Inflow and Outflow Signatures in. Flowing Wellbore Electrical-Conductivity Logs. Abstract

Transcription

1 LBNL Inflow and Outflow Signatures Flowg Wellbore Electrical-onductivity Logs hriste Doughty and h-fu Tsang Earth Sciences Division E.O. Lawrence Berkeley National Laboratory University of alifornia September 22 Abstract Flowg wellbore electrical-conductivity loggg provides a means to determe hydrologic properties of fractures, fracture zones, or other permeable layers tersectg a borehole saturated rock. The method volves analyzg the time-evolution of fluid electricalconductivity logs obtaed while the well is beg pumped and yields formation on the location, hydraulic transmissivity, and sality of permeable layers, as well as their itial (or ambient) pressure head. Earlier analysis methods were restricted to the case which flows from the permeable layers or fractures were directed to the borehole. More recently, a numerical model for simulatg flowg-conductivity loggg was adapted to permit treatment of both flow and outflow, cludg analysis of natural regional flow the permeable layer. However, determg the fracture properties with the numerical model by optimizg the match to the conductivity logs is a laborious trial-and-error procedure. In this paper, we identify the signatures of various flow and outflow features the conductivity logs to epedite this procedure and to provide physical sight for the analysis of these logs. Generally, flow pots are found to produce a distctive signature on the conductivity logs themselves, enablg the determation of location, flow rate, and ion concentration a straightforward manner. Identifyg outflow locations and flow rates, on the other hand, can be done with a more complicated tegral method. Runng a set of several conductivity logs with different pumpg rates (e.g., half and double the origal pumpg rate) provides further formation on the nature of the feed pots. In addition to enablg the estimation of flow parameters from conductivity logs, an understandg of the conductivity log signatures can aid the design of follow-up loggg activities. 1

2 1. Introduction In the study of flow and transport through fractured rocks, knowledge of the locations of fractures and their hydraulic properties is essential. Often such knowledge is obtaed usg deep boreholes penetratg the fractured rock. Among the various downhole methods of determg fracture properties that have been developed over the past few decades, flowg wellbore electrical conductivity-loggg has proved to be quite useful (Tsang et al., 199). In this method, the wellbore water is first replaced by de-ionized water or, alternatively, water of a constant sality distctly different from that of the formation water. This is done by passg the de-ionized water down a tube to the bottom of the borehole at a given rate, while simultaneously pumpg from the top of the well at the same rate, for a short time period. Net, the well is shut or pumped from the top at a constant low flow rate (e.g., tens of liters per mute), while an electric conductivity probe is lowered to the borehole to scan the fluid electric conductivity (FE) as a function of depth. This produces what is known as an FE log. With constant pumpg conditions, a series of five or si FE logs are typically obtaed over a one- or twoday period. At depth locations where water enters the borehole (the feed pots), the FE logs display peaks. These peaks grow with time and are skewed the direction of water flow. By analyzg these logs, it is possible to obta the flow rate and sality of groundwater flow from the dividual fractures. The method is more accurate than spner flow meters and much more efficient than packer tests (Tsang et al., 199). Although we often refer to feed pots as representg flow through hydraulically conductive fractures, they can just as easily represent flow through any permeable zone that tersects the wellbore section beg logged. In heterogeneous porous media such as alluvial systems composed of terspersed sand and clay lenses, flow can be just as localized as 2

3 fractured rock, and the need for identifyg permeable strata just as great. The method developed this paper is equally applicable to such heterogeneous media. Figure 1.1 shows a typical series of FE logs obtaed from a 23 m well (olog, Inc., personal communication, 1999). Key features apparent the logs clude (a) an isolated peak at a depth of 164 m with an unusually sharp upper limb; (b) several terferg peaks the depth range of m; and (c) an overall downward propagation of peaks below a depth of 164 m. The goal of the present study is to vestigate the typical signatures FE logs produced by different combations of feed pots and flow conditions. Further, tegral measures will be derived from the FE logs, to facilitate analysis of logs such as these. Eistg tools for analyzg flowg wellbore FE logs clude analytical solutions, numerical modelg, and tegral approaches. Simple analytical solutions based on mass balances can be used to fer feed-pot properties at early times before peaks terfere with each other (Tsang et al., 199); under steady-state conditions when peaks fully terfere (Tsang et al., 199); and for the special case of horizontal flow (Drost et al., 1968). These solutions provide useful formation when used as part of a more sophisticated analysis (as described Section 2), but by themselves are too simplistic for most real-world problems. The numerical model BORE (Hale and Tsang, 1988; Tsang et al., 199) and the recently enhanced version BORE II (Doughty and Tsang, 2) calculate the time evolution of ion concentration (sality) through the wellbore, given a set of feed-pot locations, strengths, and concentrations (i.e., the forward problem). BORE II broadens the range of applicability of the analytical solutions described above by considerg multiple flow and outflow feed pots, isolated and overlappg FE peaks, early-time and late-time behavior, time-varyg feed-pot strengths and concentrations, and the terplay of advection and diffusion the wellbore. The 3

