occurs at the same Reynolds numbers Re as it does in smooth pipes, which is Re. 2300 [Atkinson, 1986]. Experimental results of Lomize [1951] and, later, Louis.
WATER RESOURCES RESEARCH, VOL. 33, NO. 3, PAGES 407– 418, MARCH 1997
Observation and simulation of non-Darcian flow transients in fractured rock T. Kohl,1 K. F. Evans,1,2 R. J. Hopkirk,3 R. Jung,4 and L. Rybach1 Abstract. Two independent multirate flow experiments were conducted in 1994 in the open hole depth interval of a well bore at the hot dry rock (HDR) test site Soultz. The steady state and transient downhole pressure records gave clear indications of nonDarcian flow. A numerical model has been set up to evaluate these two measurements. An excellent fit of the transient pressure responses of the two flow tests could be achieved by assuming one model of simple geometry. The model predicts fluid transport along a conduit with substantial surface area in which fully turbulent flow is occurring. All the parameters required by our best-fit simulation fall into a physically reasonable range. Sensitivity analysis demonstrates a non-Darcian flow regime along highly conductive features. The existence of high capacity far-field faults as postulated in our model confirms earlier characterizations of the Soultz test site.
1.
Introduction
The characterization of flow behavior in fractured rock under disturbed conditions is important for the disposal of radioactive waste, geothermal utilization by hot dry rock (HDR) systems, oil and gas production from fractured reservoirs, and water production from fractured rock. Recent activity in developing HDR systems has succeeded in attracting both public and political attention and has produced new field and theoretical results that could have a significant influence on other hydrologic disciplines. HDR systems are attractive because they offer the prospect of producing electrical and thermal energy from deep rock formations in areas of high population density since they primarily require only hot (preferably T . 1508C for electrical production) crystalline rocks at drillable depths [Armstead and Tester, 1987]. The HDR concept involves essentially forced flow of cool fluid injected into a hot fractured rock between two or more boreholes. The natural permeability of the reservoir is enhanced by performing massive fluid injections. During operation, thermal energy is extracted from the underground reservoir by circulating water which transfers the energy to the surface. The experience obtained from world-wide HDR research has highlighted the need to better understand the processes that influence the flow field within the reservoir fracture system. Important HDR activities have taken place in the United States (Fenton Hill), Japan (Hijiori, Ogachi), Great Britain (Rosemanowes Quarry), and France (SoultzsousForeˆts). Commercial operation of an HDR system requires, in general, flow rates in excess of 50 L s21. At such high rates the flow velocity in the individual fractures can be extremely high (above 1 m s21) and non-Darcian effects have to be investigated. The studies which have questioned the validity of Darcy flow 1
Institut fu ¨r Geophysik, Zu ¨rich, Switzerland. Also at Solexperts, Schwerzenbach, Switzerland. Polydynamics Engineering, Ma¨nnedorf, Switzerland. 4 Bundesanstalt fu ¨r Geowissenschaften und Rohstoffe, Hannover, Germany. 2 3
Copyright 1997 by the American Geophysical Union. Paper number 96WR03495. 0043-1397/97/96WR-03495$09.00
in fractures were strongly influenced by investigations of the hydraulic behavior in pipes from the early century [Blasius, 1913; White, 1929; Forchheimer, 1930; Nikuradse, 1933]. The studies are described by Schlichting [1979]. Fracture flow studies are based mainly on laboratory tests [Lomize, 1951; Baker, 1955; Louis, 1967; Sharp, 1970; Rissler, 1977; Witherspoon et al., 1980; Atkinson, 1986]. With the exception of that of Witherspoon et al. [1980] the studies all conclude that non-Darcian flow is likely to occur at even moderate flow rates. This effect manifests itself by decreasing transmissivity with increasing flow. To our knowledge there are few reported studies of flow regime in fractured rock under in situ conditions [Baker, 1955; Jung, 1989; Tulinius et al., 1996]. Baker [1955] compared several steady state differential pressure versus production rate data from oil field production tests to theoretical drawdown curves. By means of laboratory experiments on a radial flow field he was able to define a flow-dependent critical radius around a borehole in which turbulent flow occurs. Jung [1989] conducted injection tests to study turbulent energy losses developed at the intersection of the well bore with a fracture. From pressure measurements in further observation wells he found that the pressure drop around the borehole was linearly related to injection rates up to 0.5 L s21. At higher rates the relation became nonlinear in a manner that was consistent with the development of a circular turbulent region within the fracture whose radius increased with flow rate. Its size could be explained with a model proposed by Rissler [1977]. Non-Darcian flow behavior was observed by Tulinius et al. [1996] during a series of production tests ranging from 5 to 35 L s21 which were interrupted by hydraulic stimulations. The decrease of turbulent losses due to the successive stimulations was documented by a graphical fit of the data assuming that each level reached steady state after 1 hour. Despite the evident importance of non-Darcian flow behavior, there exists only a limited number of detailed and quantitative interpretations of the nature of the problem. If deviations from Darcy’s law are admitted, they are mostly qualitatively attributed to “skin effects” [e.g., Tulinius et al., 1996] limited to a few centimetres close to the well bore. We are aware of two different approaches to simulate numerically such effects. They both examined the two-regime (Darcian/
407
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KOHL ET AL.: NON-DARCIAN FLOW TRANSIENTS
non-Darcian) steady state pressure in a radial flow field [Rissler, 1977; Atkinson, 1986]. Rissler [1977] monitored the steady state pressure field monitored in a roughened fracture through a 2-m diameter Plexiglas disc. Atkinson [1986] analyzed the flow field in a 1.5-m-diameter, deformable fracture with both surfaces in intimate contact. Both authors conclude that a significant portion of pressure differences developed across their models were due to non-Darcian flow in the vicinity of a well bore. Up to now, there seems to be no numerical analyses available on the transient behavior of non-Darcian flow field. For the purpose of investigating the transient data sets from the HDR test site Soultz s.F., the finite element code FRACTure has been extended to handle non-Darcian flow in fractures. First applications have shown the importance of including such flow assumptions in modeling these data [Kohl et al., 1995]. In this paper we focus on simulating the 1994 production and injection test series of the Soultz site.
