Chapter 7: Problems

Many of the problems listed below require calculations using the law of mass action and the charge-balance and mass-balance equations. Appropriate computational methods can be deduced from material presented in Chapters 3 and 7. We suggest that students exclude the occurrence of ion pairs and complexes from their calculations. In solutions of the problems, this approach will introduce some error that could otherwise be avoided by pursuit of more tedious calculations. The errors are generally small and the instructive nature of the problems is not significantly altered. Values of equilibrium constants can be obtained from Chapter 3, or in cases where they are not included in Chapter 3, they can be calculated from free-energy data listed in well-known texts, such as Garrels and Christ (1965), Krauskopf (1967), and Berner (1971). It can be assumed that the groundwaters referred to in the problems are situated at a sufficiently shallow depth for the effect of differences in fluid pressure from the 1-bar standard value to be neglected.

  1. A sample of rain has the following composition (concentrations in mg/\ell): K+ = 0.3, Na+ = 0.5, Ca2+ = 0.6, Mg2+ = 0.4, \ce{HCO^-_3} = 2.5, Cl = 0.2, SO42– = 15, and \ce{NO^-_3} = 1.2; pH 3.5; temperature 25°C. Can the pH of this water be accounted for by a hydrochemical model that is based on the assumption that the rain chemistry can be represented by equilibration of the water with atmospheric carbon dioxide as the dominant pH control? If it cannot, offer an alternative explanation for pH control.
  1. Rain that infiltrates into a soil zone has pH 5.7, K+ = 0.3, Na+ = 0.5, Ca2+ = 0.6, Mg2+ = 0.4, \ce{HCO^-_3} = 2.5, Cl = 0.2, and SO42– = 0.8 (concentrations in mg/\ell).
    1. In the soil zone, the water equilibrates with soil air that has a CO2 partial pressure of 10–2 Calculate the H2CO3 and \ce{HCO^-_3} concentrations and pH of the water. Assume that the water does not react with solid phases in the soil.
    2. In the soil zone, the water that initially has a dissolved oxygen content in equilibrium with the above ground atmosphere (i.e., 9 mg/\ell), has half of its dissolved oxygen consumed by oxidation of organic matter and half consumed by oxidation of iron sulfide (FeS2). Assume that the soil is saturated with water when these processes occur, that the water reacts with no other solid phases, and that oxidation of organic matter produces CO2 and H2O as reaction products. Estimate the PCO2, H2CO3, and SO42– contents and pH of the pore water. Which process exerts the dominant pH control, organic-matter oxidation or sulfide-mineral oxidation?
  2. Water with a dissolved oxygen concentration of 4 mg/\ell moves below the water table into geologic materials that contain 0.5%, by weight pyrite (FeS2). In this zone the dissolved oxygen is consumed by oxidation of pyrite. Estimate the pH of the water after the oxidation has occurred. The initial pH of the Water is 7.9. Assume that the water reacts with no other solid phases and that the groundwater zone is at 10°C.
  1. The following results were obtained from a chemical analysis of a groundwater sample (concentrations in mg/\ell); K+ = 21, Na+ = 12, Ca2+ = 81, Mg2+ = 49, \ce{HCO^-_3} = 145, Cl = 17, SO42– = 190, and Si = 12; pH 7.3; temperature 15°C.
    1. Is there any evidence suggesting that this analysis has significant analytical errors? Explain.
    2. Represent this chemical analysis using the following diagrams: bar graph, circular graph, Stiff graph, Piper trilinear diagram, Schoeller semilogarithmic diagram, and Durov diagram.
    3. Classify the water according to its anion and cation contents.
    4. On which of the diagrams would the chemical analysis be indistinguishable from an analysis of water with different concentrations but similar ionic percentages?
  1. Groundwater deep in a sedimentary basin has an electrical conductance of 300 millisiemens (or millimhos).
    1. Make a rough estimate of the total dissolved solids of this water (in mg/\ell).
    2. What is the dominant anion in the water? Explain.
  1. A sample of water from a well in a limestone aquifer has the following com- position (concentrations in mg/\ell): K+ = 1.2, Na+ = 5.4, Ca2+ = 121, Mg2+ = 5.2, \ce{HCO^-_3} = 371, Cl = 8.4, and SO42– = 19; pH 8.1; temperature 10°C.
