Thermal records from boreholes in young oceanic crust, in which water is flowing up or down, are used to assess formation and borehole flow properties using three analytic equations that describe the transient thermal and barometric influence of downhole or uphole flow. We link these calculations with an iterative model and apply Markov chain Monte Carlo (MCMC) analysis to quantify ranges of possible values. The model is applied to two data sets interpreted in previous studies, from Deep Sea Drilling Project Hole 504B on the southern flank of the Costa Rica Rift and Ocean Drilling Program Hole 1026B on the eastern flank of the Juan de Fuca Ridge, and to two new records collected in Integrated Ocean Drilling Program Holes U1301A and U1301B, also on the eastern flank of the Juan de Fuca Ridge. Our calculations indicate that fluid flow rates when thermal logs were collected were ∼2 L/s in Holes 504B, 1026B, and U1301A, and >20 L/s in Hole U1301B. The median bulk permeabilities determined with MCMC analyses are 4 to 7 × 10−12 m2 around the uppermost parts of Holes 504B, 1026B, and U1301A, and 1.5 × 10−11 m2 around a deeper section of Hole U1301B, with a standard deviation of 0.2 to 0.3 log cycles at each borehole. The consistency of permeability values inferred from these four holes is surprising, given the range of values determined globally and the tendency for permeability to be highly variable in fractured crystalline rock formations such as the upper oceanic crust.
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 Volumetric fluxes of water through ridge-flank hydrothermal systems are commensurate with global riverine discharge [Mottl, 2003], advecting enough heat to account for 70% of the deficit in the oceanic lithospheric heat budget [e.g., Stein et al., 1995]. Seafloor hydrothermal circulation impacts processes throughout the global ocean, modifying ocean chemistry [Edmond et al., 1979; Elderfield and Schultz, 1996], lithospheric heat budgets [e.g., Davis and Lister, 1977], the physical state and evolution of oceanic crust [Jacobson, 1992], and biology both above and within the crust [Edwards et al., 2011; Kelley et al., 2002]. Hydrothermal flows are controlled in large part by the permeability structure (magnitude and connectivity) of the crust, which has been tested directly at relatively few subseafloor locations [e.g., Becker, 1990].
 Direct measurements of permeability in the volcanic ocean crust are difficult and expensive to perform, requiring a borehole to be drilled through seafloor sediments and into the basement rocks below. Core recovery is often low in fractured crystalline rocks, and permeability analyses of core samples may not be characteristic of larger scales [Civan, 2001; Clauser, 1992]. Single-hole packer tests, the simplest form of in-situ test, involve actively pumping into a borehole while monitoring the pressure response in the same hole. The requirement that a ship be onsite throughout active pumping limits the duration of such tests during ocean drilling, typically restricting the radial extent to which the test is sensitive to the near-borehole environment [Becker and Davis, 2003; Fisher, 1998].
 Temperature records in flowing seafloor boreholes (i.e., those with fluid flowing downward from, or upward to, the seafloor) can also be used to infer crustal permeability [Becker and Davis, 2003; Becker et al., 1983; Fisher et al., 1997]. Thermal data can be collected months to years after a borehole is drilled, allowing assessment of permeability at a scale larger than that associated with single-hole packer tests. Thermal analyses can be used to assess additional physical characteristics in and near the borehole, such as the borehole radius, aquifer thickness, aquifer compressibility, and sediment thermal conductivity. Some prior knowledge of these parameters is required, but the thermal history can inform understanding of these terms through the use of statistically based inverse modeling techniques.
 We introduce two fundamental improvements to traditional analyses of borehole properties based on thermal logs from flowing crustal boreholes in the deep ocean. First, we connect three separate analytic heat and fluid flow models, directly linking the borehole thermal state, fluid properties, and flow to/from the surrounding formation to find simultaneous, self-consistent solutions. Second, we treat uncertain physical parameters as distributions, rather than single values, allowing development of the range of possible flow parameters and formation and borehole properties.
 In the next section of this paper, we describe the nature of boreholes drilled into the volcanic ocean crust, and introduce four borehole locations where the thermal data used in this study were acquired. We apply the linked analytic and statistical approach to data from these boreholes, and compare the results to earlier analyses of thermal data and other observations used to assess crustal conditions.
1.2. Experimental Operations and Settings
1.2.1. Borehole Configurations, Operations, and Thermal Data
 Subsea boreholes drilled into the upper volcanic crust, which is generally porous and permeable, are usually located below a conductive boundary layer of less permeable marine sediments. These sediments help to stabilize the drill string and can support a reentry cone at the seafloor that permits repeated bit changes, installation of casing, and other operations needed to achieve penetration below the upper tens of meters of volcanic rock. Surface seawater is pumped into the borehole as a drilling fluid to lubricate the bit and remove cuttings from the hole, and may also be pumped during coring, casing, and other operations. The initially warm surface seawater used as drilling fluid cools considerably during descent to the seafloor (often 2–5 km), as the drill pipe is an efficient heat exchanger. As a result, fluids pumped into the seafloor generally have a lower temperature than formation fluids adjacent to the borehole, as is common during deep crustal drilling in general [e.g., Becker et al., 1983; Jaeger, 1961; Langseth, 1990]. The imposition of a tall column of cold (dense) seawater in the borehole adjacent to warmer (less dense) formation fluid creates a positive differential pressure that can drive borehole fluid into the formation, even where the formation is naturally overpressured relative to ambient hydrostatic. In many boreholes, self-sustained downflow continues for days to years, until the borehole is plugged to prevent inflow of bottom seawater [e.g., Becker et al., 2001; Gable et al., 1992] (Figure 1).