2.
Hydraulic Flow Regimes
Laminar flow behavior in a porous medium is described by Darcy’s law, which expresses a linear relationship between pressure gradient and flow rate. Neglecting gravity effects it can be expressed by Q/A c 5 2K z ¹P
(1)
K 5 k/ m
(2)
with
where ¹P is the pressure gradient, K is the hydraulic conductivity, Q is the flow rate, A c is the cross section perpendicular to flow, k is the permeability, and m is the viscosity. Substituting Darcy’s law into the continuity equation yields SS
P 5 ¹ z ~K z ¹P! t
(3)
where S s is the specific storage coefficient and t is time. For horizontal flow in a layered medium, (3) is often expressed with a transmissivity instead of the hydraulic conductivity and with storativity instead of the specific storage coefficient. The transmissivity for flow between smooth parallel surfaces can be described by the so-called cubic law. According to Lamb [1957], this law is also valid for flow between moderately variable and curved surfaces provided that the pressure gradient is sufficiently small (smaller than r c r g with r c being the radius of curvature). Higher curvature, however, inevitably leads to deviations from (3). This restricts the validity of Darcy’s flow law to laminar flow between smooth, slightly curved surfaces. In describing flow in fractures, an analogy with flow in parallel smooth plates is often used. As pointed out by Sharp [1970], such idealizations are of limited utility and even exclude junctions of different flow paths. Such geometrical restrictions generally result in deviations from a Darcy relationship with increasing flow rate far before the onset of turbulence. Prandtl et al. [1931] explains the deviations observed in nonuniform flow channels due to a continuous increase of energy losses by increasing internal kinematic forces. Thus non-Darcian flow starts when these inertial fluid forces reach the order of magnitude of the internal viscous fluid forces. This also explains why the flow regime prevailing between rough-walled surfaces does not immediately switch from Darcy flow to turbulent flow
as velocity is increased or vice versa. The following quadratic polynomial with coefficients A and B expresses the general flow behavior in transition [Forchheimer, 1930]: ¹P 5 AQ 1 BQ 2
(4)
with A 5 1/KA c (introduced in (1)). It should be mentioned that the quadratic term in (4) is written with a different exponent by some authors (powers between 1.8 and 2.7 can be found). It should also be noted that the parameters A and B are dependent upon Reynolds number in the transition range. The Reynolds number for onedimensional (1-D) flow between parallel plates is given by Re 5 @2 r ~Q/h!#/ m
(5)
where Q/h is the flow rate per unit height of a vertical fracture. When applying flow rate boundary conditions to such 1-D flow, (5) illustrates that the parallel plate Reynolds number is apparently independent of aperture. It is generally considered that the onset of turbulent flow between two smooth plates occurs at the same Reynolds numbers Re as it does in smooth pipes, which is Re . 2300 [Atkinson, 1986]. Experimental results of Lomize [1951] and, later, Louis [1967] also indicate that nonlaminar behavior begins before full turbulence develops. Both investigators studied the pressure gradients developed by parallel flow between roughened artificial fracture surfaces. Their results were expressed as empirical formulae which describe different flow regimes in a fracture. In Louis’ [1967] formulae the relative roughness is defined as the ratio between the mean absolute height of asperities, «, and the hydraulic diameter, D h . For a planar fracture the hydraulic diameter corresponds to twice the mean aperture. Values close to zero represent smooth fractures, whereas for sinusoidal roughness profiles the fracture surfaces just touch when «/D h 5 0.5. Louis’ results for rough fracture surfaces indicate deviations from Darcian flow at equivalent parallel plate Reynolds numbers as low as 500. Lomize’s [1951] results show that this threshold may even migrate down to Re ' 100 for a rough, narrow fracture. One-dimensional flow measurements conducted by Witherspoon et al. [1980] seem to confirm the validity of the cubic law up to a maximum value of Re 5 100. However, according to literature studies performed by Gale et al. [1985], deviations from Darcy’s law have been measured in the laboratory even at Re ' 80. The consequences are obvious: Given Re 5 2300 for the onset of non-Darcian flow and the typical values for water at 1008C (r 5 970 kg m23, m 5 3 3 10204 Pa s), (5) requires that the flow per unit fracture height has to be greater than 0.35 L s21 m21. However, only 0.017 L s21 m21 is required for Re 5 100. The comprehensive study of Louis [1967] on empirical flow laws examined 1-D, parallel flow through fractures of different relative roughnesses. Fully turbulent flow between smooth surfaces («/D h 5 0.0) was found to be described by Blasius’ [1913] law (also known as Blasius resistance formula), the same flow pattern between moderate rough surfaces (up to «/D h , 0.032) by Nikuradse’s [1933] law, and fully turbulent flow through rough surfaces («/D h . 0.032) by the Louis law. For our applications purposes, only the Nikuradse and Louis laws will be reported since smooth fracture surfaces are not likely to be met in natural fractures or joints. To maintain consistency with the variables already introduced, Louis’ formula for turbulent flow between rough surfaces is written in terms of flow rate and pressure rather than
KOHL ET AL.: NON-DARCIAN FLOW TRANSIENTS
velocity and piezometric head. Making the necessary substitutions, Louis’ law becomes ˜ ~¹P! 1/ 2 Q/h 5 2aK
ditions, turbulent-like flow is likely to occur at very low parallel plate Reynolds numbers.
