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Geothermal reservoir characterization
Characterizing geothermal reservoirs draws on techniques common to petroleum reservoirs. Key differences create special challenges to gaining a good understanding of geothermal reservoirs. This article covers appropriate approaches and caveats for well testing, drawdown/buildup analyses and decline curve analysis for characterizing geothermal reservoirs.
Well testing
Geothermal well testing is similar in many respects to transient pressure testing of oil/gas wells, with some significant differences. Many geothermal wells induce boiling in the near-well reservoir, giving rise to temperature transients as well as pressure transients. Substantial phase change may also take place in the well, further complicating analysis. Pressure tools must be kept in a high-temperature environment for long periods of time, and production intervals are frequently very small portions of overall well depth. Production intervals, which are usually associated with fracture zones, may be at substantially different thermodynamic conditions. Finally, pressure and temperature changes induce fluid property changes that require correction. Nevertheless, the principles of geothermal well testing are the same as petroleum well testing. And with the caveats already noted, standard interpretation methods can be used.
Because the primary objective in geothermal well testing is to determine the ultimate productivity of a well prior to completion, injectivity testing is perhaps the most useful kind of well test. In contrast with a production test, cold water injection does not induce flashing (phase change) near the wellbore. Injection testing in an all-liquid reservoir can therefore be interpreted conventionally to yield formation transmissivity and well skin factor. For a well with a single feed zone, an injection test will yield unambiguous values for formation properties.
Most wells have multiple feed points, and it is necessary to relate the outflows (or inflows) from the well to the difference between the pressure gradient in the well and that in the reservoir. A fluid entry that accepts fluid during injection may nevertheless prove to be unproductive in a production test.^{[1]}
Because of changes in temperature (and therefore fluid density and viscosity) with time, it is imperative to obtain downhole measurements of pressure and temperature. As an example, replacement of a 2,500-m fluid column at 150°C with cold fluid at 20°C results in a pressure increase of ~20 bar at the base of the column. Pressure, temperature, spinner (PTS) logs are frequently run to obtain estimates of feed-zone locations and pressure and temperature profiles. However, convective effects in a geothermal wellbore often totally dominate the measured pressure and temperature profiles. Interpretation of such data is treated by White,^{[2]} Stefansson and Steingrimsson,^{[3]} and Grant.^{[4]}^{[5]} The heat and mass transport mechanisms involved in the intersecting wellbore and reservoir system are reviewed here to illustrate the difficulties inherent in characterizing reservoir permeability structure and fluid state from downhole measurements.
Three possible types of temperature profiles that may be observed during cold water injection into a geothermal well are shown in Fig. 1. The well in this example has two permeable horizons whose limits are marked by crosses on the depth (vertical) axis. Profile 1 is the simplest. Water enters at the wellhead (z= 0). The measured temperature increases slowly to depth z_{2}, and then it increases rapidly. The rapid increase below z_{2} indicates that the depth z_{2} is the major zone of fluid loss and that little or no cold water penetrates below z_{2}.
A break in gradient at depth z_{1}, followed by a sharp increase in gradient at z_{2}, is shown in Profile 2. This indicates some fluid loss at depth z_{1} and loss of all or nearly all of the balance of the injected fluid by depth z_{2}. Except in permeable zones, the fluid gains heat by conduction from the surrounding formation. If W is the mass flow rate down the well, T_{w}(z) is the fluid temperature, and T_{r}(z) is the formation temperature, then the conductive heating (caused by the temperature difference T_{r} –T_{w}) of the descending water is given by
where
C_{pw} | = | specific heat of the fluid (J/kgK), |
d | = | wellbore diameter (m), |
and | ||
U | = | overall heat transfer coefficient (W/m^{2}K). |
The temperature gradient dT_{w}/dz is inversely proportional to the flow rate W. Therefore, an increase in temperature gradient (see Profile 2, Fig. 1) implies a decrease in W and, hence, water loss from the well at the depth of gradient change. In many cases, flow rates are so large that dT_{w}/dz is small or, within measurement error, zero. Essentially no change in T_{w}between wellhead and datum z_{2} is indicated in Profile 1 (an "isothermal" profile).