 Prior to drilling, geothermal conditions in marine sediments above volcanic basement rocks tend to be conductive, with heat flowing from depth to the seafloor. Heat transport is usually conductive in marine sediments because sediment permeability and natural driving forces are generally too small to allow fluid flow at thermally significant rates, particularly where sediment thickness is greater than a few tens of meters [Davis, 1988; Fisher, 2004; Spinelli et al., 2004]. In contrast, conditions in boreholes drilled and cased through sediments and into volcanic crust are often nonconductive because of fluid flow up or down the borehole. If this flow were extremely rapid, it could lead to isothermal conditions in the borehole. In practice, there is generally significant lateral exchange of heat between the borehole and the surrounding formation (Figure 1). The balance of vertical heat advection and lateral heat conduction generally results in a curved thermal profile with depth, the shape of which depends on several parameters: the predrilling geothermal gradient, thermal properties of the fluid and surrounding formation, size of the borehole and casing, and the direction, rate and duration of fluid flow (Figure 1) [Becker et al., 1983; Lesem et al., 1957].
 When fluid flow is downward, formation heat flows inward, warming the water as it descends. If a natural formation overpressure exceeds the differential pressure created during downflow, the flow direction can reverse [Becker and Davis, 2004; Fisher et al., 1997; Wheat et al., 2010]. Then warm borehole fluids rising from depth lose heat to the sediments surrounding casing during ascent toward the seafloor. The resulting borehole thermal profile has opposite curvature to that caused by downward flow (Figure 1).
 The temperature records used in this study were recovered from Deep Sea Drilling Project (DSDP) Hole 504B, Ocean Drilling Project (ODP) Hole 1026B, and Integrated Ocean Drilling Project (IODP) Holes U1301A and U1301B (Figure 2 and Table 1), all of which have demonstrated clear evidence of sustained fluid flow when the holes were not sealed (Figure 3). Temperature data from Holes 504B and 1026B were analyzed in previous studies [e.g., Becker et al., 1983, 2004, 1985; Fisher et al., 1997; Gable et al., 1989; Guerin et al., 1996], but we revisit earlier interpretations to assess the impact of methods introduced in the present paper, and to compare results from multiple sites that share common characteristics. In addition, we present and interpret new borehole thermal data collected from Holes U1301A and U1301B using long-term, borehole observatories.
 DSDP Site 504 is located in 5.9 Myr old crust on the southern flank of the medium-spreading-rate Costa Rica Rift, where sediment thickness is 274 m above volcanic basement (Figure 2 and Table 1). Hole 504B was drilled in 1979 on DSDP Leg 69 to assess the physical and chemical state of young oceanic crust, and has been revisited, deepened, sampled, and monitored for decades [e.g., Anderson et al., 1982; Becker et al., 1985, 2004; Costa Rica Rift United Scientific Team, 1982]. Hole 504B was drilled much deeper than other subsea boreholes that existed at the time, eventually penetrating 2111 m below the seafloor (mbsf) and 1836 m subbasement (msb); Hole 504B holds significant historical importance as one of the first well-sampled windows into the formation, structure, and alteration of the oceanic crust [e.g., Alt et al., 1986; Anderson et al., 1982; Becker et al., 1989].
 The thermal state of Hole 504B has been measured repeatedly with downhole temperature tools. These logs show clear evidence for continued downflow of bottom water into basement rocks below casing, but changes in the thermal profile over time indicate a complex flow history. The initial thermal log from DSDP Leg 70, taken 2 months after Hole 504B was first drilled, indicated rapid downflow [Becker et al., 1983], but borehole thermal logs from approximately 2, 3, and 7 years later (DSDP Legs 83 and 92 and ODP Leg 111) indicated progressively slower downflow with time [Becker et al., 1985; Gable et al., 1989]. A borehole temperature log taken twelve years after initial drilling (ODP Leg 137) indicated faster downflow than inferred from the previous two logs [Gable et al., 1995], a transient behavior that remains enigmatic. In the present study, we interpret only the thermal record taken from DSDP Leg 70 (Figure 3), comprising 10 measurements made at 28 m intervals in the cased sedimentary section of the borehole, collected after the first 1–2 months of continuous downflow.
1.2.3. Hole 1026B
 ODP Site 1026 is located in 3.5 Myr old crust ∼100 km east of the intermediate spreading-rate Endeavor segment of the Juan de Fuca Ridge (Figure 2 and Table 1). The area is characterized by thick sediments overlying relatively young crust, due to high sedimentation rates and the infilling of turbidites above abyssal hill topography. Hole 1026B was drilled in 1996 on ODP Leg 168 as part of an east-west crustal age transect [Davis et al., 1997b]. Although the hole was initially drilled through 247 m of sediments and 52 m into volcanic rocks, upper basement was unstable. A “liner” of drillpipe was drilled into the rubble at the bottom of the hole to keep the hole from collapsing before installation of a long-term borehole observatory (CORK) [Davis et al., 1997b]. A thermal log collected 13 days later using an autonomous temperature probe lowered by wireline showed that warm formation fluid was moving rapidly up the hole from depth, indicating that the natural formation overpressure had overcome the differential pressure induced by drilling and other operations [Fisher et al., 1997] (Figure 3). This log consists of 16 measurements made at 10 m intervals in the cased hole within the sedimentary section, and represents conditions after 1–3 weeks of upflow following the flow reversal by overpressured formation fluids. Because this hole was not monitored during the interim period, the time of reversal and the duration of upflow prior to the thermal log are uncertain.