(6)
3. where ˜ K 5 4 log
S DS D 1.9 «/D h
a r
1/ 2
(7)
with a being the aperture of a fracture and r being the fluid density. Louis’ law (equation (6)) has the same form as the Forchheimer equation (equation (4)) with the linear term set to zero. Nikuradse’s law can be obtained from Louis’ law by replacing the constant 1.9 in (7) by 3.7. Thus the transient flow field is governed by the following equation: SS
409
P ˜ ~¹P! 1/ 2# 5 ¹ z @K t
(8)
We note that transient pressure fields are expected to show a different behavior under non-Darcian flow conditions (equations (8)) than under Darcy flow conditions (equations (3)). The laboratory experiments of Sharp [1970] were consistent with Darcy’s law up to a pressure gradient of 2.5 kPa m21 but deviated at higher pressure. His data on fully turbulent flow regimes obtained at pressure gradients .30 kPa m21 showed, however, that flow rate varied with pressure gradient raised to the power of 0.55 rather than 0.5, as was found by Louis [1967] and Lomize [1951]. It has to be emphasized that both the results of the latter investigators are based on a higher number of measurements than those of Sharp [1970]. In essence, the following three different flow regimes had been investigated in fractures by the above mentioned studies: (1) Darcy flow at low Reynolds numbers, Q } ¹P; (2) nonDarcian (still laminar) flow due to geometrical heterogeneities (curvature of fracture surfaces, aperture changes, surface roughness, junction of fluid paths), Q not }¹P; and (3) nonDarcian (nonlaminar), turbulent flow at high Reynolds numbers, Q } (¹P) 1/ 2 . The investigations described above were carried out on largely open fractures in which only a small percentage of the fracture surfaces are in contact. The authors mostly agree on the description of Darcy flow (equation (1)) and of turbulent flow (with a slightly different exponential term but still similar to (6)). However, the transition range is not well understood. A crucial factor represents the value of the critical Reynolds number. Since it is derived from a smooth-walled geometry, deviations have to be taken into account for the significant geometrical variation inevitably experienced along a typical fracture flow path. The preferred approach to the interpretation of field data is presented by Louis [1967]. Owing to lack of better data and the fact that these empirical formulations are derived neither from fractures in intimate surface contact nor for complex twodimensional (2-D) flow systems, the interpretation of the present data sets is based on his formulations. Implicitly, it is assumed that most of the transition range between Darcy and fully turbulent flow can be described by (8). In order to avoid misinterpretations the term “turbulent-like” will be used to describe such flow regimes. Since values for relative roughness become higher for fractures in intimate contact, Lomize’s [1951] and Louis’ [1967] data indicate that under in situ con-
Hydraulic Tests at the Soultz Test Site
The European HDR project site is located in the middle part in the Rhine-Graben at Soultz s.F., about 50 km north of Strasbourg. The area is transversed by several large N-S striking faults, parallel to the trend of the graben structure. These faults largely show sinistral movement; that is, the western part has moved southward with respect to the eastern part. The maximum horizontal stress component (N1708E) is also directed approximately parallel to the graben structure and is almost equal in magnitude to the vertical stress [Klee and Rummel, 1993]. The minimum horizontal stress is only about half as large as the vertical stress and is thus approximately 10 MPa higher than the hydrostatic water pressure at 3 km depth. The Soultz area was selected as an HDR test site mainly for its high local surface temperature gradient (;1008C km21) and heat flow (above 0.150 W m22). However, subsequent drilling of the granitic basement (top at 1400 m depth) showed that the temperature gradient reduces down to values of 88C km21 below 2000 m before increasing again at greater depth to reach 328C km21 at 3500 m [Baria et al., 1995]. These fluctuations can be explained by the existence of thermal convection cells within the granite. Model simulations show that these convection cells require local permeabilities above 10215 m2 [Le Carlier et al., 1994]. In 1992 borehole GPK1 was drilled to a depth of 3.6 km and was cased to 2850 m. In 1993 several major stimulation tests were performed during which a volume of 45,000 m3 of fresh water was injected. Subsequent hydraulic tests conducted in 1994 indicated that the injectivity (i.e., the injected flow rate per unit of downhole pressure change) of the open hole section had been increased by a factor of 20. These hydraulic tests consisted of a 12-day production test in June 1994 (94JUN16) which was followed in July by an 8-day injection test (94JUL04). Both tests were conducted on the entire open hole section and were step pressure or step rate in nature. For each step level in these tests, flow rate or pressure was held constant until steady state conditions were approached, after which a new level was established. Pressure was measured by downhole gauges whereas flow rate was measured at the well bore head. In the 94JUN16 production test (Figure 1), fluid was produced at constant drawdown levels of 1.0, 0.31, and 0.71 MPa. These drawdowns led to approximately steady state production flow rates of about 11, 5, and 7 L s21 respectively. Production was enabled by the formation artesian pressure of 0.15 MPa and by buoyancy in the borehole. This experiment was performed by controlling the venting of GPK1 with a throttle valve so as to maintain constant pressure at the casing shoe. As can be seen in Figure 1, near–steady state conditions were attained for the 0.31- and 0.71-MPa production levels. Spinner logs run at the end of each flow sequence indicated that 56% 6 5% of the extracted flow came from the uppermost 50 m of the open hole section (Figure 3). The hydrostatic pressure at 2830 m was measured before the 94JUN16 test at 28.63 MPa. During the 94JUL04 test (Figure 2) constant injection flow rates of 6, 12, and 18 l s21 were used. These injection steps led to quasi–steady state pressure increases of 0.43, 1.70, and 3.07 MPa respectively. The spinner logs performed during 94JUL04 showed a flow profile similar to that during test 94JUN16, with 60% 6 6% of the fluid entering the formation over the same
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KOHL ET AL.: NON-DARCIAN FLOW TRANSIENTS
Figure 1.