4 Appendi presents the governg equations used by BORE II. Usg BORE II to match observed FE profiles (the verse problem) requires the trial-and-error adjustment of feed-pot parameters. This can be a difficult and time-consumg process, especially for noisy data. Several tegral approaches have been developed (Tsang and Hale, 1989; Loew et al., 199) that can provide good itial guesses to BORE, greatly enhancg its ease of use. However, these methods are limited to wellbore sections contag flow pots only. Thus, for eample, they are not applicable to cases of horizontal flow across the well diameter or ternal wellbore flow. The motivation for the current vestigation of feed-pot signatures is threefold. First, we want to develop a method for makg zero-order parameter estimates for the general case which flow and outflow pots eist. These can be used as good itial guesses for BORE II, order to epedite the analysis of wellbore FE logs. Second, we want to obta physical sight to the nature of the feed pots from particular features or signatures the FE logs. Third, we want to determe the optimal scheme for collectg new data by vestigatg how the choice of pumpg rate can enhance feed-pot signatures. Differences feed-pot strengths among various permeable layers or fractures reflect differences local hydraulic transmissivity values or pressures or both, and by varyg the pumpg rate we can discrimate between these cases. In Section 2, we eame the typical signatures observed a wellbore FE log, and determe what they tell us about the feed-pot parameters. For flow pots we look at the FE profile itself, while for outflow pots we troduce an tegral analysis. Section 3 vestigates the effect of pumpg rate on these signatures, both to decide on the optimal pumpg rate for log terpretation and to vestigate the nature of feed pots. Section 4 illustrates the application of the techniques by analyzg FE logs obtaed from two field sites. Section 5 summarizes the material and presents some concludg remarks. 4

5 2. Signatures of Inflow and Outflow Pots The signatures of dividual flow feed pots and various combations of flow and outflow feed pots conta both qualitative and quantitative formation about the feed-pot parameters, which can be used to improve the itial parameter values for a BORE II simulation. In these discussions, we refer to concentration profile (), which is the ion concentration of borehole fluid as a function of depth. The conversion of an FE log, σ(), to a concentration profile, (), is described the Appendi. Boreholes that conta only flow feed pots produce the most straightforward () signatures. The signatures of outflow feed pots are generally difficult to see the () profiles, and therefore a more elaborate mass-tegral technique has been developed for cases which both flow and outflow pots eist. 2.1 oncentration Profiles The four quantities that need to be determed for each feed pot are location i, flow or outflow rate q i (positive for flow and negative for outflow), and, for flow pots, concentration i and the time t i at which feed-pot concentration first differs from the itial wellbore concentration. Often t i =, but a non-zero value can occur if de-ionized water migrates to the fracture durg the itialization phase of replacg borehole water prior to wellbore loggg, or if wellbore loggg takes place durg a tracer test which a sale tracer arrives from a nearby well or source Inflow Pots Figure 2.1 shows a series of idealized concentration profiles simulated with the numerical model BORE II, usg feed pots with constant q i >, constant i >, and t i =. Factors such as 5

6 upflow from the bottom of the wellbore section, time-dependent feed pots, flow pots with i =, and outflow pots are discussed separately later subsections. The first step is to locate the feed pots. Inflow pots can usually be located fairly accurately from early-time concentration profiles, when each flow pot produces a small, isolated concentration peak. In the calculated eample shown Figure 2.1, flow pots are apparent at depths of 6, 9, and 12 m (see also Figures 4.1 and 4.5 for field eamples). For flow pots, the estimation of q i and i is best done concurrently, because these quantities have a coupled effect on (). For early-time data, before flow peaks beg to terfere with each other, mass conservation requires that at time t, the ion mass represented by the ith concentration peak, M i (t), be given by M i (t) = q i i t, (2.1) where we assume for simplicity that =. Therefore, the area under the ith () peak can be epressed as A i M i ( t) qiit = ( ) d = =, (2.2) 2 2 πr πr where r is the wellbore radius. alculatg A i by numerically tegratg the () profile the vicity of the ith feed pot, at a series of times, can be done to estimate the product q i i and to verify that it is constant time. Very early profiles, which show small peaks, generally provide less accurate tegrals than do larger peaks. As a rule, the largest nonterferg peaks should be used to estimate the q i i product. Table 2.1 summarizes the q i i products obtaed for the peaks shown Figure 2.1. At termediate times, because the well bottom is closed to flow, peaks become skewed toward the top of the well. In prciple, such skewness can be used to separate q i and i. This 6

7 can be done by fittg to numerical results from BORE II which q i and i are varied, while keepg their product constant. At late times, the concentration profile reaches a steady-state condition consistg of a series of steps with concentration mai, each associated with an flow pot. For the lowest flow pot ( 1, q 1, 1 ) a wellbore section closed at the bottom, the steady-state concentration ma1 is equal to 1. Mig rules dictate that the second-lowest flow pot ( 2, q 2, 2, with 2 < 1 ) has ma2 given by ma 2 q q =. (2.3) q q 2 Generally, for the ith flow pot, q j j ma i =, (2.4) q j where the sums are taken over all feed pots with j i. This epression can be solved for q i, q i q j j = qk mai, (2.5) where the j sum is taken over all feed pots with j i and the k sum is taken over all feed pots with k > i. Fally, we determe i from the q i i product and q i : i ( q ) q i i =. (2.6) i Hence, observg ma1 gives us 1, and usg the value of the q 1 1 product from the early-time data determes q 1. Then, knowg q 1 1, q 2 2, q 1, and ma2 enables us to determe q 2 and 2. We contue this way up the wellbore section until all feed-pot properties are 7

8 determed. Table 2.1 summarizes the mai, q i, and i values for the feed pots shown Figure 2.1. Note that any errors troduced at lower feed pots fluence the results for shallower feed pots, so the accuracy of the properties may decrease as we move up the wellbore. A consistency check is provided by comparg the sum of all the feed-pot flow rates Σq i to the pumpg rate from the top of the wellbore section Q, which is a known quantity prescribed as part of the loggg procedure. If these two quantities do not agree, there are two possible remedies. If all the feed pots show equally good plateaus, then all the flow rates can be scaled by Q/Σq i. On the other hand, a common situation is for loggg to end before the uppermost (Nth) peak reaches steady state, which case Q can simply be used place of Σq i Equation (2.4) to determe man. Note particularly that to use Q as a constrat, care must be taken that it does not clude unknown contributions from flow to the wellbore above the logged section beg analyzed. This can easily happen actual field conditions. If we have reason to believe that the i for all flow pots are the same (say i = ), then Equation (2.4) gives mai =, implyg that steady-state concentration profiles do not provide any new formation. In fact, for this special case, it is possible to determe all the q i values and from early-time profiles only. First, the early-time profiles are used to determe the q i product for each feed pot as usual. Then, the equation Q = Σq i is multiplied by on both sides, q = Q = i qi (2.7) and solved for qi =. (2.8) Q 8

