances for  each phase present in  a geothermal system. 
A. Mass Balance 
and 
The mass  balances for steam, s, and water, w, may be written as: 
where 
- -1 v = phase average velocity, [Lt ] 
P = average density, [ML'3] 
q = source term, [ML t 1 
d = rate of vaporization, [ML 
9 = porosity, dimensionless 
S 
-3 -1 
-3t- 11 
V 
= saturation, dimensionless, where Sw t Ss = 1 
258 
B. Momentum Balance 
For  velocity, we assume that Darcy's equation for multiphase flow may 
be used (see Kef. 10) : 
and 
where 
- 
k = local intrinsic permeability tensor, [L2] 
= relative permeability, dimensionless kr 
p = dynamic viscosity, [ML-It-'] 
1-2 p = pressure,  [ML- t ] 
g = gravitational acceleration, [~t- '] 
Combining Eqs. (1) with ( 3 )  and (2 )  with (4) gives 
- 
and 
2 59 
These equations describe the flow of steam and wa te r  in a porous medium 
where a pressure difference exists between the two phases. 
difference is generally defined as capillary pres sure, pc, where 
This pressure 
(7) - Pc - Ps - Pw 
Probably the most important effect that capillary pressure has in a geothermal 
reservoir is to lower the vapor-pressure curve. Ramey, et al. (Ref. 11) point 
out that the reason for the lowering of the vapor-pressure curve is that vapor- 
pressure data found in steam tables (Refs. 12 and 13) are based on flat steam- 
water interfaces, whereas capillary pressure causes the interface in porous 
media to be curved. 
geothermal reservoir is not completely understood. 
e t  al. (Ref. 14), on consolidated rock does show a lowering of the vapor-pressure 
curve with decreased fluid saturation. 
Bilhartz (Ref. 16), however, indicate no. significant vapor pressure lowering in 
their experiments using unconsolidated sands. An important difference in the 
two results i s  that the experiments of Calhoun, et al., were made a t  a temper- 
ature of 36"C, and those conducted by Cady and Bilhartz ranged from approxi- 
mately 121 to 240°C. Further work on the importance of capillary pressure in 
geothermal reservoirs is required. In this paper, capillary pressure is 
assumed t o  be negligible. 
reduce the number of equations and unknown parameters by one: 
The amount the vapor-pressure curve is lowered in a 
The work of Calhoun, 
The efforts of Cady (Ref. 15) and 
Under this assumption Eqs. (5) and (6) combine to 
where p is  the density of the total steam water mixture, defined as: 
P = PSSS + PWSW (9) 
has also been 
dV* 
Note that in combining Eqs. (5) and (6)  the vaporization term,  
eliminated. 
C. Energy Balance 
Three energy balances, analogous to the mass  balances written for steam 
and water, can be written for steam, water, and rock. If it is assumed that the 
movement of fluid (water and steam) through porous media is sufficiently slow, 
and the surface a reas  of all phases concerned a r e  sufficiently large, then thermal 
equilibrium exists between the rock, steam, and water. This assumption 
reduces the three energy-balance equations to the following one: 
260 
where 
The t e rm described by Eq. (11) represents the compressible w o r k  term,  
arid i s  often assumed negligible and omitted from the energy balance. 
Other definitions of t e rms  in Eq. (10) include: 
hs = enthalpy of saturated steam, [ L  2 t - 2  1 
= enthalpy of saturated water, EL 2 t - 2  1 h 
T = temperature, IT] 
K = thermal dispersion tensor for the medium [MLt T 1 
p, 
W 
- -3 -1 
m 
= average rock density, [MLm3] 
= rock enthalpy, [L t 1 
= total enthalpy of the mixture, as defined by 
2 - 2  
hr 
h 
h =  S s p s h S  -!- Swpwhw 
P 
D. Assumptions 
By making the assumptions of thermal equilibrium and negligible capillary 
pressure, we a r e  left with two equations. 
ships for the remaining dependent variables as functions of these two. 
We must obtain additional relation- 
The two dependent variables specified as unknowns must uniquely define 
the thermodynamic state of the system. 
variables be commonly measured in  a field situation. 
Further, it is desirable that these 
For  these reasons, we 
261 
choose to put Eqs. (8) and (10) in terms of the dependent variables h, the total 
enthalpy of the mixture, and p, the fluid pressure.  
following assumptions: 
To do this we make the 
Porosity, 4, is a function of pressure alone. 
Rock density, p r ,  may be treated as constant. 
Total density, p,  steam density, ps,  and water density, pw, a r e  
considered functions of pres sure  and enthalpy. 
Steam enthalpy, hs, and water enthalpy, hw, can be treated as 
functions of pressure.  