Production test 94JUN16.
50-m-depth interval (Figure 3). In the following, we assume that this proportion remained constant throughout the two 1994 tests. In particular, we will focus our attention on the flow contributed by this 50 m interval and try to reproduce the transients recorded during the flow steps. The pressure “driving” the flow in these tests is the difference between the measured downhole pressure and the natural formation pressure, and we will refer to this as the downhole differential pressure, DP. Throughout the test series this pressure remained at least 5 MPa below the minimum stress component (i.e., below the pressure required for jacking). The plot of the steady state values of DP versus flow rate for both tests is shown in Figure 4. The relation is clearly not linear, as is expected for Darcy flow (equation (1)) but rather has the shape expected for fully turbulent flow with DP proportional to the square of flow rate, Q 2 (equivalent to the square of (6)). This observation has already been described by Jung et al. [1995]. Evans et al. [1996] examined the data from numerous hydraulic tests at Soultz and found that flow generally became non-Darcian at rates above 1 L s21. Such nonlinearity cannot
Figure 2.
be explained by buoyancy effects since especially during production no change with time of temperature or salinity was observed. Another important characteristic of the curves in Figures 1 and 2 is the fact that it takes days to approach steady state levels and that this period increases significantly at higher flow rate or pressure difference. In Figure 5 the time after the start of a step is plotted versus the normalized pressure increase of the first two steps in the 94JUL04 test. The normalization was performed over the maximum pressure change per step; thus, DP/DP END reaches “1” at the end of each step. For Darcy flow behavior these curves should all look similar. However, the first two steps of 94JUL04 show a strong discrepancy: the first step reaches steady state much faster than the 2nd or 3rd step. The latter two steps are nearly similar, a point which will be explained shortly. The modeling procedure described later will present a possible explanation for it. Poor control of the flow levels in the 94JUN16 test prevent the same procedure being applied to that test. These observations suggests that nonlaminar flow may be
Injection test 94JUL04.
KOHL ET AL.: NON-DARCIAN FLOW TRANSIENTS
411
common to fracture flow at moderate to high flow rates within the Soultz, and perhaps other fractured reservoirs.
4.
Numerical Tool
FRACTure is a versatile, three-dimensional (3-D) finite element program for the simulation of various processes in geoscience [Kohl and Hopkirk, 1995]. It was designed specifically for studying the coupling of different physical processes in the subsurface. Its flexible modular structure facilitates the addition of further processes and elements to the existing library and the handling of linear and nonlinear constitutive laws and the calculation of their interactions. In the current stage of development FRACTure can simulate individual hydraulic (laminar, turbulent), different transport (thermal, nonreactive solute, radioactive solutes), and elastic processes as well as special coupled interactions between these processes. Hydraulic calculations can be performed as function of the Reynolds number: Hydraulic regimes may switch from laminar to turbulent-like and vice versa. The name FRACTure reflects its functional capabilities (it is an acronym for flow, rock and coupled temperature effects) while at the same time emphasizing the fracture as the dominant hydraulic structure in crystalline rock. An iterative procedure, similar to the description given by Wollrath [1990], was chosen for the implementation of turbulent flow law in FRACTure. Therefore (8), which describes the constitutive laws for turbulent flow, was linearized: SS
P 5¹z t
S
˜ K z ¹P ~u¹Pu! 1/ 2
D
(9)
An iterative scheme is easily set up when the nonlinear expression u¹Pu 20.5 is treated explicitly. After each run through the iterative loop the appropriate nonlinear expression is actualized. Optimum convergence can be reached to update this value by applying a linear relaxation between two successive
Figure 3. Typical flowmeter log measured during 94JUN16.
Figure 4. Plot of steady state DP versus flow rate. The arrow indicates the direction of the transient [see Jung et al., 1995].
iteration steps. If the residuum of two successive calculations undergoes a certain value, the iterative loop is terminated. A verification of the implementation of nonlinear constitutive flow laws in FRACTure was performed with the steady state 1-D analysis described in section 5.2. It showed that convergence was reached after about 20 iterations by applying a minimum residuum of 1024. For the simulations from section 5.3 a time-dependent relaxation factor was used: at the beginning of each flow step, a factor of 0.5 was chosen and increased up to 0.95 when reaching nearly steady state.
5. 5.1.