1.2.4. Holes U1301A and U1301B
 Holes U1301A and U1301B were drilled in 2004 on IODP Expedition 301 into 3.5 Myr old crust on the eastern flank of the Juan de Fuca Ridge, ∼1 km south of Hole 1026B, through 262–265 m of sediments (Figure 2 and Table 1). These holes were drilled just 36 m apart, with the intent of creating sealed borehole observatories that monitor distinct basement intervals. Following completion of drilling, casing, coring, and short-term experiments, the holes were fitted with CORKs to measure pressure and temperature, and to sample formation fluids and microbial materials at depth [Fisher et al., 2005]. Unfortunately, the CORKs in Holes U1301A and U1301B were not sealed as planned, allowing cold bottom water to flow downhole through annular gaps in the casing for several years following CORK installation [Fisher et al., 2008]. The flow down Hole U1301A reversed abruptly about 3 years after CORK installation, and the flow down Hole U1301B was stopped after 5 years when the gaps between casing strings around the CORK wellhead were cemented [Fisher, 2010; Wheat et al., 2010].
 Instrument strings deployed in both CORKs included autonomous temperature probes, collecting thermal data at fixed depths during deployments of 4–5 years. These data, recovered in 2009 and 2010, show significant short-term variability in borehole temperatures resulting from tidal influences (Figure 4). However, the mean temperature was relatively consistent over the five-month period following deployment, allowing the records for each hole to be averaged and treated as a single measurement for purposes of modeling (Figure 3). In each case, we extrapolated the average of the measured borehole temperatures up to the sediment-basement interface based on the local heat flow and inferred properties in basement [Davis et al., 1997a, 1992; Fisher et al., 2005]. CORK thermal data from Hole U1301A were interpreted in studies of borehole chemistry and microbiology [Orcutt et al., 2010; Wheat et al., 2010], but have not been interpreted previously in terms of crustal hydrogeologic properties. Short-term packer experiments and the cross-hole response at nearby Hole 1027C have also provided information on hydrogeologic properties in the vicinity of these boreholes [e.g., Becker and Fisher, 2008; Fisher et al., 2008].
2. Analytic Methods
2.1. Borehole Models
 We link three analytic solutions describing the physical response to flow in a borehole (diffusive thermal exchange, hydrostatic pressure within the borehole, and radial flow to/from the formation) to relate thermal profiles to physical and hydrologic properties of interest (formation permeability, porosity, sediment thermal conductivity, regional heat flow, aquifer thickness, aquifer compressibility, borehole radius in basement, flow rate, and flow duration (Table 2)).
Table 2. MCMC Analysis Parameters, Prior Distributions, and Sources
[Wilkens et al., 1983]. Sediment samples, downhole logs.
[Pribnow et al., 2000]. Sediment samples, downhole logs.
[Fisher et al., 1997]. Thermal measurements taken in basement, drilling records.
Thermal conductivity (sediment)
Heat flow (seafloor)
Borehole radius (basement)
 Thermal profiles are modeled using an analytic solution to a thermal diffusion equation for a cylindrical flowing well with constant throughflow [Lesem et al., 1957], as a function of duration of flow (t) and depth into hole (z) (modified from Becker et al. , variables defined in Notation section),
where G is the geothermal gradient, Q is the volumetric flow rate, rs is the borehole radius, (ρc)w is the specific heat capacity of water, and κs and χs are the thermal conductivity and thermal diffusivity of sediment, respectively. Fluid properties are calculated through an empirically derived equation of state as a function of temperature and salinity [Sharqawy et al., 2010]. Salinity is held at a constant bottom water value (35 g/kg) throughout the analysis. I1 and I2 are integrals that are solved numerically with adaptive Gauss-Kronrod quadrature [Shampine, 2008], which is computationally inexpensive but must be repeated many times for each hole. The solution is derived iteratively by solving equation (1) and updating borehole temperatures and associated fluid properties after each iteration.
 The modeled thermal profile is used to calculate fluid density with depth, ρ(z), including dependence on temperature but neglecting the influence of changes in pressure or salinity. Fluid density profiles are calculated for thermal conditions in the borehole (equation (1)) and for ambient formation fluid based on a constant geothermal gradient inferred from local measurements. We assume flow in the borehole is slow enough for pressure conditions to be essentially hydrostatic, and integrate downward from the seafloor to calculate pressures in the borehole and ambient formation (Pb and Pa, respectively):
where g is the acceleration due to gravity, and ρb(z) and ρa(z) are the borehole and ambient fluid density profiles, respectively. The integral is approximated as a finite difference problem, summing pressures piecewise over 5 m intervals.