Temperature, T ,  can be treated as a function of pressure in the 
two-phase region, and can be treated as a function of pressure and 
enthalpy in the single-phase region. 
Rock enthalpy, hr,  can be considered as a function of temperature. 
Viscosities, ps and pw, a re  considered functions of temperature.  
Saturations, Sw and Ss, are  functions of enthalpy and pressure.  
Relative permeability can be treated as a function of saturation 
alone and relationships similar to those in Brooks and Corey 
(Ref. 17) may be used. 
Thermal dispersion tensor can be generalized and treated as a 
property of the medium. 
Ramey, e t  al. (Ref. 18), point out that relative permeability can also be 
a function of temperature. Using unconsolidated sand and working with oil and 
water, Poston, e t  al. (Ref. 19), observed that for increased temperatures, the 
relative permeability curves shift to the left on the saturation axis. Regarding 
the thermal dispersion tensor it should be noted that Mercer (Ref. 7 )  separates 
the medium thermal dispersion tensor into three parts: conduction in the solid 
phase, diffusion in the liquid phase, and a velocity-dependent dispersion in the 
liquid phase. 
conductivity of the medium is a function of temperature, porosity, and water 
s atur ation. 
Further,  Somerton, et al. (Ref. 2 0 ) ,  point out that the thermal 
2 62 
E. Final Equations 
Making the above assumptions, Eqs . (8) and (1 0) become: 
p P + a pw r  w(q- p-.) 
and 
+ %vhw + qshs 
263 
Equations (13) and (14) a r e  in te rms  of the dependent variables enthalpy 
and pressure.  As can be seen from Fig. 1, a pressure and enthalpy value for 
a given point in space will determine whether compressed hot water, a steam- 
water mixture or  super-heated steam exists a t  that point. 
(14) may be used to describe most types of geothermal reservoirs.  
the behavior of such a reservoir,  however, a solution for Eqs. (13) and (14) is 
r equi r e d. 
Thus, Eqs. (13) and 
To simulate 
Equations (13) and (14) a r e  nonlinear, and their solution requires the use 
of numerical techniques. 
and will be mentioned only briefly. 
the finite element method, or  some combination of the two. 
coefficients are generally handled by iteration, using, for example, the Newton- 
Raphson method (Refs. 2 and 9) .  Finally, some form of weighting of the spa- 
tially dependent variables is commonly used, upstream weighting of perme- 
ability, for example (Ref. 2). 
Such techniques a r e  beyond the scope of this paper 
They include finite difference techniques, 
The nonlinear 
III. CONCLUSIONS 
We have presented the governing equations for energy transport and the 
flow of water and steam in porous media in order to provide a basis for the 
mathematical modeling of geothermal reservoirs.  
equations is straightforward and available in various forms in the literature. 
Our objective here  was to take the general mass  and energy balances and 
reduce them to workable equations. It should be emphasized that the method 
described represents only one of many possible approaches. The develop- 
ment of a method for obtaining solutions to  these equations is a challenging 
problem and beyond the scope of this paper. 
The primitive form of these 
This particular approach to geothermal modeling has been guided by 
several  objectives: (1) the model must provide efficient and accurate solutions 
consistent with the available geological and hydrological data, (2)  it  should be 
applicable to hot-water, steam-water, and super-heated steam geothermal 
systems, and (3) the method must be e a s i l y  applied to natural geothermal 
reservoirs.  Two assumptions, peculiar to this development, were made to 
achieve these objectives: (1 ) thermal equilibrium exists between all phases, 
in steam, water, and rock; and ( 2 )  capillary pressure between the steam and 
water phases is negligible. Invoking these assumptions the basic relationships 
reduce to two partial differential equations written in te rms  of convenient field 
variables, fluid pressure and enthalpy. 
tion with well-known thermodynamic relationships provide the basis for a 
numerical model which we feel fulfills our initial objectives. 
These final two equations in conjunc- 
The suitability of this model for simulation of natural geothermal r e se r -  
voirs is limited in two general ways: (1) by the validity of the basic assump- 
tions (negligible capillary pressure,  porous media, etc. ), and (2)  by the avail- 
ability of reliable geological and hydrological and thermodynamical data for the 
reservoir to be simulated. Within these constraints, the suggested approach 
should prove useful in geothermal reservoir development and management. 
264 
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1 o2 
1 o1 
1 oo 
0 1  
A 
I 
ENTHALPY (HEAT CONTENT), cal/gm 
Fig. 1. Pressure-enthalpy diagram for pure water and vapor showing 
temperature contours (after Ref. 21). Regions a r e  for: (A) compressed 
hot water, (B) steam-water mixture, and (C) super-heated steam. 
267 


Geothermal reservoir simulation

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