Modeling General Considerations
Our intention was to set up one single model that could explain the pressure response of the two independent data sets 94JUN16 and 94JUL04 by the smallest number of degrees of freedom. Flow is assumed to take place in a relatively highly conductive zone (“conduit”) and in a poorly conductive zone (“bulk rock”). Thus, besides the borehole, only two material sets were utilized: conduit and bulk rock (Figure 6). In agreement with the stress field in 3000 m of GPK1, the conduit is assumed to be vertical. The geometry of the conduit is assumed to be rectangular with a height (h) and a length (L). It has an assigned uniform hydraulic aperture which, for reasons of simplicity, was taken as mean aperture of the conduit (a C ). According to the spinner logs, 56% of the total produced flow rate and 60% of the total injected flow rate entered/left the borehole over this 50-m section during both tests. The geometrical extension (height and length) of the conduit is, however, taken as variable in recognition of the possible vertical spreading or narrowing of the flow field within the formation or within the conduit. Flow within the conduit is assumed to be fully governed by either the Nikuradse or the Louis flow law (equation (8)). The vertical conduit connects the borehole to a far-field fault. The existence of such faults within the basement has been proposed by Jung [1992a] on the basis of the analysis of other hydraulic tests conducted at various levels within GPK1 and by Elsass et al. [1995] on the basis of geological grounds. The assumption of a vertical conduit is justified by the strong stress anisotropy present within the reservoir which favors flow within subvertical fractures striking approximately N-S [e.g.,
412
KOHL ET AL.: NON-DARCIAN FLOW TRANSIENTS
Figure 5. Normalized pressure increase of the three 94JUL04 steps.
Heinemann-Glutsch, 1994; Jupe et al., 1993]. This hypothesis is supported by the spatial distribution of the microseismic events recorded during stimulation [Jones et al., 1995] and by the fact that a highly permeable fault was intersected during the drilling of the second deep borehole, GPK2, causing the total and immediate loss of drilling mud [Baria et al., 1995]. In the present study it is assumed that such a fault has a capacity sufficiently large that it represents a constant potential boundary. Flow within the surrounding granitic bulk rock is assumed to obey Darcys’ law. The bulk rock itself is taken to be homogeneous and described by a uniform permeability, fluid viscosity and specific storage coefficient. Since the bulk rock is described by a continuum, “equivalent porous medium” approach, it includes the effect of other fractures present within the medium. The equivalent porous medium permeability of the bulk rock was allowed to vary from 3 3 10215 m2 to 3 3 10217 m2. Rummel [1992] reports matrix permeability values in the range of 10216 to 10218 m2 derived from laboratory tests on intact core samples. These values can only be considered as a rough guide to the real in situ matrix permeabilities since scale effects (introduced by macroscale, second-order fractures) will lead to higher permeabilities, and thermal and stress relief microcracking of the core samples will lead to lower permeabilities. Jung [1992b] determined an apparent in situ permeability of 2.5 3 10217 m2 for the GPK1 depth interval between 2850 and 3400 m prior to stimulation. The fluid viscosity is strongly temperature dependent and thus requires correction for the actual fluid temperature. According to Smith and Chapman [1983] the temperature dependence of fluid viscosity m can be estimated from the formula:
m 5 2.4 3 10 210 3 10 248.37/~T1133.15!
temperature of 658 and m 5 2.1 3 1024 Pa s21 for fluid at the undisturbed rock temperature of 1308C. This results at most in a 50% higher hydraulic conductivity of the bulk rock (K R ) in 94JUN16 than in 94JUL04. A crucial role in the curve fitting is played by the coefficient of specific storage of the bulk rock (S R S ) since it governs the long-term (hours to days) transient pressure response. Although fairly narrow bounds can be placed on the specific storage of intact rock [Evans et al., 1992] the presence of fractures in the reservoir demanded that we also vary the bulk value (see section 5.3). Since pressures remained 5–10 MPa below the jacking pressure during the test series, changes in conduit aperture in response to the pressure changes are likely to be small, although perhaps not insignificant, particularly at the highest injection pressure. For the purpose of this study the effects of conduit compliance on transmissivity are neglected. A series of models were run with different geometries. In the simulation process, first-order estimations of a C could be obtained by a steady state analysis. In a next step, different numerical transient forward calculations were performed with
(10)
with T, the temperature, in degrees celsius. For a produced fluid temperature of 1308C corresponding to the formation temperature at 2870 m, the dynamic fluid viscosity of 94JUN16 can be approximated as m 5 2.1 3 1024 Pa s21. In the 94JUL04 test the temperature of the injected fluid varied between 658 and 908C. Since it was intended to characterize bulk matrix properties with a single viscosity, the value chosen for the simulation of 94JUL04 could be varied between the extremum values of m 5 4.3 3 1024 Pa s21 for fluid at a
Figure 6. Schematic diagram of model geometry with a rough discretization. Flow leaves/enters the conduit from the borehole (left) and sweeps linearly across the conduit.
KOHL ET AL.: NON-DARCIAN FLOW TRANSIENTS
FRACTure to extract refined parameters on a C , K R , and the appropriate values for S R S . An option in the FRACTure code was used which allows time-flow functions to be defined so that the model’s extraction and injection history followed exactly those of the 94JUN16 and 94JUL04 data sets. 5.2.