    1. Assuming that these data represent the true chemistry of water in the aquifer in the vicinity of the well, determine whether the water is undersaturated, saturated, or supersaturated with respect to the limestone.
    2. The pH value listed in the chemical analysis was determined in the laboratory several weeks after the sample was collected. Comment on the reasonableness of the assumption stated in part (a).
    3. Assuming that the concentrations and temperature indicated in part (a) are representative of in situ aquifer conditions, calculate the pH and PCO2 that the water would have if it were in equilibrium with calcite in the aquifer.
  1. In the recharge area of a groundwater flow system, soil water becomes charged with CO2 to a partial pressure of 10–2.5 bar. The water infiltrates through quartz sand to the water table and then flows into an aquifer that contains calcite. The water dissolves calcite to equilibrium in a zone where the temperature is 15°C. Estimate the content of Ca2+ and \ce{HCO^-_3} in the water and the pH and PCO2, after this equilibrium is attained.
  2. In what types of hydrogeologic conditions would you expect \ce{HCO^-_3}-type water to exist with little or no increase in total dissolved solids along the entire length of the regional groundwater flow paths? Explain.
  1. A highly permeable carbonate-rock aquifer has natural groundwater at 5°C with the following composition (concentrations in mg/\ell): K+ = 5, Na+ = 52, Ca2+ = 60, Mg2+ = 55, \ce{HCO^-_3} = 472, Cl = 16, and SO42– = 85; pH 7.47. The water is saturated or supersaturated with respect to calcite and dolomite, It is decided to recharge the aquifer with surface water with the following composition (concentrations in mg/\ell): K+ = 2.1, Na+ = 5.8, Ca2+ = 5.2, Mg2+ = 4.3, \ce{HCO^-_3} = 48, Cl = 5, and SO42– = 3; pH 6.5. The recharge will take place by means of a network of injection wells that will receive the surface water from an aerated storage reservoir. Estimate the composition of the recharged water in the aquifer after it has achieved equilibrium with respect to calcite at a temperature of 20°C. Neglect the effects of mixing of the injection water and natural water in the aquifer. Explain why there is a major difference between the compositions of the two waters.
  1. Water charged with CO2 at a partial pressure of 10–1.5 bar in the soil zone infiltrates into a regional flow system in slightly fractured granitic rock. The water slowly dissolves albite incongruently until it becomes saturated with respect to this mineral. Assume that all other mineral-water interactions are unimportant. Estimate the water composition [Na+, Si(OH)4, \ce{HCO^-_3}, pH, and PCO2] after albite saturation is attained.
  1. Groundwater in fractured granite has the following composition (concentrations in mg/\ell): K+ =5, Na+ = 5.8, Ca2+ = 10, Mg2+ = 6.1, \ce{HCO^-_3} = 62, Cl = 2.1, and SO42– = 8.3, and Si = 12; pH 6.8. The water has acquired this composition as a result of incongruent dissolution of plagioclase feldspar in the granite. Because of the dissolution process, clay is forming on the surfaces of the fractures.
    1. Indicate the species of clay mineral or minerals that you would expect would be forming. Assume that the solid-phase reaction product is crystallized rather than amorphous in form.
    2. Would the process of incongruent dissolution cause the permeability of the fracture to increase or decrease? Explain.
  1. Estimate the water composition that results from the reaction in 1 \ell of water with 1 mmol of H2CO3, with (a) calcite, (b) dolomite, (c) albite, (d) biotite, and (e) anorthite. Express your answers in millimoles per liter and milligrams per liter. For each case, indicate the direction that the pH will evolve as dissolution occurs.
  1. Studies of a regional groundwater flow system in sedimentary terrain indicate that in part of the system there is a large decrease in SO42– and a large increase in \ce{HCO^-_3} in the direction of regional flow.