 The difference between borehole and ambient pressures (ΔP = Pb − Pa) is used to solve the pressure diffusion equations for volumetric flow rate into or out of basement (modified from Becker et al.  and Fisher et al. ; after Jaeger [1942, 1965] and Matthews and Russell ):
 In equation (3), k is the formation permeability, H is the thickness of the permeable zone, rb is the borehole radius in basement, μw is the dynamic viscosity of water, ϕ is the porosity, and τ is dimensionless time. The term β is the compressibility of the crustal aquifer system, comprising the sum of fluid and formation compressibility. We expect formation compressibility in rubbly and fractured upper oceanic crust to be considerably greater than fluid compressibility (4–5 × 10−10 Pa−1 [Fine and Millero, 1973]), and thus interpret β to represent primarily formation compressibility. The integral I3 is tabulated in Jaeger and Clarke , and can also be approximated and evaluated numerically [Becker et al., 1983]. This equation yields the volumetric flow rate (Q) from the borehole into the permeable basement zone (Q > 0, downflow case) or out of the permeable basement zone into the borehole (Q < 0, upflow case).
 Flow into or out of the formation (Q) determined by equation (3) should equal the flow along the borehole used in equation (1). The result from equation (3) is therefore used as the flow rate for equation (1), and calculations are repeated iteratively until the two rates converge to ensure a self-consistent solution. Iteration in this case is very stable, and flow rates typically converge to within 1% in fewer than five iterations.
2.2. Markov Chain Monte Carlo Analysis
 The forward model described in the last section utilizes analytic equations representing coupled fluid and heat flow to generate borehole thermal profiles based on a set of known or assumed physical parameters. We wish to characterize these parameters based on downhole thermal measurements and other observations, essentially solving an inverse problem. We do this with a Markov chain Monte Carlo (MCMC) analysis [Chib and Greenberg, 1995; Gallagher et al., 2009], which treats each of the model parameters as a statistical distribution of values, rather than a single value. This approach yields probabilistic distributions of model parameters as output.
 For MCMC analysis, a statistical distribution known as a “prior distribution” must be provided for each parameter, codifying knowledge that is not explicitly treated by the forward model. For example, the radius of the borehole wall within the open formation is often poorly known because the upper ocean crust is rubbly and unstable, frequently leading to borehole collapse, as indicated by caliper measurements showing borehole enlargement and geophysical logs (bulk density, porosity) that are of poor quality [e.g., Anderson et al., 1985; Bartetzko and Fisher, 2008; Moos, 1990]. In general, it is likely that the smallest possible borehole radius is that of the drill bit, but the actual diameter is likely to be greater and may be variable with depth. MCMC analysis is not used to determine the borehole radius per se, but provides information on how the assumed borehole radius influences estimates of other parameters of interest (e.g., formation permeability, volumetric flow rate, duration of flow). In practice, several of the key physical parameters embedded in equations (1) and (3) are poorly known, although in many cases, these values can be bounded and/or expected to fall within particular distributions. The MCMC approach uses a random process to condition each prior distribution to both the data provided and the other parameters' prior distributions. This results in a series of random samples drawn from a final statistical distribution, known as a “posterior distribution,” for each parameter (Figure 5). These are sets of values for each parameter that allow the borehole thermal observations to be fit within a range of error based on the uncertainty in the data and prior distributions. The posterior distributions are interpreted as estimates of the value of each parameter as conditioned by the full suite of available information [Tierney, 1994]. Because the MCMC approach allows all of the parameters of interest to vary simultaneously, this approach improves upon a more traditional “sensitivity analysis” based on varying one parameter at a time, which tends to underestimate their individual uncertainties.
 Our MCMC analysis uses a random-walk chain, created with a Metropolis-Hastings algorithm [Chib and Greenberg, 1995] to draw samples from posterior distributions. To generate each MCMC realization, values to be used in a single run of the forward analytic model are proposed for each parameter, starting with arbitrary choices and with subsequent steps using proposals drawn randomly from the vicinity of values used in the previous step. The resultant thermal profile from the forward model is compared to observational data (borehole thermal records) to calculate a data likelihood function [f (T|θ)], which is a measure of how closely the observed and modeled profiles agree (Figure 6). In this study, we assume independent and identically distributed normal errors in the thermal data, which simplifies the joint data likelihood function for all m thermal measurements to
 This function is used to determine the acceptance probability (αθ) for each proposed parameter value (θ*) derived from each MCMC realization (n):
 If a parameter value is accepted, θn+1= θ*; otherwise, it is rejected and θn+1= θn . Once this is done for each model parameter, new random values are proposed, and the process is repeated until sufficient samples have been generated to create a set of relatively smooth posterior distributions. The first sequence of steps is often strongly influenced by the initial arbitrary selection of the chain's starting point, and is therefore discarded. The remaining random-walk steps are downsampled to ensure independence and interpreted as samples from the posterior distribution functions for each parameter [Tierney, 1994]. The MCMC analyses in this study were run to generate >500 independent samples of posterior probability distributions for all parameters in each hole, after which mean values for all distributions converged to within 1%. In several cases, this required running >100,000 realizations. The code to perform the complete analysis, and example input and output files, are available at http://pmc.ucsc.edu/∼afisher/Research/Appen/BHT-MCMC.