9 Fally, Equation (2.6) is used to determe each value of q i from the q i product and. Another potential simplification occurs if the feed pots are far enough apart for () plateaus to develop before the peaks beg to terfere with one another, as shown for the two shallower peaks Figure 2.1a at 9 mutes and Figure 2.1b at.1 day. The isolated plateau concentration is denoted midi, and is given by an equation similar to Equation (2.4) for mai qii midi = (2.9) q j where the denomator sum runs over all j i. Epressions for q i and i analogous to Equations (2.5) and (2.6) follow directly, enablg determation of q i and i for the ith peak, given midi and the q i i products for the ith and all deeper peaks. Note that for the lowest peak, mid1 = ma1 = Upflow from Below Upflow from below can occur when the bottom of the wellbore terval beg vestigated is not the actual bottom of the well and is not sealed with a packer. If upflow from below has a distctive sality, it can be treated as any other flow pot, but if it has sality, it will not produce a peak of its own and its presence must be ferred from its fluence on the other peaks. Figure 2.2 shows the BORE II concentration profiles obtaed usg the same feed-pot parameters as for Figure 2.1, but with the addition of upflow from below at a rate q w and a concentration. omparg Figures 2.2 and Figure 2.1 shows that the peaks are all more strongly skewed upward when q w >. The most evident change occurs at the lower limb of the lowest peak, which shows a diffusive profile when q w = (Figure 2.1) and a combation of advection and diffusion when q w > (Figure 2.2). Integration under the early-time () peaks 9

10 provides q i i estimates as before, despite the asymmetric shape of peaks, but, sce the peaks terfere sooner, care must be taken the choice of the profiles to tegrate. The steady-state mig rule for the ith feed pot becomes q j j ma i =, (2.1) q + q w j where as before, the sums are taken over all feed pots with j i. If the peaks develop plateaus before they beg to terfere with one another, Equation (2.9) for midi can be modified by addg q w to the denomator. In either case, we have a system of equations with one more unknown than number of equations (if Q is not used); therefore, we cannot determe all the feed-pot parameters dividually. However, we can determe the sum q 1 + q w and dividual values of q i and i for feed pots above the lowest one. We still have mid1 = ma1, but with a non-zero q w, ma1 1. omparg the different i values might provide sight to the likely range for 1, enablg an estimate of q 1 to be made. Furthermore, if we have an dependent estimate of 1, q 1, or q w, then we can determe the remader of the feed-pot parameters. As before, wellbore pumpg rate Q may be used to constra feed-pot properties by requirg Q = q w + Σq i, where the sum runs over all feed pots. Aga, dog this care must be taken that Q does not clude unknown possible flows to the wellbore above the logged section Time-Dependent Feed Pots Time-dependent values of q i and i most commonly arise when pumpg rates are altered or tracers are troduced nearby wells. However, they can also represent actual physical or chemical variations the rock near the borehole. Figure 2.3 shows the BORE II concentration profiles for a sgle flow pot which feed-pot concentration is either constant at 1 (Figure 2.3a), creases learly from to 1 (Figure 2.3b), or decreases learly from 1 to 1

11 (Figures 2.3c and d). A decreasg feed-pot concentration provides a distctive concentration profile signature, particularly when combed with upflow. In contrast, the profiles for creasg feed-pot concentration or creasg feed-pot flow rate (not shown) may be difficult to distguish from those for a constant feed pot. For early times, before the profiles for adjacent flow pots beg to terfere with each other, we can generalize Equation (2.1) for M i (t) to obta i t M ( t) = q ( t' ) ( t' ) dt'. (2.11) i i Hence, the slope of the M i (t) versus t curve gives q i (t) i (t). This approach does not distguish between time dependencies q i or i separately, but the skewness of concentration peaks, which depends only on q i, may provide sight to q i and i time variations. If either q i or i is known to be constant, then Equation (2.11) can be used to calculate the time dependence of the other quantity. A common feed-pot time dependence, which arises if de-ionized water migrates to the fracture durg the replacement of borehole water prior to wellbore loggg, is for both q i and i to be constant after a time t i, but to have i = before t i. In this case, it is generally possible to estimate t i along with q i i by tegratg over a series of profiles and fittg the resultg M i (t) values to the lear relation M i (t) = q i i (t t i ). (2.12) Inflow Pots with i = It may happen that the itial wellbore ion concentration is similar to some of the feed pot concentrations, i.e., for some feed pots, i =. An flow pot with i = does not 11

12 produce a concentration peak of its own, but its effect on neighborg peaks may be visible. Figure 2.4 shows the BORE II concentration profiles for two flow pots, one of which has i =. In each of the four eamples Figure 2.4, the flow rate of the i = flow pot is twice that of the adjacent flow pot. If the i = flow pot is above (downstream of) the other flow pot, there is a subtle signature the form of a break slope of the concentration profiles when the dilutg effect of the i = flow is first felt (Figure 2.4a). This break slope is accentuated if an upflow from below is present (Figure 2.4b). At late times, the i = flow pot causes a distctive plateau (Figure 2.4c). If the i = flow is below (i.e., upstream of) the other flow pot, the break slope is difficult to see (Figure 2.4d). The concentration profile is skewed upward as when an upflow is present; this identifies the eistence of a i = flow, but not its location Outflow Pots Outflow may occur when the far-field heads certa fractures or permeable zones penetrated by a wellbore are different. When the well is shut or is pumpg at a very low rate, water flows to the wellbore from the higher head zone, is transmitted through the wellbore, and flows out to a lower-head zone. Figure 2.5 shows the BORE II concentration profiles for an outflow pot located adjacent to an flow pot. In each eample, the outflow strength is twice the flow strength. The outflow pots cause subtle changes the shape of the peaks, but these changes do not identify the outflow location well. The addition of an upflow changes the peak shape, but does not help p down the outflow location. An tegral method that can estimate the outflow location is presented the net section. Figure 2.6 shows the BORE II concentration profiles for flow and outflow pots located farther apart, with profiles collected frequently enough to monitor the movement of the 12