A jump in temperature at z_{1} is shown in Profile 3. This indicates fluid influx at this depth. Hot fluid enters the well at z_{1}, mixes with the cold water from the wellhead, and the entire flow is injected at z_{2}. Given the enthalpy/temperature of the inflow at z_{1}, the amount of the inflow can be calculated by a heat balance. If H(T + ) is the enthalpy in the well above z_{1}, H(T_{–}) is the enthalpy below z_{1} and H_{1} is the inflow enthalpy, then the inflow W_{1} is given by ....................(2)
Alternatively, the inflow at z_{1} can be directly measured by a spinner or other downhole flowmeter, and the inflow enthalpy can then be estimated from Eq. 2.
To quantitatively compare the fluid gain or loss at the two depths (z_{1} and z_{2}), it is essential to compare the pressure profile in the well with that in the reservoir. Two possible pressure profiles are shown in Fig. 2. A much larger difference between wellbore and formation pressure at z_{2} than at z_{1} in Profile 1 need not imply less permeability at z_{2} than at z_{1}. Pressure Profile 2 corresponding to temperature Profile 3 in Fig. 1 indicates that fluid enters the well at z_{1} and is injected into the formation at z_{2}.
A typical temperature profile in a discharging geothermal well with several liquid feed zones is shown in Fig. 3. The feed zones are indicated by discontinuous changes in temperature gradient; the isothermal intervals between the feed zones denote impermeable zones. In this figure, the middle feed zone is a zone of cooler fluid, giving the temperature reversal. The ascending water boils at some depth (flash depth) in the wellbore; above this depth, the temperature profile follows the saturation curve for water. If the inflows from the various feed zones are known (say from a spinner survey), then Eq. 2 may be used to compute the feed-zone temperatures. Location of feed zones in a well with in-situ boiling (i.e., two-phase feed zones) is somewhat involved. The interpretation of PTS surveys in wells with two-phase feeds is discussed by Kaspereit^{[6]} and by Spielman. ^{[7]}
The behavior of the well as it warms up after cold-water injection (or production with insitu boiling and the consequent drop in formation temperature around the wellbore) provides additional information regarding permeable zones. Permeability is often indicated by a marked feature, in successive surveys, such as a persistent cold zone, rapid warming, and interzonal flow. The permeable zone, which has accepted water during injection, may appear as a persistent cold feature as the well heats up (similar to the middle feed zone in Fig. 3). If the injected cold water does not move to other parts of the reservoir, it will take longer to heat this portion of the well than the impermeable sections of the well that have not accepted fluid.
Alternatively, the permeable zones may permit rapid circulation of the injected water away from the well, and there appears a marked peak in the heating surveys. Finally, the disturbed pressures may initiate flow between two permeable zones of the well. Such a transient flow during warm-up is manifested by an isothermal temperature profile and implies permeable zones at both the end points of the isothermal interval.
Because of boiling, convection, and interzonal flow in the wellbore, it is necessary to carefully interpret downhole temperature data to deduce the reservoir temperature distribution. Great care must be exercised to identify those measurements affected by convection/interzonal flow and by boiling in the wellbore; such data often mask the true formation temperatures and should be discarded. Stable temperatures measured at feed depths usually provide the most reliable measures of reservoir temperatures. Liquid feed-zone temperatures are best determined from temperature surveys recorded in discharging wells. Additionally, in impermeable sections of the borehole, it is often possible to extrapolate the measured temperatures to estimate the formation temperatures.^{[4]}
The pressure profile in a geothermal well can be measured directly by a downhole gauge. It is also possible to compute the downhole pressure from the water level data and the temperature gradient survey. Basically, this involves numerically integrating the differential equation,
together with the boundary condition p = p_{o}at z = z_{o}(e.g., wellhead pressure). Here, z_{o}denotes the water level in the well (measured downwards from the wellhead), ρ (p,T) denotes the fluid density, and g is the acceleration because of gravity. Given T(z) and p(z), density ρ (p,T) can be obtained from the thermodynamic equation-of-state data for liquid water. This procedure for calculating downhole pressures from water level data works only in single-phase (all liquid) wells. The presence of boiling conditions anywhere in the wellbore (below the water level, z_{o}) invalidates the use of this method. Experience has shown that the downhole pressures computed from water level and temperature data are often more accurate than those recorded by downhole pressure gauges.