Steady State Conditions
The assumption of non-Darcian flow conditions in a fractured zone is justified by the quadratic relationship between the steady state DP and Q data pairs obtained from the 1994 tests (see Figure 4). For given values of fracture height, h, and length, L, these steady state data can be used to place firstorder constraints on the mean conduit aperture, a C , for the case of impermeable bulk rock. This provides a useful scoping calculation for guiding the modeling procedure. The nonDarcian flow laws (equation (6)) must therefore be solved for the mean conduit aperture, and the solution must be applied to the measured pressure change at steady state conditions. Upon introducing the volumetric flow, Q, forming the square and rearranging, Eqn. 6 becomes: 1 ¹P 5 ˜ 2 C 2 2Q 2 K ~a ! h
(11)
Assuming an impermeable rock matrix, the steady state pressure change at the borehole DP inj can be obtained from the pressure gradient (¹P 5 DP inj/L). Thus L DP inj 5 ˜ 2 C 2 2Q 2 K ~a ! h
(12)
Comparing this expression to the quadratic term in (4) and introducing the variable B9 (B9 5 BL) yields L 5 B9 ˜ K 2~a C! 2h 2
(13)
Applying the modified Louis law (7) and solving for a C yields aC 5
1
Lr
B9 3 16 3
S
log
1.9 «/D h
D
2
h2
2
1/3
(14)
The value of B9 can be obtained from steady state data. The parabola described by (6) intersects the origin of the steady state pressure versus flow diagram and thus is completely determined by a single nonzero steady state data point. Taking account of the 60% portion of the injection flow, B9 can be calculated for the first step of 94JUL04 as follows: DP inj 5 B91~0.6 3 0.006@m3/s#! 2 5 0.43 3 10 6@Pa# f B91 5 3.32 3 10 10@kg/m7#
(15)
Applying the same procedure to the second step of 94JUL04 yields B92 5 3.28 3 10 10 [kg m27]. Accuracy sufficient for our considerations is obtained by taking the mean value of B9 as 3.3 3 1010 [kg m27]. Table 1 lists aperture values obtained from (14) using the mean value of B9 for various conduit lengths and heights. A fluid density r of 1000 kg m23 and two relative roughnesses considered: «/D h 5 0.05, which implies Louis’ law, and «/D h 5 0.001, which implies Nikuradse’s law. It should be noted that the constant B of Forchheimer’s equation (equation (4)) can be determined from B9 only if the a length L of the conduit is known.
413
Table 1. Appropriate Apertures to Fit the Steady State Pressure Levels With Nikuradse’s («/D h 5 0.001) and Louis’ Laws («/D h 5 0.05) Flow Law
L, m
h, m
a C, mm
Louis Nikuradse Louis Nikuradse Louis Nikuradse Louis Nikuradse Louis Nikuradse
1000 1000 1000 1000 1000 1000 200 200 200 200
10 10 100 100 200 200 100 100 200 200
1.97 1.14 0.42 0.25 0.27 0.15 0.25 0.14 0.16 0.09
5.3.
Transient Behavior
In the preceding steady state analysis, permeability effects of the matrix were ignored. When fluid exchange between the conduit and the matrix occurs, a numerical calculation is required to obtain a steady state solution of a C and for a transient simulation. Thus a numerical model was set up to simulate the transient pressure change for each step of 94JUN16 and 94JUL04 tests. The model consisted of 1300 nodes and 1200 matrix elements with linear shape functions. A rectangular grid shape (length L, width 2L) was taken for all calculations (schematically shown in Figure 6). Different lengths were simply scaled up. The grid is finest near the injection point, where the element length is L/ 2000. The conduit is modeled by 44 1-D elements which are embedded in 2-D elements. The time step was varied between 500 s at the beginning of each step to 10,000 s as the steady state level was approached. Two different sets of calculations, corresponding to different flow laws in the conduit, were performed: the Louis and the Nikuradse laws. In both cases three parameters could be constrained by the modeling: the coefficient of specific storage of the bulk rock (S R S ), the hydraulic conductivity of the bulk rock (K R ), and the aperture of the conduit (a C ). The specific storage coefficient of the conduit was taken to be the compressibility of water (S S 5 5 3 10 210 Pa21). A borehole storativity (specific storage coefficient times volume of open hole section) value of 5 3 1028 m3 Pa21 was assumed. Sensitivity analysis showed that both values remain unaffected by a parameter tuning in a range from 1026 to 10212. Although, in this first simple model, rock elasticity and fracture (conduit) compliance were not activated in the program, this band of insensitivity covers the values expected from a compliant fracture (up to 1027 m Pa21). The results of the fitting procedure are illustrated in Figures 7 and 8 for the case of a 1000-m-long conduit of 100 m height and Louis’ flow law. The parameter values used in this model run are listed in Table 2 and will be discussed below. A good fit to the observations was obtained for all production levels for 94JUN16 and for the first two injection levels of 94JUL04. Furthermore, the initial part of the third-level transient of 94JUL04 is also well fitted. As expected from the steady state analysis, the long-term change in pressure toward an asymptotic level for five flow steps is well predicted. Within the framework of the model the transient is primarily governed by fluid exchange between the conduit and the matrix. Flow within the conduit is comparatively quickly established and thus influences only the initial steep rise in the transient. Ow-
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Figure 7. Results of the simulation of the production test 94JUN16. The upper plot indicates the measured flow history and main points of the applied flow-time function (open circles) that has to be linearily interpolated. On the lower graph the measured and the modeled pressure increase can be recognized.