2.3. Inferred Property Distributions
 We provided prior distributions for eight parameters for each downflow case (Holes 504B, U1301A, U1301B) and nine for the upflow case (Hole 1026B), as the upflow case requires basement temperature as an additional parameter (Table 2). The prior distributions for each parameter required careful selection, as the number of free parameters used in the full analysis is large relative to the number of observations used to constrain acceptable results (mainly temperature measurements). Most prior distributions assigned in this study are gamma distributions, which converge to normal distributions when the specified mean is much larger than the variance, as was the case for most parameters treated. The gamma distribution was chosen over the normal distribution because it converges to zero probability at its lower bound. This behavior is consistent with most parameters used in this study, which often have no meaning below some physical limitation (e.g., bit size for borehole radius). For both permeability and flow duration, where earlier studies or data were often ambiguous, bounded uniform prior distributions were used. We emphasize discussion of borehole flow rate and formation properties in this paper, but MCMC analyses were used to generate posterior distributions for all parameters, all of which are shown as figures in the supporting information.1
 Volumetric flow rate (Q) is solved for directly through iteration of the analytic forward model and is thus not treated explicitly as a parameter by the MCMC analysis. Without an explicitly defined prior distribution to condition to thermal data, a posterior distribution for Q must instead be determined after the MCMC analysis is completed. To do this, the forward model is rerun once for each realization, using values from the previously generated posterior distribution for each parameter. The resultant volumetric flow rates form a posterior distribution that is consistent with the ensemble of MCMC results.
2.4. Model Assumptions and Limitations
 Equations (1), (2a), (2b), (3) are analytic solutions based on several assumptions and approximations. First, although the solution to the thermal response is transient, fluid flow up or down a borehole is assumed to occur at a constant rate and direction. Thus this analysis does not strictly apply to records with more complex flow histories (periods of active pumping; changing flow rates; flow reversals) [Lesem et al., 1957]. The analysis assumes laminar flow, although solutions presented later have Reynolds numbers indicating transitional or mildly turbulent flow, the most significant effect of which is likely an overestimation of driving pressures from neglecting turbulent energy losses. This effect should be small in Holes 504B and 1026B, but could be more significant in Holes U1301A and U1301B where fluid flowed through small gaps in the casing hanger. Further, equations (1) and (3) use constants for some physical properties that vary over spatial scales relevant to the problem or are otherwise difficult to constrain. Examples include the geothermal gradient and sediment thermal conductivity, both of which are treated as constants by the model but are known to vary somewhat with depth. These uncertainties and idealizations are addressed, to a large extent, by the MCMC analysis, and we have taken a conservative approach in allowing acceptance of proposed parameters even when differences between modeled and observed thermal logs are relatively large.
3.1. Summary of Model Results and Parameter Distributions
 Results from the linked models of formation properties and borehole flow and temperature, developed with MCMC analyses, are summarized in Figures 6-8, Table 3, and Figures S1 to S4 (supporting information). Figure 6 shows profiles of temperature versus depth, comparing model results to observations. The range of model results shown is wide relative to the accuracy of individual temperature measurements, because the MCMC method permits the acceptance (with low probability) of parameter sets that deviate significantly from observations. Rather than truncating these posterior distributions, we show 90% of the range of resulting temperature ensembles as a conservative indication of model confidence. Table 3 lists posterior median borehole flow rates (Q) and formation bulk permeability values (k) with their standard deviations (in log cycles), and compares these values to those published previously. Figures in the supporting information comprise comprehensive plots of all posterior parameter distributions. Results are described site by site in the rest of this section, after which we discuss their implications.
Table 3. Median MCMC Analysis Flow Velocity and Permeability Results, With Comparison to Results of Previous Studies
 MCMC analysis using the linked analytic models generated a good fit to borehole thermal data (Figure 6a). The shallowest temperatures are effectively “pinned” by the temperature of bottom water at the seafloor, whereas borehole temperatures at the base of the cased interval are more sensitive to the downhole flow rate. The posterior distribution of volumetric flow rates (Q) down Hole 504B was strongly asymmetric, having relatively high probability at lower values (≥0.2 L/s), and a long declining tail of higher values (≤60 L/s) (Figure 7a). The distribution has a median of Q = 1.6 L/s (Table 3), with 90% of the total probability between 0.5 and 3.5 L/s.
 Results for the posterior permeability distribution (k) were roughly log-normally distributed (Figure 8a). The median k value was 4.7 × 10−12 m2 (Table 3), with 90% of the probability between 1.7 × 10−12 m2 and 1.4 × 10−11 m2. The posterior distribution for flow duration showed little sensitivity across the allowed range (39–57 days) (Figure S1G; supporting information). The remaining parameters had posterior distributions with shapes that were similar to those of their prior distributions (Figure S1; v), with deviations of only 1–5% from the prior probability density for each parameter. That many of the posterior distributions closely resemble the specified prior distributions suggests that the parameter ranges chosen allow reasonable fits to the thermal observations, though this does not strictly comprise a test of distribution accuracy.