Regardless of how the downhole pressures are obtained, the pressure profiles can provide information regarding formation permeability by showing a "pivot" as the well warms. The mechanism is illustrated in Fig. 4. The well in Fig. 4 has a single entry at z_{1}. Profile 1 is during cold-water injection. As the well heats up, two physical mechanisms affect the downhole pressures: the transient decay of the pressure buildup caused by injection and the change in gradient caused by the warming of the water column in the well. For injection into a homogeneous single-phase reservoir, the time required for the pressure transient to decay is proportional to the injection time; in practice, the pressure decay is usually complete before much warming of the water column has occurred. This produces pressure Profile 2, with the coldwater pressure gradient but where the pressure at z_{1} has reached equilibrium with reservoir pressure.
Fig. 4—Schematic of a wellbore pressure profile after cold water injection. As the well heats up the hydrostatic pressure in the well falls (density is reduced). The wellbore pressure pivots about the reservoir pressure at the depth of a single feed zone. In the case of multiple feed locations, the pivot point is a weighted average. The differences in pressure gradient are exaggerated in the figure.
As the well contacts the reservoir only at its permeable point, only here does it equilibrate with the reservoir pressure. The pressures measured at other depths in the well merely reflect the weight of the fluid column present in the well. As the well warms up, the water column lightens to produce Profiles 3 and 4. The successive profiles pivot about the reservoir pressure at depth z_{1}.
The pressure pivot works best for wells in reservoirs with good permeability, where the pressure transients are small. If substantial transient effects are present, the pivot is displaced above the feed zone. As a check on the pivot, it should be defined by the intersection of more than two pressure surveys and preferably with as wide a range of temperatures as possible. Large temperature differences mean more contrast in pressure gradient. If the well has two significant permeable zones, the pressure pivot appears between them at a point weighted by the productivity ratio of the two zones. In this case, the pressure at the pivot lies between the reservoir pressures at the two zones and probably corresponds roughly to the reservoir pressure at the depth of the pivot. Having identified the well’s permeable depths, measured pressures at these different depths in the various wells can be used to construct a reservoir pressure profile.^{[4]}^{[5]}
In practice, the application of the techniques discussed here to actual field data sometimes proves to be difficult. Temperature and pressure profiles in wells of poor permeability often fail to provide any definite indications of feed zones. Geothermal wells are frequently drilled with foam or air to avoid damaging the formation; in these cases fluid gain zones often go unnoticed. Because of the economic desirability of putting a geothermal well on production quickly, long-term temperature recovery is in many cases not recorded; this makes the determination of stable reservoir temperatures very difficult. Because of variations in hole diameter and condition (slotted and blank intervals) and changes in fluid state downhole, spinner data (in the absence of simultaneous pressure and temperature surveys) may yield ambiguous interpretations. In spite of these limitations, the interpretation methods discussed herein have been used in numerous cases to successfully locate a well’s permeable horizons.^{[4]}^{[5]}^{[7]}
Pressure transient data
Pressure transient tests are conducted to diagnose a well’s condition and to estimate formation properties. The test data may be analyzed to yield quantitative information regarding:
- Formation permeability, storativity, and porosity
- The presence of barriers and leaky boundaries
- The condition of the well (i.e., damaged or stimulated)
- The presence of major fractures close to the well
- The mean formation pressure
After well completion, testing is performed by producing one or more wells at controlled rates and monitoring downhole pressure changes within the producing well itself or nearby observation wells (interference tests). A comprehensive review of techniques for analyzing pressure transient data may be found in monographs by Matthews and Russell,^{[8]} Earlougher,^{[9]} and Streltsova.^{[10]}.