13 concentration front up the wellbore. In this case, q 1 q, q 2 q out, and q w all have the same magnitude and q out = q, so the speed of the front is halved as it passes the outflow pot. By eamg the spacg between concentration profiles obtaed at known time tervals, the location of the outflow pot can be ferred Horizontal Flow Measurement of horizontal flow across the wellbore is important because it can be used to estimate the natural regional flow the hydrologic layer (Drost et al., 1968). In our case, horizontal flow can be represented with a pair of flow and outflow feed pots located at the same depth, with q 1 q = Q and q 2 q out = Q, where Q is the volumetric flow rate across the wellbore. Q can be related to the regional Darcy velocity v d the layer tercepted by the wellbore usg Q = v d 2rbα h, where r is the wellbore radius, b is the thickness of the hydrologic layer, and α h is a dimensionless convergence factor rangg from 1 to 4, which depends on well completion (Drost et al., 1968). Figure 2.7 compares the BORE II concentration profiles for several horizontal-flow cases with the correspondg case which only an flow feed pot with strength q is present. Unlike previous plots that show successive concentration profiles at equally spaced time tervals, here the time terval doubles between each successive profile, to enable both early- and late-time behavior to be illustrated a sgle plot. At early times, the concentration profiles for flow and horizontal flow are similar, both showg symmetric profiles. At later times, the horizontal-flow profiles rema symmetric, whereas the flow profiles become skewed up the wellbore. Peak concentration creases faster with flow only, but the absence of longitudal diffusion, the steady-state concentration ma would be 1 for both cases. 13

14 Figures 2.7a and 2.7b depict a typical situation which diffusion along the wellbore is moderately strong compared to horizontal-flow or flow strength (practically, the movement of the conductivity loggg tool up and down the wellbore greatly enhances the longitudal diffusion coefficient D over the value for still water). Figures 2.7c and 2.7d illustrate how the concentration profiles sharpen when D is very small. Figures 2.7e and 2.7f show the profiles for thick layers of flow and horizontal flow, modeled by placg multiple flow pots or flow/outflow pairs over a range of values. The value of D is moderate, as Figures 2.7a and 2.7b, but at the center of the flow layer it has little impact, allowg the concentration profiles to reach ma =. The black dots on the horizontal-flow profiles Figure 2.7f show the concentration given by an analytical solution (Drost et al., 1968) that considers horizontal flow only (i.e., longitudal diffusion is negligible or the hydrologic flow layer is very thick): 2tvdαh ( t) = [ ]ep. (2.13) πr The BORE II simulation of a thick horizontal-flow layer matches the analytical solution well. Figure 2.8 shows (t)/ at the center of the flow layer as a function of time for the cases shown Figures 2.7b, d, and f, as well as for a thick layer with very small D and the Drost et al. (1968) solution itself. Both thick-layer cases follow the analytical solution closely, regardless of the value of D applied, and the th-layer case with very small D shows similar behavior. However, for a th layer (such as a narrow conductg zone or a sgle fracture) with a realistic value of D, the peak concentration grows much more slowly. Diffusion plays a significant role, decreasg the concentration at the feed-pot concentration by effectively mig formation water with wellbore water. In this case, matchg with BORE II will yield a more accurate estimate of horizontal flow than Drost s solution. 14

15 2.2 Mass Integrals The eamples of concentration profiles shown the previous section dicate that general, outflow pots do not produce a strong signature that enables them to be easily located. Here, we describe an tegral procedure that enables outflow pots to be located by eamg changes ion mass the wellbore section. onsider a wellbore section with one or more outflow pots above one or more flow pots. Let us assume feed-pot strength and concentration do not vary time. The procedure is as follows. We tegrate each () profile over the entire wellbore section of terest to obta the area A(t) under the () profile at time t (cludg all peaks, whether or not they terfere). Then, we multiply A(t) by the mean wellbore cross-sectional area to determe ion mass place at time t, which we denote as the mass tegral M(t), and plot M(t) versus t. Note that before the concentration front reaches any outflow pots, M(t) is lear, with slope S early = q i i. (2.14) When the concentration front reaches an outflow pot, the slope of M(t) decreases, sce ion mass leaves the wellbore at that pot. When the concentration front passes the uppermost outflow pot, M(t) becomes lear aga, with slope S late = qii qi i = Searly out ma ma qi, (2.15) out where ma is the maimum concentration at the uppermost outflow pot. To determe the aggregate outflow rate, we rearrange Equation (2.15) to yield S Slate qi =. (2.16) out early ma Now, let us eame the () profiles and locate the times when (a) M(t) becomes nonlear and (b) M(t) becomes lear aga. The leadg edge of the concentration front at (a) identifies the 15