3.3. Hole 1026B
 The fit of MCMC results to borehole thermal data from Hole 1026B, an upflow case, is inverted relative to that from 504B, with less sensitivity to flow rate closer to the zone of discharge from basement (Figure 6b). Posterior distributions for both Q and k in Hole 1026B were also asymmetrically distributed, with probability density skewed toward lower values (Figures 7b and 8b). The median of the distribution for Q was 2.3 L/s (90% between 0.9 and 7.2 L/s), and the median of k was 3.7 × 10−12 m2 (90% between 1.4 × 10−12 m2 and 1.6 × 10−11 m2) (Table 3).
 The posterior distribution for the duration of flow at Hole 1026B shows a consistent deviation from its uniform prior distribution (Figure S2G; supporting information). Most of the allowed range of flow duration appears flat, but the probability density shows a marked drop during the time period of 0–3 days, indicating that the borehole thermal data were poorly modeled by short-duration flows. As an upflow case, the model for Hole 1026B required the specification of a distribution of basement temperatures. The posterior distribution for basement temperature has a higher mean and lower variance than did the prior distribution (Figure S2H; supporting information). The warmer basement temperatures generate modeled thermal profiles that better match the data, illustrating how posterior distributions can differ significantly from their prior distributions when the data require it. The posterior distributions of remaining parameters are similar to those selected for prior distributions (Figure S2; supporting information).
3.4. Holes U1301A and U1301B
 Observational data from Holes U1301A and U1301B, which suggest downward flow of fluid from the seafloor during the first 5 months post drilling, provide less constraint on borehole and formation properties than data from Holes 504B and 1026B, because the analysis is based on temperatures from a single depth in each hole (Figures 6c and 6d).
 The distributions of downward flow rates in Holes U1301A and U1301B are asymmetric, skewed toward lower values than true log-normal distributions (Figures 7c and 7d). The median value for Q at Hole U1301A was 2.3 L/s (90% probability between 1.4 and 4.4 L/s), whereas the median value for Q at Hole U1301B was an order of magnitude greater, 27 L/s (90% probability between 15 and 67 L/s). Posterior distributions for k in both holes are close to log-normally distributed (Figures 8c and 8d), with a median of 6.6 × 10−12 m2 in Hole U1301A (90% probability between 3.0 × 10−12 m2 and 1.7 × 10−11 m2), and a median of 1.5 × 10−11 m2 in Hole U1301B (90% probability between 7.5 × 10−12 m2 and 4.1 × 10−11 m2.) (Figures 8c and 8d). The posterior distributions of the remaining parameters are similar to prior distributions for both holes (Figures S3 and S4; supporting information).
4.1. Hole 504B
 Our median results for volumetric flow rate (Q) at Hole 504B are approximately 20% higher than the previously published estimate [Becker et al., 1983], but the difference is within the estimated uncertainty generated from the MCMC analysis (Table 3 and Figure 7). The forward model used in the earlier study was essentially the same as that applied in this paper, but without an assessment of uncertainties in the sedimentary thermal gradient, thermal properties, or the duration of flow, and no iterative analysis to assure a consistent result between the three analytic models.
 The difference in inferred bulk permeability of upper basement is considerably greater, with the median value from new analyses being 20 times larger than calculated previously based on the same thermal data (Table 3 and Figure 8). Most of this difference can be attributed to the use of a higher apparent formation underpressure, inferred from packer testing [Anderson and Zoback, 1982; Becker et al., 1983], in lieu of the borehole hydrostatic condition used in this study (equation (2)). At the time of initial packer tests in Hole 504B, there had been no direct measurements of natural pressures in upper basement rocks on ridge flanks through use of sealed borehole observatories, and a differential pressure of ∼1 MPa was not considered to be unreasonable. Since that time, natural differential pressures on the order of a few tens of kilopascals have been measured in upper basement using borehole observatories at several ridge-flank locations, including the eastern flank of the Juan de Fuca Ridge [e.g., Davis and Becker, 2002] and Hole 504B [Becker et al., 2004]. Similarly small differential pressures have also been inferred from transient numerical models of coupled fluid and heat flow on ridge flanks [e.g., Hutnak et al., 2006; Spinelli and Fisher, 2004; Stein and Fisher, 2003].
 Basement permeability inferred in the present study is more than 100 times greater than that inferred from packer testing of upper basement [Anderson et al., 1982], but this ratio is reduced to 30 after accounting for the difference in assumed basement aquifer thickness (172 m versus 40 m). This apparent difference in permeability with test type is consistent with the scaling of permeability in heterogeneous rocks, such as the upper oceanic crust, with higher values typically resulting from tests run at greater length scales [Becker and Davis, 2003; Clauser, 1992; Fisher, 1998]. Short-term, single-hole packer tests tend to produce results representative of a radial distance of a few meters, whereas flow tests lasting weeks to months are representative of a kilometer scale.
4.2. Hole 1026B
 The median volumetric flow rate up Hole 1026B determined in the present study was ∼60% greater than that estimated previously from the same thermal data (Table 3 and Figure 7), whereas median bulk permeability in upper basement was lower by 50% (Table 3 and Figure 8) [Fisher et al., 1997]. In both cases, previous estimates fall within the range indicated by 90% of the MCMC results. The differences in median flow rate and bulk permeability resulted from the iterative approach and MCMC analysis used in the present study.