ing to the different flow laws, the hydraulic impedance increases with the square of flow rate along the conduit but only linearly with flow rate into the matrix. With increasing flow rate, this results in higher pressure gradients not only along the conduit but also from the conduit into the matrix. It is this latter pressure gradient that drives the fluid exchange from the conduit to the matrix and accordingly leads to a prolongation of the transient phase with increasing flow rate. Both data sets show such characteristics and are therefore excellent examples of flow rate– dependent transients. As noted above, the long-term trend observed for the 18 l s21 injection step is the only part of the entire test sequence not well fitted by the model. The predicted pressure (Figure 8) is significantly higher than the observed for both the transient phase and the steady state asymptote level. Since the absolute fluid pressure within the conduit is greatest for this step, reaching some 7 MPa below jacking pressure, we suggest that the deviation reflects fracture dilatation. The amount required is rather small: about 8% of the total aperture. Such dilatation is unlikely to be due to compliance effects but rather due to a sudden, inelastic change such as accompanies a shear event. Further evaluation of this possibility, however, is beyond the scope of this paper. Because of the geometrical nonuniqueness of the problem, which is evident in (14), different geometries were investigated with L varied between 200 and 1000 m and h varied between 100 and 500 m. The optimum parameter set of a C , K R , and S R S for each geometry that was extracted by the forward modeling procedure is listed in Table 2. Each parameter set generated
fits to the data which were of similar quality to those shown in Figures 7 and 8. In general, of all parameters, S R S is most sensitive to the model geometry. If the crystalline rock through which leak-off occurs is considered free of open, compliant natural fractures, 211 then we would expect S R Pa21 S values of the order of 10 [Evans et al., 1992, p. 105]. However, in recognition of the presence of natural fractures, and the fact that the zone in question has been subjected to a major stimulation injection, increases up of to a factor of 10 must be considered plausible. Table 2 shows that this constraint is satisfied by models which have a comparatively large swept area (above 0.1 km2). For example, if the conduit length is taken as 1000 m, then a height larger than 100 m is required to fit the data. Smaller geometries yield values for S R S which are too high. The required values of a C and K R are linked together, with larger conduit apertures requiring lower bulk rock conductivities. Small variations up to 5% of the value of one parameter can be compensated for by adjusting the value of the other one. The effect when exceeding this bound is illustrated in Figure 9: The first production step shows the strongest sensitivity to K R ; the other production and injection steps are essentially unaffected by the 30% reduction. During the modeling procedure the K R value for the production test had to be chosen approximately 30% higher than that for the injection test. This may indicate viscosity effects due to injection of cooler fluid. Given the duration of the tests, such effects are only significant in the vicinity of the well bore where the bulk rock cools off the most. The best-fit K R value for the model with a 1000-m-long conduit
KOHL ET AL.: NON-DARCIAN FLOW TRANSIENTS
415
Figure 8. Results of the simulation of the injection test 94JUL04. At the top the measured and the assumed flow history; at the bottom, the measured and the modeled pressure history.
corresponds to a permeability of 4 3 10216 m2, which is 1 order of magnitude higher than the prestimulation in situ bulk rock permeability at GPK1 [Jung, 1992b]. The absolute value of the mean conduit aperture depends strongly upon the relative roughness of the conduit surface and remains questionable. Usage of a different relative roughness («/D h , 5 0.05 or 0.001), which implies Louis or Nikuradse flow laws, effects only a C . Best-fit solutions obtained using Louis’ flow law require nearly the double aperture of Nikuradse’s law. It can be concluded that models having comparatively large conduit surface areas result in physically reasonable bounds for the values of a C , K R , and S R S . Applying our results and the
Table 2. Fracture Dimensions and Parameter Values Required to Obtain a Satisfactory Fit to the Transient Data for Nikuradse’s («/D h 5 0.001) and Louis’ Laws («/D h 5 0.05) K R , m2 Pa21 s21 L, m
h, m
SR S, Pa21
1000 1000
200 100
2.0 3 10211 8.0 3 10211
1000 1000 200 200
200 100 500 200
2.0 3 10211 8.0 3 10211 2.0 3 10210 1.0 3 10209
94JUL04
a C, mm
Nikuradse 1.9 3 10212 1.9 3 10212
1.2 3 10212 1.2 3 10212
0.14 0.22
Louis 1.9 3 10212 1.9 3 10212 0.5 3 10212 0.5 3 10212
1.2 3 10212 1.2 3 10212 0.3 3 10212 0.3 3 10212
0.24 0.38 0.08 0.14
94JUN16
smallest flow rate (with h 5 100 m, Q 5 60% of 5 l s21, r 5 1000 kg m23, and m 5 2.2 3 10204 Pa s) to the Reynolds number definition in (5) shows that at Re ; 280, non-Darcian flow seems to occur. This supports the findings of Lomize [1951] and others, that the “parallel plate” critical Reynolds number of 2300 cannot be applied to fractured rock.
6.
Discussion of Results
It is of interest to examine whether models in which nonDarcian flow is restricted to the vicinity of the well bore can fit the data. Three different models were run in which the nonlaminar zone extended from the borehole over distances of zero (Darcy flow), 0.05L and 0.2L. The model parameters derived in fitting the first step of 94JUL04 were further used to simulate the following steps of this experiment and all pressure steps of 94JUN16. The results for the 0.05L and 0.2L runs are shown in Figure 9. Evidently, the effect of localizing the nonlaminar zone to the vicinity of the borehole is to diminish the quality of the fit to the data for the other flow steps. Owing to its simpler flow history, the 94JUL04 test is better suited to explain the results of these tests. The linear pressure increases obtained using a Darcy flow law (0.5, 1.0, and 1.5 MPa) are closely approximated in the 0.05L run (0.5, 1.1, and 1.8 MPa). The pressure increase in this run can be separated into linear increases of 0.45, 0.9, and 1.35 MPa and nonlinear increases of 0.05, 0.2, and 0.45 MPa. By applying the same procedure for the 0.2L run (with pressure increases of 0.5, 1.3, and 2.4 MPa), we obtain linear increases of 0.35, 0.7, and 1.05 MPa and nonlinear pressure increase of 0.15, 0.6, and 1.35
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KOHL ET AL.: NON-DARCIAN FLOW TRANSIENTS
Figure 9. Sensitivity study showing the effects of parameter variations for 94JUN16 and 94JUL04: The 0.05 L curve is nearly identical to the result of a fully Darcy flow model (for details see text).