16 deepest outflow pot, ma. The trailg edge of the concentration front at (b) identifies the shallowest outflow pot, m. Here, we defe the leadg edge as the location at which () =.1 ma and the trailg edge as the location at which () =.9 ma. An eample of the mass-tegral procedure for early-time concentration profiles is shown Figure 2.9. We consider a sgle flow pot located below a sgle outflow pot a wellbore section with upflow from below. The BORE II concentration profiles (Figure 2.9a) conta a mor break slope at the outflow pot, which would probably be impossible to identify real data. In contrast, the mass-tegral plot (Figure 2.9b) shows a clear divergence from learity between t =.4 and t =.6 days. Returng to the () profiles, we fd that the leadg edges of the t =.4-day and t =.6-day profiles lie at = 84 and = 82.5 m, respectively. Sce the t =.4-day tegral fits the lear M(t) trend but the t =.6-day tegral does not, we fer that the outflow pot occurs at m. The actual location of the outflow pot is 84.5 m. Hence the M(t) method, while not perfect, does provide useful formation for outflow-pot location. Another eample, this time considerg long-time concentration profiles, is shown Figure 2.1. We consider two flow pots located below two outflow pots a wellbore section with upflow from below. The BORE II concentration profiles (Figure 2.1a) show terference between two flow peaks, which makes it difficult to simply locate the outflow pots by spection. The mass-tegral plot (Figure 2.1b) shows early-time and late-time lear sections, with a slight departure from learity for t =.3 days and a return to learity by t =.7 days. The leadg edge of the t =.3-day profile lies at = 65 m, suggestg that the deepest outflow pot is just below this depth. The trailg edge of the t =.7-day profile lies at = 5 m, suggestg that the shallowest outflow pot is just below this depth. These predictions are 16

17 reasonably close to the actual locations of the outflow pots, 7.5 and 5.5 m. Usg Equation (2.16) with the values of S early, S late, and ma shown Figure 2.1 yields an aggregate outflow rate of 1.48 L/m, which agrees closely with the actual value, 1.5 L/m. Note that theory, if two outflow pots are separated by a large enough distance, a lear portion the M(t) plot will develop when the concentration fronts are between the two pots, potentially enablg the locations and strengths of the dividual pots to be determed. However, if the two pots are separated by a distance comparable to or less than the width of the concentration fronts, as Figure 2.1, the M(t) method will not be able to resolve them. 3. Effect of Pumpg Rate The previous section showed that the flow rate the wellbore terval of terest has a strong effect on the signature of the peaks. Sce we have control of the flow rate through the selection of the well pumpg rate Q for a given conductivity log, the choice of Q or the use of two loggg runs with different Q values may be used to improve the accuracy of parameter determation. To study this effect systematically, we need to understand how feed-pot strength changes when Q is modified. This dependence is derived below for several practical cases of terest, followed by eamples illustratg how Q affects () profiles and M(t) tegrals. We then discuss how modifyg Q may be used to vestigate the nature of the feed pots. 3.1 How Feed-Pot Strength Depends on Q We consider a wellbore terval contag N feed pots. The strength of the ith feed pot is q i and Σq i = Q. By convention, flow pots have positive q i and outflow pots have negative q i. Upflow from below can be absent (i.e., the lower end of the studied terval is the well bottom or 17

18 sealed by a packer) or represented by one of the N feed pots (upflow is positive and downflow negative). For each feed pot, q i and concentration i are assumed to be constant time. The strength of a feed pot is related to its hydraulic transmissivity T * i, the far-field pressure P i a distance r i away from the wellbore, and the pressure P wb at the wellbore radius r through Darcy s law. Assumg steady radial flow to the wellbore, q i * 2πTi ( Pi Pwb ) = = Ti ( Pi Pwb ), (3.1) ln( r / r) i where T i represents an effective hydraulic transmissivity, to which the constant factors volvg radial distance have been lumped. T i is troduced to simplify notation. For the special case of horizontal flow with net q i =, the log dependence on r is replaced by a lear dependence, but the right-side form (contag only T i and pressure) is unchanged. We assume that the hydraulic transmissivity with the wellbore itself is much greater than that of any flow zone, so that P wb is constant over the wellbore terval of terest. Sce Σq i = Q, we can write Q = T P P ). (3.2) i ( i wb If we now alter the pumpg rate from Q to Q', T i and P i rema unchanged but P wb becomes P wb ', and q i ' = T i (P i P wb ') (3.3) Q ' = T ( P P ' ). (3.4) i i wb We solve Equation (3.4) for P wb ' and substitute to Equation (3.3), yieldg an epression for q i ' terms of Q': ' + Ti Ti ( Q Ti Pi ) q ' = T P. (3.5) i i i 18

19 Addg and subtractg the term T i P wb to Equation (3.5) and then substitutg from Equations (3.1) and (3.2), we obta Ti ( Q' Q) qi ' = qi +. (3.6) T i Defg q i = q i ' q i, Q = Q' Q, and T tot = ΣT i yields the more compact form T Q i qi =, (3.7) Ttot which is the fundamental relationship between the change feed-pot strength q i and the change pumpg rate Q. Note that q i is directly proportional to T i, and thus the feed pots with larger hydraulic transmissivity show greater changes strength when Q is modified. In particular, if the jth feed pot has a much larger hydraulic transmissivity than all the others (T j T tot ), then q j Q and all the other feed-pot strengths will not change much. This situation might arise if the well tercepts an etensive flow zone that has not been ecluded from the loggg section by packers. There are several special cases of Equation (3.7) that are of terest. If all the T i s are the same, then T i = T tot /N, and Equation (3.7) simplifies to Q q i =, (3.8) N where N is the number of feed pots. In this case, when Q is modified, all feed-pot strengths change by the same amount. A particular case of the equal T i condition is horizontal flow across the wellbore, where Q =, N = 2, q = Q, and q out = Q. Thus, if pumpg is added to a horizontal-flow case ( Q > ), flow pots become stronger (q ' = Q + Q/2) and outflow pots become weaker (q out ' = Q + Q/2). When Q/2 > Q, q out ' changes sign and the outflow pots become flow pots. For 19