 The bulk permeability estimated in the present study is 20–40 times greater than that calculated from packer testing in the same hole [Becker and Fisher, 2000], similar to the ratio between values derived from thermal data and packer testing in Hole 504B. Pumping rates used during packer testing were similar to flow rates inferred during free flow discharge from Hole 1026B, so there should have been a comparable influence of turbulent energy losses. As with Hole 504B, the difference in permeability estimated by the two methods is likely the result of scale effects.
 The drop in the probability density in the posterior distribution for flow duration appears to be significant, indicating that short flow durations (0–3 days) are less likely to produce the observed thermal profile. These results suggest that it is more likely that the upward flow in Hole 1026B began relatively early during borehole operations, rather than in the few days immediately before the thermal log was run.
4.3. Holes U1301A and U1301B
4.3.1. Comparison With Results of Earlier Studies
 Thermal data from Holes U1301A and U1301B have not been interpreted previously, but we compare new results to basement permeability calculations from packer testing [Becker and Fisher, 2008] and from the cross-hole response to downflow in Hole U1301B detected in Hole 1027C, 2.4 km to the east [Fisher et al., 2008]. Packer tests were attempted in both Holes U1301A and U1301B on IODP Expedition 301, but were unsuccessful in Hole U1301A due to the lack of a casing seal or cement between casing and the formation. Results from packer testing in Hole U1301B indicated bulk permeability of 3 × 10−12 m2 for a crustal interval 166 m thick (the lower half of the basement hole), and 2 × 10−11 m2 for a 30 m thick interval between two packer-setting depths. This permeability range overlaps with the distribution of bulk permeability values inferred in the present study (Table 3 and Figure 8), but may not be directly comparable, because each analysis sampled different depth intervals within the upper crust. Packer testing focused on the interval 152–318 msb, whereas we have interpreted long-term downhole flow to enter the crust around Hole U1301B above the shallowest casing packer installed with the borehole observatory, located at 166 msb.
 The cross-hole response seen in Hole 1027C, resulting from 13 months of downward flow into Hole U1301B, suggested bulk permeability in the same shallow crustal interval that is an order of magnitude lower than inferred in the present study, 7 × 10−13 m2 to 2 × 10−12 m2 (Figure 8) [Fisher et al., 2008]. The use of a second borehole and the long duration of the cross-hole response would tend to sample a larger volume than a single-hole flow test based on thermal data. The larger-scale cross-hole test could have yielded a lower bulk permeability because of azimuthal anisotropy in permeability, with the orientation having the highest permeability being oblique to the direction between Holes U1301B and 1027C. This is consistent with the preferred fluid flow direction suggested by studies in the same region [e.g., Fisher et al., 2008; Hutnak et al., 2006; Wheat et al., 2000], and would not be resolved by a single-hole analysis using thermal records.
4.3.2. Comparison of Results in Holes U1301A and U1301B
 The calculated rate of downward flow into Hole U1301B is an order of magnitude greater than that down Hole U1301A (Table 3 and Figure 7). This is consistent with the nature of the tidal response in the two holes (Figure 4), which is much greater in Hole U1301A, the lower borehole temperatures in Hole U1301B, and the subsequent reversal of flow direction in Hole 1301A [Wheat et al., 2010]. The more rapid flow rate in Hole U1301B is likely the result of the difference in the total depth of these holes (370 mbsf in Hole U1301A, 582 mbsf in Hole U1301B). The depth of Hole U1301B provides greater thermal perturbation and a taller cold hydrostatic column, resulting in commensurately higher differential pressure, driving fluid into the formation surrounding the borehole at a greater rate.
 The median bulk permeability calculated from the temperature data in Hole U1301B is twice that calculated from similar data in Hole U1301A (Table 3), suggesting that the deeper crustal section around Hole U1301B is more permeable than the shallower section around Hole U1301A. This interpretation is consistent with the nature of the cross-hole pressure response in Hole 1027C [Fisher et al., 2008]: drilling, casing, and other operations in Hole U1301A had little influence on crustal pressure conditions monitored in Hole 1027C, whereas similar operations in Hole U1301B caused rapid, measurable pressure responses in Hole 1027C. A difference in permeability between the holes could result from lateral or vertical variability in crustal properties. However, given the overlap in the distributions of permeability values determined by MCMC analyses (Figure 8) the apparent difference in permeability between Holes U1301A and U1301B may not be statistically significant.
4.4. Comparison of Results From All Boreholes
 Median bulk permeability values for the four boreholes evaluated in this study vary by a factor of 4 (3.7 × 10−12 to 1.5 × 10−11 m2), despite differences in setting, seafloor age, and hole completion parameters. Given the wide range of ocean crustal permeability values seen globally [e.g., Fisher et al., 2008] and the extent of local variability in crustal properties commonly found in individual holes [e.g., Bartetzko and Fisher, 2008; Broglia and Moos, 1988; Jarrard et al., 2003], the observed consistency in hydrogeologic properties is surprising. Determining whether the consistency of permeability in these four boreholes is coincidence, or indicative of properties more broadly, will require the application of similar methods to additional boreholes.