Much of the existing literature^{[9]} deals with isothermal single-phase (water/oil/gas) and isothermal two-phase (oil with gas in solution, free gas) systems. Geothermal reservoirs commonly involve nonisothermal, two-phase flow during well testing. In addition, geothermal wells, unlike most oil/gas wells, do not usually penetrate a formation with uniform properties. In this page, these and other problems that are specific to geothermal well testing are briefly discussed.
Partial penetration
The line source solution forms the basis of most of the existing techniques for pressure transient analysis. It is assumed that the production (or injection) well fully penetrates an aquifer of uniform and homogeneous permeability. In a geothermal reservoir, the bulk of formation permeability is associated with thin stratigraphic units and/or a fracture network.
The well is open to the reservoir only at the depths where it intersects the permeable zones, and for the balance of its depth, the well penetrates impermeable rock. A geothermal well is comparable to an oil/groundwater well that only partially penetrates the permeable formation. The mathematical theory for a partially penetrating well has been developed by Nisle^{[11]} and Brons and Marting.^{[12]} Partial penetration is detectable from the shape of the buildup (or drawdown) curve. A Horner plot of the buildup data shows the existence of two straight lines. The penetration ratio is given by the ratio of the slope of the late part to that of the early part of the buildup curve. In at least some geothermal wells, the permeable interval(s) constitutes such a small fraction of the "total formation thickness" that it is not meaningful to define a penetration ratio. For small flow/shut-in times, the well in these cases exhibits a pressure response resembling that of an spherically symmetric source/sink and not a line source/sink. The mathematical theory for a geothermal well undergoing spherical flow is presented by Tang.^{[13]} For the spherical flow period, a plot of pressure drop or pressure buildup vs. t_{p}^{–0.5} orΔt^{–0.5} (t_{p}= total production time, Δt = shut-in time) yields a straight line; the slope of the straight line can be used to compute the formation permeability. One important consequence of partial penetration in geothermal systems is that the transmissivity value determined from interference tests frequently exceeds that of single-well tests.
Drawdown/buildup analysis for two-phase wells
A geothermal system may be two-phase before production begins or may evolve into a two-phase system as a result of fluid production.
Theoretical analysis of pressure drawdown and pressure buildup data from single wells in such systems has been published by Grant,^{[14]} Garg,^{[15]} Garg and Pritchett,^{[16]} Moench and Atkinson,^{[17]} and Sorey et al.^{[18]} For a constant rate of mass production, W, a Horner plot of pressure buildup vs. log [ (t + Δt)/Δt] gives a straight line at late times. (For drawdown tests, well pressure vs. logarithm of production time, t, yields a straight line.) Here, t denotes the production time, and Δt is the shut-in time. The slope m of the straight line is related to the "kinematic mobility," ....................(4)
where
In Eqs. 4 and 5, k is the absolute formation permeability, H_{t}is the formation thickness, k_{rℓ} (k_{rg})is the liquid (gas) phase relative permeability, and υ_{ℓ}(υ_{g}) is the liquid (gas) phase kinematic viscosity.
Given the specific flowing enthalpy H_{f}, it also is possible to estimate the separate liquid and vapor phase mobilities.
and
where H_{ℓ}(H_{g}) denotes the specific liquid (gas) phase enthalpy. The flowing enthalpy, H_{f}, is given by
where m_{g} is the vapor mass fraction of the fluid flow. Substituting from Eq. 9.11 into Eq. 9.9 and Eq. 9.10, it follows that
To evaluate k_{rg} and k_{rℓ} separately, an additional relation is required between k_{rg} and k_{rℓ}.