20 a thick layer, N can be any even number. In that case, itially Q =, q = 2Q /N, q out = 2Q /N, and then when pumpg at rate Q is imposed, q ' = (2Q + Q)/N and q out ' = ( 2Q + Q)/N. On the other hand, if the P i s are all the same, then we solve Equation (3.1) for T i and substitute to Equation (3.7) Qqi Qqi qi = = T ( P P ) Q. (3.9) tot i wb Note that when all P i s are the same, feed pots must be either all flow pots or all outflow pots, and there can be no horizontal or ternal flow. In this case, when Q is modified, the relative change of each feed pot q i /q i is the same and is equal to the relative change of Q, qi q i = Q. (3.1) Q Thus, feed pots can only change sign if Q changes sign (i.e., pumpg becomes jection), and then they all will change sign, aga precludg horizontal or ternal flow. Fally, if all the T i s are the same and all the P i s are the same, then accordg to Equation (3.1), all the q i s must be the same. Thus, q i = Q/N, and Equations (3.8) and (3.9) become equivalent. 3.2 Eamples of How Modifyg Q Affects () and M(t) Signatures Runng FE logs with two or more values of Q (for eample, halvg and doublg the origal pumpg rate) should provide a means to better characterize the feed pots for cases with flow pots only, cases with both flow and outflow pots, and for the special case of horizontal flow. Figure 3.1 shows the effect of creasg Q on a sgle flow pot with a constant strength and concentration. With a low Q (Figure 3.1a), only the q product can be determed. With a larger Q (Figure 3.1b), evidence of the ma plateau develops at an earlier time, enablg 2

21 determation of and q dependently. Note that, the absence of longitudal diffusion, creasg or decreasg Q would be completely equivalent to loggg for a longer or shorter period of time, respectively. Figure 3.2 shows the effect of modifyg Q when both flow and outflow pots are present and all T i s are the same. For the origal Q value (Figure 3.2a), the outflow pot produces only a subtle change slope the M(t) plot, makg the mass-tegral procedure difficult. If Q is halved (Figure 3.2b), the change slope becomes larger and the analysis becomes straightforward. On the other hand, if Q is doubled (Figure 3.2c), the outflow pot becomes an flow pot and is easily identified the early-time () profiles as another peak. Fally, if Q is reduced to zero (Figure 3.2d), then the outflow pot must capture all the flow the wellbore, and aga the () profiles would provide a strong signature of the outflow pot. Figure 3.3 shows the effect of creasg Q on a th layer with horizontal flow, where itially (Figure 3.3a) Q =, q = Q and q out = Q. The T i s for the flow and outflow pots are the same, and hence, as Q is creased from zero, the magnitude of q creases and the magnitude of q out decreases (Figures 3.3b and 3.3c). When Q = 2Q, outflow vanishes (Figure 3.3d), and when Q > 2Q, the outflow pot becomes an flow pot. Figure 3.4 shows the effect of creasg Q on the horizontal-flow case when an upflow from below q w accompanies the crease Q. We assume that when Q =, q w =, q = Q, q out = Q, and the concentration profiles are as shown Figure 3.3a. One possible approach is to consider q w as arisg from a sgle deep feed pot with a T value equal to that of the flow/outflow pair used to model horizontal flow (Figures 3.4a, c, and e). As Q is creased, q = Q/3 = Q/3 for each of the three feed pots, so the flow strength and upflow crease while the outflow strength decreases. The steady-state plateau concentration is given by 21

22 ma q ( Q + Q / 3) = =. (3.11) q + q Q + 2Q / 3 w Another approach is to associate upflow with a very large T value, which is equivalent to considerg upflow as arisg from many dividual feed pots or an etensive flow zone (Figures 3.4b, d, and f). In this case, the crease Q is largely mataed by an crease upflow, with the flow and outflow strengths remag at +/ Q, which leads to a much lower plateau concentration q Q ma = =. (3.12) q + q Q + Q w Hence by varyg Q, we can learn somethg about the nature of the upflow from below. This procedure can also be applied to dividual feed pots, as described the followg section. 3.3 Investigatg Feed-Pot Nature by Analyzg FE Logs with Two or More Q s When wellbore FE loggg is done usg a sgle pumpg rate Q, we can determe feed-pot strengths q i by vestigatg () and M(t), as described Section 2. However, without additional formation, we do not know whether differences between q i values arise from differences T i or P i or both (see Equation (3.1)). However, Equations (3.7) through (3.9) dicate that the change q i arisg from a change Q depends on how T i and P i differ between feed pots. Thus, by repeatg FE loggg usg a different value of Q, we can fer more formation about the feed pots (as well as confirmg analysis of sgle-q FE logs). For eample, if the change feed-pot strength q i is the same for all feed pots when Q is changed, then accordg to Equation (3.8), all the T i s are the same, and differences between q i s arise from differences between P i s (i.e., q i ~ P i ). A more common situation is for all the P i s to be the same and for differences between q i s to arise from differences between T i s (i.e., q i ~ 22

23 T i ). In this latter case, Equation (3.9) predicts that the relative change feed-pot strength q i /q i will be the same for all feed pots when Q is changed. If neither q i nor q i /q i are the same for all feed pots, then differences between q i s comes from differences between both T i s and P i s. Figure 3.5 illustrates this behavior. In general, we can rearrange Equation (3.7) to isolate the known quantities on the lefthand-side Q q i = Ti T tot, (3.13) which dicates that the fractional change pumpg rate ehibited by a feed pot equals the fractional hydraulic transmissivity of the feed pot. Similar formation about the differences between the P i s may be obtaed as follows. We can rewrite Equation (3.2) for Q as Q = T P P ) = T ( P P ), (3.14) where P avg, defed as i ( i wb tot avg wb i i avg =, (3.15) Ttot P T P is the hydraulic-transmissivity weighted average feed-pot pressure. Rearrangg Equation (3.14) yields Q T tot = P P, (3.16) avg wb rearrangg Equation (3.1) yields qi T i = P P, (3.17) i wb 23