 The median flow rates inferred down Holes 504B and U1301A and up Hole 1026B were also similar, but median flow rate down Hole U1301B is an order of magnitude greater (Table 3 and Figure 7). The greater flow rate down Hole U1301B relative to Hole U1301A helps to explain why Hole U1301A reversed after ∼3 years of continuous downflow [e.g., Wheat et al., 2010], whereas flow down Hole U1301B ended only when the reentry cone surrounding the wellhead was cemented [Fisher, 2010]. Hole 1026B, which had even less penetration into basement than Hole U1301A, reversed flow (down to up) within 14 days after the end of drilling and casing operations [Fisher et al., 1997] (Figure A-2G). Flow down Hole 504B slowed gradually to very low values over 22 years [Becker et al., 2004], aside from the enigmatic but brief renewal of downflow at 12 years suggested by the temperature log of Gable et al. . In contrast, in Hole 896A, located <1 km to the south of Hole 504B, initial downflow reversed quickly following drilling operations [Becker et al., 2004]. In addition to depending on total borehole depth and depth into basement, the occurrence and timing of a reversal in flow direction in ocean crustal holes that initially experience down flow may depend on basement topographic patterns beneath ridge-flank sediments [Bani-Hassan et al., 2012; Fisher et al., 1990; Hartline and Lister, 1981; Wang et al., 1997]. Holes 896A, 1026B, and U1301A/B were all drilled into sediment-covered basement highs, which tend to be naturally overpressured relative to ambient hydrostatic conditions, whereas Holes 504B and 1027C were associated with sediment-covered basement lows, which tend to be underpressured [Davis and Becker, 2002, 2004].
5. Summary and Conclusions
 We have presented a model of the thermal state of flowing seafloor boreholes that links analytic solutions to three distinct physical processes (diffusive thermal exchange, development of hydrostatic pressure within the borehole, and radial flow to/from the formation at depth). The model is run as part of a MCMC analysis to assess uncertainty in flow properties (rate, duration) and in borehole and formation physical properties (basement permeability, borehole radius, aquifer thickness, etc.). Results were presented for two thermal records that have been interpreted previously (Holes 504B and 1026B) and for two records recently recovered from long-term borehole observatories (Holes U1301A and U1301B). All of these holes were drilled into relatively young, upper oceanic crust below thick overlying sediments.
 New models suggest that the bulk permeability of upper oceanic crust around Holes 504B, 1026B, and U1301A is 4–7 × 10−12 m2. Calculated median permeability around the upper part of Hole 504B is 20 times greater than estimated previously [Becker et al., 1983], mainly because an improved estimate of the differential pressure that drives flow into the formation, and ∼50% less than estimated previously for upper crust around Hole 1026B [Fisher et al., 1997]. The median permeability of the upper ocean crust around Hole U1301B is ∼1.5 × 10−11 m2, somewhat greater than that calculated for Hole U1301A located just 36 m away. Apparent differences in bulk permeability of the ocean crust around these boreholes are relatively small when compared to the wide range of values measured globally, and the MCMC analyses generated overlapping distributions of properties for these boreholes. Given the uncertainties in borehole and formation parameters revealed through the MCMC analyses, results from the present study suggest relatively consistent bulk permeability values for the uppermost volcanic ocean crust.
 Permeability estimates in Holes 504B and 1026B based on thermal logs are 20–40 times greater than those from packer testing, which may result from the larger scale of measurement associated with longer-term thermal studies. Results from packer testing in Hole U1301B yielded permeability estimates consistent with those from this study [Becker and Fisher, 2000], but packer and thermal analyses from Hole U1301B tested different crustal depths, and so may not be directly comparable. Bulk permeability calculated from analysis of the thermal record from Hole U1301B is 10 times greater than that determined from the observed cross-hole response between Holes U1301B and 1027C, which could result from azimuthal anisotropy in permeability [Fisher et al., 2008].
data likelihood of forward model.
gravitational acceleration, m/s2.
geothermal gradient, °C/m.
thickness of permeable zone, m.
integrals required for solution to thermal diffusion equation, based on cylindrical borehole geometry, detailed in [Becker et al., 1983].
integral required for solution to pressure diffusion equation, tabulated in [Jaeger and Clarke, 1942].
permeability of basement rocks, m2.
number of thermal measurements.
MCMC realization number.
probability of proposing θ*, given the previously accepted value θn.
pressure in ambient formation, Pa.
pressure in the borehole fluid column, Pa.
volumetric flow rate up or down borehole, m3/s.
borehole radius in basement, m.
borehole radius in sediments, m.
observed borehole temperature, °C.
modeled borehole temperature, °C.
depth below seafloor, m.
acceptance probability for proposed parameter in MCMC realization.
fluid compressibility, Pa−1.
generic MCMC parameter.
thermal conductivity of sediment, W/(m°C).
dynamic viscosity of water, Pa·s.
specific heat capacity of water, J/(kg°C).
estimated uncertainty in thermal data, °C.
dimensionless time, used as argument of I3.
thermal diffusivity of sediments, m2/s.
 This research was based on data and samples provided by the Integrated Ocean Drilling Program, the Ocean Drilling Program, and the Deep Sea Drilling Project. We thank the crews, technicians, and officers of the drilling ships and of numerous oceanographic and deep submergence platforms whose skill and hard work made these studies possible. This research was supported by Consortium for Ocean Leadership (COL) projects T327A7 and T327B7 (A.T.F.) and project T327B8 (K.B.), and NSF grants OCE-0939564 and 1031808 (A.T.F.) and OCE-1030350 (K.B.). This is C-DEBI contribution 173.