The previously described analysis procedure for drawdown/buildup data is only approximate. Because of the nonlinear nature of two-phase flow, buildup and drawdown tests yield different values for kinematic mobility k/ν_{t}; this introduces an element of uncertainty in the determination of k/ν_{t}.
A second complicating factor arises in the calculation of well skin factor, S. Grant and Sorey^{[19]} showed that the compressibility of two-phase mixtures of steam and water in porous rock can be written as
where C_{pr} and C_{pw} are the specific heat capacities of rock and water respectively; ρ_{r}, ρ_{w}, and ρ_{v} are the densities of rock, water, and steam, respectively; L_{v} is the enthalpy change because of boiling; P_{sat} and T_{sat} are the saturation pressure and temperature; and Φ is the porosity. This expression does not include the compressibility of each phase; it merely accounts for the volumetric change because of phase change. For typical geothermal problems, however, this compressibility is 10^{2} larger than steam compressibility and 10^{4} larger than liquid. Grant and Sorey^{[20]} also show that the compressibility can be approximated by
where compressibility, c_{t}, is in bar_{–1}; bulk volumetric heat capacity ρC_{p} is in kJ/m^{3}-°C; and pressure, P, is in bars.
An additional complication frequently arises in practice. If the pressure gauge is not located adjacent to the major entry for a well, then the pressure data must be corrected for the pressure difference between the gauge location and the feed point. If the well contains two-phase fluid, it will generally be necessary to correct the measured pressures by different amounts for different drawdown/buildup times. In Fig. 5, taken from Riney and Garg, ^{[1]} a semilog plot is presented of the pressure buildup for Well B-20 at a depth of 1,372 m, where most of the downhole pressure recordings were made. The primary production zone for this well is located at a depth of 1,220 m. Several pressure gradient surveys made during the buildup period show that the well is two-phase. Riney and Garg^{[1]} used these pressure gradient surveys to estimate the buildup pressures at the feed-point depth of 1,220 m; the replotted Horner plot is given in Fig. 6. A comparison of Fig. 5 and 6 shows that the slope of straight line in Fig. 6 is approximately one-half of that in Fig. 5.
Fig. 5—Pressure buildup for Well B-20 following flow test 4. A pressure gauge is set at 1373 m below ground surface (bgs); primary fluid entries are at 1220 m bgs. Time-varying two-phase conditions in the wellbore require corrections to the measured pressure that also vary in time. A corrected pressure buildup for this test is given in Fig. 6. The figure is modified from Riney and Garg by permission of the American Geophysical Union (after Riney and Garg^{[1]}).
Fig. 6—Corrected pressure buildup for Well B-20 following flow test 4. Effects of transient boiling in the wellbore are corrected for in the plotted pressure at 1372 m. Note the Horner plot slope is half that of Fig. 9.6. Individual data points (e.g., S3, S4, etc.) are pressure measurements at different times. An H_{2}O-CO_{2} EOS was used for pressure-depth corrections required. The figure is modified from Riney and Garg by permission of the American Geophysical Union (after Riney and Garg^{[1]}).