24 and takg the ratio yields qit QT tot i P P P P i wb =. (3.18) avg wb Elimatg T tot /T i by usg Equation (3.13) gives qi / qi Q / Q = Pi P P P avg wb wb. (3.19) All the terms on the left-hand-side are known; hence, the relative pressure difference drivg flow to or out of the wellbore at the ith feed pot can be determed. Note from Equation (3.16) that when Q =, P avg = P wb. That is, P avg is the pressure that would be measured the wellbore when there is no pumpg, only ternal wellbore flow between feed pots with different values of P i. Determg the values of T i (Equation 3.13) and P i (Equation 3.19) relative to the values of other feed pots is the most formation we can glean usg FE analysis itself. However, if the wellbore pressure change durg FE loggg (P avg - P wb ) is monitored, then Equation (3.16) may be used to determe T tot, enablg Equation (3.13) to determe dividual T i values uniquely. If, additionally, the value of P wb itself is monitored, then the P i values may be determed from Equation (3.19). In practical field conditions, it is deed straightforward to monitor wellbore fluid pressures. The results from two fluid loggg tests with different pumpg rates potentially provide a powerful way of determg T i and P i values for all feed pots, relative to those of one feed pot. Thus, we consider two distct feed pots, i and j, write Equations (3.13) and (3.19) for each one, and take the ratio: 24

25 q q i = j Ti T j (3.2) qi q j / qi / q j = Pi P j P P wb wb (3.21) The q values and q/ q ratios on the left-hand-sides are easily identifiable Figures 3.5b and 3.5c, respectively. In practice, if T j and P j are known for one feed pot, say from a packer test either before or after the fluid loggg, then the T i and P i values for all the rest of the feed pots can be obtaed simply from Equations (3.2) and (3.21). Note that these equations consider only two feed pots at a time, as opposed to Equations (3.13) and (3.19), which clude quantities that represent all the feed pots (Q, Q, T tot, and P avg ), with more possibilities of troducg accuracies under imperfect, real-world conditions. 4. Application to Real Data 4.1 Raymond Field Site At the Raymond field site, located the foothills of the Sierra Nevada mountas alifornia, ne wells were drilled that penetrated a fractured granodiorite. The wells are 9 m deep and are cased over the upper 8 m through a sediment layer, then open below that. Many different kds of well logs and well tests have been conducted these wells, to develop and test equipment and methodologies for characterizg the hydrological behavior of fractured rock (Karasaki et al., 2). Flowg wellbore electrical-conductivity loggg was carried out seven of the ne wells (ohen, 1995) usg pumpg rates rangg from 7 to 2 L/m. Loggg was conducted while the tool was moved up and down the wellbore. FE profiles obtaed durg upward loggg showed smeared-out, less well-defed peaks that are slightly offset from those obtaed 25

26 durg downward loggg, and are not used. Si or seven downward loggg profiles for creasg time were obtaed for each well. Because the wells are quite shallow, borehole temperatures do not vary much with depth, and FE values do not need to be corrected for temperature variations (see Appendi). FE is converted to usg the quadratic relationship given Equation (A.1). Here, we present and analyze concentration profiles from the two wells at the Raymond site, labeled SW1 and W, that show the most terestg signatures. Figure 4.1 shows the concentration profiles for well SW1. Si flow pots can be identified. The diffusive shape of the lower limb of the lowest peak (peak 1) suggests that q w =. Three peaks (Peaks 2, 3, and 4) show an approach to steady state, enablg estimates of mai to be made. We ignore the shallowest peak ( = 2 8 m), which decays rather than grows, on the assumption that it is evidence of leakage around the casg rather than fracture flow. Such leakage has been confirmed by field observations (ohen, 1995). Five of the peaks are well enough separated to use the area under the dividual peaks to determe M i (t), the mass arisg from the ith feed pot as a function of time (Figure 4.2). We then fit a straight le to M i (t) and use Equation (2.12) to identify the slope of the le as q i i and the time-ais tercept as t i. In general, the late-time drop M i (t) below the lear fittg le does not identify outflow, as described Section 2, but dicates the peak reachg the edge of the tegration doma. Note Figure 4.1 that Peak 5 overlaps with Peaks 4 and 6 too early for the estimates of q 5 5 and t 5 obtaed from M 5 (t) to be reliable. For Peak 1, the q 1 1 product is well-defed, but there is no evidence of a ma1 plateau. Furthermore, the height of an isolated peak such as this is very sensitive to diffusion/dispersion strength D, which is unknown. Therefore, we search for 1 and D values by trial and error usg the BORE II code, by comparg the observed () profiles for Peak 1 to simulation 26

27 results. Once 1 and q 1 have been found, Equations (2.5) and (2.6) can be used to calculate the parameters of the upper peaks. Table 4.1 summarizes the results. Recall that sce Peak 1 properties are not uniquely determed, those of all shallower peaks are uncerta too. Unfortunately, the shallow leakage around the casg precludes the use of Q to constra the q i values. Figure 4.3 shows () profiles simulated with BORE II usg the parameters given Table 4.1. The simulated profiles match the observed ones approimately, but there is room for improvement. In particular, Peak 5 is much too small and there is generally not enough terference between the upper five peaks. Because the peaks overlap relatively early, the M i (t) tegrals cannot etend as far along the wellbore as they should. Thus, they tend to underestimate the q i i products, which turn leads to too-small values of q i. The parameter values shown Table 4.1 are then optimized by data fittg usg BORE II simulations. Figure 4.4 shows the results of this fittg process, and Table 4.2 shows the correspondg feed-pot properties. Note that flow pot 4a with i = has been added between Peaks 4 and 5, to account for the narrow peak and lower plateau above Peak 4 (compare to Figure 2.4). With just this additional flow, the upward flow through the wellbore at Peak 6 is too big ( ma6 is too small and the peak is too broad), so we also need to add outflow pot 4b just below Peak 5. Overall, the property changes required for the eistg feed pots are mor. Peak 5 is an eception, but eamation of the () profiles (Figure 4.1) made it clear a priori that this peak was too close to adjacent peaks to be well characterized. It is terestg to note, from Table 4.2, the advantages of the flowg wellbore electricconductivity loggg method. From one set of data, obtaed only about one hour (ohen, 1995), we are able to identify the locations of eight conductg fractures tersected by the well, cludg one where the flow is out of the borehole to the fractures (feed pot 4b Table 4.2). 27