MPa. Further increase of the turbulent-like zone within the conduit will result in decreasing linear impedance and in increasing nonlinear impedance. Since significant linear impedances cannot be extracted from the data, it is implied that nonlaminar flow dominates the major part of the conduit. However it cannot be excluded that intermittent zones of Darcy flow occur along the conduit. On the basis of these considerations, a dominating radial or spherical expansive flow field within the conduit seem unlikely. Such flow paths necessarily result in transition to a Darcy regime at some distance from the borehole. Our findings thus support the statement of Atkinson [1986], who suggested that in a stress field favoring vertical fractures, the nonlaminar flow regime might extend throughout the total fracture length, whereas horizontal fractures are rather submitted to Darcy flow. Consideration of “skin effects” near the borehole might lead one initially to expect that in this region a different, more complex non-Darcian flow regime could dominate. This could be described by similar nonlinear relations but by using different coefficients [Baker, 1955]. Obviously, in our case the hydraulic stimulation of that zone has been successful enough that such effects did not arise around the borehole. Another restriction that could call the results into question concerns the assumed simplicity of the model. The homogeneity of the two materials is a feature of continuum modeling procedures. In reality these sets represent different hydraulic structures which themselves consists of a series of heterogeneities. The conclusion made from these simulations is that the rock matrix adjacent to the open hole section of GPK1 can be described by two large scale representative elementary vol-
umes (REVs): the one represents a highly conductive zone (“the conduit”) and the other represents a poorly conductive zone (“the bulk matrix”). The conduit can include a main fracture system formed by multiple, nearly subparallel flow paths oriented towards the far-field fault and flow channels formed by fracture intersections. It is rather unlikely that the conduit consists of a single large fracture of uniform aperture. The bulk rock may consist of intact rock and second-order fractures that are not orientated toward the local drainage sink (i.e., the far-field fault) or which have a significant lower hydraulic conductivity. Owing to the geometrical nonuniqueness of the problem, the REVs cannot be any further refined. The location of the farfield fault necessarily needs to remain speculative. Only by means of further transport data like solute or thermal tracers can the geometry be further constrained. This will lead to a more complex model geometry but will not question the existence of large active fracture.
7.
Conclusions
The two independent multirate flow experiments, 94JUN16 and 94JUL04, conducted at the Soultz HDR test site show clear indications of non-Darcian flow. The transient pressure response of these two hydraulic production and injection experiments can be well explained by the same model assuming fluid transport along a conduit with substantial surface area in which fully turbulent flow is occurring. The parameters required to fit the data all fall into a physically reasonable range, provided that swept fracture areas above 0.1 km2 are chosen.
KOHL ET AL.: NON-DARCIAN FLOW TRANSIENTS
From a total of six flow steps, five were well fitted by a single model of simple geometry. Specifically, the steady state pressure change and the increase in the duration of the transient phase with flow rate are well predicted. Sensitivity analysis highlighted the need for a nonlaminar flow regime along highly conductive features. The far-field high-capacity faults which serve as a boundary condition in our model approach lead to a funneling of flow paths and correspond to earlier characterizations of the Soultz test site a partially open system. To our knowledge, this presents the first transient interpretation of hydraulic field measurements in fractured rock by utilizing a nonlaminar flow analysis. The present study demonstrates that the empirical relationship by Louis [1967] based on laboratory test data can be applied to field data. This emphasizes that a transition from a Darcy to a turbulent flow regime is likely to occur at equivalent parallel plate Reynolds numbers much smaller than 2300. In the present data set nonlaminar flow conditions are manifested down to flow rates of 5 l s21. The question as to whether these observations are unique to the Soultz site or are a common feature at moderate to high flow rates in fractured rock can be partly answered. Since divergent flow paths result in slower flow velocities at larger distances, the observed nonlaminar effects along large portions of fractures require quasi-parallel flow. Such conditions are met when the borehole is parallel to a fracture system (generally vertical) and when a large drainage system can be hydraulically connected. Such a drain does not necessarily need to be a far-field fault; it also might be the other leg of a duplet system. Thus our simulation strongly suggests that hydraulic tests in fractured rock be designed so that flow patterns can be derived and investigated. The importance of a general investigation of nonlaminar hydraulic flow patterns in fractured rock should lead to new efforts in designing laboratory or small field experiments with real fractures. The necessary flow laws for the transition zone between Darcy and nonlaminar flow can thereby be extended, and appropriate laws for turbulent-like flow can be confirmed or newly derived. Acknowledgments. This study is part of the investigation of the Soultz HDR site, European Hot Dry Rock project. We thank the SOCOMINE core team responsible for supervising the field experiments. The test program was performed under the financial and logistic support of DGXII of the European Commission; ADEME, BRGM, and CNRS; (France) and the BMBF (Germany). The authors would like to thank especially the Swiss “Bundesamt fu ¨r Bildung und Wissenschaft” (BBW) for generously supporting this project. This is publication 932, Institute of Geophysics, ETH Zurich.
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thesis, Inst. fu ¨r Stro ¨mungsmechan., no. 28, Univ. Hannover, Hannover, Germany, 1990. K. F. Evans, T. Kohl, and L. Rybach, Institut fu ¨r Geophysik, ETHHo ¨ nggerberg, CH-8093 Zu ¨ rich, Switzerland. (e-mail: Kohl@geo. phys.ethz.ch) R. J. Hopkirk, Polydynamics Eng., Bahngasse 3, CH 8708, Ma¨nnedorf, Switzerland. R. Jung, Bundesanstalt fu ¨ r Geowissenschaften und Rohstaffe, Skilleweg 2, D-30631, Hannover, Germany. (Received March 26, 1996; revised September 27, 1996; accepted November 8, 1996.)