Decline curve analysis
A method that has enjoyed extensive use in geothermal engineering for production forecasting is decline curve analysis. Two types of decline curve analysis are used:
- Empirical rate-time analysis using the Arps method^{[20]}
- Fetkovich-type curves^{[21]}
Empirical decline curves consist of plotting rate as a function of time in either Cartesian, semilog, or log-log coordinates. The usual goal is to establish a linear trend between rate and time and use that relationship to forecast future production schedules, abandonment rates, production cumulatives, etc. It requires a continuous history of static reservoir pressure and/or flow rates at constant flowing wellhead pressure. These data are often not available for a variety of reasons, but can be estimated from production data. First, the well mass flow rate, W, must be normalized against a standard flowing wellhead pressure, P_{std}.^{[22]}
where W is the measured flow rate, p is the estimated (or measured) static wellhead pressure, and P_{wf} is the measured well flowing pressure. This relationship was developed for steam wells; for liquid-dominated wells, the appropriate equation is
Having normalized the rate against a reference pressure, decline analysis can then be used. It is important to note the dangers in extrapolating rates too far into the future, given that a phase change, for example, may lead to orders of magnitude change in density, kinematic viscosity, compressibility, etc. The normalized rate can then be analyzed with either Arps-type decline curves or Fetkovich-type curves.
Decline-curve analysis is based on the empirical rate decline equations originally given by Arps. ^{[20]} The general rate-time equation can be written as
D_{i} is decline rate, b is the Arps exponent (0 ≤ b ≤ 1), and t is time. Depending on the value of b, the following forms of rate decline can be identified.
These decline equations can be used to estimate abandonment flow rates or time.
Fetkovich-type curves^{[21]} can also be used to estimate decline rates and reservoir properties. These type curves were originally developed to provide a theoretical basis for decline curve analysis and are used to estimate the decline parameters D_{i} and b. The type curves also provide estimates of permeability-thickness product and wellbore skin properties. The Fetkovich decline equations can be used with the relevant changes in units (e.g., from volumetric flow rate to mass flow rates).
and
For vapor-dominated reservoirs, one can also use a pseudo-pressure approach (e.g., Faulder^{[23]}). The application of Fetkovich-type curves for geothermal well analysis is similar to that of oil/ gas wells, with a few caveats. Most importantly, if reservoir conditions are two-phase, or if boiling is induced in the vicinity of the wellbore, the effective compressibility follows from Grant and Sorey. ^{[19]} If conditions change (e.g., if boiling is induced, or if the reservoir becomes superheated), the compressibility is discontinuous, changing by more than two orders of magnitude. Also, if phase conditions change, the well decline rate will also be incorrect. It is thus dangerous to predict geothermal well behavior too far into the future if such phase change possibilities exist.
With the above caveats, one applies the Fetkovich-type curves in the following fashion:
- Normalize the well flow rates against a standard flowing well pressure using the backflow equations (Eqs. 12 or 13) as appropriate. The flow rates may have to be renormalized occasionally, if for example, substantial disruption of production occurs and transient conditions again prevail.
- Plot the normalized rate vs. time on log-log tracing paper of the same size as the type curve to be used.
- Shift the tracing paper, keeping the axes aligned, to obtain agreement between the real data and the type curve. A match point can be selected from the overlay, and reservoir properties (kh, r_{e}, and S) can be determined from the match point.
- From the pseudo-steady-state portion of the production, the decline parameter, b, can be determined. Note that an attempt to estimate b from the transient data may either give a nonunique^{[21]} or nonphysical value^{[24]} for b.