28 The sality of water from the seven flow pots varies by a factor of two, ecept for one pot that has very low sality. Sce we have the flow rates q i for these flowg fractures, their hydraulic transmissivities can be directly calculated, usg the pressure drawdown the wellbore measured durg pumpg. Table 4.2 shows that variation of the hydraulic transmissivity (proportional to q i ) covers a range of almost two orders of magnitude. To obta all this formation usg conventional packer methods (i.e., pump tests conducted packed-off sections of the borehole) would require considerable more time and effort. The second set of data from the Raymond site analyzed with our methods are from Well W. Figure 4.5 shows the concentration profiles for this well. Five peaks are apparent. The lowest peak (Peak 1) does not show the upward skewg of a normal flow pot, but the more symmetric appearance of horizontal flow. Peaks 2 and 3 terfere after a short time. The longtime behavior of Peak 4 suggests that there is a i = flow pot above it (denoted feed pot 4a) that causes a decrease from ma4. We ignore the non-uniform itial condition at shallow depths, because it is likely to represent sedimentary layers and leakage around the casg. Figure 4.6 plots M i (t) versus t for Peaks 1 and 5 dividually, for Peaks 2 and 3 combed, and for Peaks 4 and 4a combed. Results are shown Table 4.3. Because we consider Peak 1 to represent horizontal flow, which cludes both flow and outflow at the same depth, the M 1 (t) values obtaed from the area under the peak do not directly determe the q 1 1 product. Furthermore, because no ma1 plateau is visible, we need to fit q 1 and 1 by trial and error usg BORE II simulations. In contrast, for Peaks 2 and 3, we vary q 2,3 q 2 + q 3 and 2,3 2 = 3 while matag their product equal to the q 2,3 2,3 value obtaed from M 2,3 (t). We then vary the manner which q 2,3 is allocated between Peak 2 and Peak 3. 28

29 For Peak 4, we itially assumed that feed pot 4a contributes a negligible amount to the q 4 4 product, because is small. However, when we used q 4 and 4 values consistent with the q 4 4 product given Table 4.3 for a BORE II simulation, we got much too large a peak (profiles not shown). This suggests that the q 4 4 product is not all associated with the feed pot 4 location. If we look carefully at the early-time profiles the vicity of feed pot 4a, we see some evidence for flow with >, which would contribute to the q 4 4 product of the combed peaks. The magnitude of the possible peak is much too small to produce a reliable estimate of the q 4a 4a product, so we proceed as follows. First, we determe q 4 and 4 by trial and error with Bore II simulations, then multiply them to get the true value of q 4 4, which is subtracted from the combed q value given Table 4.3. The remader is q 4a 4a, which is used along with ma4a to determe q 4a and 4a from Equations (2.5) and (2.6). Peak 5 is then analyzed usg Equations (2.5) and (2.6). The resultg parameters are shown Table 4.3 and the correspondg () profiles are shown Figure 4.7. Above the feed pot 4a, flow up the wellbore should be bigger and the early-time match for Peak 5 is not very accurate, owg to complications from the non-uniform itial conditions. Overall, however, the procedure works reasonably well. As for well SW1, additional trial-and-error parameter variations could be used conjunction with BORE II simulations to improve the match. Now let us eame Peak 1 more carefully. Figure 4.8 shows the same concentration profiles as Figure 4.7, but zooms on the region around Peak 1. It is apparent that the horizontal-flow model does not capture all the features of the observed data (Figure 4.8a). If we move the outflow pot a few meters below the flow pot, then the pair represents ternal downward flow through the wellbore rather than horizontal flow. This feed-pot configuration produces a slight downward skewg of the concentration peak at late times, consistent with the 29

30 observed data (Figure 4.8b), but it does not significantly alter the early-time concentration profiles. That is because the early-time simulated profiles reflect a short residence time the wellbore, and the downward flow does not have time to affect them. Figure 4.6 dicates that the M 1 (t) data can be fit equally well by a curve as by as a straight le. Assumg a constant value of q 1, the slope of the M 1 (t) fittg curve determes 1 (t). Usg a variable 1 (t) BORE II simulations yields a better match to both early- and late-time profiles for Peak 1 (Figure 4.8c). The gradual crease i implied by the nonlear curve fit suggests a diffuse boundary between the native groundwater and the de-ionized water that entered the fracture durg wellbore flushg prior to loggg. 4.2 olog Field Site The FE logs shown Figure 1.1 are proprietary and the site geologic formation is not made known to us. Nevertheless they provide a good eample of usg the mass-tegral method to identify the location and strength of an outflow pot. The FE logs are converted to concentration profiles usg the quadratic relationship given Equation (A.1). Figure 4.9a shows the () profiles at a series of 12 times, and Figure 4.9b shows the correspondg M(t) tegral for the entire wellbore section from = 146 to = 226 m, denoted M 226. The first two pots on the M 226 curve essentially represent the itial conditions, so they are not cluded the early-time data fit to a straight le, which yields S early =.81 kg/hr and t =.22 hr. The itial deviation from learity occurs at a time t = 1.6 hr. Accordg to Figure 4.9a, the concentration front at this time is at a depth of = 212 m, which is ferred to be the outflowpot location, denoted out. Fittg the subsequent data to a straight le yields S late =.59 kg/hr. Note that after a time of t = 2.6 hr, the concentration profiles do not show complete peaks, 3