    1. What geochemical processes could cause these changes in anion concentrations?
    2. Indicate other chemical characteristics of the water that should show trends that would support your explanation.
  1. Groundwater in a sandstone bed within a layered sedimentary sequence comprised of shale, siltstone, lignite, and sandstone, all of continental depositional origin, has the following composition (concentrations in mg/\ell): K+ = 1.2, Na+ = 450, Ca2+ = 5.8, Mg2+ = 7.9, \ce{HCO^-_3} = 1190, SO42– = 20, and Cl = 12; pH 7.5; temperature 15°C. What combination of hydrogeochemical processes could account for this type of water chemistry? Write the chemical reactions that form the framework of your answer.
  1. Groundwater moves into a clayey stratum that is characterized by a selectivity coefficient of 0.7 with respect to the Mg–Ca exchange reaction described by Eqs. (3.105) and (3.107). The cation exchange capacity is 10 meq/100 g. The mole fractions of adsorbed Ca2+ and Mg2+ are both 0.5. The groundwater entering the clayey stratum has a Ca2+ concentration of 120 mg/\ell and a Mg2+ concentration of 57 mg/\ell. Assume that the concentration of other cations is negligible. Estimate the equilibrium concentrations of Ca2+ and Mg2+ that will occur after the water composition is altered by the Ca–Mg exchange reaction.
  1. Groundwater at a temperature of 25°C moves through a limestone bed where it attains saturation with respect to calcite. It then moves into strata which contain considerable gypsum. Estimate the composition of the water after it has attained equilibrium with respect to gypsum. Assume that the rate of calcite precipitation is very slow relative to the rate of gypsum dissolution. Prior to moving into the gypsiforous strata, the water has the following composition (concentrations in mg/\ell): K+ =3, Na+ = 8.1, Ca2+ = 110, Mg2+ = 9.2, \ce{HCO^-_3} = 310, Cl = 12, and SO42– = 36.
  1. Prepare a graph that shows the relation between the uncorrected (unadjusted) 14C age, corrected (adjusted) 14C age, and the parameter designated as Q in Sections 3.8 and 7.6. For a specified value of Q, are the differences between the corrected and uncorrected ages largest at young ages or old ages? Explain why the 14C method is generally not useful for dating groundwater that is younger than several thousand years.
  1. As water passes through the soil zone, it acquires, as a result of open-system dissolution, a \ce{HCO^-_3} content of 96 mg/\ell and a pH of 6.1. The water then moves below the water table into a dolomite aquifer. In the aquifer the \ce{HCO^-_3} content rapidly increases to 210 mg/\ell.
    1. What will be the value of Q for use in the adjustment of 14C dates of water from the aquifer?
    2. The water has an uncorrected age of 43,300 years. What is the corrected age based on the Q value obtained from part (a)?
  1. A horizontal sandstone aquifer occurs between two thick beds of shale. The sandstone is composed of quartz and a small percent of feldspar. Water in the sandstone is not capable of acquiring an appreciable concentration of dissolved solids by reaction with the aquifer minerals. Pore water in the shale, however, has high concentrations of dissolved solids. By considering various combinations of aquifer thickness, velocity of groundwater in the aquifer, concentration gradients in the aquitards, and diffusion coefficients, determine conditions under which the water chemistry in the aquifer would be controlled by the vertical flux by molecular diffusion of dissolved solids from the shale into the aquifer. Assume that dissolved solids that enter the aquifer are distributed uniformly over the aquifer thickness as a result of dispersion. Do you think such conditions could occur in nature?
  1. Groundwater A, at PCO2 = 10–2, has a composition that results from the open-system dissolution of siderite (FeCO3) in a stratum with no calcite or dolomite. Groundwater B, at the same PCO2, has a composition that results from open-system dissolution of calcite in a stratum with no siderite or dolomite. These two waters, each having been in equilibrium with their respective solid phase, are intercepted by a well in which they are mixed in equal proportions as pumping occurs. The system has a temperature of 25°C.
    1. Compute the cation and anion concentrations and pH of each of these waters.
    2. Compute the composition of the mixed water in the well.
    3. Is this mixture capable of producing calcite or siderite by precipitation?
    4. After discharge from the well into an open-air storage tank at 25°C, would calcite and/or siderite precipitate?