Nomenclature
b | = | Arps exponent (0 ≤ b ≤ 1) |
c_{ij} | = | arc cost function for the travel arc between a given injection and production well pair (e.g., along a streamline) |
c_{r} | = | rock compressibility |
c_{w} | = | liquid compressibility |
Cl_{p} | = | produced chloride for a given well |
C_{pr}, C_{pw} | = | specific heat capacities of rock and water, respectively |
c_{t} | = | total compressibility, including effects of phase change |
C_{T} | = | tracer concentration by weight |
d | = | wellbore diameter |
D | = | depth |
D_{i} | = | decline rate |
g | = | acceleration because of gravity |
H_{t} | = | formation thickness |
H_{f} | = | flowing enthalpy |
H_{ℓ}(H_{g}) | = | liquid (vapor) phase enthalpy, respectively |
i | = | injector |
k | = | absolute formation permeability |
k_{rℓ} (k_{rg}) | = | liquid- (gas-) phase relative permeability |
L_{v} | = | enthalpy change because of boiling |
m | = | slope of pressure buildup or drawdown test |
m_{g} | = | vapor mass fraction |
P | = | estimated (or measured) static wellhead pressure |
P_{sat} | = | saturation pressure |
P_{std} | = | standard flowing wellhead pressure to normalize flow rates for decline curve analysis |
P_{wf} | = | measured well flowing pressure |
P_{I} | = | initial formation pressure |
q* | = | the component of heat flux that originates from the lower crust or mantle |
q | = | total heat flux |
q_{Dd} | = | dimensionless flow rate W/W_{I} |
q_{i} | = | chloride injection rate in well i |
q_{ri} | = | re-injection rate at well i |
Q_{L,V} | = | mass rate of fluid (liquid or steam) |
Q_{T} | = | tracer injection mass rate |
r_{e} | = | effective radius |
r_{w} | = | wellbore radius |
S | = | well skin factor |
S_{w} | = | liquid saturation |
t | = | production time |
t_{Dd} | = | dimensionless time used in decline curve analysis |
t_{p} | = | total production time |
T | = | temperature at depth |
T_{r}(z) | = | formation temperature |
T_{sat} | = | saturation temperature |
T_{w}(z) | = | fluid temperature |
U | = | overall heat transfer coefficient |
v_{T} | = | velocity of temperature front |
W | = | mass flow rate |
W_{I} | = | initial mass flow rate |
z | = | vertical distance |
β | = | mixing rule parameter for geothermometers |
Δt | = | shut-in time (buildup test) |
κ | = | pressure diffusivity |
Γ | = | temperature gradient |
μ | = | dynamic viscosity |
ρ (p,T) | = | fluid density |
ρ_{r}, ρ_{w}, ρ_{v} | = | densities of rock, water, and steam, respectively |
υ_{ℓ}(υ_{g}) | = | liquid- (gas-) phase kinematic viscosity |
υ_{t} | = | kinematic viscosity |
Φ | = | porosity |
References
- ↑ ^{1.0} ^{1.1} ^{1.2} ^{1.3} ^{1.4} Riney, T.D. and Garg, S.K. 1985. Pressure Buildup Analysis for Two-Phase Geothermal Wells: Application to the Baca Geothermal Field. Water Resources Research 21 (3): 372.
- ↑ White, D.E. et al. 1975. Physical Results of Research Drilling in Thermal Waters of Yellowstone National Park, Wyoming. US Geological Survey, Menlo Park, California, professional paper No. 892.
- ↑ Stefansson, V. and Steingrimsson, B. 1980. Production Characteristics of Wells Tapping Two-Phase Reservoirs at Krafla and Namafjall Paths. Proc., Sixth Workshop on Geothermal Reservoir Engineering, Stanford University, Stanford, California, 49.
- ↑ ^{4.0} ^{4.1} ^{4.2} ^{4.3} Grant, M.A., Donaldson, I.G., and Bixley, P.F. 1982. Geothermal Reservoir Engineering, 76-107. New York City: Academic Press.
- ↑ ^{5.0} ^{5.1} ^{5.2} Grant, M.A., Garg, S.K., and Riney, T.D. 1984. Interpretation of Downhole Data and Development of a Conceptual Model for the Redondo Creek Area of the Baca Geothermal Field. Water Resources Research 20 (10): 1401 Cite error: Invalid
<ref>
tag; name "r5" defined multiple times with different content Cite error: Invalid<ref>
tag; name "r5" defined multiple times with different content - ↑ Kaspereit, D.H. 1990. Enthalpy Determination Using Flowing Pressure-Temperature Surveys in Two-Phase Wellbores in the Coso Geothermal Field. Geothermal Resources Council Trans. 14: 1211.
- ↑ ^{7.0} ^{7.1} Spielman, P. 1994. Computer Program to Analyze Multipass Pressure-Temperature-Spinner Surveys. Proc., Nineteenth Workshop on Geothermal Reservoir Engineering, Stanford University, Stanford, California, 147.
- ↑ Matthews, C.S. and Russell, D.G. 1967. Pressure Buildup and Flow Tests in Wells, 1. Richardson, Texas: Monograph Series, SPE.
- ↑ ^{9.0} ^{9.1} Earlougher, R.C. 1977. Advances in Well Test Analysis, 5. Richardson, Texas: Monograph Series, SPE.
- ↑ Streltsova, T.D. 1988. Well Testing in Heterogeneous Formations. New York City: John Wiley & Sons, Inc.
- ↑ Nisle, R.G. 1958. The Effect of Partial Penetration on Pressure Buildup in Oil Wells. Trans., AIME 213: 85.
- ↑ Brons, F. and Marting, V.E. 1961. The Effect of Restricted Fluid Entry on Well Productivity. J Pet Technol 13 (2): 172–174. SPE-1322-G. http://dx.doi.org/10.2118/1322-G.
- ↑ Tang, R.W. 1988. A Model of Limited-Entry Completions Undergoing Spherical Flow. SPE Form Eval 3 (4): 761-770. SPE-14310-PA. http://dx.doi.org/10.2118/14310-PA.
- ↑ _
- ↑ Garg, S.K. 1980. Pressure Transient Analysis for Two-Phase (Water/Steam) Geothermal Reservoirs. SPE J 20 (3): 206-214. SPE-7479-PA. http://dx.doi.org/10.2118/7479-PA.
- ↑ Garg, S.K. and Pritchett, J.W. 1984. Pressure Transient Analysis for Two-Phase Geothermal Wells: Some Numerical Results. Water Resources Research 20 (7): 963.
- ↑ Moench, A.F. and Atkinson, P.G. 1978. Transient Pressure Analysis in Geothermal Steam Reservoirs with an Immobile Vaporizing Liquid Phase. Geothermics 7 (2–4): 253.
- ↑ Sorey, M.L., Grant, M.A., and Bradford, E. 1980. Nonlinear Effects in Two-Phase Flow to Wells in Geothermal Reservoirs. Water Resources Research 16 (4): 767.
- ↑ ^{19.0} ^{19.1} Grant, M.A. and Sorey, M.L. 1979. The Compressibility and Hydraulic Diffusivity of a Water-Steam Flow. Water Resources Research 15 (3): 684.
- ↑ ^{20.0} ^{20.1} ^{20.2} Arps, J.J. 1945. Analysis of Decline Curves. In Petroleum Development and Technology 1945, Vol. 160, SPE-945228-G, 228–247. New York: Transactions of the American Institute of Mining and Metallurgical Engineers, AIME Cite error: Invalid
<ref>
tag; name "r20" defined multiple times with different content Cite error: Invalid<ref>
tag; name "r20" defined multiple times with different content - ↑ ^{21.0} ^{21.1} ^{21.2} Fetkovich, M.J. 1980. Decline Curve Analysis Using Type Curves. J Pet Technol 32 (6): 1065–1077. SPE 4629-PA. http://dx.doi.org/10.2118/4629-PA.
- ↑ Sanyal, S.K. et al. 1991. A Systematic Approach to Decline Curve Analysis for the Geysers Steamfield, California. Geothermal Resources Council Special Report 17: 189.
- ↑ Faulder, D.D. 1996. Permeability-Thickness Determination from Transient Production Response at the Southeast Geysers. Geothermal Resources Council Trans. 20: 797.
- ↑ Enedy, S.L. 1987. Applying Flow Rate Type Curves to Geysers Steam Wells. Proc., Twelfth Workshop on Geothermal Reservoir Engineering, Stanford University, Stanford, California, 29.
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See also
Geothermal reservoir engineering
Modeling geothermal reservoirs